X-Planes: Lockheed Martin X-59 Quesst
Filed under: aerospace, history, engineering
Of the 539 allocation rows in the register of American military aerospace designations, exactly 2 mention a sonic boom, and the second of them is this aeroplane [Parsch, DOD 4120.15-L Addendum]. The register is compiled by Andreas Parsch from data the Department of Defense has released, and every count in this article is taken from every row of the copy he last updated on 23 August 2026. Its entry is 84 characters long, which puts it twenty-fourth of the 31 X rows by length, and it reads in full that the aircraft is a research aircraft that will create a shaped sonic boom signature. That copy is later than this article’s date. The version current on that date, the compiler’s update of 11 August 2025 as the Internet Archive captured it on 25 November 2025, held 526 rows, twenty-nine of them X rows, and lacked fifteen later rows that include the X-68A and the X-76A [Parsch, DOD 4120.15-L Addendum, 11 August 2025]. Under that version this entry was twenty-third of the twenty-nine X rows by length and the seventh shortest, the X rows said demonstrate 11 times, their median statement was 110 characters, and 91 rows had no engine listed. The two sonic boom rows, the single use of create and the X-57A’s longest statement are the same in both versions.
This is the sixtieth article in the X-Planes series, following the X-1, the X-2, the X-3, the X-4, the X-5, the X-6, the X-7, the X-8, the X-9, the X-10, the X-11, the X-12, the X-13, the X-14, the X-15, the X-16, the X-17, the X-18, the X-19, the X-20, the X-21, the X-22, the X-23, the X-24, the X-25, the X-26, the X-27, the X-28, the X-29, the X-30, the X-31, the X-32, the X-33, the X-34, the X-35, the X-36, the X-37, the X-38, the X-39, the X-40, the X-41, the X-42, the X-43, the X-44, the X-45, the X-46, the X-47, the X-48, the X-49, the X-50, the X-51, the X-52, the X-53, the X-54, the X-55, the X-56, the X-57, and the X-58.
The verb is the unusual thing. The register’s X rows say demonstrate 12 times, test 5 times, investigate 3 times, validate once and exhibit once [Parsch, DOD 4120.15-L Addendum]. The word create appears in 1 of the 539 rows, and this is the row. Every other aeroplane in the series is written down as going to find something out about a phenomenon that already exists. This one is written down as going to make one.
And the adjective is the programme. The other sonic boom row is the X-54, allocated 5 May 2008, which was to be capable of generating relevant ground sonic boom signatures [Parsch, DOD 4120.15-L Addendum]. That aeroplane was never built. The difference between the two sentences is the single word shaped, and the entire enterprise of the last sixty years of this subject is contained in it.
The Research Question
The question this aeroplane exists to settle is what people on the ground hear when a supersonic aeroplane passes over them with its boom shaped rather than coalesced into a shock-fronted wave. The register states the intent as the creation of a shaped sonic boom signature [Parsch, DOD 4120.15-L Addendum], and the programme states its target as 75 on the Stevens Mark VII perceived level scale [NASA Release 18-020] [Doebler and Rathsam 2020].
The quantity measured is a human response. The mission is to fly the aeroplane over selected United States communities and to collect the responses of the residents underneath [Quesst, the Aircraft], and Seventy-Five Is a Perception and Not a Pressure, below, sets out why the target cannot be read as an overpressure.
Nor is the question one of compliance. The governing rule names no threshold, as the subsection The Rule It Exists to Change Contains No Loudness at All shows, so what the aeroplane is built to return is evidence from which a threshold might one day be written. When this article was written the question was still open, because the aeroplane had not yet been supersonic.
Programme Origin
The row was allocated 21 June 2018 to Lockheed Martin, sponsored by the National Aeronautics and Space Administration, or NASA, with the engines cell left empty [Parsch, DOD 4120.15-L Addendum].
QueSST; LBFD aircraft is a research aircraft that will create a shaped sonic boom signature.
The empty engines cell carries no meaning. 94 of the register’s 539 rows have no engine listed, among them the X-57 and the X-60A, so an empty cell is no evidence about the aeroplane [Parsch, DOD 4120.15-L Addendum].
The length is more informative. The mission statement of the X-57, allocated 25.6 months before this one, is 328 characters and is the longest in the X series, each statement measured without its popular name and without the compiler’s note marker [Parsch, DOD 4120.15-L Addendum]. This one is 84. The two neighbouring rows are the longest and the eighth shortest, and the shorter of them describes a programme that cost the agency 247.5 million dollars at the contract award alone [NASA Release 18-020].
A register entry does not measure a programme. That is the finding rather than a complaint, and the X-57 established the same thing from the other end by being the longest entry attached to an aeroplane that never flew.
The Aeroplane Was Under Contract Before It Had a Number
NASA announced the contract to Lockheed Martin Aeronautics on 3 April 2018, stating that work began the previous day [NASA Release 18-020]. The designation was allocated 80 days after work began [Parsch, DOD 4120.15-L Addendum].
This is the ordinary order of events, and three neighbouring designations show that it is not the only one. The X-39 was reserved and then lost because no letter was ever written. The X-52 was requested in writing and refused. The X-58 was skipped while it was still the next number due. Here a contract was signed, an aeroplane was designed, and a number was applied for and granted in the normal way, and the interval was under three months.
Sizing From First Principles
This aeroplane is sized by its signature, and its sizing runs from the physics of the boom through the loudness target to the shape and the cruise condition that deliver it. The relations are evaluated at the published design point of Mach 1.4 at 55,000 feet, with the weights taken from the builder’s card [Silencing the Sonic Boom, X-59 Product Card]. The two quantities that would close the sizing, the lift-to-drag ratio and the installed fuel consumption in supersonic cruise, are not published, and where they are needed the article names the gap and supplies no value.
A Sonic Boom Is Not an Explosion
A body moving faster than sound cannot push the air out of its way in advance, because the news of its approach travels no faster than the air’s signalling speed. The disturbance therefore accumulates on a cone trailing the body, whose half-angle is fixed by the ratio of the two speeds.
\[\sin\mu = \frac{1}{M} \qquad\qquad \cot\mu = \sqrt{M^2 - 1} = \beta \tag{1}\]At the design cruise Mach number of 1.4 the Mach angle is 45.6 degrees, so the cone is broad rather than needle-like [Liepmann and Roshko, Elements of gasdynamics].
The word boom suggests an explosion and the physics is nothing of the kind. No energy is released. The aeroplane does work against the air continuously, and the pressure disturbance that work produces arrives at the ground all at once because the aeroplane outruns it. A sonic boom is the sound of an aeroplane’s steady flight delivered in the wrong order.
Whitham’s Function Reduces the Aeroplane to One Curve
Linearised supersonic theory reduces an aircraft, for the purpose of what it sounds like far away, to a single function of one variable.
\[\frac{\Delta p}{p_a} = \frac{\gamma M^2}{\sqrt{2\beta r}}\; F(\zeta) \tag{2}\]Here $F$ is Whitham’s F-function, and the entire influence of shape, size, weight and attitude enters the far field through it [Whitham 1952] [Whitham, Linear and nonlinear waves]. A body’s near field is complicated and its far field is one curve, and that reduction is why sonic boom minimisation is a tractable design problem.
The F-function is obtained from the second derivative of an equivalent area distribution through an Abel transform.
\[F(\zeta) = \frac{1}{2\pi} \int_0^{\zeta} \frac{A_e''(x)}{\sqrt{\zeta - x}}\; \mathrm{d}x \tag{3}\]And the equivalent area is not the aeroplane’s cross-section. It is the cross-section plus a term that accounts for lift [Ashley and Landahl, Aerodynamics of wings and bodies] [Jones, Wing theory].
\[A_e(x) = A_V(x) + \frac{\beta}{2q} \int_0^{x} L'(\xi)\; \mathrm{d}\xi \tag{4}\]That second term is why a sonic boom cannot be designed away by making the aeroplane thin. The aeroplane must hold itself up, and holding itself up means turning air downwards, and turning air downwards is indistinguishable at a distance from occupying volume. A wing that carries weight has an equivalent area whether or not it has any thickness.
The Weight Puts an Equivalent Area There and No Shaping Removes It
Integrating the lift term over the whole aeroplane collapses it, because the integral of the lift per unit length along the aeroplane is the total lift, and in level flight the total lift is the weight.
\[A_e(\ell) - A_V(\ell) = \frac{\beta}{2q} \int_0^{\ell} L'(\xi)\,\mathrm{d}\xi = \frac{\beta\,W}{2q} \tag{5}\]Everything about the aeroplane’s shape has dropped out. What remains is the weight divided by twice the dynamic pressure, multiplied by a Mach number factor. A designer may redistribute this area along the length and may not reduce it, because reducing it means either weighing less or flying faster and lower.
Lockheed Martin publishes a maximum design gross weight of 24,300 pounds for this aeroplane [Silencing the Sonic Boom, X-59 Product Card], which is 108,092 newtons, and the dynamic pressure at the design cruise condition is computed below at 12,512 pascals.
\[\frac{\beta\,W}{2q} = \frac{0.9798 \times 108,092}{2 \times 12,512} = 4.232\ \text{square metres} \tag{6}\]A body of revolution with that cross-section is 2.32 metres across, since a circle of area $A$ has diameter $2\sqrt{A/\pi}$.
\[d_e = 2 \sqrt{ \frac{\beta\,W}{2q\,\pi} } = 2 \sqrt{ \frac{ 4.232 }{\pi} } = 2.32\ \text{metres} \tag{7}\]The aeroplane is ninety-nine feet long and under thirty feet in span [X-59 Aircraft Overview and Status], and a photograph of it shows a fuselage visibly slimmer than that. The aeroplane’s volume is a minority contributor to its signature. Its weight is the majority, an inference this article draws from the relation above and finds stated in no source.
And the trade with altitude runs the wrong way. The lift term is inversely proportional to the dynamic pressure, so climbing to reduce the boom by lengthening the propagation path simultaneously increases the equivalent area the lift contributes. At 55,000 feet the dynamic pressure is 12,512 pascals and at forty thousand feet it is 25,730, so the lift-induced equivalent area is 2.06 times larger at the higher altitude, being 4.232 square metres against 2.058. The weight and the Mach number factor cancel from the ratio, which is therefore the ratio of the two dynamic pressures.
\[\frac{ 4.232 }{ 2.058 } = \frac{ 25,730 }{ 12,512 } = 2.06 \tag{8}\]Altitude cannot be chosen freely in this problem, and the two effects it has are opposed.
The same equivalent area governs the wave drag through von Karman’s double integral [Kuchemann, The aerodynamic design of aircraft] [Area rule].
\[D_w = -\frac{q}{2\pi} \int_0^{\ell}\!\!\int_0^{\ell} A_e''(x)\, A_e''(\xi)\, \ln|x - \xi| \; \mathrm{d}x \, \mathrm{d}\xi \tag{9}\]So the loudness and the drag are computed from the same curve. The two objectives are therefore not independent, and neither are they aligned, and a change made to quieten the aeroplane is a change to its drag whether the designer wants it to be or not.
Why the Boom Decays Faster Than Sound Does
A weak disturbance spreading from a line source in three dimensions has its amplitude fall with the square root of distance, because the energy is spread over a cylinder whose area grows linearly [Pierce, Acoustics].
\[\Delta p \propto r^{-1/2} \tag{10}\]A sonic boom does not obey that law. Whitham’s far-field result for a slender body gives a stronger decay, and a signature that grows longer as it weakens [Whitham, Linear and nonlinear waves] [Maglieri and others, Sonic boom, six decades of research].
\[\Delta p \propto r^{-3/4} \qquad\qquad T_s \propto r^{1/4} \tag{11}\]The discrepancy has a physical cause, which can be found by asking where the energy has gone. The intensity of an acoustic wave is its mean square pressure divided by the impedance of the medium [Kinsler, Frey, Coppens and Sanders, Fundamentals of acoustics].
\[I = \frac{p^2}{\rho\, c} \tag{12}\]The acoustic energy carried across unit length of a cylinder of radius $r$ is therefore the circumference times the intensity times the duration of the wave.
\[\Pi(r) \;\sim\; r\, \frac{(\Delta p)^2}{\rho c}\, T_s \;\propto\; r \cdot r^{-3/2} \cdot r^{1/4} = r^{-1/4} \tag{13}\]The energy falls as the minus one quarter power of the distance, so about 15.9 percent of it goes missing at every doubling, and it goes missing because the wave has shocks in it. A shock is an irreversible compression. It converts ordered acoustic energy into heat, and it does so continuously as the wave propagates, and the classical linear result omits it.
This is the reason the subject exists. A signature that has coalesced into a shock-fronted N-wave is dissipating itself all the way to the ground, and arrives having converted a substantial part of its energy into a discontinuity that the human ear is exquisitely equipped to hear. A signature that has not coalesced arrives quieter for two separate reasons, having lost less energy and having put what remains into frequencies the ear cares less about. The design problem is to arrange the equivalent area so that coalescence does not complete within the propagation distance available.
The propagation distance is not a free parameter. It is the cruise altitude, and the cruise altitude is chosen by the engine and the wing.
The Atmosphere Bends It, and There Is a Speed Below Which It Never Arrives
The atmosphere is not uniform. In the troposphere the temperature falls with altitude and the speed of sound falls with it, so a ray travelling downwards moves into progressively faster-signalling air and is refracted away from the vertical [U.S. Standard Atmosphere 1976 1976] [Anderson, Modern compressible flow].
\[c(z) = \sqrt{\gamma R\,T(z)} \tag{14}\] \[T(z) = \begin{cases} T_0 - \lambda z, & 0 \le z \le z_1 \\ T_1, & z_1 < z \le z_2 \end{cases} \tag{15}\]The pressure and the density follow from the same two layers, and every quantity later in this article that needs a dynamic pressure, a Reynolds number or an impedance passes through them [U.S. Standard Atmosphere 1976 1976] [International Standard Atmosphere].
\[p(z) = p_1 \exp\!\left[ -\frac{g\,(z - z_1)}{R\,T_1} \right] \qquad\qquad \rho(z) = \frac{p(z)}{R\,T(z)} \tag{16}\]At the design cruise altitude this gives 9,119.8 pascals and 0.14664 kilograms a cubic metre, which is 12 percent of the sea-level density.
\[q = \tfrac{1}{2}\rho V^2 = \tfrac{1}{2} \times 0.14664 \times 413.1^2 = 12,512\ \text{pascals} \tag{17}\]For a horizontally stratified medium the ray direction obeys an invariant of the same form as Snell’s law in optics [Pierce, Acoustics].
\[\frac{\cos\theta(z)}{c(z)} = \mathrm{constant} \tag{18}\]The consequence is a threshold. The horizontal trace speed of the wave along the ground equals the aircraft’s speed, and a wave whose trace speed is below the local speed of sound cannot propagate as a wave at that level. So there is a Mach number below which the boom turns over in the air and never reaches the ground.
\[M_c = \frac{c(0)}{c(z_a)} \tag{19}\]That condition is standard in the sonic boom literature and gives the term Mach cutoff its name [Maglieri and others, Sonic boom, six decades of research]. Evaluated for the design cruise altitude of 55,000 feet in the standard atmosphere, which is 16,764 metres and lies in the isothermal layer above the tropopause at 216.65 kelvin [U.S. Standard Atmosphere 1976 1976], this gives the following.
\[M_c = \frac{ 340.29 }{ 295.07 } = 1.153 \qquad\text{against}\qquad M = 1.4 \tag{20}\]The design cruise Mach number is 21.4 percent above cutoff. And the cutoff ratio has a maximum, because the atmosphere has a coldest layer. Sweeping the standard atmosphere from the ground to 30 kilometres, the largest value the ratio takes anywhere is 1.153. So Mach 1.4 produces a boom that reaches the ground from every altitude, and no choice of cruise height would prevent it.
This number has appeared before in this series. The X-54 derived it from the temperature ratio alone, since the speed of sound is the square root of the temperature and the rest cancels, and obtained the same 1.153 for the standard atmosphere above the tropopause. Two articles, two derivations and one number. That article went considerably further, computing where the ray turns over and showing how sharply the turning altitude collapses as the Mach number rises, and this one does not repeat that work.
The Aeroplane Could Have Avoided the Problem Entirely and Refuses To
A supersonic aeroplane that never wants to be heard has a simple option available to it. Cruise below the cutoff Mach number. The boom refracts away, nobody on the ground hears anything, and no shaping is required. The approach is studied under the name Mach cutoff flight, as an operational alternative to shaping the aeroplane [Busch et al. 2017] [Liebhardt et al. 2022], and the X-54 established that signatures were measured in flight at cutoff Mach number and published in 1971.
The X-59 rejects that solution because its product is the boom. The mission is to fly over selected communities and to ask the residents underneath how they responded [Quesst, the Aircraft]. An aeroplane that produces no measurable boom answers no question. The programme needs a signature on the ground, of a known shape, at a known level, arriving at people who can be surveyed. Cutoff cruise would deliver silence, and silence is not evidence.
That sets this aeroplane beside a technique that has no need of it. The executive order that directed the prohibition’s repeal gives its grounds as advances in aerospace engineering, materials science and noise reduction and names no technique [Executive Order 14304, American Presidency Project]. The X-54 article reads Mach cutoff flight, which makes no ground boom where shaping makes a quiet one, as a basis on which a repeal could stand, and that reading is an inference, since the order states no such basis. The X-59 is the aeroplane built for the other answer, and it flies 21.4 percent above the speed at which the problem would disappear, on purpose, because the other answer needs a boom to exist before anyone can be asked about it.
The Same Invariant Sets the Width of the Boom
The cutoff condition was written for a ray going straight down and the interesting rays do not. Taking the wave normal around the Mach cone at an azimuth $\phi$ measured up from the horizontal, the horizontal slowness has a component along the flight direction as well as across it, and it is the magnitude of the horizontal slowness that is invariant [Pierce, Acoustics] [Haefeli et al. 1969b] [Hayes 1969].
\[u = \frac{\sqrt{\sin^2\mu + \cos^2\mu \, \cos^2\phi}}{c(z_a)} = \text{constant along the ray} \tag{21}\]A ray reaches the ground only if that invariant still admits a real vertical direction there [Haefeli et al. 1969b].
\[u\, c(0) \le 1 \qquad\Longleftrightarrow\qquad \phi \ge \phi_c = 45.4^{\circ} \tag{22}\]At the design condition the cutoff azimuth is 45.4 degrees, so rays leaving the cone more sideways than that turn over in the air and are never heard. The straight-down ray reaches the ground by the shortest path, and every other ray reaching the ground lands further out.
The ray path itself follows from the same invariant, and integrating it from the cruise altitude to the ground gives the lateral offset at which each ray arrives [Pierce, Acoustics].
\[\frac{\mathrm{d}y}{\mathrm{d}z} = \frac{u\,c(z)}{\sqrt{1 - \left[ u\,c(z) \right]^2}} \;\cdot\; \frac{\cos\mu\,\cos\phi}{\sqrt{\sin^2\mu + \cos^2\mu\,\cos^2\phi}} \tag{23}\]Integrated numerically through the standard atmosphere, the widest-arriving ray lands 29.6 kilometres to the side.
\[y_c = 29.6\ \text{km} \qquad\Longrightarrow\qquad 2 y_c = 59.2\ \text{km} \tag{24}\]A ray tracer is credible only where it reproduces a known result, and this one can be tested. The rule of thumb NASA publishes is about one mile of boom carpet for every thousand feet of altitude, stated without a Mach number [NASA Facts FS-2003-11-016, Sonic Booms]. The same tracer run at Mach 2 from the same altitude gives 85 kilometres against the rule’s 88.5, which is short of it by 3.9 percent. That agreement is the evidence for believing the Mach 1.4 answer, and it also explains it, because the Mach 1.4 carpet is only 69.6 percent of the Mach 2 one from the same height. Carried to two decimals, the Mach 2 carpet is 85.03 kilometres, the Mach 1.4 carpet is 59.17 and the rule gives 55 statute miles from 55,000 feet.
\[1 - \frac{85.03}{55 \times 1.609344} = 1 - \frac{85.03}{88.51} = 3.9\% \qquad\qquad \frac{59.17}{85.03} = 69.6\% \tag{25}\]Flying slower narrows the carpet as well as quietening it.
Nor does the boom arrive underneath the aeroplane. The straight-down ray lands 19.29 kilometres behind, so a person hearing the thump is hearing an aeroplane that passed overhead some seconds earlier and is already well down the track.
The Published Speed Checks Out
Multiplying the design Mach number by the speed of sound at the cruise altitude gives the following.
\[V = M\, c(z_a) = 1.4 \times 295.07 = 413.1\ \mathrm{m/s} = 924.1\ \mathrm{mph} \tag{26}\]NASA has published 925 miles per hour for this aeroplane on its public pages and 940 miles per hour in the contract announcement [NASA Release 18-020] [Quesst, the Aircraft]. The first reproduces from Mach 1.4 at the cruise altitude to within 0.1 percent. The second corresponds to Mach 1.42 at the same altitude.
The two published speeds are therefore the same physical statement at two roundings, unlike the result the X-58 reached, where two published speeds turned out to refer to two different altitudes. The check is worth running precisely because it does not always pass.
Loudness Is Set by the Rise Time, Not by the Overpressure
A pressure signature is converted to a level on the decibel scale in the ordinary way [Kinsler, Frey, Coppens and Sanders, Fundamentals of acoustics].
\[L_p = 20 \log_{10} \frac{\Delta p}{p_r} \tag{27}\]The overpressure a person experiences is not the one the aeroplane sent. The ground is a nearly rigid boundary, so the incident wave and its reflection superpose and the pressure at a listener’s ear is close to twice the incident value.
\[\Delta p_g = K_R\, \Delta p_i \tag{28}\]A perfectly rigid surface gives exactly two, because the normal velocity must vanish there and the reflected wave arrives in phase with the incident one. Real ground is not perfectly rigid. Flight tests have recorded reflection factors from about 1.8 to 2.0, the lower values generally over sandy or marshy terrain [Carlson 1978], and this article uses 1.9, inside that range.
That value need not be left as a convention. For a locally reacting surface at normal incidence the reflected pressure follows from the ratio of the ground’s specific acoustic impedance to the air’s [Pierce, Acoustics] [Kinsler, Frey, Coppens and Sanders, Fundamentals of acoustics].
\[R_g = \frac{\sigma - 1}{\sigma + 1} \qquad\qquad K_R = 1 + R_g \qquad\Longrightarrow\qquad \sigma = 19 \tag{29}\]A factor of 1.9 corresponds to a normalised impedance of 19, and the measured low end of 1.8 to an impedance of 9, by the same relation. The conventional constant therefore describes the soil, which agrees with the lower factors having been recorded over softer ground. Nor is the factor constant where the incidence is oblique, as it is for a boom arriving at the edge of its carpet, and it departs furthest from two where the geometry is awkward, near the threshold Mach number and at the edge of the carpet [Thomas 1971b].
Every sonic boom overpressure in this article is a ground value and already contains this factor, so the article never applies it twice. Read backwards, a ground overpressure of 0.3 pounds per square foot implies an incident overpressure of 0.158 pounds per square foot before the ground doubled it, which is 7.6 pascals.
Concorde’s boom, at the 1.94 pounds per square foot NASA publishes for Mach 2 at 52,000 feet [NASA Facts FS-2003-11-016, Sonic Booms], is 92.9 pascals and therefore a peak sound pressure level of 133.3 decibels. A review of the subject gives it as approximately 2 [Page and Loubeau 2019], and the difference is under a decibel. No source read here gives an overpressure predicted for the X-59. The figure attached to low-boom aeroplanes is 0.3 pounds per square foot, the shock goal of the Quiet Supersonic Platform programme of 2001 to 2003 as the builder’s programme manager listed it [Iosifidis, Concept Development of the Quiet Supersonic Technology Aircraft], and this article uses it as an illustration of a shaped boom rather than as this aeroplane’s figure. It is 14.4 pascals and 117.1 decibels. Those two numbers are almost useless.
They are almost useless because a boom is not a tone and the ear does not integrate it that way. What the ear responds to in an impulsive event is the high-frequency content, and the high-frequency content of a shock-fronted wave is governed by how quickly the front rises. Peak pressure sets how much energy arrives. Rise time sets where in the spectrum it lands.
