The register of designations allocated since August 1998, which Andreas Parsch compiles from data the Department of Defense has released, holds 539 rows in the copy the compiler last updated on 23 August 2026, and exactly one of their mission statements names a regulation. It belongs to the X-54 [Parsch, DOD 4120.15-L Addendum, MDS Designators Allocated After 19 August 1998]. 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. Every register finding in this article holds in that version too, including the X-54 row as the only one to name a regulation [Parsch, DOD 4120.15-L Addendum, 11 August 2025].

This is the fifty-fifth 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, and the X-53.

It was never built, and the regulation is being repealed without it. [Gulfstream X-54]

Both halves of that sentence are load-bearing and the second is deliberately in the present tense. No airframe exists, no flight test report was written, and no title in this article’s survey of its field names the aeroplane [Gulfstream X-54]. The prohibition it was designated to help lift was ordered repealed by the President of the United States on 6 June 2025, with a deadline that falls four days after the date on this article [Executive Order 14304, Leading the World in Supersonic Flight]. The order names no technique. The technique that keeps a boom off the ground without a new airframe differs from the one the X-54 was designed around, and it predates the prohibition. [Henderson et al. 1971]

The Research Question

The X-54 existed to settle one question, and its registry entry states it. Could an aeroplane shaped all the way to the tail generate a shaped sonic boom signature that survives to the ground, so that the people beneath it could be asked about it and a regulatory change process could use what they said? When the number was allocated, a shaped forward signature had been measured on the ground and the Quiet Spike’s near field had been probed, and no ground signature of an aeroplane shaped all the way to the tail had been measured [Pawlowski et al. 2005] [Smolka et al. 2007].

The question was never put to an X-54 in flight, because no airframe was built [Gulfstream X-54]. The entry that poses the question and the physics that makes it a question at all are set out under the headings The Registry Entry Is the Primary Document and Why a Sonic Boom Is Loud.

The Registry Entry Is the Primary Document

On 5 May 2008 the joint designation registry recorded the following against X-54A [Parsch, DOD 4120.15-L Addendum, MDS Designators Allocated After 19 August 1998].

Supersonic experimental aircraft to conduct flight research. Capable of generating relevant ground sonic boom signatures to gather scientific data in support of NASA and a regulatory change process.

Read the last five words again. Regulatory appears once in the whole register, and certification, rulemaking and policy appear in none of the 539 entries at all. The contractor is Gulfstream Aerospace, the sponsor is NASA, the National Aeronautics and Space Administration, and the powerplant is two Rolls-Royce Tay 651 engines [Gulfstream Aerospace] [Rolls-Royce Tay]. Those are ordinary registry fields. The mission statement is not, because it names an administrative outcome rather than a technical one.

The neighbouring entries show how unusual this one is. The entry immediately above it in the register is allocated to the aeroplane that was refused the number X-52 and given X-53 instead [Missing USAF and DOD Aircraft Designations]. The X-51A is described as a hypersonic vehicle to flight test a scramjet engine. The X-53A is described as a modified F/A-18A to flight test active aeroelastic wing technology [Parsch, DOD 4120.15-L Addendum, MDS Designators Allocated After 19 August 1998] [X-Planes: Boeing X-53 Active Aeroelastic Wing]. Every one of them names a technology and a flight condition. The X-54A entry names a technology, a flight condition, and then a rulemaking.

One word in the entry carries the argument, and that word is ground. Not near-field, not signature, not overpressure. Ground. By 2008 the near field of a spike had been measured and the ground signature of a whole shaped aeroplane had not, and the difference between those two facts is what this aeroplane was for [Smolka et al. 2007].

Parsch obtained the data in this addendum from official Department of Defense sources under the Freedom of Information Act, because the department stopped publishing the list itself for almost six years after 1998 [Allocation of Official Aerospace Vehicle MDS Designations] [DOD 4120.15-L, Model Designation of Military Aerospace Vehicles].

Why a Sonic Boom Is Loud

An aeroplane in supersonic flight does not make a noise. It makes a shape, and the shape becomes a noise on the way down.

Every point on the airframe that displaces air sends out a disturbance. Above the speed of sound those disturbances cannot run ahead, so they pile onto a cone trailing the aeroplane at the Mach angle [Mach Number] [Liepmann and Roshko, Elements of gasdynamics]

\[\mu \;=\; \arcsin\!\left(\frac{1}{M}\right), \tag{1}\]

where the speed of sound in a perfect gas depends on temperature and on nothing else [Anderson, Modern compressible flow],

\[a \;=\; \sqrt{\gamma R T}. \tag{2}\]

At Mach 1.4 the Mach angle is 45.6 degrees and at Mach 1.8 it is 33.7 degrees, so a faster aeroplane drags a narrower cone behind it.

The wave does not travel straight down and that matters later. A ray leaves normal to the cone surface, so it makes the Mach angle with the vertical, and the distance it covers from cruise altitude $h$ to the ground is [Haefeli et al. 1969a]

\[r \;=\; \frac{h}{\cos\mu} \;=\; \frac{h}{\sqrt{1 - M^{-2}}}. \tag{3}\]

At Mach 1.4 from 45,000 feet that is 19,598 metres of air against an altitude of 13,716, a path 42.9 percent longer than the altitude suggests.

Close to the aeroplane the disturbance is not one shock but many, one for the nose, one for the canopy, one for each wing leading edge, one for the inlets, one for the tail. The near-field signature is a rippled thing with structure in it [Pawlowski et al. 2005].

What turns that structure into a boom is that the strong parts travel faster than the weak parts. A weak shock does not propagate at the speed of sound. It propagates slightly faster, by an amount proportional to its own strength [Whitham, Linear and nonlinear waves] [Hamilton and Blackstock, Nonlinear acoustics]

\[\frac{U}{a} \;=\; 1 \;+\; \frac{\gamma+1}{4\gamma}\,\frac{\Delta p}{p} \;+\; \mathcal{O}\!\left(\frac{\Delta p}{p}\right)^{2}. \tag{4}\]

So the front of the signature catches the middle and the back of it catches the middle from behind. Given enough distance the structure collapses into two shocks with a linear expansion between them, which is the N-wave everybody on the ground actually hears [Sonic Boom] [N-Wave] [Pierce, Acoustics, an introduction to its physical principles and applications].

The shape it decays into has a name and a form. For an N-wave duration $T_{N}$ the waveform is a straight line between two jumps, the symbol carrying a subscript because $T$ is already the temperature in this article [Darden 1976] [Application of sonic-boom minimization concepts in supersonic transport design],

\[\Delta p(t) \;=\; \Delta p_{\max}\left(1 - \frac{2t}{T_{N}}\right), \qquad 0 \le t \le T_{N}, \tag{5}\]

whose positive-phase impulse follows immediately, and which matters because loudness depends on the shape and not only on the peak [McCurdy 1994],

\[I \;=\; \int_{0}^{T_{N}/2}\Delta p(t)\,\mathrm{d}t \;=\; \frac{\Delta p_{\max} T_{N}}{4}. \tag{6}\]

The crucial point here is easy to miss. The N-wave is not what the aeroplane makes. The N-wave is what any shape decays into, and two isolated shocks with nothing between them make it loud, because they present the ear with two near-instantaneous pressure steps.

That suggests the question at the centre of the field. If the N-wave forms because strong shocks overtake weak ones, could an aeroplane be shaped so that the overtaking never finishes?

Programme Origin

The X-54 did not begin with its designation in 2008, and its record runs back through three lines of earlier work. The first is the minimisation theory, which was complete and validated in a wind tunnel by 1979 [Mack and Darden]. The second is flight evidence, being the Shaped Sonic Boom Demonstration of 2003 and the Quiet Spike flights of 2006 and 2007, together with the Gulfstream studies, patent and papers from 2003 onward [A Summary of Recent Supersonic Vehicle Studies at Gulfstream Aerospace]. The third is the prohibition of 1973 that the registry entry names as its purpose [14 CFR 91.817, Civil Aircraft Sonic Boom].

The record reviewed here shows how the case for the aeroplane was argued, and it does not contain the decision that turned the case into an allocation. The subsections gathered under this heading follow those three lines in turn.