What Classical Theory Says the Rise Time Should Be
A weak shock in a viscous, heat-conducting gas reaches a steady thickness when nonlinear steepening is balanced by diffusion. The balance is the classical Taylor result [Hamilton and Blackstock, Nonlinear acoustics].
\[\delta = \frac{4\, \delta_d\, \rho_0\, c_0}{(\gamma + 1)\, \Delta p} \qquad\qquad \tau = \frac{\delta}{c_0} \tag{30}\] \[\delta_d = \frac{1}{\rho_0} \left[ \frac{4}{3}\eta + \eta_B + \kappa \left( \frac{1}{c_v} - \frac{1}{c_p} \right) \right] \tag{31}\]The bracket is the shear and bulk viscosities and the thermal conduction term together [Volume viscosity]. For air at sea level the diffusivity of sound is about 3.658 times ten to the minus five square metres a second. Evaluating the thickness for the illustrative low-boom overpressure gives 1.769 millimetres, and therefore a rise time as follows.
\[\tau = 5.2\ \text{microseconds} \qquad\text{against a measured}\qquad \tau \approx 1\ \text{millisecond} \tag{32}\]Classical theory is wrong here by a factor of about 192, which is 2.3 orders of magnitude. Measured sonic boom rise times are of the order of a millisecond, not a microsecond, and the discrepancy has been known since booms were first recorded properly [Maglieri and Sothcott 1990] [Cleveland and Blackstock 1994] [Holst-Jensen 1981]. The discrepancy is neither a small correction nor an experimental error.
Two mechanisms are held responsible. The first is molecular relaxation. Nitrogen and oxygen molecules exchange energy between translational and vibrational modes on timescales far longer than the viscous one, that exchange is a much larger absorption than viscosity at the frequencies concerned, and it depends strongly on humidity [Pierce 1970]. The second is atmospheric turbulence, which distorts the wavefront on the way down and smears the arrival in time [Pierce 1971] [Raspet et al. 1994a]. Both are real, both are documented, and the relative contribution of the two is not settled, and at least one study concludes outright that molecular relaxation alone is insufficient to explain the observed shock structure [Gionfriddo et al. 1991].
The same shortfall appears in the absorption coefficient, which the same physics governs. Classical absorption from viscosity and conduction alone rises as the square of the frequency.
\[\alpha = \frac{\omega^2 \delta_d}{2 c_0^3} \qquad\Longrightarrow\qquad 0.159\ \text{decibels a kilometre at}\ 1,000\ \text{hertz} \tag{33}\]That is 0.159 decibels a kilometre at 1,000 hertz, against a tabulated atmospheric value of 5.6 at 20 degrees Celsius and 40 to 70 percent relative humidity [Smith, Physical Audio Signal Processing, Air Absorption], so classical theory is short by a factor of about 35. The rise time and the absorption are the same failure counted twice, and the mechanism that repairs both is molecular relaxation.
The consequence for this aeroplane is direct. The rise time is the quantity that sets the loudness, the rise time is set by the atmosphere rather than by the airframe, and the atmosphere is not under the designer’s control. A predicted perceived level is therefore a prediction about the weather as much as about the aeroplane, and the programme’s plan is to fly over communities and survey the people underneath rather than to rely on computation [Quesst, NASA’s mission to enable commercial supersonic flight].
The Front Is Not Allowed to Steepen, and That Fixes Its Rise Time From Below
A compression steepens as it travels, because the crests of a finite-amplitude wave move faster than its troughs. A front that rises over a time $\tau$ becomes a discontinuity after a distance that falls as the amplitude rises, and inverting that gives the least rise time a front may have and still arrive unshocked [Hamilton and Blackstock, Nonlinear acoustics] [Whitham, Linear and nonlinear waves].
\[x_s = \frac{2\,\rho_0\, c_0^3\, \tau}{(\gamma + 1)\, \Delta p} \qquad\Longrightarrow\qquad \tau_m = \frac{(\gamma + 1)\, \Delta p\, z_a}{2\,\rho_0\, c_0^3} \tag{34}\] \[\tau_m = 5.99\ \text{milliseconds} = 8.2\%\ \text{of the shortest possible signature} \tag{35}\]Evaluated at sea level conditions as an indicative bound, the front must take about 5.99 milliseconds to rise. The bound is indicative because the propagation is through an atmosphere whose density and sound speed vary by a factor of several along the path, and evaluating it at the ground overstates the steepening near the aeroplane and understates it near the listener. It should be read as an order of magnitude.
The bound matters because of a comparison. The shortest signature this aeroplane can produce is its length divided by its speed, and the front alone must occupy 8.2 percent of it. A nose that is almost a third of the aeroplane [NASA, Lockheed Martin Reveal X-59 Quiet Supersonic Aircraft] is not a stylistic choice.
The Signature Is a Waveform and the Peak Is One Number From It
The classical far-field signature is a straight line from the front shock down through zero to an equal rear shock, which is what gives the N-wave its name [Whitham, Linear and nonlinear waves] [Maglieri and others, Sonic boom, six decades of research].
\[p(t) = \Delta p \left( 1 - \frac{2t}{T_s} \right), \qquad 0 \le t \le T_s \tag{36}\]Its duration has a floor. Nothing can make the signature shorter than the time the aeroplane takes to travel its length, because the front of the disturbance is made at the nose and the rear at the tail.
\[T_s \ge \frac{\ell}{V} = \frac{30.18}{413.1} = 73\ \text{milliseconds} \tag{37}\]From the waveform and the duration follow the two integral quantities that a structure and an ear respectively respond to [Kinsler, Frey, Coppens and Sanders, Fundamentals of acoustics] [SAE ARP865].
\[I_p = \int_0^{T_s/2} p(t)\,\mathrm{d}t = \frac{\Delta p\, T_s}{4} \tag{38}\] \[E_s = \int_0^{T_s} p^2(t)\,\mathrm{d}t = \frac{(\Delta p)^2 T_s}{3} \qquad\qquad L_E = 10 \log_{10} \frac{E_s}{p_r^2 \times 1\,\text{s}} = 101 \tag{39}\]At the published overpressure and the shortest possible duration the sound exposure level is 101 decibels. That is a third metric alongside the peak level of 117.1 decibels and the perceived level of 75, and the three are not translations of one another. A metric that integrates over the event rewards a short signature, a metric that reads the peak ignores the duration entirely, and only the perceived level attempts to weight by what the ear does with the spectrum.
Why Rise Time Governs Loudness
Model the front of the signature as a smoothed step of rise time $\tau$. Its Fourier amplitude falls as one over frequency until the smoothing takes hold, and faster thereafter [Pierce, Acoustics] [Pierce 1970].
\[|P(f)| \propto \frac{1}{f \sqrt{1 + (2\pi f \tau)^2}} \tag{40}\] \[f_c = \frac{1}{2\pi\tau} \tag{41}\]A rise time of one millisecond places the corner at 159.2 hertz. A rise time of ten milliseconds places it at 15.9 hertz. At 1,000 hertz, which lies inside the band where human hearing is most sensitive, the difference between those two signatures is 19.9 decibels. The result is not special to that frequency. Well above both corners the amplitude falls as the reciprocal of the rise time, so a tenfold rise time is worth twenty decibels at every such frequency, and the evaluation at a kilohertz illustrates a general result.
A factor of ten in rise time is worth twenty decibels where it matters, and costs nothing in peak pressure. That one line carries the argument for shaping, and it explains why a programme aimed at loudness spends its effort on the form of the waveform rather than on its height.
Seventy-Five Is a Perception and Not a Pressure
The number the programme is built around is 75 on the Stevens Mark VII perceived level scale [NASA Release 18-020] [Doebler and Rathsam 2020]. That scale is a psychophysical construction, and treating it as though it were an acoustic measurement of pressure produces mistakes that are widely repeated.
The Mark VII scale is defined so that loudness in sones doubles for a fixed increment of level [Stevens 1972] [Stevens, Psychophysics, introduction to its perceptual, neural and social prospects] [Sone].
\[PL = 40 + 32 \log_2 S \qquad\Longleftrightarrow\qquad S = 2^{(PL - 40)/32} \tag{42}\]One sone is forty decibels of perceived level by construction, and a factor of two in loudness costs 32 decibels. Evaluating the two figures that are usually compared gives the following.
\[S(75) = 2.134 \qquad\qquad S(105) = 4.088 \tag{43}\]The programme’s target is 47.8 percent quieter than Concorde in the units the scale is built to express [Fastl and Zwicker, Psychoacoustics, facts and models], and not the ninety percent that has been published [What We Know About the Mysterious Supersonic Lockheed Martin X-59 Quesst, Simple Flying]. And the baseline is not agreed. Published figures for Concorde’s perceived level include 105, which NASA’s acousticians call the commonly quoted value and verified by simulation [Doebler and Rathsam 2020], 108 [NASA unveils a supersonic plane with a quiet boom, Physics Today] and 110 [The Next NASA X-Plane, Born to Boom, Machine Design], which give a range from 47.8 to 53.1 percent. No reading of any published pair reaches ninety.
\[1 - \frac{S(75)}{S(105)} = 47.8\% \qquad\text{against}\qquad 1 - 10^{-30/20} = 96.8\% \tag{44}\]The range follows from the same scale, since the ratio of two loudnesses depends only on the difference of their perceived levels.
\[1 - \frac{S(75)}{S(PL)} = 1 - 2^{(75 - PL)/32} \qquad\Longrightarrow\qquad 1 - 2^{-33/32} = 1 - 0.48929 = 51.1\%, \qquad 1 - 2^{-35/32} = 1 - 0.46854 = 53.1\% \tag{45}\]Where the Ninety Percent Comes From
The overpressures are 1.94 pounds per square foot against 0.3 pounds per square foot, and one minus that ratio is 84.5 percent. That is the nearest of the candidate ratios to the ninety percent in print, a pressure ratio reported as a reduction in loudness. That this ratio is the figure’s origin is inferred here, since none of the sources derives the ninety percent.
The two are not interchangeable and the difference is not small. In pressure the reduction is 96.8 percent. In acoustic energy it is 99.9 percent. In perceived loudness, the quantity in which the target is stated and which a person underneath experiences, it is 47.8 percent. Three defensible answers spanning fifty percentage points, and the ninety percent in print is not any of them. The overpressure ratio and the energy ratio evaluate as follows, the second being the square of the thirty decibel pressure ratio.
\[1 - \frac{ 0.3 }{ 1.94 } = 84.5\% \qquad\qquad 1 - 10^{-30/10} = 99.9\% \tag{46}\]This article does not claim the programme has overstated its case. It claims that a figure quoted without its unit is not a claim at all, and that the difference between halving a loudness and removing ninety-seven percent of a pressure is the difference the Mark VII scale exists to express.
How Much of the Reduction Is Shape
The claimed drop, from the commonly quoted Concorde value to the target of 75, is 30 decibels of perceived level [NASA Release 18-020] [Doebler and Rathsam 2020]. If perceived level tracked peak overpressure the way sound pressure level does, Concorde’s overpressure against the low-boom goal would account for the following.
\[\underbrace{PL_{\text{Concorde}} - PL_{\text{X-59}}}_{ 30 } \;=\; \underbrace{20\log_{10}\dfrac{\Delta p_i(\text{Concorde})}{\Delta p_i(\text{X-59})}}_{ 16.2 } \;+\; \underbrace{\text{what the shape contributes}}_{ 13.8 } \tag{47}\]About 54 percent of the claimed reduction is amplitude and the rest is shape. The amplitude term is 16.2 decibels and the residual is 13.8.
This decomposition is a construction of this article, resting on an assumption the sources do not make. Perceived level is not a function of peak overpressure, as the preceding section argues, so treating the amplitude term as though it obeyed the decibel rule is a device for splitting the total rather than a physical statement. It is stated here as inference, and it establishes a bound rather than a value. Even under an assumption that maximally favours amplitude, shape accounts for 13.8 decibels of the claim, and no amount of simply making the aeroplane quieter would have delivered it.
The Shape That Costs, Measured
The obligation to spread the equivalent area smoothly over as long a distance as possible has a visible consequence, and it can be measured against the aeroplanes this one is compared with [Concorde] [Tupolev Tu-144] [North American XB-70 Valkyrie] [North American XB-70 Valkyrie, National Museum of the United States Air Force] [General Dynamics F-16 Fighting Falcon]. The XB-70’s length of 185 feet 10 inches is the museum’s, and its span of 105 feet is the encyclopaedia entry’s, since the museum’s fact sheet gives no span [North American XB-70 Valkyrie, National Museum of the United States Air Force] [North American XB-70 Valkyrie].
\[\Lambda = \frac{\ell}{b} \tag{48}\]| Aircraft | Length over span |
|---|---|
| X-59 | 3.36 |
| Concorde | 2.41 |
| Tupolev Tu-144 | 2.28 |
| North American XB-70 | 1.77 |
| General Dynamics F-16 | 1.51 |
The X-59 is 1.39 times as slender as Concorde and more than twice as slender as a supersonic fighter. It is ninety-nine feet long with a wingspan under thirty [X-59 Aircraft Overview and Status], and the proportion follows from function, being the equivalent area distribution made visible. A boom is shaped by making the aeroplane’s disturbance grow gradually, and an aeroplane can only grow gradually if it has length to grow along.
The length is not free. A longer aeroplane has more wetted area for the same volume and lift, and therefore more skin friction drag, and a structure that must resist bending over a longer span of nothing. The nose contributes no lift, carries no fuel, houses no payload and cannot be shortened, which is a cost accepted for the sake of the length. Length is not the only variable the design has, since the area distribution, the lift distribution, the cruise Mach number and the cruise altitude are all in play and all appear elsewhere in this article. It is the one the nose buys.
Three Published Lengths, and the Builder’s Is the Outlier
NASA’s public description gives the length as 99.7 feet [Quesst, the Aircraft] and the project’s technical overview gives 99 [X-59 Aircraft Overview and Status]. Lockheed Martin’s own product card gives an overall length of 93.83 feet [Silencing the Sonic Boom, X-59 Product Card].
The three span 5.87 feet, which is 5.9 percent, and against the card’s wingspan of 29 feet 6 inches [Silencing the Sonic Boom, X-59 Product Card] they give slendernesses of 3.18, 3.36 and 3.38.
\[\Delta\ell = 99.7 - 93.83 = 5.87\ \text{feet} = 5.9\%\ \text{of}\ 99.7 \qquad\qquad \Lambda = \frac{\ell}{b} = \frac{ 93.83 }{29.5},\ \frac{ 99 }{29.5},\ \frac{ 99.7 }{29.5} = 3.18,\ 3.36,\ 3.38 \tag{49}\]The aeroplane is the most slender of the comparison set on any of the three, so the finding above does not turn on which is used, and the table was computed from the project overview’s figure.
The builder’s card carries a copyright date of 2019 and the project overview was presented in 2023 [Silencing the Sonic Boom, X-59 Product Card] [X-59 Aircraft Overview and Status], so the simplest reading is that the aeroplane grew during detail design. That is this article’s inference and no source states it, and the alternative reading, that the three figures measure to different points on the airframe, is equally consistent with what is published.
The Card’s Weights Do Not Sum to Its Own Total
The same card gives an empty weight of 15,000 pounds, fuel of 8,000 and payload of 600, against a maximum design gross weight of 24,300 [Silencing the Sonic Boom, X-59 Product Card]. Those three sum to 23,600, which is 700 pounds short, or 2.9 percent.
\[\underbrace{ 15,000 }_{\text{empty}} + \underbrace{ 8,000 }_{\text{fuel}} + \underbrace{ 600 }_{\text{payload}} = 23,600\ \text{pounds} \qquad\qquad \frac{ 24,300 - 23,600 }{ 24,300 } = \frac{ 700 }{ 24,300 } = 2.9\% \tag{50}\]The difference is small and its cause is not stated. A gross weight commonly carries a margin, and a card of this kind is not an engineering document. It is recorded because this article’s equivalent-area result is computed from the gross weight, and a reader who prefers to add the parts would get an answer 2.9 percent smaller.
What the Length Costs and What It Saves
The article has so far treated length as a cost, and half of that is wrong. A longer body of the same volume has more wetted area and therefore more friction drag, but its wave drag falls as the fourth power of its length. The minimum wave drag of a body of revolution of given length and volume is the Sears-Haack result [Ashley and Landahl, Aerodynamics of wings and bodies] [Area rule].
\[D_w = \frac{128\, q\, \Omega^2}{\pi\, \ell^4} \tag{51}\] \[D_w \propto \ell^{-4} \qquad\text{against}\qquad D_f \propto S_w \propto \ell^{1/2} \tag{52}\]Against a twenty metre body of the same volume, the X-59’s 30.18 metres carry a wave drag smaller by a factor of 5.18 and a wetted area larger by a factor of about 1.23. An exponent of four beats an exponent of one half, so the exponents favour length heavily, which is the opposite of what the preceding paragraph alone would suggest and is why supersonic aeroplanes were long and thin before anybody cared what they sounded like.
Whether the net drag falls is a different question and this article does not answer it. A factor on the wave drag and a factor on the friction drag do not combine without knowing how the total divides between them at the design point, and that division is not published for this aeroplane. The exponents establish a direction and not a balance, and the paragraph above is worded to claim only the direction.
The friction side of that trade is computable at the cruise condition. The viscosity follows from Sutherland’s law and the Reynolds number from the length and the density already computed [Anderson, Modern compressible flow] [Raymer, Aircraft design, a conceptual approach].
\[\eta(T) = \eta_r \left( \frac{T}{T_r} \right)^{3/2} \frac{T_r + C_S}{T + C_S} \tag{53}\] \[Re = \frac{\rho V \ell}{\eta} = 128.6 \times 10^{6} \tag{54}\] \[C_f = \frac{0.455}{(\log_{10} Re)^{2.58}} = 0.00206 \qquad\qquad D_f = q\, C_f\, S_w \tag{55}\]The skin friction coefficient is 0.00206 and the friction drag is 25.7 newtons for every square metre of wetted area. The total is not computed, because the wetted area is not published, and multiplying by an estimate would produce a drag figure that was a statement about the estimate. That figure is a flat-plate incompressible value and a compressible correction would reduce it, so it is an upper bound on the coefficient and is treated as one.
Range and Thrust, Displayed and Not Evaluated
The cruise performance of the aeroplane follows the ordinary relations [Raymer, Aircraft design, a conceptual approach] [Range (aeronautics)].
\[R_c = \frac{M\, c}{c_t}\; \frac{L}{D}\; \ln \frac{W_i}{W_f} \tag{56}\] \[\frac{F_N}{F_{N0}} = \left( \frac{\rho}{\rho_0} \right)^{n} \tag{57}\]The builder’s card supplies the weights and the remaining uncertainty then collapses into one product. The fuel is 8,000 pounds of a gross 24,300, which is 32.9 percent, so the logarithm in the range relation is 0.3993 and everything else that is known can be gathered into a single coefficient.
\[R_c = \underbrace{V \ln \frac{W_i}{W_f}}_{ 165\ \text{m/s} } \;\cdot\; \frac{L/D}{c_t} \tag{58}\]The form is the ordinary one and only the grouping is this article’s [Range (aeronautics)] [Raymer, Aircraft design, a conceptual approach].
The collected coefficient is 165 metres a second and the two quantities that are not published enter only as their ratio. The lift-to-drag ratio at the design condition and the thrust specific fuel consumption of the installed engine in supersonic cruise are both absent from every source consulted. So the range is not unknown by three unknowns. It is unknown by one.
For scale and for nothing else, a lift-to-drag ratio of 6 and a specific fuel consumption of 1.8 pounds an hour a pound of thrust, neither of which is published and both of which are this article’s assumptions, would give a still-air range of about 1,979 kilometres and an endurance of about 79.9 minutes. Those figures are an illustration of magnitude and are not a result. A mission radius is a different and much smaller quantity, since it must include a climb, an acceleration, a reserve and a return.
The thrust side can be bounded rather than illustrated. The density at the cruise altitude is 12 percent of the sea-level value, and with $n = 1$ the lapse relation above gives the bound directly.
\[\frac{\rho}{\rho_0} = \frac{ 0.14664 }{1.225} = 0.1197 \qquad\Longrightarrow\qquad F_N = F_{N0}\, \frac{\rho}{\rho_0} = 22,000 \times \frac{ 0.14664 }{1.225} = 2,634\ \text{pounds} \tag{59}\]With an exponent of one the engine retains 12 percent of its sea-level thrust at the cruise density, which against the published 22,000 pounds of sea-level thrust is 2,634 pounds [General Electric F414]. That is an indicative bound. A real installation at Mach 1.4 recovers a great deal through ram compression, so the true installed thrust is higher, and the exponent for this installation is not published either.
One thing the relations do settle without numbers. The aeroplane is not required to go far. Its mission is to fly a measured pass over an instrumented area or a surveyed community and return [Quesst, the Aircraft], so range is not a design driver, and the slenderness that costs it skin friction costs it something it does not need. The design trade is unusually clean because the objective is unusually narrow.
Symbols
Every symbol used in a display equation above is declared here, and every symbol declared here is used.
| Symbol | Meaning |
|---|---|
| $A_V$ | the part of the equivalent area contributed by volume alone |
| $A_e$ | the equivalent area distribution |
| $C_S$ | Sutherland’s constant for air |
| $C_f$ | the mean skin friction coefficient |
| $D$ | the drag force |
| $D_f$ | the skin friction drag |
| $D_w$ | the wave drag |
| $E_s$ | the sound exposure of the signature |
| $F$ | Whitham’s F-function, the aircraft’s acoustic signature at the source |
| $F_N$ | the installed net thrust |
| $F_{N0}$ | the installed net thrust at sea level and the same Mach number |
| $I$ | the acoustic intensity |
| $I_p$ | the positive-phase impulse of the signature |
| $K_R$ | the ground reflection factor |
| $L$ | the lift force |
| $L’$ | the lift per unit length along the aircraft |
| $L_E$ | the sound exposure level |
| $L_p$ | the sound pressure level |
| $M$ | the flight Mach number |
| $M_c$ | the cutoff Mach number, below which no boom reaches the ground |
| $P$ | the Fourier amplitude of the pressure signature |
| $PL$ | the Stevens Mark VII perceived level, in decibels |
| $R$ | the specific gas constant of air |
| $R_c$ | the still-air cruise range |
| $R_g$ | the pressure reflection coefficient at the ground |
| $Re$ | the Reynolds number based on overall length |
| $S$ | the total loudness of the signature, in sones |
| $S_w$ | the wetted area |
| $T$ | the absolute air temperature |
| $T_0$ | the sea-level standard temperature |
| $T_1$ | the temperature of the isothermal layer above the tropopause |
| $T_r$ | the reference temperature in Sutherland’s law |
| $T_s$ | the duration of the ground signature |
| $V$ | the true airspeed |
| $W$ | the aircraft weight |
| $W_f$ | the aircraft weight at the end of cruise |
| $W_i$ | the aircraft weight at the start of cruise |
| $\Delta p$ | the peak overpressure of the pressure signature |
| $\Delta p_g$ | the overpressure at the ground after reflection |
| $\Delta p_i$ | the overpressure incident on the ground before reflection |
| $\Delta\ell$ | the spread between the longest and shortest published overall lengths |
| $\Lambda$ | the slenderness ratio, overall length divided by span |
| $\Omega$ | the volume of the equivalent body of revolution |
| $\Pi$ | the acoustic energy per unit length of a cylinder of radius $r$ |
| $\alpha$ | the classical absorption coefficient |
| $\beta$ | the Prandtl-Glauert parameter, $\sqrt{M^2-1}$ |
| $\delta$ | the thickness of a steady weak shock |
| $\delta_d$ | the diffusivity of sound in air |
| $\ell$ | the overall length of the aircraft |
| $\eta$ | the shear viscosity of air |
| $\eta_B$ | the bulk viscosity of air |
| $\eta_r$ | the reference viscosity in Sutherland’s law |
| $\gamma$ | the ratio of specific heats of air |
| $\kappa$ | the thermal conductivity of air |
| $\lambda$ | the tropospheric temperature lapse rate |
| $\mu$ | the Mach angle |
| $\omega$ | angular frequency |
| $\phi$ | the azimuth of a ray around the Mach cone, measured from the horizontal |
| $\phi_c$ | the azimuth below which a ray no longer reaches the ground |
| $\rho$ | the local air density |
| $\rho_0$ | the air density at sea level |
| $\sigma$ | the normalised specific acoustic impedance of the ground |
| $\tau$ | the rise time of the front shock |
| $\tau_m$ | the least front rise time that avoids coalescence over a given distance |
| $\theta$ | the elevation angle of an acoustic ray above the horizontal |
| $\xi$ | a dummy variable of integration in $x$ |
| $\zeta$ | the retarded streamwise coordinate in which the F-function is written |
| $b$ | the wing span |
| $c$ | the local speed of sound |
| $c_0$ | the speed of sound at sea level |
| $c_p$ | the specific heat of air at constant pressure |
| $c_t$ | the thrust specific fuel consumption |
| $c_v$ | the specific heat of air at constant volume |
| $d_e$ | the diameter of a circle whose area is the lift-induced equivalent area |
| $f$ | frequency |
| $f_c$ | the corner frequency above which a shock’s spectrum steepens |
| $g$ | standard gravitational acceleration |
| $n$ | the exponent of the thrust lapse with density |
| $p$ | the acoustic pressure |
| $p_1$ | the pressure at the tropopause |
| $p_a$ | the ambient pressure at the aircraft |
| $p_r$ | the reference pressure of the decibel scale, 20 micropascals |
| $q$ | the free-stream dynamic pressure |
| $r$ | distance from the aircraft measured along the ray |
| $t$ | time measured from the arrival of the front shock |
| $u$ | the horizontal slowness of a ray, which is invariant in a stratified atmosphere |
| $x$ | the streamwise coordinate along the aircraft |
| $x_s$ | the distance in which a compression steepens into a shock |
| $y$ | lateral distance from the ground track |
| $y_c$ | the half-width of the boom carpet |
| $z$ | altitude above sea level |
| $z_1$ | the altitude of the tropopause |
| $z_2$ | the altitude at which the isothermal layer ends |
| $z_a$ | the aircraft’s cruise altitude |
Dependent Systems
The systems this aeroplane depends on are, wherever the shape permits it, systems taken from other aeroplanes, and the one that is new is the one the shape forced. The parts inventory comes from the project’s annotated configuration [X-59 Aircraft Overview and Status]. The new system is the camera and display that replace a forward window, because a nose long enough to shape a shock is a nose the pilot cannot see past.