The Mathematics of Choosing a Shape

The answer is yes in principle and the theory is older than the prohibition [George and Seebass 1971].

Whitham showed in 1952 that the far-field disturbance of a slender supersonic body is governed by a single function of one variable, computed from the body’s cross-sectional area distribution [Whitham 1952] [Ashley and Landahl, Aerodynamics of wings and bodies]

\[F(y) \;=\; \frac{1}{2\pi}\int_{0}^{y} \frac{A_e''(\xi)}{\sqrt{y-\xi}}\;\mathrm{d}\xi . \tag{7}\]

The subscript matters. $A_e$ is not the geometric cross-section. It is the equivalent area, which adds to the physical area a term representing the lift the aeroplane is carrying, because a lifting surface displaces the air whether or not it is thick [Darden 1976]

\[A_e(x) \;=\; A(x) \;+\; \frac{\beta}{2q}\int_{0}^{x} \ell(\xi)\,\mathrm{d}\xi , \qquad \beta = \sqrt{M^{2}-1}. \tag{8}\]

An aeroplane’s boom is therefore partly a consequence of its weight, and all of the lift enters the equivalent area by the tail,

\[A_{L}(l) \;=\; \frac{\beta W}{2q}, \tag{9}\]

so a heavier aeroplane at the same length is louder for reasons no amount of shaping can remove [Area Rule].

The penalty follows a square root, a milder law than proportionality. A fully aged N-wave amplitude goes as the square root of the F-function amplitude, and the F-function amplitude is linear in the lift [McLean 1965] [Application of sonic-boom minimization concepts in supersonic transport design], so

\[\Delta p \;\propto\; W^{1/2}. \tag{10}\]

Doubling the aeroplane’s weight costs only 41.4 percent in overpressure. No other exponent in this subject is generous.

Before nonlinear steepening takes hold, the pressure disturbance at distance $r$ follows from $F$ directly [On the propagation of weak shock waves],

\[\frac{\Delta p}{p_\infty} \;=\; \frac{\gamma M^{2}}{\sqrt{2\beta}}\,\frac{F(y)}{\sqrt{r}} , \tag{11}\]

and once the wave has aged into an N-wave the amplitude and the duration take the classical exponents [On the propagation of weak shock waves] [Haefeli et al. 1969a]

\[\Delta p \;\propto\; r^{-3/4}, \qquad \Delta t \;\propto\; r^{1/4}. \tag{12}\]

The obvious reading of those two exponents is wrong. Doubling the distance costs 40.5 percent of the overpressure and buys 18.9 percent more duration, which sounds like a great deal. The trouble is that an aeroplane cannot double its cruise altitude. Climbing from 45,000 feet to 50,000 feet is an 11.1 percent increase in distance and buys 7.6 percent off the overpressure, and halving the boom by climbing alone would require 113,393 feet.

So altitude is a weak lever not because the exponent is small but because the range is short, which is why the subject is about shape.

The design result came in 1971 and 1972. George and Seebass showed that for a given length, weight, Mach number and altitude there exists an equivalent area distribution that minimises the shock strength, and that the optimum differs from the smooth body an aerodynamicist would draw [George and Seebass 1971] [Seebass and George 1972]. Darden then relaxed the requirement for an infinitely sharp nose, which had made the optimum unbuildable, and produced the family of shapes the low-boom discipline still uses [Darden 1979].

Hayes supplied the other half in 1972, showing how the signature propagates through an atmosphere whose density and sound speed vary with height [Hayes and Runyan 1972].

The mathematics also contains a serious catch. The wave drag of a slender body in linearised supersonic flow is a functional of the same second derivative that sets the F-function [Ashley and Landahl, Aerodynamics of wings and bodies],

\[D \;=\; -\frac{q}{2\pi}\int_{0}^{l}\!\!\int_{0}^{l} A_e''(x_{1})\,A_e''(x_{2})\, \ln\lvert x_{1}-x_{2}\rvert \;\mathrm{d}x_{1}\,\mathrm{d}x_{2}. \tag{13}\]

So the boom and the drag are two different functionals of one function, and the distribution that minimises the first is not the distribution that minimises the second. A low-boom aeroplane pays for its quietness in cruise efficiency, and the design literature that follows accordingly treats the two objectives as a trade.

And the theory was checked against a model before anybody flew it. A wind-tunnel investigation published alongside the nose-bluntness work in 1979 measured the signature of a body designed by the minimisation method and found it behaved as the method said it would [Mack and Darden], while a companion study worked the minimisation through a real atmosphere rather than an isothermal one [Darden 1975b] [Darden 1975a].

By 1979 the theory was complete, validated in a tunnel, and reduced to design charts [Darden et al. 1989] [Application of sonic-boom minimization concepts in supersonic transport design]. The X-54 was designated twenty-nine years later.

What Had Already Been Proved by 2008

One thing had been proved and it had been proved convincingly.

On 27 August 2003 a Northrop Grumman F-5E with a reshaped nose glove flew supersonic over Edwards Air Force Base, and the pressure signature measured on the ground was not an N-wave [Pawlowski et al. 2005] [Shaped Sonic Boom Demonstration]. It was a flat-topped signature, and it had kept its flat top all the way down, a result the agency’s Virginia laboratory had spent four decades of wind-tunnel work predicting [NASA Langley Research Center] [Haering et al. 2005a] [Haering and Murray 2004].

That result is the reason anybody took the X-54 seriously. The Shaped Sonic Boom Demonstration, run under the Defense Advanced Research Projects Agency’s Quiet Supersonic Platform programme [Wlezien and Veitch 2002] [Defense Advanced Research Projects Agency], established in flight and in a real atmosphere that a deliberately shaped signature can survive the trip to the ground. Follow-on flights measured how far that persistence extended [Morgenstern et al. 2005] [Hartman et al. 2001].

What it did not establish is that a useful aeroplane can be built that way. The F-5E was modified in its forebody only, because showing that the positive phase of the signature could be shaped was enough to test whether a shaped signature persists [Pawlowski et al. 2005], and it was not a design anybody would fly passengers in. The demonstration proved the physics and left the engineering open.

What Had Not Been Proved

Gulfstream had been arguing for this aeroplane in public since 2003 and the argument is on the record. The company published a summary of its supersonic vehicle studies that year [A Summary of Recent Supersonic Vehicle Studies at Gulfstream Aerospace], a shaping method that does not assume an axisymmetric body [Howe 2003], and a case for the small supersonic civil aircraft as a class [Henne 2005]. The extendable spike was patented in March 2004 [US Patent 6,698,684 B1, Supersonic Aircraft with Spike for Controlling and Reducing Sonic Boom], and the minimisation argument for it was published the following year [Howe 2005] [Simmons and Freund 2005a].

The hardware flew in 2006, and it deliberately did not attempt the ground signature [Smolka et al. 2007].

The Quiet Spike was a telescoping composite nose boom, which the flight test report’s drawing dimensions at 14.15 feet retracted and 24.31 feet deployed and its text calls thirty feet long, mounted on the radar bulkhead of a NASA F-15B at the flight research centre in the California desert [Smolka et al. 2007] [NASA Armstrong Flight Research Center] [Quiet Spike] [McDonnell Douglas F-15 Eagle]. Its conical segments were intended to place a series of weak shocks ahead of the aeroplane instead of one strong bow shock. Thirty-two flights were flown in the Quiet Spike configuration between August 2006 and February 2007, reaching Mach 1.8 at a design dynamic pressure of 685 pounds per square foot, and the near-field shock structure was probed by a second aeroplane at Mach 1.4 on 13 December 2006 [Haering et al. 2007] [Moua et al. 2007].

The programme documented its work thoroughly, and the problem its documents record is structural. The aerodynamics had been confirmed in the agency’s supersonic wind tunnel at Langley in August 2002 [Smolka et al. 2007]. What had not been settled was whether a twenty-four-foot telescoping tube could be flown at all.

The joint stiffness between segments turned out to govern the design, which a quarter-scale dynamic model established before the flight article was drawn [Simmons et al. 2007] [Howe et al. 2008]. A pneumatic extension mechanism was built, found unable to control position and speed precisely, and replaced by an electrically driven cable and pulley system on a half-scale prototype [Smolka et al. 2007].

The schedule then would not allow a conventional ground vibration test of the mated aeroplane before the flutter analysis had to be finished, so the programme built the modal model up in stages instead [Spivey et al. 2007] [Herrera and Pak 2007]. Structural mode interaction testing then found a lightly damped pitch oscillation between ten and thirteen hertz with the spike retracted, and the answer was to clear the envelope with the spike extended and restrict angle of attack [Brenner and Kukreja 2007] [Kukreja 2008].

The spike was predicted to cost the host aeroplane stability, by three to five percent subsonically and by between three and twenty-four percent supersonically in the lateral-directional axis, and that prediction was the programme’s chief worry going into flight test [Cumming et al. 2008] [Freund et al. 2008]. Flight measurement contradicted it. Directional stability flattened out by Mach 1.4 and the aeroplane was more stable than a standard F-15 by Mach 1.8 [Smolka et al. 2007]. The limit was set by pitch damping, which fell away sharply above Mach 1.7 and left little margin for going faster [Freund et al. 2007] [Knight et al. 2008].

None of that is about sonic boom. It is the price of carrying a boom-shaping device on an aeroplane not designed for one, and it was the second reason for wanting a purpose-built airframe. The company went on publishing propagation work for years after the aeroplane it wanted was not built [Salamone 2009] [Howe and Henne 2006].

The programme’s report records why no ground measurement was attempted, a decision taken while the flight test was being planned [Smolka et al. 2007].

It was quickly determined that ground signature measurement would not be possible as in the Supersonic Boom Demonstrator (SSBD) program. Combining the Quiet Spike with any existing supersonic aircraft would result in a traditional N-wave ground signature as the stronger shocks of today’s aircraft would overtake the spike’s weak shocks within a short distance below the flight path. Only in-flight near-field shock probing could be obtained for further aero-acoustic validation.

That paragraph is the case for the X-54, written by the people who would have built it and published about seven months before the number was allocated. [Smolka et al. 2007] The shaping of a nose is undone by the aeroplane behind the nose. To measure a shaped signature on the ground you need an aeroplane that is shaped all the way to the tail, and no such aeroplane existed.

The registry entry then asks for the thing the Quiet Spike report says cannot be got any other way. Capable of generating relevant ground sonic boom signatures.

The Rule

The prohibition the X-54 was meant to help lift is one sentence of the regulation and it dates from 1973 [14 CFR 91.817, Civil Aircraft Sonic Boom].

It forbids a civil aircraft from exceeding Mach 1 over the United States, except under a special flight authorisation issued case by case [14 CFR 91.818, Special Flight Authorizations for Supersonic Aircraft]. It limits speed and sets no limit on noise. The agency chose a speed limit because it judged that the monitoring and control technology of the time could not support anything more discriminating [Civil Aircraft Sonic Boom 1973]. The rule was published on 28 March 1973 and took effect on 27 April 1973 [Civil Aircraft Sonic Boom 1973].

The regulatory problem comes down to that distinction. A speed limit cannot be met by a quiet aeroplane, because the rule cannot see how quiet an aeroplane is. Any aeroplane that wants to fly supersonic over land needs the rule replaced and not satisfied, and replacing it requires evidence about people rather than evidence about aeroplanes.

The rule outlived the aeroplanes it was written for. Concorde never operated supersonically over the United States, and the American supersonic transport was cancelled in 1971, two years before the prohibition took effect [Concorde] [Boeing 2707].

On 6 June 2025 the President ordered it repealed [Executive Order 14304, Leading the World in Supersonic Flight]. The order directs the Federal Aviation Administration to take the necessary steps to repeal the prohibition within 180 days and to establish an interim noise-based certification standard in its place [Supersonic Flight, Federal Aviation Administration]. The order names no technique. Its stated ground is that advances in aerospace engineering, materials science and noise reduction now make supersonic flight possible [Executive Order 14304, Leading the World in Supersonic Flight].

That deadline expires on 3 December 2025, four days after this article is dated. The prohibition had been in force for 52.1 years when the order came, and the order came 17.1 years after an aeroplane was designated to gather evidence for this outcome.

Sizing From First Principles

There was no X-54 to size. No airframe was built, and the registry records only the contractor, the sponsor and two Rolls-Royce Tay 651 engines [Gulfstream X-54] [Parsch, DOD 4120.15-L Addendum, MDS Designators Allocated After 19 August 1998]. The cruise Mach number, altitude and perceived-level target attributed to the intended aeroplane come from the secondary record, and Dependent Systems sets them out.

What can be sized from first principles is the boom. The subsections gathered under this heading compute the atmosphere the article works in, the distance over which a shaped nose signature is consumed by the aeroplane behind it, what the atmosphere does to a signature on the way down, the units a boom is measured in, and the cutoff Mach number below which no boom reaches the ground. Each of those numbers is computed here and none is taken from a source, and Epistemic State lists them.

The Standard Atmosphere

Every altitude, Mach number and dynamic pressure in this article is computed in the standard atmosphere, and the relations are set out here so that none of them is assumed.

Below the tropopause the temperature falls linearly and hydrostatic equilibrium in a perfect gas fixes the pressure [U.S. Standard Atmosphere],

\[T(z) \;=\; T_{0} - Lz, \qquad p(z) \;=\; p_{0}\left(\frac{T(z)}{T_{0}}\right)^{g/LR}, \tag{14}\]

with $L = 0.0065$ kelvin per metre giving an exponent of 5.2559. Above it the temperature is constant to a first approximation and the pressure falls exponentially with a scale height of 6,341.6 metres [U.S. Standard Atmosphere 1976 1976],

\[p(z) \;=\; p_{11}\exp\!\left[-\frac{g\,(z - z_{11})}{R\,T_{11}}\right]. \tag{15}\]

The dynamic pressure follows from the pressure and the Mach number alone, and this form is the useful one because a flight test report states both [Liepmann and Roshko, Elements of gasdynamics],

\[q \;=\; \tfrac{1}{2}\rho V^{2} \;=\; \frac{\gamma}{2}\,p\,M^{2}. \tag{16}\]

That relation is also a check on the record and the record passes it. The Quiet Spike report gives a design dynamic pressure of 685 pounds per square foot and a design Mach number of 1.8 at 45,000 feet. Inverting the relation at that altitude gives Mach 1.782, which agrees with the stated figure to within the rounding of the inputs.

The Arithmetic the Report Set Out and Did Not Perform

A short distance below the flight path is a quantity, and weak-shock theory gives it.

Two shocks separated by a distance $s$ along the ray close on each other at a rate set by the difference in their strengths [On the propagation of weak shock waves]. The separation is written $s$ because $\ell$ is already the lift per unit length above, and the coalescence distance is written $\mathcal{L}$ to keep it clear of the lapse rate $L$. Substituting the propagation speed,

\[\Delta U \;=\; a\,\frac{\gamma+1}{4\gamma}\,\frac{\Delta p}{p}, \tag{17}\]

so the stronger overtakes the weaker after a travel distance

\[\mathcal{L} \;=\; \frac{a\,s}{\Delta U_{1}-\Delta U_{2}} \;=\; s\,\frac{4\gamma}{(\gamma+1)}\left[\frac{\Delta p_{1}-\Delta p_{2}}{p}\right]^{-1}. \tag{18}\]

The speed of sound cancels, so the result depends on altitude only through the pressure ratio, and that independence is the reason to make the estimate at all.

The expression is an old one. It is the aging length of nonlinear acoustics, being the distance over which a wave of length $s$ and strength $\Delta p / p$ distorts into its asymptotic form,

\[r_{\text{age}} \;\sim\; s\,\frac{4\gamma}{(\gamma+1)}\,\frac{p}{\Delta p}. \tag{19}\]

So the question of whether a spike survives to the ground and the question of when a signature becomes an N-wave are the same question, asked with different words, and the classical theory answers both [Whitham, Linear and nonlinear waves] [Hamilton and Blackstock, Nonlinear acoustics].

With the spike extended to 24.31 feet, and taking the difference in strength between the aeroplane’s own shocks and the spike’s shocklets as a fraction of ambient pressure, weak-shock theory gives a coalescence distance of 1.73 kilometres at the weakest shock difference in the range and 0.35 kilometres at a typical one.