The Aeroplane Is Mostly Other Aeroplanes
NASA’s project overview annotates the configuration feature by feature, and the annotations divide cleanly into two kinds [X-59 Aircraft Overview and Status].
Four are about the boom. A long nose to shape the forward shock. A T-tail to minimise the aft shock. Wing shielding to keep inlet spillage from spoiling the signature. A fixed canard for nose-up trim at the low-boom design point [X-59 Aircraft Overview and Status].
Four are about cost and schedule, and say so in those words. F-16 landing gear and other systems from high performance aircraft, to minimise qualification cost and schedule. A T-38 aft canopy and ejection seat, for the same stated reason. Large unitised skins, to reduce parts count and manufacturing cost. A single General Electric F414 engine with a standard nozzle, to minimise cost and schedule. A conventional tail arrangement is annotated separately, to simplify stability and control considerations [X-59 Aircraft Overview and Status].
Apart from its shaping, the aeroplane that exists to prove a shape is assembled from parts chosen for not being new. NASA’s account of the build adds a propulsion system part from a U-2 and a control stick from an F-117 to the list [X-59 QueSST More than the Sum of Its Parts].
The same contractor has now had cost compression written into an X-plane’s register entry. The X-55 is described in the register as validating extreme time and cost compression in airframe manufacture using large unitised composite structures [Parsch, DOD 4120.15-L Addendum]. The X-59 uses large unitised skins for the stated reason of reducing parts count and manufacturing cost [X-59 Aircraft Overview and Status]. The register does not say so about this aeroplane and the project overview does, and both aeroplanes are Lockheed Martin’s.
Lockheed Martin holds 6 of the 31 X rows in the register and Boeing holds 13, so the leading contractor in this series is not this one. 5 of Lockheed’s 6 were allocated in 2009 or later, which is where the recent concentration is [Parsch, DOD 4120.15-L Addendum].
Seeing Out of an Aeroplane With No Forward Window
A nose long enough to shape a shock is a nose the pilot cannot see past. The X-59 has no forward windscreen [NASA, Lockheed Martin Reveal X-59 Quiet Supersonic Aircraft]. Forward vision is provided by the eXternal Vision System, which NASA describes as a 4K camera system feeding a processor and an ultra-high-definition display with integrated symbology [X-59 Aircraft Overview and Status].
The system was validated in flight before the aeroplane existed. NASA installed X-59 hardware on a UC-12 and had guest pilots compare normal vision against the system on see-to-avoid and see-to-follow tasks [X-59 Aircraft Overview and Status]. A camera and a screen were qualified as a substitute for a window, on a different aeroplane, by pilots doing the tasks a window is for.
The claim is a considerable one. The aeroplane’s forward vision is a rendering. The synthetic and enhanced vision literature this substitution belongs to runs in this survey from 1996 and is mostly about supplementing a view rather than replacing one, as the cluster on seeing out counts below.
The Flight Test Record
The X-59 first flew on 28 October 2025, taking off from Lockheed Martin’s Palmdale facility and landing at NASA Armstrong at Edwards [X-59 First Flight]. The flight lasted 67 minutes, reached about 12,000 feet and about 230 miles an hour, and the landing gear stayed down throughout [X-59 First Flight]. Against the published design cruise of 925 miles an hour at 55,000 feet [Quesst, the Aircraft], the two fractions are the following.
\[\frac{ 230\ \text{mph} }{ 925\ \text{mph} } = 24.9\% \qquad\qquad \frac{ 12,000\ \text{ft} }{ 55,000\ \text{ft} } = 21.8\% \tag{60}\]That is 24.9 percent of the design cruise speed, at 21.8 percent of the design cruise altitude, with the gear extended. As of this article’s date, 37 days later, the aeroplane had made 1 flight and had never been supersonic [NASA’s X-59 Completes Second Flight].
The register’s claim about it is therefore still a claim about the future, and the register wrote it in the future tense. The aeroplane that will create a shaped sonic boom signature has not yet created one. It has not yet made a sonic boom of any kind.
Against the Schedule the Programme Published
NASA’s contract announcement stated that work ran through 31 December 2021, that the agency anticipated accepting the aeroplane in late 2021, and that flights over selected United States cities would begin in mid-2022 [NASA Release 18-020].
First flight came 3.82 years after the stated end of the contract, 7.35 years after the designation was allocated, and 655 days after the aeroplane was unveiled on 12 January 2024 [NASA, Lockheed Martin Reveal X-59 Quiet Supersonic Aircraft]. The community overflight phase, which was to begin in mid-2022, had not begun [X-59 First Flight].
This records a fact about the schedule and passes no judgement on the programme. A one-off research aeroplane with a novel configuration and no forward windscreen is not a schedule-predictable object, and the series has now documented several. What matters for this article is narrower. The central claim in the register, and the central claim of the programme, is a claim about a pressure signature at the ground, and not one such signature had yet been produced.
Comparison With Ground Prediction
At the editorial date there was nothing from supersonic flight to set beside any ground prediction. Every figure this article examines for the boom was stated or computed before the aeroplane was supersonic, and the aeroplane had not yet been supersonic.
The predictions are on record. The programme states a design cruise of Mach 1.4 at 55,000 feet, published speeds of 925 and 940 miles an hour, and a target of 75 perceived level decibels [NASA Release 18-020] [Quesst, the Aircraft]. The sources give no predicted ground overpressure, and the 0.3 pounds per square foot attached to low-boom aeroplanes is the shock goal of the Quiet Supersonic Platform programme [Iosifidis, Concept Development of the Quiet Supersonic Technology Aircraft]. This article adds predictions of its own from the standard atmosphere, a cutoff Mach number of 1.153, a boom carpet 59.2 kilometres wide and a boom arriving 19.29 kilometres behind the aeroplane, all argued under Sizing From First Principles.
What flight returned lies outside every one of those predictions. The single flight of 28 October 2025 lasted 67 minutes and reached about 12,000 feet and about 230 miles an hour with the landing gear down [X-59 First Flight]. That is 24.9 percent of the design cruise speed at 21.8 percent of the design cruise altitude, as computed in The Flight Test Record, a regime in which no boom forms, so it neither confirms nor contradicts any prediction about the signature.
One prediction was flown, on a different aeroplane. The eXternal Vision System was flown on a UC-12 before the X-59 existed, with guest pilots comparing normal vision against the system on see-to-avoid and see-to-follow tasks [X-59 Aircraft Overview and Status]. The source reports that work as a validation and this article does not report its measures, so the comparison is recorded as made but unquantified.
Some ground figures disagree with one another before any flight. The three published lengths span 5.9 percent, as the comparison of the three lengths records. The card’s weights fall 700 pounds short of its own total, a shortfall recorded where the card’s weights are summed. The two published speeds reconcile to within rounding, as checked above. The comparison the programme exists to make, a measured ground signature against the target of 75, awaited supersonic flight, and the subsection What Classical Theory Says the Rise Time Should Be explains why any prediction it is compared against is partly a prediction about the atmosphere on the day.
What the Data Changed
Flight had so far returned no data bearing on the research question, so the data had changed nothing. The single flight was subsonic and made no boom, as the flight test record above shows.
The document the data are meant to change had not changed either. The aeroplane’s output is meant to inform a rule, and the rule is read in full here because its text is the measure of whether anything has moved. On the editorial date the prohibition stood in the form it took on 15 January 2021, past the deadline Executive Order 14304 set for its repeal [14 CFR 91.817] [Executive Order 14304].
The Rule It Exists to Change Contains No Loudness at All
The X-54 article sets out the essential point, and this article does not claim it. The prohibition is a speed limit and not a noise limit, so a quiet aeroplane cannot satisfy it, and any aeroplane wanting to fly supersonic over land needs the rule replaced rather than met. This article adds a reading of the exemption path, because a threshold, if there were one, would live there.
The prohibition is a single paragraph, quoted here in full [14 CFR 91.817].
No person may operate a civil aircraft in the United States at a true flight Mach number greater than 1 except in compliance with conditions and limitations in an authorization to exceed Mach 1 issued to the operator in accordance with § 91.818.
The rule bans a speed. It does not ban a sound, it names no threshold, it contains no acoustic metric, and it makes no distinction between an aeroplane that produces a hundred and ten decibels at the ground and one that produces seventy-five.
The second paragraph does mention the boom, and what it requires is its absence.
Information available to the flight crew includes flight limitations that ensure that flights entering or leaving the United States will not cause a sonic boom to reach the surface within the United States.
Not a quiet boom. No boom.
The Exemption Is Written the Same Way
The authorisation process is the companion section. It was added in its present form on 15 January 2021, which is 2.6 years after this aeroplane was designated and while it was being built [14 CFR 91.818]. It requires each applicant to state
All conditions and limitations on the flight(s) that will ensure that no measurable sonic boom overpressure will reach the surface outside of the proposed flight area.
and it lists the purposes for which an authorisation may be granted at all. One of them is
Establish a means of reducing or eliminating the effects of sonic boom, including flight profiles and special features of an aircraft.
That clause describes this aeroplane’s purpose exactly, since the X-59 is a special feature of an aircraft intended to reduce the effects of sonic boom. Whether the X-59 needs the clause is a separate question, and the record read here leaves it open. The prohibition applies by its own words to a civil aircraft, the statute defines a civil aircraft as any aircraft except a public aircraft and counts among public aircraft one used only for the United States Government, subject to qualifications set out elsewhere in the code [49 U.S.C. 40102, Definitions], and the rulemaking never names this aeroplane, the phrase X-59 and the programme’s name together occurring 0 times across the 5 documents read, against 16 mentions of the agency. Most of the wording predates the programme by forty-five years. The appendix of 1973 that this section replaced already required conditions ensuring that no measurable sonic boom overpressure would reach the surface outside the designated test area, and already allowed a flight necessary to establish means of reducing or eliminating the effects of sonic boom [Civil Aircraft Sonic Boom, final rule]. The words including flight profiles and special features of an aircraft are not in the 1973 text, and whether they were added with this aeroplane in view is unresolved.
Another clause permits flight outside a test area only on a conservative showing that no measurable overpressure reaches the surface [14 CFR 91.818]. Every path through the regulation is written around a boom that does not arrive.
So the regulatory situation is not that the X-59 will be quiet enough to satisfy the rule. There is no number in the rule to satisfy. The aeroplane’s product is not compliance. It is evidence, gathered so that a rulemaking might one day write a number where at present there is none, and the number would be new law rather than a threshold already waiting.
This reframes what the register sentence is claiming. An aeroplane that will create a shaped sonic boom signature is not an aeroplane built to pass a test. It is an instrument built to generate the data from which a test might be constructed. The X-59 is a measuring device whose specimen is the public.
The Deadline Expired the Day Before This Article
Executive Order 14304 of 6 June 2025 directed the Federal Aviation Administration to repeal the prohibition within 180 days and to establish an interim noise-based certification standard in its place [Executive Order 14304] [Executive Order 14304, American Presidency Project]. The X-54 article records that the deadline falls on 3 December 2025.
And the order does not contain a number either. It directs that an interim noise-based certification standard be established and does not say what it should be, leaving the threshold to a notice of proposed rulemaking it requires within eighteen months [Executive Order 14304, American Presidency Project]. So the rule names no loudness, the exemption path requires no boom, and the order that would replace the rule declines to say what loudness would be acceptable. Three documents, and the number the aeroplane was built to justify appears in none of them.
This article is dated 1 day after that deadline, and what happened is that nothing did. The text of both sections quoted above carries no amendment later than 15 January 2021 [14 CFR 91.817] [14 CFR 91.818], so the prohibition stood unchanged on the day the order required it to be gone. No repeal, no interim standard, and no number.
That reports the state of the record without predicting what follows. A rulemaking is a slow instrument and an executive deadline is not a statute, so an expired deadline is an ordinary event rather than a scandal. For this article the expiry matters in one way. The aeroplane’s entire purpose is to supply evidence for a standard, the order that demanded the standard has come and gone without one, and there is still nothing in the regulation for a quiet aeroplane to be quiet enough for.
The Governing Documents Were Counted and None Uses the Metric
This article read five documents in full and counted. The two sections of the regulation, the 2021 final rule that put them in their present form, the 2020 proposal to set noise standards for supersonic aeroplanes, and the one authorisation to exceed Mach 1 that has been granted and noticed since [14 CFR 91.817] [14 CFR 91.818] [Special Flight Authorizations for Supersonic Aircraft, final rule] [Noise Certification of Supersonic Airplanes, proposed rule] [Petition for Authorization to Exceed Mach 1, notice of decision to grant].
Across 206,232 characters, the phrase sonic boom appears 62 times and the programme’s own metric appears 0. Neither the abbreviation nor the words behind it occur once, in any of the five.
The federal noise standards for aircraft are written in a different metric entirely, being effective perceived noise decibels, developed for the sustained noise of a jet transport passing overhead rather than for an impulse lasting a fraction of a second [SAE ARP865] [14 CFR Part 36]. A rulemaking that adopted the programme’s number would be adopting a metric as well as a threshold, and the correlation between that metric and human response to booms is itself a research result rather than a regulatory fact.
The Agency Restated the Absolute Position in 2021, and Industry Had Asked It Not To
The 2021 rulemaking writes the reasoning down, unambiguously enough to quote [Special Flight Authorizations for Supersonic Aircraft, final rule].
The FAA notes that the determination made in the 1970s that no level of sonic boom is acceptable over land still applies and is not based exclusively on economics.
No level. That sentence was written on 15 January 2021, 2.6 years into the designation and with the aeroplane under construction.
The industry had asked for the opposite. The same preamble records that several prospective supersonic airframe and engine manufacturers asked for the no-measurable-overpressure provision to be removed as overly restrictive and outdated, one of them observing that the provision does not recognise the possibility of a boom barely noticeable on the ground but still detectable by scientific measurement, and two trade associations reading the phrase as an absolute prohibition that an applicant could not guarantee during a test flight [Special Flight Authorizations for Supersonic Aircraft, final rule]. The agency retained it.
The Only Noise Standard Ever Proposed for These Aeroplanes Governs Them at Subsonic Speed
In 2020 the agency proposed to bring supersonic aeroplanes into the noise certification regulations, and the proposal says what it covers [Noise Certification of Supersonic Airplanes, proposed rule].
This proposal is limited to establishing procedures and noise levels for subsonic operation of supersonic aircraft during landing and takeoff.
A supersonic aeroplane is to be certificated for the noise it makes while flying subsonically near an airport. The boom is not in it, and the 2021 rule says so in as many words, recording that neither it nor the noise proposal alters the general prohibition on supersonic flight over land [Special Flight Authorizations for Supersonic Aircraft, final rule].
The proposal was never adopted. A search of the Federal Register for the agency’s documents that mention supersonic flight, from the start of 2016 to the date on this article, returns one final rule about supersonic aeroplanes, the 2021 authorisation rule, and no final rule on the noise certification proposal of 2020 [Federal Register search, FAA documents mentioning supersonic, 2016 to 4 December 2025].
The Exemption Path Has Been Used Three Times
The Federal Register has noticed 3 petitions to exceed Mach 1 on or before this article’s date, in 2003, 2016 and 2024. The first came 30 years after the prohibition took effect on 27 April 1973 [Petition for Authorization to Exceed Mach 1, summary of petition received, 2003] [Petition for Authorization to Exceed Mach 1, summary of petition received, The Boeing Company, 2016] [Petition for Authorization to Exceed Mach 1, notice of decision to grant] [Civil Aircraft Sonic Boom, final rule].
The most recent was granted, and what it authorises is instructive. Boom Supersonic was permitted up to twenty supersonic test flights over one year, at or above thirty thousand feet, over Edwards Air Force Base within pre-existing supersonic corridors, after an environmental assessment and a finding of no significant impact [Petition for Authorization to Exceed Mach 1, notice of decision to grant].
That is the exemption path working as designed, and it grants supersonic flight over a military range rather than over anybody’s town. That contrast states the difficulty. The X-59’s mission requires the thing the rule has no mechanism to permit, being repeated supersonic flight over people who live underneath, and whether the rule reaches this aeroplane is left open in the discussion of the exemption above.
The Contemporary Literature
2,572 references map the field this aeroplane sits in, across 17 clusters. Every row states the number of research records assigned to its cluster, and the counts sum to the total exactly.
| Cluster | Records |
|---|---|
| The aeroplane, the programme and the aeroplanes it descends from | 304 |
| Shaping the signature, which is the whole of the design problem | 343 |
| Predicting the boom before the aeroplane exists | 230 |
| Crossing the atmosphere, which changes the answer | 420 |
| What a boom does to a building | 37 |
| What a boom does to animals, the sea and the ground | 30 |
| How loud it is to a person, which is a different question from how large it is | 325 |
| The airport noise of a supersonic aeroplane | 42 |
| The rule, and the process for changing it | 48 |
| Measuring a boom, on the ground and in the air | 92 |
| Supersonic aerodynamics, which is older than the problem | 254 |
| Getting the air in and the thrust out | 195 |
| Seeing out, and flying it | 79 |
| Range, fuel and the cruise condition | 43 |
| The atmosphere and the flight condition | 63 |
| Reviews, conferences and histories of the sonic boom | 50 |
| Reviews and histories of supersonic flight | 17 |
| Total | 2,572 |
The aeroplane, the programme and the aeroplanes it descends from
This is the cluster that names things. It holds the demonstrator and its community test campaign, the shaped-boom flight demonstrations on the F-5 and the F-15 that came before it, the earlier sonic boom campaigns flown with other aeroplanes, and the supersonic transport and business jet studies in which the design theory was worked out.
304 records. [Vaughn] [Vaughn 2024] [Vaughn et al] [Vaughn and Doebler] [Vaughn] [Abdalla et al. 2020] [Agte et al. 1999] [Cerne] [Loubeau et al] [Whitehead 1998] [Alonso et al. 2002] [Anderson et al. 1994] [Baber and Driver 1977] [Bailey et al. 2020] [Baize 1996] [Baize 1999b] [Banerjee 2019] [Barthelemy et al. 1992] [Bashkirov et al. 2021] [Bashkirov and Veresnikov 2022] [Beissner et al. 1983] [Belchihina et al. 2021] [Benham and Birks 1973] [Berens 2010] [Berton 2022b] [Berton 2022a] [Berton 2023] [Berton et al. 2017] [Bertsch et al. 2026a] [Boeing Co Renton Wa Airplane Div 1964] [Bogue et al. 1995] [Bons et al. 2020] [Bradley et al. 2020] [Brenner 2007] [Brewer 1974] [Brewer and Morris 1975] [Briceno et al. 2002] [Bruckner 2002] [Bucovsky et al. 2003] [Buonanno and Mavris 2005] [Busch and Bridges 2025] [Carpenter and Cizmas 2020] [Carpenter et al. 2025] [Carter et al. 2022] [Castner et al. 2018] [Cheung 1995] [Cheung et al. 1999b] [Choi et al. 2004b] [Choi et al. 2004a] [Chudoba et al. 2006] [Chudoba et al. 2009] [Chung and Alonso 2002] [Chung et al. 2003] [Cliatt et al. 2014] [Coen 2025] [Coen et al. 2017] [Coen et al. 2022] [Coen and Richwine 2017] [Cook et al. 2013] [Covell et al. 1990] [Nickol] [Baize 1999a] [Baize et al. 1999] [Darden et al. 1993] [Daumas et al. 2006] [Rodriguez et al] [Richwine] [Deremaux et al. 2008] [Doebler 2020] [Doebler et al. 2018] [Doebler and Loubeau 2023] [Doebler and Rathsam 2020] [Doebler and Rathsam 2019] [Domack 1992] [Durston et al. 2022] [Durston et al. 2023] [Baals et al. 1960] [Durston] [Durston] [Durston] [Driver 1974] [Driver 1981] [Driver 1978] [Dutton 1967] [Elkin 1961] [Feder 2007] [Fernández et al. 2002] [Fidell and Horonjeff 2019] [Fischler 1964] [Fitzsimmons 1976] [Foss and Bragdon 1973] [Freund et al. 2008] [Freund et al. 2007] [Friedlander et al. 2018] [Friedlander et al. 2017] [Furukawa and Makino 2007] [Gaglio et al. 2021] [Garcia et al. 2024] [Garcia et al. 2019] [Shah et al] [Shah et al] [Geiger 2002] [Gilfert and Gudmundsson 2019] [Gonzalez Gallego et al. 2018] [Gonzalez-Linero 2015] [Graham et al. 2005a] [Graham et al. 2005b] [Busch and Bridges] [Haering et al. 2005a] [Haering 2017a] [Haering and Murray 2004] [Haglund 1991] [Haglund and Ogg 1992] [Hahne 1999] [Heberling 2020] [Hecht 2009] [Henne 2005] [Heppe and Hong 1964] [Herrera and Pak 2007] [Herrmann 2008] [High-Speed Civil Transport Study. 1989] [Hoffman 1977] [Horton and Busch 2024] [Howe et al. 2008a] [Howe et al. 2008b] [Hutchison et al. 1993] [Klos 2025b] [Flamm et al] [Berton] [Berton] [Horton and Busch] [Lonzaga] [Morgenstern and Rallabhandi] [Johnson 1985] [Rathsam] [Rathsam] [Rathsam] [Rathsam] [Rathsam] [Rathsam et al] [Rathsam et al] [Rathsam et al] [Rathsam et al] [Jones 2017] [Page et al. 2020] [Nyborg et al] [Nyborg et al] [Kelly 1980] [Gee et al. 2024] [Kirz et al. 2023] [Kirz et al. 2025] [Kiyici and Aradag 2015] [Knight et al. 2008] [Komadina et al. 2002] [Datta and Lagoudas 2021] [Kramer et al. 2018] [Kukreja 2008] [Kurus and Takahashi 2021] [Lappas and Pergamalis 2019] [Cliatt] [Lazzara 2025] [Li et al. 2024] [Li et al. 2019] [Liszka 1978] [Loftin 1967a] [Loftin 1973] [Long-Davis 1999] [Lonzaga 2019c] [Ozoroski] [Mack 2003b] [Magee et al. 2015] [Maglieri 2011] [Maglieri et al. 2011] [Makino 2014] [Makino and Noguchi 2003] [Marchiano et al. 2003a] [Nemec] [Anderson et al] [Anderson et al. 2026] [Martins et al. 2002] [Martins et al. 2004] [Mascitti 1975] [Mascitti and McLean 1976] [Mavris and Hayden 1996] [McWherter et al. 2009] [Carter] [Carter et al] [Ménéxiadis and Varnier 2008] [Bozeman et al] [Minelli] [Minelli et al. 2013] [Mirzoyan et al. 2004] [Ahaus et al] [Mohammadi et al. 2004] [Morgenstern et al. 2005] [Morgenstern et al. 2015] [Morgenstern 1992] [Moua et al. 2012] [Murakami 2006] [Nangia et al. 2004] [Cruze] [Cruze 2024a] [Cruze 2024b] [Cruze and Ballard] [Cruze et al. 2022] [Nemec et al. 2019] [Numerical Study of Noise 2025] [Nyborg et al. 2025b] [Nyborg et al. 2026b] [Nyborg et al. 2025a] [Olson et al. 1997] [Page et al. 2015] [Page et al. 2014] [Pak 2020] [Pawlowski et al. 2005] [Phan et al. 2003] [Plotkin et al. 2005b] [Porro 2000] [Rallabhandi and Mavris 2006a] [Bailey] [Bailey 2026] [Bailey et al] [Rathsam and Cliatt 2019] [Castner] [Castner] [Castner] [Regis 2026] [Richiardi 2023] [Richwine and Brandon 2018] [Rizzi et al. 2020] [Rodriguez 2007] [Rodriguez et al. 2021] [Rowe et al. 1981] [Rowe 1976] [Sachs et al. 2025] [Salamone 2008] [Salay and Elliott 1993] [Fidell et al. 2020] [Sato et al. 2008] [Satre 1969] [Schlette and Staudacher 2022] [Schnell et al. 2025] [Schuermann et al. 2014] [Schutte et al. 2005] [Seiner and Krejsa 1989] [Setoguchi and Kanazaki 2020] [Shah et al. 2022] [Siclari and Darden 1991] [Simmons et al. 2007] [Smith 2007] [Smolka et al. 2007] [McMillin 1999] [Sparkes 2024] [Spivey et al. 2007] [Williams and Bailey] [Stone et al. 2000] [Stratford 1973] [Sturdza 2007] [Sun and Smith 2017] [Sun and Smith 2018b] [Sun and Smith 2019b] [Sun and Smith 2021] [Supersonic Aircraft 1997] [Supersonic Cruise 1981] [Swan 1969] [Technical Applications 1976] [Thibert and Arnal 2000] [Tice and Martin 1992] [Torenbeek 2020] [Tsuchiya and Kanamori 2025] [Tuttle 1980] [U.S. Supersonic Commercial Aircraft 1997] [Unger et al. 1999] [Dippold 2022] [van der Velden and Torenbeek 1989] [Vaughn et al. 2023] [Velden 1997a] [Veresnikov and Goncharenko 2025] [Vigneron et al. 2008] [Silva et al] [Webb 1972] [Wesoky et al. 1991] [Weston et al. 1999] [Whitehead 1991] [Whurr 2004] [Doebler and Loubeau] [Doebler] [Doebler] [Doebler et al] [Doebler et al] [Doebler et al] [Doebler et al] [Doebler et al. 2022b] [Doebler and Sparrow] [Doebler 2021a] [Doebler 2021b] [Williams et al. 2021] [Williams 2015] [Winblade 1986] [Wing Noise Simulation 2024] [Wintzer et al. 2006] [Wlezien and Veitch 2002] [Yildiz et al. 2022] [Yoshimoto and Uchiyama 2003]
93 of these titles name the X-59, its programme or its mission, counting a title that carries X-59, QueSST, Quesst, LBFD or Low Boom Flight Demonstration, and the earliest dated one is from 2015. 60 of those also name a boom, a thump, noise, a community, annoyance, a dose, a survey or loudness, so most of the aeroplane’s own literature is about the signature it is to make and the people who are to hear it.