The distance that matters is the ray path and not the altitude. The ray leaves normal to the Mach cone, so it makes the Mach angle with the vertical, and its path from 45,000 feet is 19,598 metres rather than 13,716, which is 42.9 percent longer. Against the ray path the coalescence occupies 8.8 percent of the journey at the weakest assumption and 1.8 percent at a typical one.

Strength difference $\Delta(\Delta p)/p$ Coalescence distance As a share of the ray path
0.01 1,729 m 8.8 percent
0.02 864 m 4.4 percent
0.03 576 m 2.9 percent
0.05 346 m 1.8 percent
0.10 173 m 0.9 percent

The assumed shock-strength difference is an assumption and this article will not pretend otherwise. The record does not publish the near-field overpressures of the F-15B with the spike fitted in a form this calculation can consume directly. On this plane-wave estimate the conclusion looks independent of the assumption, since the coalescence completes in the first tenth of the journey across the whole range. The next subsection shows that appearance to be an artefact of the plane-wave assumption.

The Spreading Correction

The estimate above is a plane-wave result and a real boom spreads. Both shocks weaken with distance, roughly as $r^{-1/2}$ before the wave has aged, so the closing rate falls as they travel and the coalescence takes longer than the plane-wave figure.

Writing the strength difference as $\delta(r) = \delta_{0}\sqrt{r_{0}/r}$ and integrating the separation to zero [Darden et al. 1989],

\[\frac{\mathrm{d}s}{\mathrm{d}r} \;=\; -\frac{\gamma+1}{4\gamma}\,\delta_{0}\sqrt{\frac{r_{0}}{r}} \qquad\Longrightarrow\qquad \sqrt{r_{c}} \;=\; \sqrt{r_{0}} \;+\; \frac{\mathcal{L}}{2\sqrt{r_{0}}}, \tag{20}\]

where $\mathcal{L}$ is exactly the plane-wave answer above and $r_{0}$ is the distance at which $\delta_{0}$ is quoted, taken here as one aeroplane length.

The correction changes the answer at one end of the range and not at the other. Integrating the weakening gives a corrected distance of 1.9 kilometres at a typical near-field shock strength, being 9.7 percent of the ray path, against 0.35 kilometres uncorrected.

At the weakest strength in the range it does not complete at all. The corrected distance there is 40.18 kilometres against 19,598 metres of ray path, so the shocks would still be separate when they arrived. The plane-wave form of the estimate hides that, because dividing a fixed closing rate into a fixed separation gives an answer that scales the same way at every strength. The conclusion is therefore conditional. It holds for strength differences of 0.02 and above and fails below that. Where an aeroplane the size of an F-15 sits in that range is an assumption, because no source in this article publishes its near-field shock strengths in a form this calculation can use.

So the programme’s stated reason holds for its own aeroplane if that aeroplane’s strength difference is 0.02 or more, but it does not generalise to every spike. A weaker host, or a longer one, would push the coalescence further out, and the same observation explains why a purpose-built low-boom airframe works at all.

Strength difference at one length Plane wave With spreading As a share of the ray path
0.01 1,729 m 40,177 m never completes
0.02 864 m 10,491 m 53.5 percent
0.03 576 m 4,866 m 24.8 percent
0.05 346 m 1,902 m 9.7 percent
0.10 173 m 577 m 2.9 percent

The finding stands where it matters without being universal, and the table shows both ends of it.

Whether the Shape Survives the Trip

Even for an aeroplane shaped from nose to tail, the ground signature is not the design signature.

The atmosphere is stratified, so rays bend. It is turbulent, so the wavefront is distorted by eddies it passes through. And it absorbs, so the shock front is not a discontinuity at all [State of the art of sonic boom modeling] [Plotkin and Maglieri 2003].

The bending is Snell’s law with a continuously varying index and it has one conserved quantity. For a horizontally stratified medium the ray parameter is invariant along a ray [Pierce, Acoustics, an introduction to its physical principles and applications],

\[\frac{a(z)}{\cos\theta(z)} \;=\; V_{t} \;=\; \text{constant}, \tag{21}\]

where $\theta$ is the ray’s angle below the horizontal and $V_{t}$ is the horizontal trace velocity of the wavefront. This subsection and The Alternative That Never Needed the Aeroplane both follow from that line, because a ray turns horizontal exactly where $a(z)$ reaches $V_{t}$, and it turns before the ground whenever the air below is fast enough.

Of the three, absorption decides loudness. In a real gas a shock has a finite thickness, set by a steady balance between nonlinear steepening and diffusion, and the profile that balance produces is a hyperbolic tangent [Blackstock, Fundamentals of physical acoustics],

\[\Delta p(t) \;=\; \frac{\Delta p_{\max}}{2}\left[1 - \tanh\!\left(\frac{t}{\tau}\right)\right], \tag{22}\]

whose width varies inversely with the amplitude,

\[\tau \;\propto\; \frac{1}{\Delta p}. \tag{23}\]

A weaker boom therefore rises more slowly as well as less far, and both changes reduce the perceived loudness, which is a happy accident the discipline relies on [Blackstock, Fundamentals of physical acoustics].

Turbulence is the unhappy one. Passage through the atmospheric boundary layer distorts a shaped signature, sometimes sharpening it into something closer to an N-wave and sometimes not, and the variability between two nominally identical passes is large [Loubeau] [Raspet et al. 1994] [Plotkin 1994]. A low-boom aeroplane produces a distribution of booms, and the medium causes the spread, whatever the design.

The propagation codes that carry this work are the space agency’s own, and they are documented in the open [Lonzaga et al. 2025].

Manoeuvring makes it worse in a well-studied way. An accelerating or turning aeroplane focuses its rays, and where they cross, the ground receives a superboom several times the level of the steady-flight carpet [Page et al. 2015]. A regulation written around steady cruise has to say something about the corners.

What the Boom Is Measured In

A pressure is not a loudness, and the rule is about loudness.

A units problem comes first. An overpressure becomes a level in the ordinary acoustic way [A loudness calculation procedure applied to shaped sonic booms],

\[L_{p} \;=\; 20\log_{10}\!\left(\frac{\Delta p}{p_{\mathrm{ref}}}\right), \qquad p_{\mathrm{ref}} = 20\ \mu\mathrm{Pa}, \tag{24}\]

with the reference pressure 20 micropascals. A Concorde at Mach 2 and 52,000 feet laid down 1.94 pounds per square foot [NASA Facts FS-2003-11-016, Sonic Booms], which is 92.9 pascals, or 133.3 decibels peak. As a fraction of the pressure the air is already at,

\[\frac{\Delta p}{p_{\infty}} \;=\; 9.17 \times 10^{-4}, \tag{25}\]

which is one part in 1,090. A boom is a very small pressure change arriving very fast, and that second half alone makes it loud.

No pressure target for a shaped signature is quoted here, and the omission is deliberate. A low-boom design goal is stated in perceived level rather than in overpressure, and the two are not interconvertible, because a perceived level depends on the shape of the waveform and an overpressure does not [Doebler and Rathsam 2020].

A ground measurement is also not the incident wave. The surface reflects, and the two arrivals superpose, so a gauge on the ground reads, with $\mathcal{R}$ distinguished from the gas constant $R$ [Stansbery et al. 1999],

\[\Delta p_{\text{measured}} \;=\; (1 + \mathcal{R})\,\Delta p_{\text{incident}} \;\approx\; 1.9\,\Delta p_{\text{incident}}, \qquad \mathcal{R} \approx 0.9, \tag{26}\]

for a hard surface. A rule written about the measured quantity is therefore written about roughly twice the wave the aeroplane sent. The doubling is a measurement convention and makes no physical claim, but it needs stating once.

The metric the discipline settled on is the perceived level in decibels, computed by Stevens’ Mark VII procedure from the one-third-octave spectrum of the waveform [Perceived Level of Noise by Mark VII and Decibels] [Calculation of the perceived level of noise (PLdB) using Stevens’ method (Mark VII)]. It was not invented for booms. It was a general loudness metric that turned out to rank boom waveforms in an order that matched what listeners reported, which is the only property a metric of this kind needs [Leatherwood and Sullivan 1992b].

The sound exposure level, being the energy of the event referred to one second, is the other quantity in constant use,

\[\mathrm{SEL} \;=\; 10\log_{10}\!\left[\frac{1}{t_{\mathrm{ref}}}\int \frac{p^{2}(t)}{p_{\mathrm{ref}}^{2}}\,\mathrm{d}t\right], \qquad t_{\mathrm{ref}} = 1\ \mathrm{s}. \tag{27}\]

Neither number settles the question, because only what people report can. The space agency has run human response programmes on this question since the 1960s, and it has run them in laboratories, in purpose-built simulators, and in people’s own houses [Shepherd 1992] [McCurdy et al. 2004] [Shepherd et al. 1992].

The house matters. A boom is heard indoors by most of the people it reaches, and a building is a filter that resonates and rattles, so the indoor event and the outdoor event are different sounds [Brown and Sutherland 1992b] [Carden et al. 1969]. A low-amplitude boom combined with the rattle it excites can be more annoying than the boom alone [Loubeau et al. 2013b].

Sixty years of laboratory and field work stand behind that sentence. Subjects have been asked to rate booms of different shapes, rise times and durations [McCurdy 1994], booms against aircraft flyovers [Leatherwood and Sullivan 1994b], and simulated booms in their homes rather than in a booth [McCurdy et al. 1994]. The loudness procedure the discipline uses was calibrated against those ratings [A loudness calculation procedure applied to shaped sonic booms] [Leatherwood and Sullivan 1992b].

The community response programme that a low-boom rule needs was run as a pilot in 2011 and it did not use a low-boom aeroplane, because there was not one. It used a fighter in a dive, producing a quieter boom on the ground by geometry rather than by shaping [Page et al. 2014] [Cliatt et al. 2014] [Lee et al. 2019].

Three years after the X-54 was designated to supply the stimulus, the stimulus was being faked with a dive.

The Alternative That Never Needed the Aeroplane

There is a second way to fly supersonic without booming anybody, and it involves no shaping.

The speed of sound rises with temperature, and the atmosphere is warmer near the ground than at cruise altitude. A ray leaving the aeroplane on a downward path therefore enters progressively faster air and refracts, turning away from the vertical. If the aeroplane is slow enough, every ray turns before it arrives, and the boom never reaches the surface at all.

The condition is that the shock’s intersection with the ground, which travels at the aeroplane’s speed, must outrun sound at ground level. Since $a=\sqrt{\gamma R T}$, the Mach number falls out as a temperature ratio and nothing else,

\[M_{\mathrm{co}} \;=\; \frac{a_{\text{ground}}}{a_{\text{cruise}}} \;=\; \sqrt{\frac{T_{\text{ground}}}{T_{\text{cruise}}}}. \tag{28}\]

For the standard atmosphere above the tropopause this gives 1.153, a value very nearly independent of altitude, because the temperature above 11 kilometres is constant [U.S. Standard Atmosphere] [Anderson, Modern compressible flow].

The ray invariant also locates the turning, and the location is uncomfortably sensitive. Below the tropopause the temperature is linear in height, so setting $a(z)$ equal to the trace velocity and squaring gives a turning altitude that is a straight line in the square of the Mach number [Coulouvrat 2002],

\[z_{\text{turn}} \;=\; \frac{T_{0} - M^{2}\,T_{\text{cruise}}}{L}. \tag{29}\]

At Mach 1.10 the ray turns at 4,001 metres, and at Mach 1.15 it turns at 251 metres. A five-hundredths increase in Mach number collapses the shadow zone from 13,125 feet to 823, which is the margin an operator is managing.

So an aeroplane may fly at Mach 1.153 and make no boom on the ground whatever its shape. This was known, and flown, before the rule existed. Signatures were measured in flight at cutoff Mach number and published in 1971 [Henderson et al. 1971], which is two years before the prohibition took effect and 54 years before the order to repeal it. Companion work measured what happens at the edge of the carpet, where the same refraction cuts the boom off laterally [Flight test measurements and analysis of sonic boom phenomena near the shock wave extremity] [Downing 1992].

The space agency measured it again in 2012 with modern instrumentation, flying an F-18B so that its cutoff caustic lay at varying heights above a 60-microphone array 2,248 metres long and mapping the acoustic field in the shadow zone below the cutoff altitude [Cliatt et al. 2016b] [Cliatt et al. 2016a]. What arrives is not silence but an evanescent rumble, and the programme measured it and named a metric for it [Haering et al. 2005b].

The catch is in the temperature ratio, and there is a second term the still-air form leaves out. Wind adds to the propagation speed along the ray, so the quantity that matters is an effective sound speed, which adds the wind component to the thermodynamic one [Haefeli et al. 1969a],

\[c_{\mathrm{eff}}(z) \;=\; a(z) + \mathbf{u}(z)\cdot\hat{\mathbf{n}}, \qquad M_{\mathrm{co}} \;=\; \frac{c_{\mathrm{eff}}(0)}{a(z_{\text{cruise}})} . \tag{30}\]

A ground temperature ten kelvin warmer or cooler moves the cutoff Mach number by 0.02 either way, a spread of 0.04, on its own, and a tailwind at the surface moves it further. So the technique is not a chart but a forecast, and an operator using it is betting on an atmosphere it did not measure.

The same ratio also bounds the boom carpet. A ray leaving at azimuth $\phi$ from the track carries a horizontal trace velocity reduced by the cosine, so it reaches the ground only while

\[M\cos\phi \;\ge\; M_{\mathrm{co}} \qquad\Longrightarrow\qquad \phi_{\max} \;=\; \arccos\!\left(\frac{M_{\mathrm{co}}}{M}\right). \tag{31}\]

At Mach 1.4 that is 34.5 degrees, and it widens to 50.2 degrees at Mach 1.8. A supersonic aeroplane does not boom everything beneath it, and the width of what it does boom is set by the same temperature ratio as the cutoff [Cliatt et al. 2016a].

Mach cutoff needs no low-boom aeroplane, no shaped signature and no new airframe. It needs a temperature profile and a willingness to fly slowly. It answers the same question more cheaply, and it was already available.

Dependent Systems

The registry gives the engines and almost nothing else [Parsch, DOD 4120.15-L Addendum, MDS Designators Allocated After 19 August 1998].

Two Rolls-Royce Tay 651 turbofans, a Gulfstream airframe, a NASA sponsor. The nearest thing to a public statement of what the aeroplane was for is not the registry entry but a journal paper, which sets out the case for a small supersonic civil aircraft as a class and the boom constraint that governs it [Henne 2005]. Anything further comes from the secondary record, and this article marks it as such. Those sources describe an aeroplane cruising above Mach 1.4 and above 50,000 feet and producing a ground signature under 75 perceived-level decibels [Gulfstream X-54].

At 50,000 feet and Mach 1.4 exactly, which is the floor of that range, the true airspeed is 413.1 metres per second and the Mach cone half-angle is 45.6 degrees. The cruise Mach number sits 0.247 above the cutoff Mach number, and that margin is the point. An aeroplane designed for Mach cutoff would fly below 1.153 and make no ground boom at all. The X-54 was designed to fly well above it and make a ground boom on purpose, quietly, so that somebody could be asked what they thought of it.

Why it was not built is not in the record this article can reach. A trade press account holds that the agency regarded the designation as a placeholder and was not actively working with the manufacturer on the aircraft [Gulfstream X-54]. The claim is secondary and comes from a single reported source, and this article repeats it on those terms. No primary document reviewed for this article states a cancellation, a funding decision or a programme termination, and the defensible position is that the number was allocated and nothing followed it.

The Flight Test Record

The X-54 never flew, and there is no flight test record of it to report. No airframe exists and no flight test report was written [Gulfstream X-54]. The flights that bear on its question were flown by other aeroplanes, and the article reports them as evidence about the question rather than about the X-54.

The Shaped Sonic Boom Demonstration supplied the only shaped ground signature measured before the designation. A modified F-5E flew supersonic over Edwards Air Force Base on 27 August 2003, and the signature measured on the ground kept its flat top [Pawlowski et al. 2005].