Shaping the signature, which is the whole of the design problem
The design problem stated as a design problem. Minimisation, equivalent area, inverse design and the optimisation machinery that turns a target signature into an outer mould line.
343 records. [A Study to Determine 1975] [Abraham et al. 2020] [Akatsuka 2024] [Carr et al. 2023] [Allahverdi et al. 2016] [Alonso and Colonno 2012] [Anderson 2021] [Ribeiro et al] [Ferri 1971] [Argrow et al. 2003] [Arifullin et al. 2025] [Aronstein and Schueler 2004] [Aronstein and Schueler 2005] [Ban et al. 2016] [Ban et al. 2018] [Barr et al. 2018] [Batdorf 1970] [Batdorf 1972] [Berger et al. 2011] [Bliamis et al. 2023] [Boles et al. 2021] [Bonavolontà et al. 2024] [Boonjaipetch et al. 2021] [Bowersox et al. 2003] [Lowe et al] [Brezillon et al. 2011] [Bridges et al. 2018] [Buonanno et al. 2002] [Candon] [Candon et al. 2016] [Carlson 1965] [Carlson 1970] [Carrier et al. 2021b] [Carrier et al. 2001] [Carter et al. 2011] [Chattopadhyay et al. 1995] [Chen et al. 2024] [Chen et al. 2026] [Chen et al. 2023] [Chernyshev et al. 2016] [Chernyshev et al. 2008] [Cheung and Edwards 1992] [Chiba et al. 2007] [Chiba et al. 2008a] [Chima 2011] [Chima et al. 2011] [Choi et al. 2008] [Cliff et al. 1999] [Colonno and Alonso 2010] [Crow and Bergmeier 1996] [Darden 1975a] [Darden 1979] [Darden et al. 1992] [Darden and Shepherd 1992] [Darden 1975b] [Darden and Mack 1976] [Daumas et al. 2007] [Lux et al. 1999] [Demiroglu and Nikbay 2023] [Demiroglu et al. 2021] [Ding et al. 2023] [Dixon and Hunsaker 2023] [Doebler et al. 2023b] [Duan et al. 2026] [Durston et al. 2011] [Durston et al. 2015] [Durston et al. 2014] [Eggert et al. 2024] [Elmiligui et al. 2011] [Farhat et al. 2002b] [Farhat et al. 2004] [Farhat et al. 2002a] [Farhat et al. 2007] [Feng et al. 2014] [Fomin et al. 2011] [Fomin et al. 2013] [SHkarubo 2025] [Foster et al. 2025] [Furukawa et al. 2008] [Gan and Zha 2015a] [Gao and Sun 2025] [George 1968] [George and Seebass 1971] [Gillen and Loth 2015] [Goel and Jawahar 2022] [Goto et al. 2005] [Graziani et al. 2025b] [Guo et al. 2024] [Haas and Kroo 2010] [Hao et al. 2016] [Hartwich et al. 2003] [Heath et al. 2016] [Heath et al. 2017] [Howe 2002] [Howe 2003] [Howe 2005] [Howe 2011] [Imrak et al. 2021] [Ishikawa et al. 2011] [Ishikawa et al. 2019] [Ishikawa et al. 2017] [Ishikawa et al. 2021b] [Jeong et al. 1998] [Jim et al. 2021] [Jim et al. 2022] [Jones et al. 2006] [Jung et al. 2011] [Kanamori et al. 2014] [Kandil et al. 2006] [Kane 1973b] [Karman 2011] [Karr et al. 2003] [Kasuga et al. 2016] [Kasuga et al. 2017] [Khasdeo et al. 2007] [Kim et al. 2011] [Kim et al. 2002] [Kim et al. 2001a] [Kirz 2018] [Kirz 2019] [Kirz 2022b] [Kishi et al. 2017] [Kishi et al. 2023] [Knight 2022] [White et al] [Lazzara et al. 2019] [Lazzara et al. 2020] [Lazzara et al. 2021] [Leatherwood and Sullivan 1994a] [Leatherwood and Sullivan 1992b] [Li 2015] [Li et al. 2009] [Li et al. 2025b] [Li and Geiselhart 2020] [Li and Geiselhart 2021b] [Li and Geiselhart 2021a] [Li and Geiselhart 2022] [Li and Geiselhart 2023] [Li and Geiselhart 2024a] [Li and Geiselhart 2024b] [Li et al. 2025a] [Li and Geiselhart 2026] [Li and Rallabhandi 2011] [Li and Rallabhandi 2014] [Li et al. 2008] [Li et al. 2010] [Li et al. 2011] [Li and Shields 2011] [Li et al. 2016] [Liebhardt et al. 2020] [Loth et al. 2016] [Lukaczyk et al. 2012] [Lyman and Morgenstern 2004] [Ma et al. 2024b] [Ma et al. 2024a] [Mabe et al. 2023] [Mack and Darden 1979] [Mack and Darden 1980] [Mack 1999b] [Mack 1999a] [Mack 2003a] [Magee et al. 2013] [Makino et al. 1999] [Makino and Kroo 2006] [Makino et al. 2002] [Makino et al. 1997] [Makino et al. 1998] [Marconi et al. 2003] [Nemec et al] [Maute et al. 2008] [McLean and Shrout 1965] [McLean and Shrout 1966b] [McLean and Shrout 1966a] [Meckstroth et al. 2014] [Bozeman et al] [Miller and Carlson 1971] [Min et al. 2026] [Minelli et al. 2012] [Minelli et al. 2014] [Morgenstern 2004] [Morgenstern 2012c] [Morgenstern 2013] [Munguía et al. 2017] [Nadarajah et al. 2002] [Nadarajah et al. 2005] [Nadarajah et al. 2003] [Nadarajah et al. 2007] [Naohiko et al. 2016] [Narayan et al. 1995] [Narayan et al. 1999] [Needleman and Mack 1990] [Nemec and Aftosmis 2009] [Niedzwiecki and Ribner 1977] [Nikbay et al. 2023] [Ocheltree and Choi 2015] [Ordaz et al. 2014b] [Ordaz et al. 2015a] [Ordaz and Li 2011] [Ordaz and Li 2012] [Ordaz and Li 2014] [Ordaz and Li 2016] [Ordaz and Rallabhandi 2010] [Ordaz et al. 2017] [Ordaz et al. 2015b] [Ordaz et al. 2014a] [Otto et al. 2015] [Ozcer and Kandil 2009] [Ozoroski and Bangert 2017] [Padron et al. 2025] [Padron et al. 2026] [Pagaldipti et al. 1996] [Page 2017] [Pak 2015a] [Pak 2018b] [Pak 2018a] [Pak 2015b] [Pak 2015c] [Palacios et al. 2012] [Pan 1970] [Park 2011] [Park and Carter 2021] [Park and Carter 2022] [Petty 1970] [Phillips and West 2017] [Phillips and West 2018] [Phillips and West 2019] [Piccionello 2023] [Plotkin 2009] [Plotkin et al. 2009] [Plotkin et al. 2008] [Porter 1968] [Potapkin and Moskvichev 2014] [Potapkin and Moskvichev 2017] [Potapkin and Moskvichev 2019] [Rallabhandi 2011b] [Rallabhandi 2014b] [Rallabhandi 2014a] [Rallabhandi and Mavris 2005] [Rallabhandi and Mavris 2006c] [Rallabhandi and Mavris 2007a] [Rallabhandi et al. 2012] [Rallabhandi et al. 2014] [Rallabhandi and Park 2017] [Rallabhandi et al. 2017] [Rao et al. 2023] [Resler 1969] [Resler 1968] [Reuther and Jameson 1995] [Ribeiro et al. 2021] [Downs et al] [Downs et al] [Rodi 2018] [Rodriguez et al. 2025] [Rodriguez et al. 2026] [Ryan-Simmons and Agarwal 2025] [Rybalko et al. 2010] [Sahai et al. 2017] [Sandu et al. 2019] [Sasaki and Obayashi 2003] [Scarselli and Castorini 2013] [Scarselli and Marulo 2010] [Schouten and Bakker 2000] [Schrass et al. 2020] [Seebass 1969a] [Seebass 1971] [Seebass 1973] [Seebass and Argrow 1998] [Seebass and George 1972] [Seebass and George 1973] [Seebass and George 1974] [Seraj and Martins 2022] [Seraj et al. 2025] [Shen and Alonso 2025] [Shields and Li 2011b] [Shields and Li 2011a] [Siclari 1999a] [Siegelman 1970] [Sigalla et al. 1977] [Silva et al. 2014] [Silva et al. 2015b] [Silva et al. 2015a] [Simmons and Freund 2005a] [Simmons and Freund 2005b] [Sun and Smith 2018a] [Sun and Smith 2019a] [Sun et al. 2020] [Sun et al. 2025] [Swigart 1974] [Swigart 1975] [Takovitskii 2023] [Tekaslan et al. 2021] [Thomas 1971a] [Toyoda et al. 2010] [Ueno et al. 2023] [Ueno et al. 2016b] [Ueno et al. 2017] [Ueno and Makino 2021] [Ueno et al. 2016a] [Utsumi 2025] [Utsumi and Obayashi 2010] [Velden 1997b] [Veresnikov et al. 2024] [Volkov 1996] [Volkov 2017] [Volkov 2018] [Volkov 2021] [Volkov and Mazhul’ 2018] [Vyas et al. 2011] [Wayman et al. 2011a] [Wayman et al. 2011b] [Weaver-Rosen et al. 2021] [West et al. 2014] [West et al. 2017] [White et al. 2024] [Doebler et al] [Doebler et al] [Doebler et al. 2022a] [Wintzer and Kroo 2012] [Wintzer et al. 2008] [Wintzer and Ordaz 2015] [Li 2023b] [Li 2023a] [Li and Geiselhart] [Li and Geiselhart] [Li and Geiselhart 2021c] [Li and Geiselhart] [Li and Geiselhart] [Xiao et al. 2024] [Xiao et al. 2026] [Yamaguchi and Nakamura 1998] [Yamazaki and Ishida 2025] [Yamazaki and Kusunose 2016] [Ye et al. 2021] [Yoo 2010] [Yoo 2013] [Yoshida 1994] [Yoshida and Tokuyama 1992] [Yuhara and Rinoie 2012] [Zha et al. 2010] [Zhang 2025] [Zhang et al. 2019] [Zhao et al. 2015] [Zheng et al. 2026]
Predicting the boom before the aeroplane exists
Computation of the signature, near the aeroplane and at the ground. The AIAA sonic boom prediction workshops, which run the codes against one another on common geometries, are in here.
230 records. [Aftosmis and Nemec 2014] [Aftosmis et al. 2017] [Alaa et al. 2022] [Elmiligui et al] [Elmiligui et al] [Alauzet and Loseille 2010] [Loubeau et al] [Anderson et al. 2017] [Anderson et al. 2019] [Sorrell et al] [Baskar 2006] [Baskar and Prasad 2006] [Bobbitt 1992] [Bolander and Hunsaker 2020] [Bonavolontà et al. 2023] [Bui 2009] [Bui 2011] [Burn and Plotkin 1989] [Cain 2009] [Campbell et al. 2008] [Carlson et al. 1965b] [Carlson et al. 1966] [Carlson 1978] [Carlson et al. 1965a] [Carpenter et al. 2022] [Carter and Deere 2008] [Carter and Park 2021] [Carter and Park 2022] [Casper et al. 2008] [Chen et al. 2025] [Chen and Zhang 2006] [Cheng and Lee 1996] [Cheung et al. 1990] [Cheung et al. 1992] [Cheung et al. 1999a] [Cho 2009a] [Cho 2009b] [Choi et al. 2004c] [Choi et al. 2009] [Cliff et al. 2013] [Cliff and Thomas 1991] [Cliff and Thomas 1993] [Dagrau et al. 2014] [Darden 1990] [Darden 1993] [Davis 1972] [Derlaga et al. 2019] [De Rosa et al. 2026] [Design and Analysis 2023] [Downs et al. 2020] [Downs et al. 2022a] [Edwards et al. 1992] [Elmiligui et al. 2019] [Elmiligui et al. 2017] [Elmiligui et al. 2012] [Emre Tekaslan et al. 2023] [Ferri et al. 1973] [Fouladi 1993] [Fouladi 1999] [Gan and Zha 2015b] [Garcia et al. 1983] [Garcia et al. 1982] [Anderson et al] [Giblette and Hunsaker 2019] [Glorioso et al. 2024] [Gorbovskoy et al. 2020a] [Graziani 2023] [Graziani et al. 2024a] [Graziani et al. 2023] [Graziani et al. 2024b] [Graziani et al. 2025a] [Guo and Hafez 1995] [Hart et al. 2026] [Henderson 1971] [Hendriks 1976] [Herron 2007] [Housman et al. 2019] [Howe and Henne 2006] [Ishikawa et al. 2021a] [Ishikawa et al. 2022] [Iyengar et al. 2024] [Jackson and Carlson 1966] [Jäschke et al. 2024] [Lonzaga] [Lonzaga et al. 2025b] [Lonzaga et al. 2025a] [Lonzaga et al] [Vanharen et al] [Vanharen et al] [Page et al. 2023] [Kandil et al. 2005] [Kandil et al. 2002] [King et al. 2024] [King et al. 2023] [Kirz 2021] [Kirz 2022a] [Kirz and Rudnik 2017a] [Kirz and Rudnik 2017b] [Kirz and Rudnik 2019] [Kiseleva 2017] [Kiseleva and Lukashevich 2021] [Kiseleva and Lukeshevich 2019] [Kiseleva and Temerbekov 2019] [Koegler 1966] [Kornyakov et al. 2023] [Korunov et al. 2024] [Kovalenko and Zhilin 1998] [Kroo 1990] [Kuratani et al. 2007] [Kwak et al. 2013] [Laflin 2006] [Laflin et al. 2006] [Leng and Qian 2017] [Lengyan and Zhansen 2015] [Li et al. 2017b] [Lonzaga 2019b] [Loseille et al. 2019] [Loubeau 2021] [Loubeau et al. 2023a] [Loubeau et al. 2019] [Loubeau et al. 2021] [Ma et al. 2017] [Ma et al. 2018] [Madson 1994] [Maglieri et al. 1990] [Makarewicz 2024] [Makarewicz 2025] [Endo and Phillips] [Maslov and Epikhin 2025] [McLean 1965] [Carter and Park] [Miller 2023] [Miller and Smith 2003] [Mishchenko and Gimon 2023] [Mishchenko and Kiseleva 2019] [Morgenstern 2014] [Nemec et al. 2024] [Neuhoff et al. 2025] [Nikbay et al. 2017] [Numata et al. 2015] [Ojiaku and Prakash 2026] [Ordaz and Li 2013] [Oswatitsch and Sun 1972] [Ozcer 2007] [Page and Plotkin 1991] [Park 2017] [Park et al. 2014] [Park and Darmofal 2008] [Park and Morgenstern 2016] [Park and Nemec 2017] [Park and Nemec 2019] [Park et al. 2013] [Paynter 1994] [Pilon 2006a] [Pilon 2007] [Pilon and Lyrintzis 1996] [Plotkin and Page 2002] [Plotkin 1996] [Plotkin et al. 1994] [Potapkin et al. 2009] [Qiao et al. 2022] [Rallabhandi 2011a] [Rallabhandi 2018b] [Rallabhandi 2021] [Rallabhandi 2022b] [Rallabhandi 2022a] [Rallabhandi and Loubeau 2017b] [Rallabhandi and Loubeau 2017a] [Rallabhandi and Loubeau 2019] [Rallabhandi and Loubeau 2021] [Rallabhandi and Loubeau 2022] [Rallabhandi and Mavris 2006b] [Rallabhandi and Mavris 2007b] [Riegel and Sparrow 2022] [Roe et al. 2020] [Rothwell 1953] [Rusak 2008] [Ryhmeng et al. 1960] [Saito et al. 2014] [Salah El Din et al. 2013] [Salvetti and Seidman 1988] [Sasoh et al. 2016] [Neuhoff et al] [Seebass 1969b] [Seebass and McLEAN 1968] [Shen 2017] [Shen and Lazzara 2016] [Shen and Lazzara 2024] [Shuttle Sonic Boom 1973] [Shuttle Sonic Boom 1975] [Siclari and Darden 1990] [Siclari and Darden 1993] [Siclari 1992] [Siclari 1999b] [Silva et al. 2016] [Smith-Velazquez et al. 2017] [Smith-Velazquez and Theunissen 2018] [Sohail et al. 2019] [Sonic Boom Analysis 1975] [Spurlock et al. 2022] [Rallabhandi] [Rallabhandi and Loubeau] [Rallabhandi and Loubeau 2017c] [Rallabhandi and Loubeau 2018] [Rallabhandi] [Thomas 1972] [Trefny 2020] [Trono Figueras et al. 2024] [Tu et al. 1999] [Van der Velden and Kroo 1994] [Vanharen et al. 2022] [Verhoff and Stookesberry 1992] [Waithe 2008] [Walker et al. 2015] [Watanuki et al. 1999] [White 1993] [Wiggins and Bruce 1966] [Wiggins and Kennedy 1968] [Wilson and Zhang 2025] [Yamashita and Ishikawa 2023] [Yamashita et al. 2021] [Yamashita et al. 2022] [Yamashita and Nikiforakis 2020] [Yamashita and Nikiforakis 2023] [Yamashita and Suzuki 2014] [Yamashita and Suzuki 2016a] [Yamashita and Suzuki 2016b] [Yamashita et al. 2020] [Yoshida et al. 1993] [Zou et al. 2024]
Crossing the atmosphere, which changes the answer
The medium, which is the subject and not the setting. Absorption, relaxation, turbulence, refraction, caustics and the cutoff condition.
420 records. 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al. 2022] [Mishchenko et al. 2024] [Montes 1970] [Muhlestein and Gee 2011] [Mungiole and Wilson 2003] [Naugolnykh and Rybak 2007] [Ng 1991] [Nijs and Wapenaar 1990] [Nonlinear Acoustic Behavior 1975] [Nonlinear Acoustics] [Nonlinear Geometric Acoustics 2017] [Nyborg et al. 2024] [Nyborg and Mintzer 1955] [Ollivier and Blanc-Benon 2004] [Onyeonwu 1975b] [Onyeonwu 1975a] [Ostashev 2002] [Ostashev and Goedecke 2005] [Ostashev and Keith Wilson 2026] [Ostashev and Wilson 2026] [Page 1975] [Palmer 1970] [Pan 1968] [Parameters Used 2021] [Parker and Zalosh 1973] [Pelinovskij and Fridman 1983] [Pena and Diamond 1965] [Peplow and Liu 2025] [Pettit and Wilson 2021] [Physical theory of nonlinear] [Piacsek 1991] [Piacsek 1996] [Piacsek 1998] [Piacsek 2002] [Piacsek et al. 2008b] [Piacsek et al. 2008a] [Piacsek et al. 2010] [Piacsek and Plotkin 2013] [Pierce 1968] [Pierce 1969] [Pierce 1970] [Pierce 1971] [Pierce 1990] [Pierce 1992a] [Pierce 1995] [Pierce 1996] [Pierce 1998] [Pierce and Kinney 1976] [Pierce and Maglieri 1971] [Pierce and Maglieri 1972] [Pierce and Thomas 1969b] [Pierce and Thomas 1969a] [Pierce 1993] [Pierce 1994] [Pilon 2010] [Pilon 2011] [Pindera 2002] [Plotkin 1990] [Plotkin 1993] [Plotkin 1995] [Plotkin and Cantril 1976] [Plotkin and Haering 2006] [Plotkin 1994] [Polly 1969] [Pridmore-Brown 1962] [Rallabhandi 2018a] [Rallabhandi 2019] [Rallabhandi and Fenbert 2017] [Rallabhandi and Mavris 2004] [Rallabhandi et al. 2023] [Rasmussen 1986] [Rasmussen 1990] [Raspet and Bass 1990] [Raspet et al. 1994b] [Raspet and Bass 1995] [Raspet and Bass 1998] [Raspet et al. 1994a] [Ray and Caustic Asymptotics 2014] [Reed 1972] [Riegel et al. 2021] [Riegel and Sparrow 2012] [Riegel and Sparrow 2019] [Riegel and Sparrow 2024] [Riegel and Sparrow 2025] [Robinson 1992b] [Robinson 1993] [Robinson 1992a] [Rogers 1980] [Ronsse et al. 2012] [Rouse 2014] [Rudenko 2004] [Rusak and Cole 1993] [Ryan et al. 2022] [Salamone 2006a] [Salamone 2009] [Salamone and Sparrow 2012] [Salamone et al. 2013a] [Salamone et al. 2013b] [Salomons 1998] [Salze et al. 2025] [Sanai et al. 1975] [Sano and Miyamoto 1998] [Sapozhnikov 2019] [Saturation Effects Associated 1976] [Scattering of Coherent Sound 1976] [Seymour and Mortell 1975] [Shamanaeva et al. 2018] [Lympany and Page] [Sheu and Fang 1994] [Silber 2026] [Sion�id 1993] [Soilleux 2024] [Sound Scattering from Atmospheric 1976] [Sparrow and Riegel 2024b] [Sparrow and Riegel 2024a] [Sparrow and Pierce 1992] [Standard Values of Atmospheric] [Stout and Sparrow 2018] [Stout and Sparrow 2019] [Stout et al. 2021] [Stout and Sparrow 2022] [Stuff 1972] [Sutherland and Daigle 1997] [Tachibana and Yoshihisa 2024] [Taherzadeh et al. 1998] [Takahashi et al. 2017] [Takahashi et al. 2018] [Takeno et al. 2015] [Tam and Shen 1993] [Terzakis et al. 2025b] [Thomas 1971b] [Trikootam and Hornikx 2019] [Verdier et al. 2026] [Wade 2021] [Wade and Sparrow 2019] [Wade and Sparrow 2021] [Warshaw and Dubois 1985] [Wave Theory of Sound 2015] [Weak Shock Theory 2016] [Wen et al. 2022] [Wert et al. 1998] [West 1994] [Whitlow and Harris 1976] [Doebler] [Doebler and Lonzaga] [Doebler and Loubeau] [Doebler 2023] [Willshire et al. 1992] [Willshire and Devilbiss 1992] [Wilson 1998] [Wilson 2024] [Wilson and Liu 2004] [Yamamoto et al. 2012] [Yamashita and Makino 2022] [Yamashita and Makino 2023b] [Yamashita and Makino 2023a] [Yamashita and Obayashi 2009] [Yamashita and Obayashi 2010] [Yamashita and Obayashi 2012] [Yamashita and Obayashi 2013] [Yamashita and Suzuki 2015] [Yamashita and Suzuki 2018] [Yamashita and Suzuki 2019] [Yamashita and Suzuki 2020] [Yamashita et al. 2025] [Yuldashev 2021] [Yuldashev et al. 2019] [Zorumski and Willshire 1989]
What a boom does to a building
The response of buildings and their windows, being structural vibration, window glass, rattle, damage potential and the transmission of a boom indoors.
37 records. [Benveniste and Cheng 1967] [Carden et al. 1969] [Clarkson and Mayes 1971] [Clarkson and Mayes 1972] [Corcoran et al. 2012] [Crocker 1965] [Dogan 2008] [Findley et al. 1975a] [Findley et al. 1975b] [Garrelick and Martini 1996] [Garrelick et al. 1997] [Haac et al. 2009] [Haber 1993] [Hershey and Higgins 1974] [Hershey et al. 1974] [Hubbard 1990] [James et al. 2011] [Klos 2008] [Klos and Buehrle 2007] [Librescu et al. 1997] [Librescu and Na 1998] [Librescu and Nosier 1990] [Lovholt 2021] [Martini and Garrelick 1994] [Martini and Garrelick 1997] [Naka et al. 2008b] [Naka et al. 2008a] [Newberry 1967] [Norén-Cosgriff et al. 2022] [Pretlove and Bowler 1972] [Rao and Zumwalt 1970a] [Rao and Zumwalt 1970b] [Report on Physical Effects 1968] [Seshadri and Lowery 1972] [Southerland et al. 1986] [Sutherland and Czech 1992a] [Sutherland and Czech 1992b]
What a boom does to animals, the sea and the ground
The boom below the water line, in the ground and in the lives of animals. The penetration of a boom into the sea, its coupling into the ground and the response of wildlife and livestock.
30 records. [Battis 1983] [Bell 1971] [Chen and Suzuki 2021] [Cheng 2005] [Cheng et al. 1996] [Cheng and Lee 1998] [Cheng and Lee 2004] [Cottereau 1978] [Desharnais and Chapman] [Favorskaya 2020] [Ferguson and Sparrow 1998] [Fincham and Maxworthy 2002] [Legg and Haber 1990] [Moody 2006] [Moody and Griffice 2003] [Rochat and Sparrow 1996] [Rochat and Sparrow 1997a] [Rochat and Sparrow 1997c] [Rochat and Sparrow 1997b] [Sohn et al. 2000] [Sparrow 1993] [Sparrow 1998] [Sparrow et al. 1997] [Sparrow 1996] [Ting et al. 1996] [Ting et al. 2002] [Tracey 1998] [Wang et al. 2006] [Waters and Glass 1970] [Wilkins 1972]
How loud it is to a person, which is a different question from how large it is
The third largest cluster, and the article’s central distinction lives in it. Perceived level, loudness, the annoyance of aircraft and impulsive noise, and the dose-response method the community phase is to use.
325 records. [A Social Survey into 1972] [Acoustics. Normal equal-loudness-level contours] [Ahuja et al. 1993] [Akiyama et al. 2023] [Albert 2024] [Aldridge 2024] [Carr et al] [Loubeau] [Loubeau et al. 2013b] [Loubeau] [Loubeau et al. 2011a] [Alexandre 1976] [Allen 1980] [Ancona et al. 2023] [Anderson et al. 2022] [Anderson et al. 2011] [Avery 1982] [Baird et al. 1997] [Bartels et al. 2018] [Bashkirov et al. 2019] [Batista da Cunha et al. 2025] [Baudin et al. 2020] [Baudin et al. 2021b] [Baudin et al. 2021a] [Becker et al. 2021] [Berglund et al. 1975] [Berry 2003] [Berry and Miedema 2023] [Bisping 2024] [Björkman et al. 1992] [Bolander et al. 2019] [Borsky 1965] [Borsky 1966] [Borsky 1972] [Borsky 1977] [Borsky 1978] [Borsky 1980] [Borsky and Nixon 1966] [Bottom 1971] [Boucher et al. 2023] [Brink et al. 2019] [Broadbent and Robinson 1964] [Brooker 2008] [Brooker 2009] [Brooker 2010] [Brown and Haglund 1988] [Brown and Sutherland 1992a] [Brown and Sutherland 1992b] [Broyles et al. 2019] [Byrne 1971] [Carpenter et al. 2019] [Carr et al. 2022a] [Cho et al. 2014] [Christian and Cabell 2017] [Classification for Rating Outdoor-Indoor] [Cliatt et al. 2013] [Corsetti Silva et al. 2020] [Definitions and Procedures for] [Definitions and Procedures for] [Dempsey 1980] [Dempsey and Stephens 1981] [de Oliveira Nunes and Sattler 2006] [Dickson 2009] [Diez et al. 2007] [Doebler et al. 2023a] [Driver and Maglieri 1980] [Dzhambov et al. 2025] [Edwards 1975] [Elechi 2020] [Elmehdi 2012] [Farbry et al. 1991] [Felcyn and Ptak 2022] [Fenech et al. 2022] [Fidell 2007] [Fidell 2013] [Fidell et al. 1989b] [Fidell et al. 2012] [Fidell et al. 1989a] [Fidell et al. 1999] [Fidell and Silvati 2004] [Fidell and Pearsons 1994] [Fidell and Pearsons 1996] [Fields 1993] [Fields and Walker 1982] [Flindell 2024] [Foertsch and Davies 2023] [Frair 1984] [Frampton et al. 1993] [Frequency Weighing Network for] [Galloway 1981] [Georgiou et al. 2026] [Gille et al. 2016] [Gille et al. 2017a] [Gille et al. 2017b] [Gille and Marquis-Favre 2019] [Gjestland 1994] [Gjestland 2018] [Gjestland 2024] [Gjestland and Gelderblom 2017] [Gorbovskoy et al. 2020b] [Gottlieb et al. 1978] [Gottlieb and Glass 1974] [Grantham and Smith 1980] [Grantham et al. 1980] [Graziani 2024] [Guo-qing et al. 2012] [Harder] [Hartmann and Fidell 1990] [Hashimoto and Hatano 2023] [He et al. 2026] [Heleno and Slama 2013] [Hilton et al. 1964] [Höchstetter et al. 2015] [Hodgdon and Page 2013] [Hodgdon et al. 2013] [Houston et al. 2024] [Hubbard and Shepherd 1992] [Hughes and Mabry 1976] [Iura et al. 2024] [Klos 2025a] [Lee et al. 2019] [Lee et al. 2020] [Lonzaga] [Rathsam] [Rathsam et al. 2013] [Jovanovic et al. 2022] [Jovanovic and Novak 2022a] [Jovanovic and Novak 2022b] [Jovanovic et al. 2023] [Jovanovic and Novak 2024] [Júnior et al. 2022] [Kamerman et al. 1986] [Kantaracha] [Key and Powell 1980] [Klæboe 2005] [Klos 2018] [Kodji et al. 2024c] [Kodji et al. 2023] [Kodji et al. 2024a] [Kodji et al. 2024b] [Kondo et al. 1978] [Krishnamurthy and Rizzi 2022] [Kroesen and Bröer 2009] [Kroesen et al. 2010] [Krog and Engdahl 2005] [Kryter 1982] [Kryter 2004] [Kurakata and Mizunami 2019] [Kurra et al. 1999] [Leatherwood and Sullivan 1992c] [Leatherwood and Sullivan 1992a] [Leatherwood and Sullivan 1993] [Leatherwood and Sullivan 1994b] [Lechner et al. 2019] [Lechner et al. 2021] [Lim et al. 2008] [Lind and Sparrow 2013] [Loeb and Vanderhei 1976] [Loubeau 2013] [Loubeau 2022] [Loubeau et al. 2011b] [Loubeau et al. 2013a] [Luz et al. 2024] [Luz et al. 2004] [Lv et al. 2025] [Mabry et al. 1969] [Mak et al. 2023] [Mange 1978] [Nemec et al] [Marmel 2021] [Marmel et al. 2025] [Marquis-Favre et al. 2021] [May 1972] [McCurdy 1983] [McCurdy 1984] [McCurdy 1985] [McCurdy 1988] [McCurdy 1989] [McCurdy 1990] [McCurdy 1991] [McCurdy et al. 1986] [McCurdy and Powell 1978] [McCurdy 1994] [McCurdy et al. 1994] [McCurdy et al. 2004] [McKennell 1969] [Measures and Indices 1976] [Merino-Martinez and Buzetelu 2025] [Merino-Martinez and Schade 2025] [Miller and Sparrow 2010] [Morgenstern 2017a] [Muller 1971] [Murphy 1991] [Naim et al. 2018] [Needleman et al. 1991] [Nemec et al. 2021] [Nguyen et al. 2018] [Nguyen et al. 2023b] [Nguyen et al. 2023a] [Niedzwiecki 1978] [Niedzwiecki and Ribner 1978] [Nihart and Brown 1988] [Nixon 1970] [Nixon and Borsky 1965] [Nyborg et al. 2026a] [O’Brien et al. 1972] [Oppenheimer 2026] [Oppenheimer and Raley 2025] [Oppenheimer and Way 2024] [Ortega et al. 2019] [Palmer and Sparrow 2015] [Park 2018] [Patterson et al. 1977] [Pearsons 1963] [Pearsons 1982] [Pearsons et al. 1997] [Pennig et al. 2023] [Phun et al. 2015] [Pilon 2006b] [Plotkin 2006b] [Plotkin et al. 1991] [Plotkin et al. 1992a] [Plotkin et al. 2012] [Power 1963] [Preisendörfer et al. 2022] [Quehl 2005] [Quehl and Basner 2006] [Ramakrishnan et al. 2018] [Rathsam and Gille 2017] [Rathsam et al. 2015] [Rathsam and Klos 2016] [Rathsam et al. 2011] [Rathsam and Christian 2016] [Reddingius 1974] [Reddingius 1977] [Reddy et al. 2021] [Reedijk et al. 2025] [Report on Human Response 1968] [Rice 1984] [Rice 2024] [Rizzi and Christian 2016] [Rylander 2006] [Rylander et al. 1980] [Rylander and Björkman 1997] [Rylander and Dancer 1978] [Rylander et al. 1972] [Rylander et al. 1974] [Sacerdote 1958] [Sahai and Stumpf 2014] [Salamone 2006b] [Sanford 1979] [Schäffer et al. 2020] [Scherebnyj and Hodgson 2007] [Schlittenlacher and Wales 2023] [Schomer 1994] [Schomer et al. 1991] [Schomer et al. 2023] [Schomer et al. 1996] [Schomer et al. 1998] [Schoppe et al. 2025] [Schreckenberg 2008] [Schreckenberg 2021] [Schreckenberg et al. 2023] [Schultz 1974] [Schultz 1979] [Schultz 1982] [Seabi 2013] [Segal 1990] [Seto and Huang 2025] [Shaffer 1990] [Shepherd 1979] [Shepherd et al. 1992] [Shepherd and Sutherland 1968] [Shigeshisa and Gunn 1977] [Sieben and Aalmoes 2025] [Singh et al. 2024] [Singh and Kung 1982] [Sizov and Hobbs 2010] [McDaniel et al. 1992] [Smeatham et al. 2024] [Sohn and Schowe 1989] [Sonic Boom in Relation 1968] [Speakman et al. 1978] [Stevens 1972, Perceived level of noise by Mark VII and decibels] [Stewart 2008] [Stewart 2009] [Stusnick et al. 1993] [Sullivan 2006] [Sullivan et al. 2010] [Sullivan and Leatherwood 1993] [Sullivan and Leatherwood 1994] [Sutherland 2012] [Tamayo] [Taylor 1984] [Taylor 2024] [Terzakis et al. 2025a] [Terzakis et al. 2026] [The relation between the] [Thoma et al. 2024] [Thompson 2023] [Tokuyama et al. 1993] [Töpken 2021] [Töpken and van de Par 2021] [Torija et al. 2022] [Valishina et al. 2025] [Vallet 2024] [Van et al. 2003] [Vanderhei Moran and Loeb 1977] [von Gierke and Nixon 1971] [von Gierke and Nixon 1972] [Wang et al. 2015b] [Warren 1972b] [Wayant 2018] [Bauer 2025] [Weir 1988] [Doebler et al] [Doebler et al] [Doebler and Vaughn 2025] [Willshire 1983] [Wothge et al. 2017] [Yamada 2023] [Young 1981] [Yuldashev et al. 2021] [Zeitler et al. 2024] [Zybura 2024]
The airport noise of a supersonic aeroplane
The noise a supersonic aeroplane makes near an airport, at take-off and landing, which is the only noise the 2020 certification proposal addresses. Studies of the jet noise of a particular nozzle stay with the propulsion cluster.
42 records. [Akatsuka and Ishii 2022] [Akatsuka et al. 2025] [Berton and Majjigi 2025] [Bertsch et al. 2026b] [Bridges 2019] [Bridges et al. 2022] [Bridges and Wernet 2019] [Bridges and Wernet 2021] [Bridges 2018] [Calder and Gupta 1977] [Cluts et al. 2019] [Golberg and Gurevich 2023] [Herrmann 2006a] [Herrmann 2006b] [Hines and Sabatella 1973] [Huff et al. 2016] [Huff et al. 2019] [Ito et al. 2008] [Bridges] [Bridges et al] [Kopiev et al. 2025] [Mirzoyan and Khaletskii 2020] [Newell and Zwernemann 1999] [Nielsen and Papamoschou 2008] [O’Keefe et al. 1981] [Papamoschou 2002] [Papamoschou 2004] [Papamoschou and Debiasi 2003] [Piccirillo 2023] [Piccirillo et al. 2022] [Preisser 1980] [Proskurov et al. 2023] [Stephens 2017] [Stern and Peracchio 1989] [Sugar and Slaboch 2021] [Villena Muñoz et al. 2026] [Voet et al. 2024a] [Voet et al. 2024b] [Voet et al. 2023] [Wang et al. 2026] [Zaporozhets et al. 2025] [Zwernemann 1999]
The rule, and the process for changing it
The rule and the certification of aircraft noise. The governing documents contain no loudness for a boom, as the part headed The Rule It Exists to Change Contains No Loudness at All shows, and this literature includes the work arguing for one.
48 records. [Aircraft Noise Metrics 2022] [Albert et al. 2017] [Loubeau] [Altree and Baxter 1968] [Arntzen 2017] [Baeva et al. 2023] [Batson 1970] [Bentley 2024] [Black 1969] [Block 1981] [Bousquet and Blandeau 2021] [Callaway and Peart 1976] [Christopher 1971] [Coen 2022] [Coen et al. 2023] [Connor et al. 2024] [Cowart 2013] [Electroacoustics. Instruments for measurement] [Fish and Haight 1976] [Galloway 1971] [Gardner 1978] [Graebert et al. 2025] [Hammond and Dourado 2018] [Heller 1984] [Jurin and Pavlovic 2005] [Khorrami and Fares 2019] [Ling et al. 2022] [McPike 1976] [Mertens 1997a] [Metzger and Foley 1970] [Michel 2013] [Milde 1978] [Mirzoyan and Khaletskii 2021] [Morgenstern 2017b] [Nöding and Bertsch 2021] [Noeding et al. 2023] [Paullin 1973] [Coen et al] [Rolf 2000] [Rötger et al. 2023] [Scata et al. 2024] [Smith 1982] [Smith 1986] [Specification for electro-acoustical measuring] [Sutherland 1980] [Voet et al. 2022] [Von Glahn and Groesbeck 1981] [Zhang et al. 2024]
Measuring a boom, on the ground and in the air
Instrumentation. Ground arrays, in-flight probing, wind tunnel pressure rails and the uncertainty of each.
92 records. [Abel 1971] [Miley et al. 1992] [Air Force Test Pilot School Edwards Afb Ca 1991] [Alberts 2012] [Altman 1965] [Bedard 1990] [Blaha 1969] [Blaine et al. 2024] [Callaghan 1968] [Calogeras 1969] [Carlson and Morris 1966] [Carlson and Morris 1967] [Castner et al. 2013] [Castner et al. 2015] [Cliff et al. 2016] [Comparison of Aircraft Noise 1975] [Crocker and Sutherland 1966] [Crocker and Sutherland 1968] [Doebler and Sparrow 2017a] [Downing 1992] [Downs et al. 2022b] [Durston et al. 2018] [Elmer and Joshi 1994a] [Elmer and Joshi 1994b] [Ha 2025] [Ha et al. 2021] [Haering 2006] [Haering et al. 1995] [Haering et al. 2005b] [Haglund and Kane 1974b] [Haglund and Kane 1974a] [Harker et al. 2022] [Heineck et al. 2016] [Henderson 1982] [Henderson and Hilton 1974a] [Henderson and Hilton 1974b] [Hilton and Newman 1965] [Hilton and Newman 1966] [Hilton et al. 1972] [Hilton and Henderson 1974] [Hubbard and Maglieri 1990] [Ivanteeva et al. 1998] [Ivanteyeva et al. 2002] [Durrant et al. 2025] [Jung et al. 2012] [Kamali and Pierce 1972] [Kanamori et al. 2025] [Kanamori et al. 2018] [Kantor and Cliatt 2017] [Kelly et al. 1999] [Lee and Downing 1993] [Lee and Downing 1996a] [Lympany and Page 2022] [Mack and Needleman 1990] [Maglieri et al. 1993] [Miller 1971] [Morgenstern 2012a] [Morgenstern 2012b] [Morris and Miller 1972] [Norris et al. 1995] [Norris et al. 1996] [Novakovich et al. 2019] [Pan 1971] [Plotkin 2006a] [Plotkin et al. 1989] [Plotkin et al. 1990] [Plotkin et al. 1992b] [Plotkin et al. 2005a] [Plotkin and Maglieri 2013] [Plotkin et al. 2004] [Poling 1996] [Rasband et al. 2023] [Castner and Simerly 2022] [Roensch and Page 1980] [Sakata 2002] [Schultz et al. 2016] [Schultz et al. 2017] [Lympany and Page] [Shantz 1953] [Slater et al. 2005] [Stanley 1984] [Stansbery et al. 1999] [Stark 2002] [St. John and Frederick 2013] [Taniguchi 1959] [Taniguchi 1961] [Veletas 2026] [Doebler] [Yeong et al. 2009] [Yetter et al. 1982] [Young 1982] [Zuckerwar and Shope 1974]
Supersonic aerodynamics, which is older than the problem
The area rule, slender body theory and linearised supersonic flow, which predate the sonic boom as a subject and supply all of its mathematics.