The Quiet Spike supplied near-field data and no ground data. It flew thirty-two flights on a NASA F-15B between August 2006 and February 2007, reached Mach 1.8, and was probed in the near field by a second aeroplane at Mach 1.4, and its report states that a ground measurement was not attempted [Smolka et al. 2007]. The community response pilot of 2011 used a fighter in a dive rather than a shaped aeroplane, as described earlier in What the Boom Is Measured In.

The flight the X-54 was designated to make is assigned to the X-59. By the date of this article that aeroplane had flown once, below the speed of sound, and What Followed Instead gives the detail.

What Followed Instead

The ground signature demonstration was reassigned, to a different number and a different contractor, and at this article’s date that demonstration had not yet been flown. The X-59 was designated in 2018 to Lockheed Martin for the low-boom flight demonstration mission, described in its own registry entry as a research aircraft that will create a shaped sonic boom signature [Parsch, DOD 4120.15-L Addendum, MDS Designators Allocated After 19 August 1998] [Lockheed Martin X-59 Quesst] [Rathsam] [Cliatt].

Ten years separate the two allocations and the mission statements are almost the same sentence.

The later programme’s design target is a perceived level, and its community response campaign is the purpose of the enterprise [Doebler and Rathsam 2020] [Rathsam et al].

That aeroplane was unveiled on 12 January 2024 [NASA, Lockheed Martin Reveal X-59 Quiet Supersonic Aircraft] and flew for the first time on 28 October 2025, a sixty-seven minute flight from Palmdale to the flight research centre at Edwards, below the speed of sound at 230 miles per hour and about 12,000 feet with the landing gear down [X-59 First Flight Media Resources, NASA] [NASA’s X-59 Completes First Flight, Prepares for More Flight Testing].

The X-54’s mission survived by being reassigned. Ten years passed between the two allocations, five and a half more before the replacement aeroplane was unveiled, and another twenty-one months before it flew. It first flew one month before the date on this article, and seventeen years after the number that was supposed to do its job was allocated to somebody else.

Comparison With Ground Prediction

The X-54 offers nothing to compare, because it never flew. The only prediction attached to the intended aeroplane is the secondary record’s ground signature under 75 perceived-level decibels [Gulfstream X-54], and no flight ever returned a measurement against it.

The programmes the X-54 descended from do supply comparisons, and the article argues each where it arises. The 1979 tunnel investigation found that a body designed by the minimisation method behaved as the method said it would [Mack and Darden], as set out in The Mathematics of Choosing a Shape. The Shaped Sonic Boom Demonstration measured on the ground the flat-topped signature that four decades of wind-tunnel work at the agency’s Virginia laboratory had predicted [Haering et al. 2005a], under the heading What Had Already Been Proved by 2008.

The Quiet Spike’s stability prediction is the clearest case of flight overturning a ground estimate. The spike was predicted to cost the host aeroplane three to five percent of its stability subsonically and between three and twenty-four percent supersonically in the lateral-directional axis. Flight found the aeroplane more stable than a standard F-15 by Mach 1.8, and the limit was set instead by pitch damping, which fell away sharply above Mach 1.7 [Freund et al. 2007]. What Had Not Been Proved gives the full account. The design dynamic pressure of 685 pounds per square foot is consistent with the stated Mach 1.8 at 45,000 feet, a check carried out in The Standard Atmosphere.

The coalescence estimate is a ground prediction that no flight has tested. The Quiet Spike report predicted that the host aeroplane’s shocks would overtake the spike’s shocklets within a short distance, and it attempted no ground measurement. The calculation in The Arithmetic the Report Set Out and Did Not Perform supports that prediction for strength differences of 0.02 and above and not at 0.01, and no flight datum in this article confirms or refutes either result.

The comparison the X-54 existed to make now falls to the X-59, a predicted shaped ground signature beside a measured one from a fully shaped aeroplane. That aeroplane had first flown a month before the article’s dateline, and the article reports no such comparison.

What the Data Changed

No data exist for the X-54, so what changes is the reading of the record around it, and four results change it. Three come from arithmetic the sources did not perform and one from the date of a measurement.

The first is the coalescence distance. The flight test report states that the aeroplane’s own shocks would overtake the spike’s shocklets within a short distance. That statement is the input to a finding, and the finding still has to be computed. Weak-shock theory turns a short distance into a number. A source that has done the measuring has not necessarily done the arithmetic.

The second is that the ray path, longer than the altitude, is the distance to divide by. A boom does not travel straight down. The ray leaves normal to the Mach cone and makes the Mach angle with the vertical, so the path from 45,000 feet at Mach 1.4 is 19,598 metres and not 13,716. A share computed against the altitude is too large by 42.9 percent, and the error lies in the direction that flatters the argument.

The third is that the conclusion is narrower than the report’s statement. The plane-wave estimate makes the finding look independent of the assumed shock strength, and adding geometric spreading shows that it is not. At the weakest strength in the range the coalescence never completes, reaching 40.18 kilometres against 19,598 metres of ray path. It completes at 0.02 and above, the range this article assumes for an aeroplane the size of an F-15. On that assumption the finding stands for the aeroplane the report was written about, though it does not generalise to spikes as a class.

The fourth is that the technique which needs no airframe to meet a noise-based rule was flown before the rule existed. The space agency measured it with modern instrumentation in 2012, but signatures at cutoff Mach number were measured in flight and published in 1971 [Henderson et al. 1971], which is two years before the prohibition took effect. That changes the shape of the story from a new technique overtaking an old aeroplane to an old technique outlasting one.

The Contemporary Literature

Cluster Records
What it does to the people underneath 447
The wave itself 406
The aeroplane that makes the wave 333
Getting it to the ground, and whether the shape survives the trip 294
Shaping the wave, which is the premise the aeroplane existed to test 289
The other noise a supersonic aeroplane makes, which is the one that got a standard 244
Computing the wave before there is an aeroplane to make it 169
Measuring a wave that arrives once and is gone 84
The rule, and the standards written to replace it 50
The atmosphere the wave travels through 41
Total 2,357

What it does to the people underneath

The largest cluster, and the reason is the whole argument of this article. Annoyance, community response, dose-response relationships, subjective testing and the surveys that produce them. The rule the X-54 was designated to support is a rule about people, so the discipline that settles it is not aerodynamics. This cluster also carries the community-noise literature of railways, roads and wind turbines, because exposure-response methodology is one field and aircraft are one of its cases.

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[Persson Waye and Rylander 2001] [Pilon 2006b] [Plotkin et al. 1992a] [Plotkin et al. 2012] [Poulsen 2003a] [Poulsen 2003b] [Poulsen 2004] [Poulsen 2005] [Preis and Golebiewski 2004] [Prels 2024] [Price 1972] [Quehl and Basner 2006] [Sivasubramanian et al. 2020] [Radosz 2021] [Rajala and Hongisto 2020] [Rallabhandi and Park 2017] [Rasmussen and Ekholm 2021] [Rathsam 2014] [Rathsam and Gille 2017] [Rathsam and Klos 2016] [Rathsam et al. 2011] [Rathsam et al. 2015b] [Rathsam et al. 2015a] [Reddingius 1974] [Reddingius 1977] [Reddy et al. 2021] [Remillieux 2012] [Remillieux et al. 2013] [Report on Human Response 1968] [Rice 1984] [Rice 1994] [Rice 2024a] [Rice 2024b] [Risks of Road Traffic 2000] [Rizzi and Rafaelof 2022] [Robinson 2024] [Rochat et al. 2021] [Rylander 2006] [Rylander and BjÖRkman 2002] [Rylander and Dancer 1978] [Rylander et al. 1980] [Ryu and Jeon 2011] [Ryu and Song 2019] [McDaniel et al. 1992] [Neuhoff et al] [Sanford 1979] [Scannell 1989] [Scannell 2024b] [Scannell 2024a] [Schad 2026] [Schlittenlacher and Wales 2023] [Schmitz and Yu 1986] [Scholten et al. 2023] [Scholten et al. 2024] [Schomer 1981] [Schomer 1982] [Schomer 1997] [Schomer 2005] [Schomer and Wagner 1996] [Schomer et al. 1996] [Schomer et al. 1998] [Schomer et al. 2023] [Schoppe et al. 2025] [Schreckenberg 2021] [Schreckenberg and Großarth 2021] [Schreckenberg et al. 2010] [Schreckenberg et al. 2018] [Schreckenberg et al. 2023] [Schultz 1979] [Schäffer et al. 2020] [Seong et al. 2023] [Seshadri and Lowery 1972] [Seto and Huang 2025] [Shaffer 1990] [Shield and Zhukov 2024] [Shield et al. 2024] [Shigehisa and Gunn 1976] [Shigehisa and Gunn 1977] [Shu et al. 2017] [Sizov and Hobbs 2010] [Smeatham et al. 2024] [Sneddon et al. 2003] [Socher et al. 2024] [Sohn and Schowe 1989] [Sommerhoff et al. 2004] [Song et al. 2016] [Chen et al. 2022] [Brown and Sutherland 1992b] [Speakman et al. 1978] [Stansfeld et al. 2020] [Lee et al. 2019] [Stusnick et al. 1993] [Suau-Sanchez et al. 2011] [Subedi et al. 2005] [Leatherwood and Sullivan 1992b] [McCurdy et al. 2004] [Sullivan 2005] [Sullivan and Leatherwood 1993] [Sullivan and Leatherwood 1994] [Sun et al. 2006] [Sutherland 2006] [Sutherland and Czech 1992a] [Sutherland and Czech 1992b] [Swearingen and White 2005] [Sörensen 1977] [Tahir Akhtar and Mitsuhashi 2009] [Tam 1983a] [Tambiki 2019] [Taylor 2024b] [Taylor 2024a] [Thackray 1972] [Thanos et al. 2023] [The Design 2016] [The relation between the] [Trollé et al. 2015] [Turner and Burns 1977] [Twardella and Ndrepepa 2011] [Töpken 2021] [Töpken and van de Par 2021] [Valishina et al. 2025] [Vallet 2024] [Van den Berg and Van der Eerden 2016] [Vasudevan and Gordon 1977] [Vasudevan and Leventhall 1982] [Vasudevan and Leventhall 1989] [Vera and Bell 2026] [VI the Metropolitan Philadelphia 1971] [von Gierke and Nixon 1971] [von Gierke and Nixon 1972] [Vos 1992] [Vos 1997] [Vos 2006] [Vrhovnik and Prislan 2024] [Bauer 2025] [Walker and Fields 2025] [Wang] [Wang and Hansman 2024] [Wang et al. 2023] [Wayant 2018] [Waye et al. 2023] [Wesler 1973] [Wessels et al. 2017] [White et al. 1993] [Widmann 2024] [Wiggins and Bruce 1966] [Woodroof 2024] [Wszołek et al. 2023] [Wunderli and Brink 2023] [Yamada 2023] [Yamashita et al. 2024] [Yano 2023] [Yano et al. 2012] [Yano et al. 2024] [Yokoshima et al. 2024] [Yonemura et al. 2021] [Young 1981] [Zeitler et al. 2024] [Zepler and Harel 1965] [Zhang and Ma 2021] [Zhao et al. 2025] [Zhou et al. 2013]

The wave itself

The sonic boom considered as an object, rather than as a design problem or a nuisance. N-waves, overpressure, shock structure and the measurements that characterise them. This is where the classical literature lives, and the oldest of it predates the rule by more than twenty years.