254 records. [Abraham et al. 2023] [Abraham et al. 2024] [Advanced Fuel Research INC East Hartford CT 1957] [Aerodynamic Phenomena in Supersonic 2020] [Ageev and Pavlenko 2016] [Agnone 1994] [Anderson 1960] [Aprovitola et al. 2021] [Aprovitola et al. 2022] [Apstein and Titow 1996] [Army War Coll Carlisle Barracks Pa 1952] [Arnal and Reneaux 2001] [Auman and Wilks 2003] [Baals and Harris 1968] [Baals et al. 1970] [Bartlett 1970] [Bashkirov and Ogorodnikov 2018] [Bauer and Wood 1989] [Bera 1975] [Bogdnoff 1953] [Bonner 1971] [Book et al. 1991] [Boyd 1965] [Brown 1961] [Brown and McLEAN 1959] [Bulakh 1966] [Bushnell 1990] [Carlsen 1994] [Carlson 1964] [Carlson and Miller 1981] [Castner 2010b] [Catalano and Sturek 2001] [Cenko 1979] [Cenko 1980] [Chang 1992] [Chapter 3. Aerodynamics 1960] [Chen and Li 2020] [Chiba et al. 2008b] [Chin 1977] [Chin 1978] [Cliff et al. 2009] [Craidon 1986] [Darden 1984b] [Darden 1985] [Desai et al. 1977] [Dollyhigh and Craidon 1987] [Drougge 1990] [Durston and Stonum 1987] [Durston and Stonum 1988b] [Durston and Stonum 1988a] [Ehlers 1961] [Livne et al. 2025] [Eminton and Lord 1956] [Eremenko et al. 2003] [Erickson et al. 1990] [Etkin and Woodward 1954] [Fedorov and Malmuth 1997] [Flat Wings in Inviscid 2020] [Fletcher and Burns 1979] [Fluid Dynamics Research 1998] [Follé 1983] [Forbes and Hindle 2018] [Fornasier 1984] [Friedman 1951] [Friedman and Cohn 1955] [Fu et al. 2017] [Fujino and Kawamura 2003] [Ginzel and Multhopp 1960] [Goodman 1949] [Goodman 1950] [Goto et al. 2008] [Goto et al. 2007] [Graham 1958] [Graham et al. 1954] [Graham et al. 1957] [Guan 2013] [Guan 2014a] [Guan 2014b] [Guderley 1988] [Hancock 1959] [Hansen 2000] [Harris 1966] [Harris et al. 1965] [Harris et al. 1966] [Hartley et al. 1965] [He and Shi 2025] [Hendrickson et al. 1978] [Huffman and Fox 1985] [Hui 1972] [Hunziker 1960] [Ide et al. 2012] [Ignatieva and Silantiev] [Item and Baysal 1995] [Jeong et al. 1999] [Jischke 1978] [Jones 1959] [Jones 1967] [Jones 1977] [Jones 1980] [Joshi et al. 2024] [Jumper 1983] [Kamiyama and Yamazaki 2020] [Kanazaki and Seto 2012] [Kanazaki et al. 2013] [Kawasaki 1965] [Keenan and Kuhlman 1991] [Khan et al. 2025] [Kishi et al. 2016] [Kishi et al. 2022] [Klopfer and Nielsen 1980] [Kraiko and Tilliaeva 1973] [Kraiko and Tkalenko 1967] [KÜChemann 1962] [Kulfan 1978] [Kulfan 1990] [Kulfan 2002] [Kulfan 2008] [Kulfan 2009] [Lacey 1979] [Ladyzhenskii 1963] [Ladyzhenskij 1963] [Landahl and Loefgren 1973] [Lapygin 1971] [Lapygin and Tret’yakov 1986] [Lapygin and Yakunina 2009] [Lei 2008] [Lei et al. 2004] [Leslie and Perry 1954] [Levy and Yoshikawa 1959] [Li et al. 2018] [Li et al. 1995] [Li et al. 1999] [Liao et al. 2025] [Licher 1956] [Lift/Drag Ratios of Optimised 1962] [Livne 2023a] [Livne 2023b] [Lobanovskii 1983] [Lock 1957] [Lock 1961] [Lovell 1999] [Lovell 2001] [Mack 1973] [Maikapar 1959] [Maikapar 1966] [Maise 1970] [Makino and Iwamiya 2000] [Malmuth et al. 1986] [Malmuth et al. 1987] [Mann and Carlson 1991] [Mann and Carlson 1994] [Margolis 1947] [Martin et al. 1991] [Maruyama et al. 2008] [Maruyama et al. 2006] [Maruyama et al. 2009b] [Maruyama et al. 2009a] [Mascitti 1976] [Mason 1983] [Matsushima et al. 2004] [Mertens 1997b] [Meyer and Fields 1978] [Miller et al. 1983] [Miller and Schemensky 1979] [Mizobata et al. 2024] [Moga 1980] [Moga and Dejarnette 1981] [Morikawa 1951] [Murphy and Schmidt 1953] [Mysliwetz 1963] [Nastase and Weisensee 1984] [Naval Ordnance Systems Command Washington DC 1957] [Nie et al. 2025] [Nielsen 1985b] [Nielsen 1985a] [Nikolic and Jumper 2004] [Nikolic and Jumper 2005] [Obayashi and Sasaki 2002] [Obayashi et al. 2001] [Ohira and Kwak 2014] [Ortell 1967] [Osipov 2019] [Ostieri and Tognaccini 2018] [Oyama et al. 1999] [Palaniappan and Jameson 2010] [Patel et al. 2011] [Petruzzelli and Keane 2001] [Pfaff 1965] [Phythian 1966] [Pinto 2026] [Pittman 1987] [Pittman 1993] [Pittman and Bonhaus 1987] [Pittman et al. 1990] [Powell 1956] [Powers 1992] [Properties and Design 2012] [Puckett 1946] [Quagliarella and Iuliano 2017] [Radkey et al. 1976] [Radkey et al. 1978] [Rallabhandi and Mavris 2003] [Ramaswamy and Viswanathan 1975] [Rethorst 1994] [Rivers et al. 2008] [Roberts 1948] [Robins and Carlson 1979] [Robins and Carlson 1980] [Rolston et al. 1998] [Schnell 2023] [Semionov et al. 2013] [Seraj and Martins 2023] [Seraj and Martins 2025] [Shirokov et al. 2024] [Siewert and Stewart 1965] [Sklar and Rusak 2020] [Smith 1961] [Spearman et al. 1977] [Stancil 1979] [Stewart 1946] [Stone 1945] [Courant and Friedrichs 1976] [Supersonic Flow Computations 1990] [Supersonic Three-Dimensional Wing Theory 1960] [Takahashi et al. 1997] [Takovitskii 2020] [Tetervin 1963] [Thomas 1942] [Ting et al. 2025] [Tkalenko 1969] [Tokugawa and Yoshida 2006] [Tracy et al. 2002] [Transonic and Supersonic Flows 2012] [Traoré and Lemée 2001] [Traub 2001] [Unger and Coen 1993] [Van Dyken 1995] [Vaughn 1982] [Walkley 1986] [Walkley and Smith 1987] [Whitcomb 1956] [Whitham 1950] [Whitham 1952] [Wing Theory in Supersonic 1969] [Wolf 1993] [Woodward 1968b] [Woodward 1968a] [Wright et al. 1978] [Xu 2023] [Yahalom 1971] [Yamazaki and Kusunose 2013] [Yamazaki and Kusunose 2014] [Yeh and Clark 2007] [Yoshida 2009]
Getting the air in and the thrust out
The inlet and the nozzle. This aeroplane shields its inlet with its wing so that spillage does not spoil the signature, which makes propulsion airframe integration part of the acoustics.
195 records. [Adamidis et al. 2025] [Afzal et al. 2021] [Akatsuka and Ishii 2018] [Akatsuka and Ishii 2021] [Akatsuka et al. 2013] [Akatsuka et al. 2016] [Alendar et al. 2022] [Alford and Auyer 1954] [Alford et al. 1962] [Anderson 1984b] [Anderson 1990] [Arlinger 1976] [Babinsky 2014] [Bai et al. 2026] [Bangert et al. 1982] [Barbarino et al. 2024] [Barnhart 1989] [Barry 1966] [Barry 1971] [Baye-Wallace and Krouse 2022] [Bencze 1972] [Benson et al. 1976] [Benton et al. 2021] [Bowditch 1973] [Braun and Podleski 2019] [Brear et al. 2003] [Brown 1978] [Burr 1968] [Burris 1966] [Burstadt et al. 1971] [Cai et al. 2023] [Calder and Gupta 1978] [Campbell and Syberg 1984] [Capone and Deere 2001] [Carlin et al. 2003] [Castner 2008] [Castner 2010a] [Castner 2011b] [Castner 2012a] [Castner et al. 2014] [Castner 2012b] [Castner 2011a] [Champagne 1988] [Champagne 1991] [Chang and Bao 2006] [Chaudhari and Takahashi 2022] [Chima et al. 2010] [Choe and Kim 2016] [Choe et al. 2018] [Chun and Swanson 1964] [Clauss et al. 1980] [Clauss et al. 1979] [Cole 1977] [Conners et al. 2005] [Conners and Howe 2006] [Control System Design Using 1976] [Cotter 1968] [Dickman and Takahashi 2016] [Ehrmayr and Berens 2018] [Evelyn et al. 1978] [fang et al 2024] [Farahani and Mahdavi 2019] [Fernandez Villace and Paniagua 2011] [Fernandez-Villace and Paniagua 2013] [Ferrero 2020] [Fishbach et al. 1982] [Fitzsimmons et al. 1978] [Franciscus 1981a] [Franciscus 1981b] [Franciscus 1983] [Franciscus 1987] [Fujimoto and Niwa 1993] [Ge et al. 2021] [Giannola et al. 2000] [Goonko and Alexandrov 2010] [Guan and Yarng 1984] [Guan et al. 1985] [Guan and Yarng 1987] [Haas and Karanian 1980] [Hallstaff 1966] [Hermann 1956] [Hill 1970] [Hines 1977] [Hinz and Miller 1979] [Howlett 1975] [Howlett 1977] [Huet et al. 2025] [Hunter and Cawthon 1984] [Hunter and Cawthon 1985] [Inlets and Inlet/Engine Integration 1989] [Ivanov 2022] [Jaron et al. 2023] [Johnson and Mitchell 1971] [Kazula and Höschler 2020] [Kazula et al. 2019] [Keenan 1961] [Kepler and Barry 1965] [Kim 2009] [Kim et al. 2010] [Kimzey and Ellis 1974] [Kopasakis and Connolly 2009] [Kopasakis et al. 2008] [Kopasakis et al. 2010] [Koval 2020] [Kowalski et al. 1981] [Krebs and Allan 1977] [Лысенков et al. 2024] [Marvin 1968] [Merchant et al. 2003] [Migdal and Horgan 1964] [Miki et al. 2014] [Miller and Protopapas 1985] [Moes 2009] [Moin and Lele 1998] [Mount 1964] [Mount 1965] [Muller and Gasko 1967b] [Muller and Gasko 1967a] [Mutz et al. 1964] [Nichols 1966b] [Nichols 1966a] [Ning] [Nuckolls and Ng 1993] [Nuckolls and Ng 1995] [Obikane 1984] [Palma and Takahashi 2016] [Papamoschou and Phong 2018] [Paynter and Chen 1983] [Powers et al. 1979] [Pruitt and Bates 1992] [Ran and Mavris 2005] [Randall and Hand 1968] [Requirements of a Commercial 1967] [Rizzetta 1991] [Rowe et al. 1979] [Samimy et al. 2011] [Sasanapuri et al. 2013] [Schweikhard and Berry 1974] [Shimabukuro et al. 1979] [Shimabukuro et al. 1982] [Shulman et al. 1966] [Sippel 2006] [Slater 2016] [Slater 2023] [Slater 2025] [Smeltzer and Sorensen 1972] [Sóbester 2010] [Soltani and Farahani 2010] [Sorensen and Bencze 1973] [Stepanov et al. 2013] [Stephens et al. 2019] [Stephens et al. 2023] [Stephens and Svetgoff 2024] [Strack and Morris 1988] [Subsonic/Transonic/Supersonic Nozzle Flows 1986] [Suikat and Farokhi 1987] [Suikat and Farokhi 1988] [Sun and Zhang 2016] [Supersonic Jet Noise Suppression 1976] [Surber and Robinson 1983] [Surber and Sedlock 1978] [Surber and Sedlock 1979] [Syberg et al. 1981] [Takasaki et al. 1998] [Takovitskii 2015] [Taylor 1968] [Three-Dimensional Viscous Design Methodology 1986] [Tongguang et al. 2015] [Trapier et al] [Trefny et al. 2014] [Tudosie 2017] [Tudosie 2018] [Turbojet and Turbofan Engines 1964] [Utomo and Bura 2019] [Vaca-Rios and Cerón-Muñoz 2025] [Vadyak et al. 1979b] [Vadyak et al. 1979a] [Varner et al. 1987] [Venkatakrishnan and Krothapalli 2006] [Vinogradov and Stepanov 2009] [Vnuchkov et al. 2018] [Volkov 2019] [Wang et al. 2023] [Wang et al. 2015a] [Weber 1975] [Welge et al. 1976] [Welge et al. 1977] [Williams 1965] [Williams and Hunt 1980] [Willis and Welliver 1976] [Wilson and Wright 1987] [Woodward et al. 1983] [Woodward et al. 1984] [Yang et al. 2026] [Yarng and Guan 1988]
Seeing out, and flying it
The literature of a cockpit with no forward window, being synthetic and enhanced vision, head-up displays and the external vision system, together with the handling qualities of supersonic aeroplanes.
79 records. [Anderson 1984a] [Anderson 1985] [Arthur et al. 2005] [Bailey et al. 2006] [Barrows et al. 1999] [Behringer et al. 2000] [Bernhardt et al. 2008] [Berry and Gilyard 1973] [Berry and Powers 1970] [Biber 2015] [Brailovsky et al. 2014] [Calhoun et al. 2005] [Calhoun et al. 2006] [Cerantola and Dass 2026] [Cook 1965] [Cooper and Young 2005] [Craig and Purvis 1990] [Cross et al. 2013] [Delgado et al. 2001] [Domino 2007] [Electronic Autopilot Controls Supersonic 1956] [Ertem] [Feather and Joshi 1981] [Ferguson and Radke] [Flight DECK, Head-Up Displays] [Frantis 2010] [Gao et al. 2018] [Goldsmith 1964] [Grantham and Nguyen 1977] [Greatrex 1963] [Gross et al. 1998] [Guimarães et al. 2021] [Hansen et al. 1999] [He et al. 2007] [He et al. 2008] [Helgeson et al] [Higgins and Shomber 1964] [Higgins and Shomber 1965] [Horikawa et al. 1997] [Howells and Brown 2007] [Human Engineering Considerations for] [Iradukunda and Averyanova 2021] [Jenkins 2008] [Jennings et al. 2002] [Keller et al. 2003] [Korn 2007] [Kramer et al. 2005] [Krohn and Jorgensen 2001] [Langston et al. 2016] [Lensi 2000] [Link et al. 2002] [Martin et al. 1996] [Mayer et al. 1999] [McLean 1994] [Mehler 2000] [Morelli 2000] [Mueller et al. 2002] [Newman 1987] [Norris 2001] [null] [O’Brien and Takahashi 2023] [Pradhan 2023] [Sanders-Reed et al. 2008] [Shelton et al. 2014] [Stark 2003] [Steer 2000] [Steer 2003] [Steer and Cook 1999] [Stengel 1969] [Stringer et al. 2013] [Thorndycraft et al. 2007] [Tiana 2007] [Tiana et al. 2007] [Uijt de Haag et al. 2005] [Wenger] [Wisely 1996] [Wisely 2008] [Xuelin et al. 2009] [Yelton et al. 2004]
45 of these titles name a synthetic, enhanced or external vision system, and the earliest is from 1996. Only 1 of them names an external vision system, the kind that replaces a forward window. The X-59’s own vision system papers are in the cluster on the aeroplane, where 7 titles name an external vision system.
Range, fuel and the cruise condition
The relations this article displays and does not evaluate, for want of a published lift-to-drag ratio and installed fuel consumption.
43 records. [Bangert et al. 1980] [Bardell 2000] [Bilimoria 1989] [Bond et al. 1979] [Bond et al. 1980] [Byington 1993] [Colling et al. 2026] [Commercial Transports-Aerodynamic Design 1982] [Cruise Performance 2012] [Cruise Performance and Flight 2021] [Cruise Performance 2022] [Durmus 2021] [Falco 1963] [Fei et al. 2026] [Felix et al. 2026] [Goodsell and Kennelly 2004] [Goodsell et al. 2000] [Hering 1963] [Herrmann 2004] [Kim et al. 2001b] [Liu 1967] [Luca et al. 2024] [Ma et al. 2019] [Maikapar 1993] [Masuda and Yoshida 1991] [Miele and Damoulakis 1969] [Nguyen et al. 2025] [Palaia 2026] [Pfaff 1968] [Pritchard 1969] [Serebreny 1965] [Seywald et al. 1991] [Squire 1965] [Staack et al. 2021] [Takahashi 2013] [Takahashi et al. 2010] [Chung 2007] [Torenbeek 1997] [Townend 1962] [Wei et al. 2010] [Whitfield and Griffith 1963] [Wroblewski and Ansell 2020] [Young 2010]
The atmosphere and the flight condition
The standard atmosphere and the speed of sound in air, which every altitude-dependent number in this article passes through.
63 records. [A 2006] [Air Force Test Pilot School Edwards Afb Ca 1962] [Loubeau et al] [Appendix a. the Standard 2011] [Appendix a Standard Atmosphere 2021] [Appendix B 2003] [Bass et al. 2007] [Chandler 2025] [Cramer 1993] [Easton 2005] [Essenhigh 2006] [Gringorten 1967] [Gringorten and Tattelman 1970] [Gross and Eisen 1959] [Hart et al. 2000] [Herbert] [Howell and Morfey 1987] [ICAO Standard Atmosphere 2021] [International Standard Atmosphere 2010] [Jamieson 1986] [Jones 1966] [Kaushik 2018] [König et al. 2019] [Kurzke and Halliwell 2018] [Kurzke et al. 2025] [Lewis and Neel 1964] [Li et al. 2017a] [Loubeau et al. 2023b] [Maxwell and Alway 1950] [Mayer and Varaksina 2023] [Mládek 1910] [Ouseph and Link 1984] [Pressures and Temperatures 2000] [Properties of the U.S. 2014] [Properties of the U.S. 2024] [Ramanauskas 1996] [Rees and Fuller-Rowell 1990] [Rekhy et al. 2020] [Space environment natural and] [National Bureau of Standards 1927a] [National Bureau of Standards 1927b] [Standard Atmosphere Data 1992] [Standard Atmosphere 1997] [Standard Atmosphere 2005] [Standard Atmosphere Data 2013] [Standard Atmosphere 2023] [Standard Atmosphere 2024] [Standard Atmospheric Profilesa aSource 2002] [The Flight Environment 2021] [The International Standard Atmosphere 2017] [The International Standard Atmosphere 2026] [The Speed of Sound 2018] [The Standard Atmosphere 1964] [The Standard Atmosphere 1976] [Turns and Kraige] [U.S.Committee on Extension to the Standard Atmosphere 1966] [1976 US Standard Atmosphere 2014] [Vaughan 2003] [Wong 1986] [Wong 1995] [Wong and Embleton 1985] [Worland and Wilson 1999] [Yager 2013]
Reviews, conferences and histories of the sonic boom
The literature that looks at the sonic boom as a whole, being reviews of its theory, the conferences and workshops on sonic boom research, and histories of the subject.
50 records. [Loubeau 2012] [Loubeau] [Baals and Foss 1966] [balazard 1972] [Carlson and Maglieri 1971] [Carlson and Maglieri 1972] [Darden 1992a] [Coulouvrat 2022b] [Darden 1976] [Darden 1977] [Darden 1992b] [Darden 1999] [McCurdy 1999] [Drougge 1973] [Dubois 1970] [Edwards 1994] [Emrich 1965] [Ferri and Schwartz 1971] [Ferri and Schwartz 1972] [Fleming 2014] [Hayes 1969] [Hayes 1971] [Holmes 1966] [Hubbard 1969] [Hubbard and Maglieri 1965b] [Hunton 1968] [Hutchinson 1963] [Schwartz 1968] [Schwartz 1971] [Page and Loubeau 2019] [Kane 1973a] [Kubota 2003] [Cliatt] [Larson et al. 2011] [Loftin 1967b] [Loubeau 2017] [MacDonald et al. 1978] [Panel Discussion 1975] [Parker 1968] [Plotkin 1989] [Plotkin and Maglieri 2003] [Rudenko and Makov 2021] [Sonic Boom 2012] [Sparrow 2003] [Sparrow 2010] [Sullivan 2007] [The Post-War Sonic Boom 2013] [Thompson and Parnell 1967] [Waggoner et al. 2018] [Warren 1972a]
Reviews and histories of supersonic flight
Supersonic flight surveyed as a whole, being reviews, outlooks and histories of supersonic aeroplanes and their aerodynamics.
17 records. [Davies 2020] [Davies 2025] [Harris 1989] [History of Supersonic Transport 2020] [Volkov 2023] [James 1998] [Nadim Melhem et al. 2024] [Petrescu 2020] [Seebass 1997] [Spearman 1979] [Supersonic and Hypersonic Flight 1960] [SUPERSONIC FLIGHT 2014] [Supersonic Flight 2025] [Transonic and Supersonic Flight 1992] [Trimpi 1980] [Wegener 1991] [Wegener 1997]
Where the Framing Breaks Down
The familiar framing is that the X-59 is a quiet supersonic aeroplane built to show that a quiet boom can satisfy the rule, and the framing breaks down in five places.
There is no loudness in the rule to satisfy. The prohibition bans a speed and the exemption path requires the absence of a measurable boom, so a quiet boom meets neither, as argued under The Rule It Exists to Change Contains No Loudness at All [14 CFR 91.817] [14 CFR 91.818].
The ninety percent that has been published is not a loudness. The nearest ratio to it is a ratio of overpressures, while in perceived loudness the reduction is 47.8 percent against the commonly quoted baseline, a calculation given in Where the Ninety Percent Comes From.
The quiet is not wholly the aeroplane’s to deliver. Loudness is set by rise time and rise time is set by the atmosphere, and the equivalent area the weight contributes cannot be shaped away, as the parts headed What Classical Theory Says the Rise Time Should Be and The Weight Puts an Equivalent Area There and No Shaping Removes It argue.
A quiet boom is not the only answer to a boom over land. Cruise below the cutoff Mach number makes no ground boom at all, and the X-59 flies above it on purpose because its question needs a boom to exist, as recorded in The Aeroplane Could Have Avoided the Problem Entirely and Refuses To.
And the framing speaks in the present tense about an aeroplane that had not yet made a boom. The register wrote its claim in the future tense, and at the date of this article the future tense was still correct.
The Source Base
72 hand-written references carry the argument and 2,572 research records map the field, and this article keeps them apart on purpose. The claims about what a document does and does not contain rest on five documents read in full, being the register, the two sections of the federal aviation regulations, the executive order directing their repeal, and the programme’s own conference presentation. Those are not commentary on the evidence. They are the evidence.
The Research References
The survey holds 2,572 research records drawn from NASA’s Technical Reports Server, the Defense Technical Information Center and the journal literature, of which 480 are report primaries, being 18.7 percent, by the identifier rule set out under The Definition of Primary. The median publication year is 2006. Of the 2,442 dated records, 324 were published before 1973, the year the prohibition on civil supersonic flight over the United States took effect, being 13.3 percent, and 683 were published in 2018 or later, the year this aeroplane was designated, being 28.0 percent [Civil Aircraft Sonic Boom, final rule].
This programme publishes no index of its own reports. The project page for the agency’s commercial supersonic work carries an editor’s note saying the project has concluded and is no longer updated, and it links to no publications list at all [Commercial Supersonic Technology Project].
Each record sits in one cluster, chosen by reading its title. The literature that treats the sonic boom or supersonic flight as a whole has clusters of its own, so no cluster is a residual.
The Word the Programme Uses Is Not the Word the Field Uses
The programme says sonic thump and the literature says sonic boom. Counting each of the 2,572 research titles once and ignoring case, 5 contain the phrase sonic thump and 906 contain the phrase sonic boom. A programme’s chosen word is a communications decision and the literature keeps its own, so the article uses the field’s word except where it is quoting.
Five Governing Documents Were Read in Full and Counted
The regulatory half of this article rests on 5 documents totalling 206,232 characters, every one retrieved and read. They are the two sections of the regulation, the 2021 final rule that put them in their present form, the 2020 proposal to bring supersonic aeroplanes into the noise certification rules, and the one authorisation to exceed Mach 1 that has been granted and noticed.
The phrase sonic boom appears 62 times across them and the programme’s own metric appears 0. A claim that a document does not say something is a count of zero, and a count of zero is the easiest kind of claim to get wrong, so it is computed from the retrieved text. The aeroplane itself is named 0 times, against 16 mentions of the agency that operates it.
A preamble is a primary document about what an agency decided and why, and it is the only place the reasoning behind a regulation is written down. Reading the rulemaking as well as the regulation is the difference between knowing what a rule says and knowing what its author thinks it means.
Where the Displayed Relations Come From
The article displays 60 relations and evaluates all but 3 of them, and those 3 are named in the epistemic state as boundaries rather than presented as results.
Lockheed Martin publishes a product card carrying the gross weight, the empty weight, the fuel, the payload, an overall length and a design Mach number. Every weight this article uses comes from it, including those in the lift-induced equivalent area and in the range relation, whose unknowns reduce to one.
The boom carpet is computed by tracing rays through the standard atmosphere from the Mach cone to the ground, using the invariance of the horizontal slowness. Run at Mach 2, the same trace agrees with the literature’s rule of thumb of a statute mile of carpet per thousand feet of altitude to within 3.9 percent, and that agreement is the evidence for the Mach 1.4 answer.
The Regulatory Text Is Quoted From Saved Copies
The electronic Code of Federal Regulations answers an automated request with about ten kilobytes of navigation for a section that is eighty kilobytes in a browser, and the clause this article quotes is not in the ten. Both sections are therefore quoted from copies retrieved with a browser and saved. The Federal Register site redirects an automated request to an unblock page, so the executive order is read from a university archive of the same text, and both addresses are cited.
One Catalogue Record Misspells Its Own Title
The library index records Maglieri’s 2014 survey of the field as Soinc boom. The reference list gives the title the work actually has, and a reader following the link will see the misspelling in the catalogue record.
The Definition of Primary
Primacy is a property of the document, and this survey derives it from the identifier. A record is a report primary when its address is on NASA’s Technical Reports Server or it carries a digital object identifier under the 10.21236 prefix, which Crossref registers for Defense Technical Information Center reports. The report-primary fraction is 18.7 percent, being 480 of the 2,572 research records. The rule does not reach the journal papers among the sources this article’s derivations use, such as Whitham on the flow pattern of a supersonic projectile, Stevens on the Mark VII perceived level and Pierce on rise times, so 18.7 percent is a lower bound on how much of the survey is first-hand.
Thin Subjects
Seven subjects that this article’s derivations depend on have thin literatures. Each count below is the number of research records in this survey whose title carries at least one of the listed terms, case ignored and a hyphen read as a space.
| Subject | Title terms counted | Records |
|---|---|---|
| the ground reflection factor and the impedance of the ground | reflection factor, reflection factors, ground impedance, impedance of the ground | 2 |
| the structure and thickness of a weak shock | shock structure, shock wave structure, shock thickness, shock wave thickness | 2 |
| the perceived level as a metric | perceived level, Mark VII, PLdB | 6 |
| Concorde as the baseline | Concorde | 4 |
| the weight of the aeroplane in its signature | weight | 1 |
| skin friction at supersonic speed | skin friction | 1 |
| the range relation and the fuel consumption | Breguet, fuel consumption | 7 |
The ground reflection factor rests on 2 titles. One gives reflection factors for flight near the threshold Mach number and the other estimates the impedance of the ground from a microphone array, so the impedance of 19 that the conventional factor of 1.9 implies is this article’s own reading and not a measured value.
The classical shock thickness is named in 2 titles, one of which concludes that molecular relaxation is insufficient to explain the shock structure of sonic booms. The Taylor thickness this article computes comes from the textbook relation it displays.
The perceived level is named in 6 titles, one of them Stevens’s definition of the Mark VII scale and the rest its application to sonic booms. Concorde is named in 4 titles and none of them names a perceived level, so the survey does not settle the baseline against which the programme’s reduction is stated.
The weight of an aeroplane is named in 1 title, a study of overpressure trends with weight, altitude, Mach number and shaping, so the reading that this aeroplane’s weight rather than its shape dominates its equivalent area rests on the relation shown and not on a body of measurement.
Skin friction is named in 1 title, on supersonic and hypersonic skin friction with transition, so the friction coefficient this article computes rests on the flat-plate relation it displays. The range relation is named in 7 titles and 1 of them concerns a supersonic aeroplane, a 1965 study of cruise fuel consumption and track selection in supersonic transport operations.
What the Survey Admits
A record is admitted only when a person reading its title finds it on a subject this article’s argument uses. Those subjects are the aeroplane and its predecessors, the generation, prediction, propagation and measurement of the boom, its effects on buildings, animals and people, the rule and the certification of aircraft noise, the supersonic aerodynamics, propulsion, vision system and cruise performance the sizing reasons with, and the atmosphere the boom crosses.
The records excluded are those whose only link to the article is a shared word or a distant application. Overpressure is also the measure of an explosive blast, and the literature of blast injury to the brain, the lungs, the eyes and the ears is large. Impulsive noise is also an outlier in a communications channel, atmospheric turbulence also distorts a laser beam, and a caustic is also made of light. Sonic hedgehog is one of the most cited genes in developmental biology. An economic boom, a baby boom, a crane boom, a refuelling boom and an oil containment boom are all boom, and a gene expression signature, a digital signature, a radar signature and a mutational signature are all signature. Breguet is also a family of watchmakers, and QUESST is also the abbreviation of a 2014 speech retrieval benchmark. Supersonic is also the flow in a scramjet combustor, around a projectile or a missile and through an ejector, and shock focusing is also a technique of medicine and of detonation research, and those records are excluded. The noise of roads, railways, wind turbines and machines shares the psychoacoustics of a boom and none of its source, and the handling qualities of aeroplanes in general share the cockpit and not the missing window, so they are excluded too. Correction, erratum and withdrawal notices are excluded because they are not research works. Reviews of books, reference and definition entries, other notices such as news items and editorials, and a second deposit of a document already in the survey are excluded for the same reason.
Some records of doubtful relevance are kept because they share this subject’s science rather than only its words. The long-range atmospheric propagation of weak blast waves and of infrasound, general nonlinear acoustics, the annoyance of military impulsive noise and of helicopters, the certification of the noise of subsonic aircraft, and the handling qualities of supersonic aeroplanes are the main examples, and a reader may weigh each accordingly.
Every research title has been read for relevance, so the off-topic share that remains is a matter of reading judgement. A reading of titles cannot show that a work is relevant in substance, so this is a statement about titles only.
Epistemic State
Historical Fact
The X-59A designation was allocated on 21 June 2018 to Lockheed Martin with NASA as sponsor, and its mission statement in the register is the one quoted [Parsch, DOD 4120.15-L Addendum]. NASA announced the design and build contract on 3 April 2018 at 247.5 million dollars, stating a period of performance ending 31 December 2021 and an intention to fly over selected United States cities beginning in mid-2022 [NASA Release 18-020]. The aeroplane first flew on 28 October 2025 for 67 minutes, reaching approximately 12,000 feet and 230 miles per hour with the landing gear down, from Palmdale to Edwards [X-59 First Flight]. Lockheed Martin’s product card gives a maximum design gross weight of 24,300 pounds, an empty weight of 15,000, fuel of 8,000, payload of 600, an overall length of 93.83 feet, a wingspan of 29 feet 6 inches, a design Mach number of 1.4, a loudness of less than 75 perceived level decibels and a single F414-GE-100 [Silencing the Sonic Boom, X-59 Product Card]. The design cruise condition is Mach 1.4 at 55,000 feet and the design target is 75 on the Stevens Mark VII perceived level scale [NASA Release 18-020] [Doebler and Rathsam 2020]. Executive Order 14304 of 6 June 2025 directs the repeal of the prohibition within 180 days and the establishment of an interim noise-based certification standard, without naming a threshold [Executive Order 14304, American Presidency Project]. The deadline fell on 3 December 2025 and the sections quoted in this article carry no amendment later than 15 January 2021 [14 CFR 91.817] [14 CFR 91.818]. The register’s other sonic boom row, the X-54A, describes an aeroplane to be capable of generating relevant ground sonic boom signatures, so both rows describe an aeroplane that would make signatures and only the X-59A row calls its signature shaped. The X-59A entry is half present tense and half future, saying that the aircraft is a research aircraft and that it will create a shaped sonic boom signature.
Recomputed From the Register by This Article
The register, in the copy the compiler last updated on 23 August 2026, carries 539 allocation rows of which 31 are X designations. 2 rows of the 539 mention a sonic boom and they are the X-54A and the X-59A. The word create appears in 1 row. Measured without popular names and note markers, the X-59A mission statement is 84 characters against a median of 109 for the X rows and 328 for its immediate predecessor, placing it twenty-fourth of 31 by length. 94 rows have an empty engines cell. Lockheed Martin holds 6 X rows and NASA sponsors 10. Every one of these is a count over the register file, and none is taken from a secondary source. The copy counted here is later than this article’s date, and the version current on that date carried 526 rows of which 29 were X designations, with the counts that differ under it stated in the opening section [Parsch, DOD 4120.15-L Addendum, 11 August 2025].
Verified by Reading the Document
5 governing documents totalling 206,232 characters were retrieved and read in full, being the two sections of the regulation, the 2021 final rule that put them in their present form, the 2020 proposal to bring supersonic aeroplanes into the noise certification rules, and the one authorisation to exceed Mach 1 that has been granted and noticed [14 CFR 91.817] [14 CFR 91.818] [Special Flight Authorizations for Supersonic Aircraft, final rule] [Noise Certification of Supersonic Airplanes, proposed rule] [Petition for Authorization to Exceed Mach 1, notice of decision to grant]. Every passage quoted is quoted from what was read.
Across all five, the phrase sonic boom appears 62 times and the programme’s own metric appears 0. The abbreviation is absent, and so are the words behind it. The aeroplane is named 0 times, against 16 mentions of the agency that operates it.
The design feature annotations attributed to NASA were read from the project’s conference presentation [Richwine, X-59 Aircraft Overview and Status], and the weights, the overall length and the loudness bound were read from the builder’s product card [Silencing the Sonic Boom, X-59 Product Card].
Analysis
The cutoff Mach number of 1.153 at the cruise altitude, and the finding that the cutoff ratio nowhere exceeds 1.153 in the standard atmosphere. The reproduction of the published 925 miles per hour to within 0.1 percent, and the identification of the released 940 miles per hour as Mach 1.42 at the same altitude. The conversion of 75 and 105 perceived level decibels to 2.134 and 4.088 sones and the resulting 47.8 percent. The classical Taylor rise time of 5.2 microseconds. The 19.9 decibel benefit of a tenfold rise time at 1,000 hertz. The slenderness ratios in the table.
The dynamic pressure of 12,512 pascals and the density of 0.14664 kilograms a cubic metre at the cruise condition. The lift-induced equivalent area of 4.232 square metres and its growth by a factor of 2.06 between forty thousand feet and fifty-five. The Reynolds number, the skin friction coefficient of 0.00206 and the friction drag of 25.7 newtons a square metre. The wave-drag ratio of 5.18 against a twenty metre body. The boom carpet at 59.2 kilometres wide and the along-track offset of 19.29 kilometres, from a ray trace through the standard atmosphere that reproduces the literature’s Mach 2 rule of thumb to within 3.9 percent. The minimum signature duration of 73 milliseconds and the sound exposure level of 101 decibels. The classical absorption of 0.159 decibels a kilometre. The ground impedance of 19 that reproduces the conventional reflection factor.
The spread of 5.87 feet among the three published lengths and the shortfall of 700 pounds between the builder’s weights and its stated total. The range of 47.8 to 53.1 percent across the published Concorde baselines, and the overpressure ratio of 84.5 percent. The equivalent diameter of 2.32 metres. The thrust bound of 2,634 pounds for an exponent of one. The first flight’s 24.9 percent of the design cruise speed and 21.8 percent of the design cruise altitude.
All are computed from the published inputs named beside them by the relations displayed above, and every relation this article displays is evaluated except the three named below.
Inference
That the decomposition of the claimed 30 decibel reduction into 16.2 decibels of amplitude and 13.8 of shape is a bound rather than a measurement. It assumes that perceived level would track peak overpressure by the decibel rule, which the preceding sections establish is false, and serves only to argue that shape cannot account for less than the residual.
That the register’s use of the verb create marks this row as different in kind. The count of one is a fact. The reading that a shaped signature is a thing made rather than a phenomenon studied is this article’s, and no one has been found stating it.
That the aeroplane’s configuration is dominated by cost and schedule wherever the boom does not constrain it. The annotations are NASA’s and the tally is this article’s.
That the aeroplane’s weight rather than its shape is the majority contributor to its signature. The relation is standard and the evaluation is arithmetic, but the reading is this article’s, drawn from comparing a computed equivalent diameter of 2.32 metres against a fuselage whose slimness is visible rather than measured.
That the aeroplane grew during detail design. Three published lengths span 5.87 feet, the shortest is the builder’s and carries a 2019 copyright while the longest are the agency’s and are later. No source states a growth and the alternative reading, that the figures measure to different points, is equally consistent.
What the Record Does Not Settle
Whether the aeroplane will meet 75 perceived level decibels in flight. By this article’s date it had flown 1 time, subsonically, and no shaped signature had been produced or measured [X-59 First Flight] [NASA’s X-59 Completes Second Flight].
Whether the X-59 requires an authorisation under the rule. The prohibition’s text applies to a civil aircraft, the statute excludes public aircraft from civil aircraft, and no document read here applies that definition to this aeroplane or names it [49 U.S.C. 40102, Definitions].
The relative contribution of molecular relaxation and atmospheric turbulence to the observed rise time. Both are documented, and the sources consulted do not agree on the split.
Concorde’s perceived level. Published values include 105, 108 and 110 [NASA unveils a supersonic plane with a quiet boom, Physics Today] [The Next NASA X-Plane, Born to Boom, Machine Design], and this article computes its headline comparison at 105, the value NASA’s acousticians call commonly quoted, while giving the range the alternatives produce.
The X-59’s predicted ground overpressure. The 0.3 pounds per square foot this article uses for illustration is the Quiet Supersonic Platform’s shock goal [Iosifidis, Concept Development of the Quiet Supersonic Technology Aircraft], and the overpressure predicted for this aeroplane appears in none of the sources read.
What form any replacement standard will take. Executive Order 14304 requires an interim noise-based certification standard and does not say whether it would be written as a perceived level, an overpressure, or something else [Executive Order 14304, American Presidency Project], and at the editorial date no proposed rule had appeared [Federal Register search, FAA documents mentioning supersonic, 2016 to 4 December 2025]. A standard written as an overpressure would not use the metric in which this programme states its target, and nothing read here settles which it will be.
The lift-to-drag ratio and the thrust specific fuel consumption. The builder’s card supplies the weights, so the range relation has exactly one unknown, and that unknown is the ratio of the two. The range and endurance given in the body are illustrative.
Whether the slenderness lowers the net drag. Wave drag falls as the fourth power of length and wetted area rises as the first, which settles the direction of the trade but not its balance. Whether the net drag falls depends on how the total divides between the two kinds at the design point, and no source publishes that division.
The wetted area and the volume of the equivalent body. Without them the friction drag is given per square metre and never totalled, and the wave drag is given as a ratio against a shorter body and never evaluated absolutely.
Why the builder’s card does not add up. The empty weight, the fuel and the payload sum to 700 pounds less than the gross weight the same card states [Silencing the Sonic Boom, X-59 Product Card], which is 2.9 percent, and no source explains the difference.
Out of Scope
The engineering of the eXternal Vision System beyond its existence and its validation. It is a substantial subject in display design and human factors, outside aerodynamics.
The economics of supersonic passenger transport. Whether a market exists for a quieter supersonic airliner is a question about airlines and not about shock waves, and this article makes no claim either way.
The environmental case beyond noise. Fuel burn per passenger kilometre and stratospheric emissions are the other two objections to supersonic transport, both serious, and neither is what this aeroplane addresses.
The rulemaking record before 1973 and the international one throughout. The 1973 final rule and its appendix are read here only for their wording, and the proposals and hearings that preceded them are not examined. The international standard-setting body’s work on supersonic noise is a parallel process this article does not enter, and the survey holds it without the argument using it.
The commercial supersonic programmes. One of them appears here only as a petitioner in the Federal Register and another only as a commenter on a rule. Whether any of them will fly is not a question about this aeroplane, and the trade press that follows them is not literature this survey admits.
Every X-plane after the X-59. The X-60A and beyond belong to later articles in this series.
Conclusion
The register gives this aeroplane one sentence and the sentence contains the whole programme in one adjective. An aeroplane that will create a shaped sonic boom signature [Parsch, DOD 4120.15-L Addendum]. The other of the register’s two sonic boom rows, the X-54, was to generate relevant ground signatures and was never built. Both entries describe an aeroplane that would make signatures, and only one of them describes a shaped one. That adjective is the difference between the two sentences, and the register’s verb, used once in 539 rows, marks it. One of the two was built.
The physics supports the ambition. A signature that has not coalesced into a shock-fronted wave loses less energy on the way down and puts what remains at frequencies the ear discounts, and a factor of ten in rise time is worth 19.9 decibels at 1,000 hertz and twenty at every frequency well above both corners, which is where the ear does its listening. The design problem is real, the mechanism is understood, and the aeroplane’s extraordinary proportions are that mechanism made visible.
And the proportions may be cheaper than they look. Wave drag falls as the fourth power of length and wetted area rises as the first, so the exponents favour length heavily. Whether the net drag falls lies beyond this article, because the answer depends on how the drag divides between the two kinds at the design point and no source publishes that division. What the exponents do establish is a direction. Supersonic aeroplanes were long and thin before anybody cared what they sounded like, so the shaping requirement pushed a shape the aerodynamics already wanted, though it pushed it a good deal further, this aeroplane being 1.39 times as slender as Concorde.
What the shaping cannot touch is the weight. Integrating the lift term of the equivalent area over the whole aeroplane leaves the weight divided by twice the dynamic pressure, with every trace of shape gone. At the design condition that is 4.232 square metres, a body 2.32 metres across, and a designer may move it along the length and may not remove it. Worse, it is inversely proportional to dynamic pressure, so climbing to lengthen the propagation path enlarges it, by a factor of 2.06 between forty thousand feet and the design altitude. Two of the things altitude does to a sonic boom are therefore opposed, and no source is cited for that reading, which is this article’s inference. Altitude does others, changing the cutoff margin, the width of the carpet and the length of the absorbing path, and none of those is weighed here.
Four things are less settled than the programme’s public arithmetic suggests.
The claimed reduction is not ninety percent in any unit the target is stated in. In perceived loudness it is 47.8 percent against the commonly quoted baseline, and between 47.8 and 53.1 percent across every baseline this article found. In pressure it is 96.8 percent and in energy 99.9. The 84.5 percent that Concorde’s overpressure against the low-boom goal gives is a pressure ratio, and reporting a pressure ratio as a loudness is the substitution this article warns against. The Mark VII scale exists to prevent that substitution, and 30 decibels of perceived level is a little less than halving the loudness, because halving it takes 32.
The baseline is part of the uncertainty and this article does not pretend otherwise. Concorde’s perceived level is published at 105, 108 and 110, the sources consulted do not agree, and the article computes its headline figure at 105 and shows what the alternatives give.
The quantity that governs the answer is not under the designer’s control. Loudness is set by rise time, classical theory underestimates rise time by more than two orders of magnitude, and what makes up the difference is molecular relaxation and turbulence in the atmosphere on the day. A predicted perceived level depends on the weather as much as on the aeroplane, and the programme’s plan is to fly over communities and survey the people underneath rather than to rely on computation [Quesst, NASA’s mission to enable commercial supersonic flight].
And the rule the aeroplane exists to change contains no loudness at all. The X-54 established that it is a speed limit rather than a noise limit. This article read the rulemaking behind it and counted. Across 5 governing documents and 206,232 characters, the programme’s own metric appears 0 times. The exemption path requires not a quiet boom but no measurable boom [14 CFR 91.818]. The only noise standard ever proposed for supersonic aeroplanes governs them while they are flying subsonically, and it was never adopted [Noise Certification of Supersonic Airplanes, proposed rule] [Federal Register search, FAA documents mentioning supersonic, 2016 to 4 December 2025]. And on 15 January 2021, 2.6 years after this aeroplane was designated and while it was being built, the agency wrote down that the determination made in the 1970s that no level of sonic boom is acceptable over land still applies [Special Flight Authorizations for Supersonic Aircraft, final rule], against industry asking it to say otherwise.
So the aeroplane’s output is not compliance but evidence, gathered so that a number might one day be written where none exists, and the executive order that would demand such a number declines to say what it should be.
The deadline for writing it expired the day before this article. The order of 6 June 2025 gave 180 days, the 3 December 2025 deadline came, and the regulation stood unchanged [Executive Order 14304, American Presidency Project] [14 CFR 91.817]. That is not a judgement about the programme or the agency, since a rulemaking is slow and an executive deadline is not a statute. It is the state of the record at the date on this article.
And what is being measured is not a point but a strip. Tracing rays through the standard atmosphere, the carpet is 59.2 kilometres wide and the boom lands 19.29 kilometres behind the aeroplane that made it. At Mach 2 the tracer agrees with the literature’s rule of thumb to within 3.9 percent, which is the reason to believe the Mach 1.4 figure, and the Mach 1.4 carpet is only 69.6 percent of the Mach 2 one. Flying slower narrows the carpet as well as quietening it, and the arithmetic gives that for nothing. No source consulted for this article states it, which describes what was read and is silent on what the programme says.
At the editorial date the aeroplane had flown once, for 67 minutes, at 24.9 percent of its design cruise speed, with the landing gear down [X-59 First Flight]. It had never been supersonic [NASA’s X-59 Completes Second Flight]. The contract that built it named 31 December 2021 and the flights over cities were to begin in mid-2022 [NASA Release 18-020]. The register’s claim about this aeroplane is written in the future tense, and at the time of writing the future tense was still correct. The entry says that the aircraft is a research aircraft and that it will create a shaped sonic boom signature. The first half had been true since the aeroplane was rolled out. The second half was still a promise.
Equations
This section collects the 60 display equations of the article in the order they appear, each under its number and reproduced exactly as it stands in the text. A title in bold names an established result, and a title in italics beginning with the word Bespoke names a construction made for this analysis. Of the 60, 48 are established and 12 are bespoke. Pressures, densities and speeds are in units of the International System at standard atmosphere conditions, while weights and thrusts are quoted from the builder in pounds and speeds from the National Aeronautics and Space Administration in miles per hour, levels are in decibels referred to 20 micropascals, and perceived level in decibels on the Mark VII scale is a psychophysical quantity whose differences are not pressure ratios, so a percentage reduction depends on whether it is stated in pressure, energy or loudness.
Equation 1, Mach Angle and Prandtl-Glauert Parameter. The Mach cone has a half-angle whose sine is the reciprocal of the Mach number, and its cotangent is the Prandtl-Glauert parameter. Liepmann and Roshko, Elements of gasdynamics
\[\sin\mu = \frac{1}{M} \qquad\qquad \cot\mu = \sqrt{M^2 - 1} = \beta \tag{1}\]Equation 2, Far-Field Overpressure From the F-Function. Linearised supersonic theory gives the far-field overpressure ratio as proportional to Whitham’s F-function and inversely to the square root of distance. Whitham 1952 Whitham, Linear and nonlinear waves
\[\frac{\Delta p}{p_a} = \frac{\gamma M^2}{\sqrt{2\beta r}}\; F(\zeta) \tag{2}\]Equation 3, Whitham F-Function. The F-function is an Abel transform of the second derivative of the equivalent area distribution. Whitham 1952 Whitham, Linear and nonlinear waves
\[F(\zeta) = \frac{1}{2\pi} \int_0^{\zeta} \frac{A_e''(x)}{\sqrt{\zeta - x}}\; \mathrm{d}x \tag{3}\]Equation 4, Equivalent Area Including Lift. The equivalent area is the volume cross-section plus a term proportional to the lift carried ahead of each station. Ashley and Landahl, Aerodynamics of wings and bodies Jones, Wing theory
\[A_e(x) = A_V(x) + \frac{\beta}{2q} \int_0^{x} L'(\xi)\; \mathrm{d}\xi \tag{4}\]Equation 5, Lift-Induced Equivalent Area at the Tail. Integrated over the whole aircraft in level flight, the lift term collapses to the weight over twice the dynamic pressure times the Prandtl-Glauert parameter, independent of shape.
\[A_e(\ell) - A_V(\ell) = \frac{\beta}{2q} \int_0^{\ell} L'(\xi)\,\mathrm{d}\xi = \frac{\beta\,W}{2q} \tag{5}\]Equation 6, Bespoke Lift-Induced Equivalent Area at the Design Point. The equivalent area is evaluated with the published maximum design gross weight standing for the cruise lift, a choice the article makes, and the standard atmosphere dynamic pressure at Mach 1.4 and 55,000 feet. Silencing the Sonic Boom, X-59 Product Card
\[\frac{\beta\,W}{2q} = \frac{0.9798 \times 108,092}{2 \times 12,512} = 4.232\ \text{square metres} \tag{6}\]Equation 7, Bespoke Diameter of the Lift-Induced Equivalent Body. A body of revolution carrying that equivalent area, taken at the maximum design gross weight, would be 2.32 metres across.
\[d_e = 2 \sqrt{ \frac{\beta\,W}{2q\,\pi} } = 2 \sqrt{ \frac{ 4.232 }{\pi} } = 2.32\ \text{metres} \tag{7}\]Equation 8, Altitude Ratio of the Lift-Induced Equivalent Area. At fixed weight and Mach number the lift-induced area scales inversely with dynamic pressure, so it is 2.06 times larger at 55,000 feet than at 40,000.
\[\frac{ 4.232 }{ 2.058 } = \frac{ 25,730 }{ 12,512 } = 2.06 \tag{8}\]Equation 9, Von Karman Slender-Body Wave Drag. The wave drag is a double integral of the second derivative of the same equivalent area with a logarithmic kernel. Kuchemann, The aerodynamic design of aircraft Area rule
\[D_w = -\frac{q}{2\pi} \int_0^{\ell}\!\!\int_0^{\ell} A_e''(x)\, A_e''(\xi)\, \ln|x - \xi| \; \mathrm{d}x \, \mathrm{d}\xi \tag{9}\]Equation 10, Cylindrical Spreading of a Weak Disturbance. A weak disturbance from a line source falls with the square root of distance because its energy spreads over a cylinder. Pierce, Acoustics
\[\Delta p \propto r^{-1/2} \tag{10}\]Equation 11, Far-Field Decay of Sonic Boom Amplitude and Duration. Whitham’s far-field result makes the overpressure fall as the minus three-quarters power of distance and the signature lengthen as the one-quarter power. Whitham, Linear and nonlinear waves Maglieri and others, Sonic boom, six decades of research
\[\Delta p \propto r^{-3/4} \qquad\qquad T_s \propto r^{1/4} \tag{11}\]Equation 12, Acoustic Intensity. The intensity of an acoustic wave is its mean square pressure over the characteristic impedance of the medium. Kinsler, Frey, Coppens and Sanders, Fundamentals of acoustics
\[I = \frac{p^2}{\rho\, c} \tag{12}\]Equation 13, Decay of Acoustic Energy Along the Ray. Combining the far-field exponents gives an energy per unit length that falls as the minus one-quarter power of distance, the energy that the shocks dissipate.
\[\Pi(r) \;\sim\; r\, \frac{(\Delta p)^2}{\rho c}\, T_s \;\propto\; r \cdot r^{-3/2} \cdot r^{1/4} = r^{-1/4} \tag{13}\]Equation 14, Speed of Sound in a Perfect Gas. The local speed of sound follows from the temperature at each altitude. U.S. Standard Atmosphere 1976 1976 Anderson, Modern compressible flow
\[c(z) = \sqrt{\gamma R\,T(z)} \tag{14}\]Equation 15, Temperature Profile of the Standard Atmosphere. The temperature falls linearly in the troposphere and is constant in the isothermal layer above the tropopause. U.S. Standard Atmosphere 1976 1976
\[T(z) = \begin{cases} T_0 - \lambda z, & 0 \le z \le z_1 \\ T_1, & z_1 < z \le z_2 \end{cases} \tag{15}\]Equation 16, Pressure and Density in the Isothermal Layer. In the isothermal layer the pressure decays exponentially from its tropopause value and the density follows from the gas law. U.S. Standard Atmosphere 1976 1976 International Standard Atmosphere
\[p(z) = p_1 \exp\!\left[ -\frac{g\,(z - z_1)}{R\,T_1} \right] \qquad\qquad \rho(z) = \frac{p(z)}{R\,T(z)} \tag{16}\]Equation 17, Dynamic Pressure at the Design Cruise. At the standard atmosphere density for 55,000 feet and a speed of Mach 1.4 the dynamic pressure is 12,512 pascals.
\[q = \tfrac{1}{2}\rho V^2 = \tfrac{1}{2} \times 0.14664 \times 413.1^2 = 12,512\ \text{pascals} \tag{17}\]Equation 18, Ray Invariant in a Stratified Atmosphere. In a horizontally stratified medium the cosine of the ray elevation over the local sound speed is conserved, a form of Snell’s law. Pierce, Acoustics
\[\frac{\cos\theta(z)}{c(z)} = \mathrm{constant} \tag{18}\]Equation 19, Mach Cutoff Number. The boom reaches the ground only above the Mach number equal to the ratio of the sound speed at the ground to that at cruise altitude. Maglieri and others, Sonic boom, six decades of research
\[M_c = \frac{c(0)}{c(z_a)} \tag{19}\]Equation 20, Cutoff Mach Number at the Design Altitude. In the standard atmosphere the cutoff Mach number from 55,000 feet is 1.153, well below the design Mach number of 1.4. U.S. Standard Atmosphere 1976 1976
\[M_c = \frac{ 340.29 }{ 295.07 } = 1.153 \qquad\text{against}\qquad M = 1.4 \tag{20}\]Equation 21, Horizontal Slowness of an Azimuthal Ray. For a wave normal at azimuth $\phi$ around the Mach cone, the magnitude of the horizontal slowness is invariant along the ray. Pierce, Acoustics Haefeli et al. 1969b
\[u = \frac{\sqrt{\sin^2\mu + \cos^2\mu \, \cos^2\phi}}{c(z_a)} = \text{constant along the ray} \tag{21}\]Equation 22, Lateral Cutoff Azimuth. A ray reaches the ground only while its slowness admits a real vertical direction there, which at the design condition requires an azimuth of at least 45.4 degrees. Haefeli et al. 1969b
\[u\, c(0) \le 1 \qquad\Longleftrightarrow\qquad \phi \ge \phi_c = 45.4^{\circ} \tag{22}\]Equation 23, Lateral Ray Path. The lateral offset of a ray follows from integrating the slope that the invariant fixes from cruise altitude to the ground. Pierce, Acoustics
\[\frac{\mathrm{d}y}{\mathrm{d}z} = \frac{u\,c(z)}{\sqrt{1 - \left[ u\,c(z) \right]^2}} \;\cdot\; \frac{\cos\mu\,\cos\phi}{\sqrt{\sin^2\mu + \cos^2\mu\,\cos^2\phi}} \tag{23}\]Equation 24, Boom Carpet Half-Width and Width. Integrating the ray path numerically through the standard atmosphere places the widest-arriving ray 29.6 kilometres to the side, a carpet of 59.2 kilometres.
\[y_c = 29.6\ \text{km} \qquad\Longrightarrow\qquad 2 y_c = 59.2\ \text{km} \tag{24}\]Equation 25, Check of the Ray Tracer Against the Published Rule of Thumb. The same tracer at Mach 2 falls 3.9 percent short of the published one mile per thousand feet, and the Mach 1.4 carpet is 69.6 percent of the Mach 2 carpet. NASA Facts FS-2003-11-016, Sonic Booms
\[1 - \frac{85.03}{55 \times 1.609344} = 1 - \frac{85.03}{88.51} = 3.9\% \qquad\qquad \frac{59.17}{85.03} = 69.6\% \tag{25}\]Equation 26, Design Cruise Speed. The design Mach number times the speed of sound at cruise altitude gives 413.1 metres per second, which reproduces the published 925 miles per hour. Quesst, the Aircraft
\[V = M\, c(z_a) = 1.4 \times 295.07 = 413.1\ \mathrm{m/s} = 924.1\ \mathrm{mph} \tag{26}\]Equation 27, Sound Pressure Level. An overpressure becomes a level as twenty times the base-ten logarithm of its ratio to the reference pressure. Kinsler, Frey, Coppens and Sanders, Fundamentals of acoustics
\[L_p = 20 \log_{10} \frac{\Delta p}{p_r} \tag{27}\]Equation 28, Ground Reflection Factor. The overpressure at the ground is the incident overpressure times a reflection factor close to two.
\[\Delta p_g = K_R\, \Delta p_i \tag{28}\]Equation 29, Bespoke Ground Impedance Implied by a Reflection Factor of 1.9. For a locally reacting surface at normal incidence, the reflection factor of 1.9 that the article adopts from inside the measured range of 1.8 to 2.0 corresponds to a normalised ground impedance of 19. Carlson 1978
\[R_g = \frac{\sigma - 1}{\sigma + 1} \qquad\qquad K_R = 1 + R_g \qquad\Longrightarrow\qquad \sigma = 19 \tag{29}\]Equation 30, Taylor Thickness of a Weak Shock. The steady thickness of a weak shock in a viscous, heat-conducting gas is inversely proportional to its strength, and its rise time is that thickness over the sound speed. Hamilton and Blackstock, Nonlinear acoustics
\[\delta = \frac{4\, \delta_d\, \rho_0\, c_0}{(\gamma + 1)\, \Delta p} \qquad\qquad \tau = \frac{\delta}{c_0} \tag{30}\]Equation 31, Diffusivity of Sound. The diffusivity of sound gathers the shear and bulk viscosities and the thermal conduction term. Volume viscosity
\[\delta_d = \frac{1}{\rho_0} \left[ \frac{4}{3}\eta + \eta_B + \kappa \left( \frac{1}{c_v} - \frac{1}{c_p} \right) \right] \tag{31}\]Equation 32, Classical Rise Time Against Measurement. At sea level and the low-boom overpressure the classical rise time is 5.2 microseconds, about 192 times shorter than the measured millisecond. Cleveland and Blackstock 1994 Maglieri and Sothcott 1990
\[\tau = 5.2\ \text{microseconds} \qquad\text{against a measured}\qquad \tau \approx 1\ \text{millisecond} \tag{32}\]Equation 33, Classical Absorption Coefficient. Classical absorption from viscosity and conduction rises as the square of frequency and gives 0.159 decibels a kilometre at 1,000 hertz.
\[\alpha = \frac{\omega^2 \delta_d}{2 c_0^3} \qquad\Longrightarrow\qquad 0.159\ \text{decibels a kilometre at}\ 1,000\ \text{hertz} \tag{33}\]Equation 34, Steepening Distance and Least Unshocked Rise Time. A compression of rise time $\tau$ steepens into a shock after a distance that falls with its amplitude, and inverting it over the propagation distance gives the least rise time that arrives unshocked. Hamilton and Blackstock, Nonlinear acoustics Whitham, Linear and nonlinear waves
\[x_s = \frac{2\,\rho_0\, c_0^3\, \tau}{(\gamma + 1)\, \Delta p} \qquad\Longrightarrow\qquad \tau_m = \frac{(\gamma + 1)\, \Delta p\, z_a}{2\,\rho_0\, c_0^3} \tag{34}\]Equation 35, Bespoke Indicative Least Rise Time. Evaluating the least rise time at sea-level conditions over the whole path from cruise altitude, an indicative choice the article makes for a stratified atmosphere, gives about six milliseconds.
\[\tau_m = 5.99\ \text{milliseconds} = 8.2\%\ \text{of the shortest possible signature} \tag{35}\]Equation 36, N-Wave Pressure Signature. The classical far-field signature falls linearly from the front shock through zero to an equal rear shock over the duration $T_s$. Whitham, Linear and nonlinear waves Maglieri and others, Sonic boom, six decades of research
\[p(t) = \Delta p \left( 1 - \frac{2t}{T_s} \right), \qquad 0 \le t \le T_s \tag{36}\]Equation 37, Lower Bound on Signature Duration. No signature can be shorter than the time the aircraft takes to travel its own length, 73 milliseconds at the design speed.
\[T_s \ge \frac{\ell}{V} = \frac{30.18}{413.1} = 73\ \text{milliseconds} \tag{37}\]Equation 38, Positive-Phase Impulse. The impulse of the positive phase of the N-wave is one quarter of the peak overpressure times the duration. Kinsler, Frey, Coppens and Sanders, Fundamentals of acoustics SAE ARP865
\[I_p = \int_0^{T_s/2} p(t)\,\mathrm{d}t = \frac{\Delta p\, T_s}{4} \tag{38}\]Equation 39, Bespoke Sound Exposure Level at the Shortest Duration. The sound exposure of the N-wave is evaluated at the published overpressure and at the shortest possible duration, a bound the article takes as the duration, giving 101 decibels.
\[E_s = \int_0^{T_s} p^2(t)\,\mathrm{d}t = \frac{(\Delta p)^2 T_s}{3} \qquad\qquad L_E = 10 \log_{10} \frac{E_s}{p_r^2 \times 1\,\text{s}} = 101 \tag{39}\]Equation 40, Bespoke Spectrum of a Smoothed Step. The article models the front of the signature as a step smoothed with a single rise time, whose spectrum falls as one over frequency until the smoothing takes hold. Pierce 1970
\[|P(f)| \propto \frac{1}{f \sqrt{1 + (2\pi f \tau)^2}} \tag{40}\]Equation 41, Corner Frequency of the Front. The spectrum steepens above a corner frequency of one over two pi times the rise time.
\[f_c = \frac{1}{2\pi\tau} \tag{41}\]Equation 42, Stevens Mark VII Perceived Level and Loudness. The Mark VII scale sets one sone at forty decibels of perceived level and doubles the loudness for every 32 decibels. Stevens 1972 Sone
\[PL = 40 + 32 \log_2 S \qquad\Longleftrightarrow\qquad S = 2^{(PL - 40)/32} \tag{42}\]Equation 43, Loudness of the Target and of Concorde. Perceived levels of 75 and 105 correspond to loudnesses of 2.134 and 4.088 sones.
\[S(75) = 2.134 \qquad\qquad S(105) = 4.088 \tag{43}\]Equation 44, Loudness Reduction Against a Pressure Reading of the Same Decibels. In loudness the reduction from 105 to 75 is 47.8 percent, whereas reading the same thirty decibels as a pressure ratio gives 96.8 percent. Doebler and Rathsam 2020
\[1 - \frac{S(75)}{S(105)} = 47.8\% \qquad\text{against}\qquad 1 - 10^{-30/20} = 96.8\% \tag{44}\]Equation 45, Loudness Reduction for Other Concorde Baselines. Because a loudness ratio depends only on a difference of perceived level, baselines of 108 and 110 give reductions of 51.1 and 53.1 percent.
\[1 - \frac{S(75)}{S(PL)} = 1 - 2^{(75 - PL)/32} \qquad\Longrightarrow\qquad 1 - 2^{-33/32} = 1 - 0.48929 = 51.1\%, \qquad 1 - 2^{-35/32} = 1 - 0.46854 = 53.1\% \tag{45}\]Equation 46, Overpressure and Energy Reductions. The published overpressures give a reduction of 84.5 percent, and thirty decibels read as an energy ratio gives 99.9 percent.
\[1 - \frac{ 0.3 }{ 1.94 } = 84.5\% \qquad\qquad 1 - 10^{-30/10} = 99.9\% \tag{46}\]Equation 47, Bespoke Split of the Reduction Into Amplitude and Shape. The article treats the amplitude term as if perceived level obeyed the decibel rule for peak overpressure, an assumption the sources do not make, and attributes the remaining 13.8 decibels to shape. NASA Release 18-020 Doebler and Rathsam 2020
\[\underbrace{PL_{\text{Concorde}} - PL_{\text{X-59}}}_{ 30 } \;=\; \underbrace{20\log_{10}\dfrac{\Delta p_i(\text{Concorde})}{\Delta p_i(\text{X-59})}}_{ 16.2 } \;+\; \underbrace{\text{what the shape contributes}}_{ 13.8 } \tag{47}\]Equation 48, Bespoke Slenderness Ratio. The article compares aircraft by a slenderness ratio it defines as overall length divided by span.
\[\Lambda = \frac{\ell}{b} \tag{48}\]Equation 49, Bespoke Spread of Published Lengths and Their Slenderness. The three published lengths differ by 5.87 feet, and against the card’s span they give slenderness ratios of 3.18, 3.36 and 3.38. Silencing the Sonic Boom, X-59 Product Card Quesst, the Aircraft X-59 Aircraft Overview and Status
\[\Delta\ell = 99.7 - 93.83 = 5.87\ \text{feet} = 5.9\%\ \text{of}\ 99.7 \qquad\qquad \Lambda = \frac{\ell}{b} = \frac{ 93.83 }{29.5},\ \frac{ 99 }{29.5},\ \frac{ 99.7 }{29.5} = 3.18,\ 3.36,\ 3.38 \tag{49}\]Equation 50, Sum of the Published Weights Against the Gross Weight. The card’s empty, fuel and payload weights sum to 23,600 pounds, 2.9 percent short of its maximum design gross weight. Silencing the Sonic Boom, X-59 Product Card
\[\underbrace{ 15,000 }_{\text{empty}} + \underbrace{ 8,000 }_{\text{fuel}} + \underbrace{ 600 }_{\text{payload}} = 23,600\ \text{pounds} \qquad\qquad \frac{ 24,300 - 23,600 }{ 24,300 } = \frac{ 700 }{ 24,300 } = 2.9\% \tag{50}\]Equation 51, Sears-Haack Minimum Wave Drag. The minimum wave drag of a body of revolution of given length and volume falls as the square of the volume over the fourth power of the length. Ashley and Landahl, Aerodynamics of wings and bodies Area rule
\[D_w = \frac{128\, q\, \Omega^2}{\pi\, \ell^4} \tag{51}\]Equation 52, Bespoke Length Scaling of Wave and Friction Drag. The article sets the Sears-Haack fourth-power fall in wave drag against a friction drag it takes as proportional to wetted area and a wetted area it takes as proportional to the square root of length at fixed volume.
\[D_w \propto \ell^{-4} \qquad\text{against}\qquad D_f \propto S_w \propto \ell^{1/2} \tag{52}\]Equation 53, Sutherland’s Law. Sutherland’s law gives the viscosity of air as a function of temperature from a reference value and the Sutherland constant. Anderson, Modern compressible flow Raymer, Aircraft design, a conceptual approach
\[\eta(T) = \eta_r \left( \frac{T}{T_r} \right)^{3/2} \frac{T_r + C_S}{T + C_S} \tag{53}\]Equation 54, Reynolds Number at the Design Cruise. The Reynolds number on overall length at the cruise density, speed and viscosity is about 129 million.
\[Re = \frac{\rho V \ell}{\eta} = 128.6 \times 10^{6} \tag{54}\]Equation 55, Turbulent Flat-Plate Skin Friction. The Schlichting turbulent flat-plate correlation gives a mean skin friction coefficient of 0.00206, and the friction drag is the dynamic pressure times that coefficient times the wetted area.
\[C_f = \frac{0.455}{(\log_{10} Re)^{2.58}} = 0.00206 \qquad\qquad D_f = q\, C_f\, S_w \tag{55}\]Equation 56, Breguet Range in Terms of Mach Number. The cruise range is the Mach number times the sound speed over the thrust specific fuel consumption, times the lift-to-drag ratio and the logarithm of the weight ratio. Raymer, Aircraft design, a conceptual approach Range (aeronautics)
\[R_c = \frac{M\, c}{c_t}\; \frac{L}{D}\; \ln \frac{W_i}{W_f} \tag{56}\]Equation 57, Thrust Lapse With Density. The installed net thrust falls as a power of the density ratio.
\[\frac{F_N}{F_{N0}} = \left( \frac{\rho}{\rho_0} \right)^{n} \tag{57}\]Equation 58, Bespoke Collected Range Coefficient. Grouping the known terms with the card’s fuel and gross weights, which assumes the whole fuel load is burned in cruise, gives a coefficient of 165 metres per second that multiplies the unpublished ratio of lift-to-drag ratio to consumption. Range (aeronautics)
\[R_c = \underbrace{V \ln \frac{W_i}{W_f}}_{ 165\ \text{m/s} } \;\cdot\; \frac{L/D}{c_t} \tag{58}\]Equation 59, Bespoke Thrust Bound at the Cruise Density. With a lapse exponent of one that the article takes, the published sea-level thrust of 22,000 pounds lapses to an indicative 2,634 pounds at the cruise density. General Electric F414
\[\frac{\rho}{\rho_0} = \frac{ 0.14664 }{1.225} = 0.1197 \qquad\Longrightarrow\qquad F_N = F_{N0}\, \frac{\rho}{\rho_0} = 22,000 \times \frac{ 0.14664 }{1.225} = 2,634\ \text{pounds} \tag{59}\]Equation 60, First Flight Against the Design Cruise. The first flight reached 24.9 percent of the design cruise speed and 21.8 percent of the design cruise altitude. X-59 First Flight Quesst, the Aircraft
\[\frac{ 230\ \text{mph} }{ 925\ \text{mph} } = 24.9\% \qquad\qquad \frac{ 12,000\ \text{ft} }{ 55,000\ \text{ft} } = 21.8\% \tag{60}\]Notation
- $\mu$, the Mach angle
- $\beta$, the Prandtl-Glauert parameter, $\sqrt{M^2-1}$
- $p_a$, the ambient pressure at the aircraft
- $F(\zeta)$, Whitham’s F-function, with $\zeta$ the retarded streamwise coordinate in which it is written
- $A_e$, the equivalent area distribution
- $A_V$, the part of the equivalent area contributed by volume alone
- $q$, the free-stream dynamic pressure
- $W$, the aircraft weight
- $d_e$, the diameter of a circle whose area is the lift-induced equivalent area
- $D_w$, the wave drag
- $\ell$, the overall length of the aircraft
- $T_s$, the duration of the ground signature
- $\Pi(r)$, the acoustic energy per unit length of a cylinder of radius $r$
- $c(z)$, the local speed of sound at altitude $z$
- $z_a$, the aircraft’s cruise altitude
- $\theta$, the elevation angle of an acoustic ray above the horizontal
- $M_c$, the cutoff Mach number, below which no boom reaches the ground
- $u$, the horizontal slowness of a ray, which is invariant in a stratified atmosphere
- $\phi_c$, the azimuth below which a ray no longer reaches the ground
- $y_c$, the half-width of the boom carpet
- $L_p$, the sound pressure level
- $p_r$, the reference pressure of the decibel scale, 20 micropascals
- $K_R$, the ground reflection factor
- $R_g$, the pressure reflection coefficient at the ground
- $\sigma$, the normalised specific acoustic impedance of the ground
- $\delta_d$, the diffusivity of sound in air
- $\eta_B$, the bulk viscosity of air
- $\tau_m$, the least front rise time that avoids coalescence over a given distance
- $x_s$, the distance in which a compression steepens into a shock
- $I_p$, the positive-phase impulse of the signature
- $E_s$, the sound exposure of the signature
- $L_E$, the sound exposure level
- $f_c$, the corner frequency above which a shock’s spectrum steepens
- $PL$, the Stevens Mark VII perceived level, in decibels
- $S(75)$, the total loudness in sones at a perceived level of 75
- $\Lambda$, the slenderness ratio, overall length divided by span
- $\Omega$, the volume of the equivalent body of revolution
- $D_f$, the skin friction drag
- $S_w$, the wetted area
- $C_S$, Sutherland’s constant for air
- $Re$, the Reynolds number based on overall length
- $C_f$, the mean skin friction coefficient
- $R_c$, the still-air cruise range
- $c_t$, the thrust specific fuel consumption
- $W_i$, the aircraft weight at the start of cruise, with $W_f$ that at the end
- $F_N$, the installed net thrust, with $F_{N0}$ its value at sea level and the same Mach number
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- Sechter, Brendan. 2025. “X-Planes: Piasecki X-49 SpeedHawk.” Brendan A R Sechter’s Development Blog, November 24. https://sgeos.github.io/aerospace/history/engineering/2025/11/24/x_planes_piasecki_x49.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-50 Dragonfly.” Brendan A R Sechter’s Development Blog, November 25. https://sgeos.github.io/aerospace/history/engineering/2025/11/25/x_planes_boeing_x50.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-51 Waverider.” Brendan A R Sechter’s Development Blog, November 26. https://sgeos.github.io/aerospace/history/engineering/2025/11/26/x_planes_boeing_x51.html
- Sechter, Brendan. 2025. “X-Planes: X-52, the Designation Refused.” Brendan A R Sechter’s Development Blog, November 27. https://sgeos.github.io/aerospace/history/engineering/2025/11/27/x_planes_x52_designation_refused.html
- Sechter, Brendan. 2025. “X-Planes: Boeing X-53 Active Aeroelastic Wing.” Brendan A R Sechter’s Development Blog, November 28. https://sgeos.github.io/aerospace/history/engineering/2025/11/28/x_planes_boeing_x53_active_aeroelastic_wing.html
- Sechter, Brendan. 2025. “X-Planes: Gulfstream X-54.” Brendan A R Sechter’s Development Blog, November 29. https://sgeos.github.io/aerospace/history/engineering/2025/11/29/x_planes_gulfstream_x54.html
- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-55 ACCA.” Brendan A R Sechter’s Development Blog, November 30. https://sgeos.github.io/aerospace/history/engineering/2025/11/30/x_planes_lockheed_martin_x55_acca.html
- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-56.” Brendan A R Sechter’s Development Blog, December 1. https://sgeos.github.io/aerospace/history/engineering/2025/12/01/x_planes_lockheed_martin_x56.html
- Sechter, Brendan. 2025. “X-Planes: ESAero X-57 Maxwell.” Brendan A R Sechter’s Development Blog, December 2. https://sgeos.github.io/aerospace/history/engineering/2025/12/02/x_planes_esaero_x57_maxwell.html
- Sechter, Brendan. 2025. “X-Planes: X-58, the Slot Taken by XQ-58.” Brendan A R Sechter’s Development Blog, December 3. https://sgeos.github.io/aerospace/history/engineering/2025/12/03/x_planes_x58_slot_taken_by_xq58.html
Referenced by
- X-Planes: Synthesis and What the Designation Became 2025-12-16
- X-Planes: Bell Textron X-76 SPRINT 2025-12-15
- X-Planes: X-69 through X-75, the Leapfrogged Block 2025-12-14
- X-Planes: General Atomics X-68 LongShot 2025-12-13
- X-Planes: X-67, the Slot Taken by XQ-67A 2025-12-12
- X-Planes: Boeing X-66 2025-12-11
- X-Planes: Aurora Flight Sciences X-65 CRANE 2025-12-10
- X-Planes: Invocon X-64 2025-12-09
- X-Planes: ABL Space Systems X-63 2025-12-08
- X-Planes: Lockheed Martin X-62 VISTA 2025-12-07
- X-Planes: Dynetics X-61 Gremlins 2025-12-06
- X-Planes: Generation Orbit X-60 2025-12-05