406 records. [A Study to Determine 1975] [Cumming et al. 2008] [Ahuja 1992] [Aizik et al. 2001] [Elmiligui et al] [Alauzet and Loseille 2010] [Loubeau 2012] [Arifullin et al. 2025] [Raspet et al. 1994] [Sorrell et al] [balazard 1972] [Banakh and Sukharev 2016] [Banakh et al. 2014] [Banerjee 2010] [Baskar 2006] [Batdorf 1970] [Batdorf 1972] [Battis 1983] [Bell 1971] [Benham and Birks 1973] [Benveniste and Cheng 1967] [Berger et al. 2011] [Blaine et al. 2024] [Blanc-Benon et al. 2005] [Blanchard et al. 2022] [Blumrich et al. 2005] [Bobbitt and Maglieri 2003] [Boeing Co Renton Wa Airplane Div 1964] [Bucovsky et al. 2003] [Bugliarello et al. 1976] [Buonanno et al. 2002] [Byrne 1971] [Cain 2009] [Cain 2010] [Callaghan 1968] [Carlson 1965] [Carlson et al. 1965] [Carlson et al. 1966] [Casper et al. 2008] [Castner 2008] [Castner 2010a] [Castner 2012a] [Castner 2012b] [Castner 2011] [Chen and Suzuki 2021] [Cheng 2005] [Cheng and Lee 1996] [Cheng and Lee 1998b] [Cheng and Lee 1998a] [Cheng and Lee 2004] [Cheung et al. 1999a] [Cho 2009a] [Cho 2009b] [Darden 1992] [Clarkson and Mayes 1971] [Clarkson and Mayes 1972] [Cliff et al. 1999] [Coen 2010] [Colombant and Gardner 1975] [Cook et al. 2013] [Cottereau 1972] [Cottereau 1978] [Coulouvrat 1998b] [Coulouvrat 2006] [Coulouvrat 2022] [Coulouvrat 2023] [Crocker 1965] [Crocker and Sutherland 1966] [Crocker and Sutherland 1968] [Crow and Bergmeier 1996] [Darden 1976] [Darden 1977] [Davis 1972] [Davy and Blackstock 1969] [DeGolia and Loubeau 2017] [Desharnais and Chapman] [Distortions of Laser Beams 2016] [Doebler 2020a] [Doebler 2020b] [Doebler and Rathsam 2019] [Dogan 2008] [Downing 2017] [Downs 2021] [Downs et al. 2022b] [Dragna 2021] [Drobzheva et al. 2021] [Dubois 1970] [Durrant et al. 2021] [Durrant et al. 2022a] [Durrant et al. 2022b] [Ehernberger et al. 1994] [Ellis et al. 1975] [Elmer and Joshi 1994] [Elmiligui et al. 2019] [Elmiligui et al. 2017] [Emmanuelli et al. 2019] [Emrich 1965] [Enflo 1985] [Falco 1963] [Fasulo et al. 2025] [Favorskaya 2020] [Ferguson and Sparrow 1998] [Ferri and Ismail 1969] [Fincham and Maxworthy 2002] [Fleming 2014] [Fomin et al. 2011] [SHkarubo 2025] [Frampton et al. 1993] [Ganjehi et al. 2008] [Gao and Sun 2025] [George 1969a] [Gottlieb and Ritzel 1988] [Gottlieb et al. 1978] [Graziani et al. 2024b] [Graziani et al. 2025b] [Graziani et al. 2025c] [Ha et al. 2021] [Haber 1993] [Haglund and Kane 1972] [Haglund and Kane 1974b] [Hall 1978] [Hammond and Dourado 2018] [Harker et al. 2022] [Harris 1967] [Hartwich et al. 2003] [Hayes 1969] [Hayes 1971] [Hecht 2009] [Heimann 2001] [Henderson 1982] [Henderson and Hilton 1974a] [Henderson and Hilton 1974b] [Herron 2007] [Hilton 1971] [Hilton and Newman 1965] [Hilton and Newman 1966] [Hilton et al. 1972] [Holmes 1966] [Doebler and Rathsam 2020] [Howe 2002] [Hubbard 1969] [Hubbard et al. 1986] [Hutchinson 1963] [Howe and Henne 2006] [Schwartz 1968] [Iyengar et al. 2024] [Page and Loubeau 2019] [Jackson and Carlson 1966] [Jung et al. 2020] [Jäschke 2021] [Jäschke and Liebhardt 2021] [Jäschke et al. 2024] [Kamali and Pierce 1971] [Kamerman et al. 1986] [Kanamori et al. 2017b] [Kandil et al. 2005] [Kandil et al. 2006] [Kane 1973b] [Kane 1973a] [Kantor and Cliatt 2017] [Kaouri et al. 2008] [Kawamura 1952] [Khasdeo et al. 2007] [Kim et al. 2010] [King et al. 2023] [King et al. 2024] [Kiseleva 2017] [Kiseleva and Temerbekov 2019] [Klos 2017] [Klos et al. 2011] [Klos 2012] [Korunov et al. 2024a] [Korunov et al. 2024b] [Kovalenko and Zhilin 1998] [Kovalenko et al. 1998] [Krenzel 1959] [Kroo 1990] [Kubota 2003] [Kunukkasseril and Ramakrishnan 1974] [Larson et al. 2011] [Lazzara 2025] [Lee and Downing 1993] [Lee and Downing 1996b] [Lee and Downing 1996a] [Legg and Haber 1990] [Leng and Qian 2017] [Li et al. 2024] [Librescu and Na 1998] [Librescu and Nosier 1990] [Librescu et al. 1997] [Liebhardt 2019] [Liebhardt 2025] [Liebhardt 2026] [Liebhardt et al. 2013] [Liebhardt et al. 2014] [Lin 1971] [Loubeau et al. 2018] [Lower Bounds for Sonic Bangs] [Jones 1967a] [Ludlow and Morgan 1972] [Lympany and Page 2022] [Lynch and Speake 1978] [Лысенков et al. 2024] [MacDonald et al. 1978] [Maglieri and Parrott 1962] [Maglieri and Parrott 1963] [Makarewicz 2024] [Makino 2014] [Makino and Noguchi 2003] [Makino et al. 1999] [Marconi et al. 2003] [Marmel 2021] [Martini and Garrelick 1994] [May and Sohn 1991] [Mazumdar and Coleby 1976] [Maglieri et al. 2011] [Merzkirch 1969] [Meshcheryakov 2021] [Miller and Sparrow 2010] [Mishchenko and Gimon 2024] [Moody 2006] [Moody and Griffice 2003] [Morfey and Fisher 1970] [Morgenstern 2017b] [Morgenstern 2021] [Simmons and Freund 2005a] [Moskvichev and Potapkin 2019] [Naka et al. 2008b] [Needleman and Mack 1990] [Newberry 1967] [Nikbay et al. 2017] [Nixon 1970] [Nixon and Borsky 1965] [Novakovich et al. 2019] [Nozzle Plume Effects 2013] [Numata et al. 2015] [Plotkin 1994] [On the propagation of weak shock waves] [Oswatitsch and Sun 1972] [Ou and Jinhuan 2025] [Ozcer and Kandil 2009] [Page 1975] [Palmer 1970] [Palmer and Sparrow 2015] [Pan and Varner 1972] [Pan 1968] [Panel Discussion 1975] [Parker 1968] [Parker and Zalosh 1973] [Parker and Zalosh 1974] [Pawlowski et al. 2005] [Pfaender 2023] [Piacsek and Plotkin 2013] [Pierce 1968] [Pierce 1998] [Pierce and Thomas 1969] [Pilon 2006a] [Pilon 2007] [Plotkin 1989] [Plotkin 1990] [Plotkin 1993] [Plotkin 2006a] [Plotkin and Cantril 1976] [Plotkin et al. 1989] [Plotkin et al. 1990] [Plotkin et al. 1992b] [Plotkin et al. 2005a] [Plotkin et al. 2005b] [Plotkin et al. 2008b] [Porter 1968] [Potapkin and Moskvichev 2017] [Potapkin and Moskvichev 2018] [Potapkin and Moskvichev 2020] [Potapkin and Moskvichev 2021a] [Potapkin and Moskvichev 2021b] [Power 1963] [Power 1965] [Pretlove and Bowler 1972] [Simmons et al. 2007] [Rachami and Page 2010] [Rallabhandi 2019] [Rallabhandi 2022b] [Rallabhandi and Mavris 2004] [Rallabhandi et al. 2010] [Rao and Zumwalt 1970b] [Rasband et al. 2023] [Rathsam and Ely 2012] [Ravetta et al. 2010] [Loubeau] [Reinis 1978] [Report on Physical Effects 1968] [Resler 1969] [Rethorst 1994] [Carlson and Maglieri 1972] [Rice 1972] [Richards and Rylander 1972] [Rochat and Sparrow 1997a] [Rodi 2018] [Rusak 2008] [Ryhmeng et al. 1960] [Rylander 1972] [Rylander et al. 1972] [Rylander et al. 1974] [Sahoo and Banerjee 2019] [Salah El Din et al. 2013] [Salamone 2006] [Salvetti and Seidman 1988] [Sandu et al. 2019] [Scarselli and Castorini 2013] [Schultz et al. 2016] [Schultz et al. 2017] [Sears 1969] [Seebass 1969b] [Seebass 1969a] [Seebass 1973] [Seebass and George 1974] [Seebass and McLEAN 1968] [Seebass 1974] [Shao et al. 2025] [Haering and Murray 2004] [Shuttle Sonic Boom 1973] [Shuttle Sonic Boom 1975] [Sigalla et al. 1977] [Silva et al. 2016] [Singh and Kung 1982] [Sizov 2017] [Sohn 1992] [McLean 1965] [Sonic Boom 2012] [Shepherd et al. 1992] [Sonic Boom in Relation 1968] [ЦАГИ et al. 2017] [Plotkin and Maglieri 2003] [Sonic Boom Results 1975] [Soroka 1963] [Sparrow 1995a] [Sparrow 1995b] [Sparrow 1997] [Sparrow 1998a] [Sparrow 1998b] [Sparrow 2003] [Sparrow 2010] [Sparrow et al. 1996] [Sparrow et al. 1997] [Stanley 1984] [Stanley et al. 1989] [State of the art of sonic boom modeling] [Darden et al. 1989] [Stewart 1993] [Stratford 1974] [Stuff 1972] [Suisman 2015] [Sukharev and Kuskov 2017] [Sullivan et al. 2008] [Sullivan 2007] [Sun and Smith 2018a] [Swigart 1974] [Swigart 1975] [Tan et al. 2026] [Taniguchi 1959] [Taniguchi 1961] [Telenin and Lipnitskii 1966] [The influence of airplane configuration on sonic-boom characteristics] [The Post-War Sonic Boom 2013] [Schwartz 1971] [Thomas 1971] [Thompson and Parnell 1967] [Ting et al. 1996] [Ting et al. 2002] [Tokuyama et al. 1993] [Toyoda et al. 2010] [Tracey 1998] [Trefny et al. 2014] [Vaidya 1972] [Van der Velden and Kroo 1994] [Volkov 1996] [Volkov 2017] [Volkov 2018] [Volkov 2019] [Volkov 2021] [Volkov and Mazhul’ 2018] [Walker et al. 2015] [Warren 1972b] [Waters and Glass 1970] [Weber 1972] [Wen et al. 2022] [Doebler] [Wu et al. 2011] [Yamashita and Nikiforakis 2020] [Yamashita and Nikiforakis 2023] [Yamashita and Obayashi 2010] [Yamashita and Obayashi 2012] [Yamashita and Obayashi 2013] [Yamashita and Suzuki 2014] [Yamashita and Suzuki 2015a] [Yamashita and Suzuki 2016] [Yamashita and Suzuki 2020] [Yamashita et al. 2020] [Yamazaki and Ishida 2025] [Yamazaki and Kusunose 2016] [Yoo 2010] [Yoo 2013] [Young 1982] [Zha et al. 2010] [Zhang 2025] [Zou et al. 2024]

The aeroplane that makes the wave

The third-largest cluster, and the one an X-Planes reader expects. Supersonic aerodynamics, wave drag, the area rule, slender-body theory and the configurations built on them, with the inlets of low-boom and civil designs. The X-54 would have belonged here and it never did, and this shelf is what the aeroplane was to have been designed against.

333 records. [A Summary of Recent Supersonic Vehicle Studies at Gulfstream Aerospace] [Abarbanel 1977] [Abdalla et al. 2020] [Abraham et al. 2023] [Abraham et al. 2024] [Advanced Fuel Research INC East Hartford CT 1957] [Aerodynamic Phenomena in Supersonic 2020] [Aerodynamics of Arbitrary Wing 1976] [Ageev and Pavlenko 2016] [Agte et al. 1999] [Alonso et al. 2002] [Anderson 1959] [Anderson 1960] [Andrew 1981] [Application of Slender-Body Theory 2025] [Army War Coll Carlisle Barracks Pa 1952] [Baals and Harris 1968] [Baals et al. 1970] [Ban et al. 2016] [Bannink and Reyn 1968] [Bauer and Wood 1989] [Benedikt 1953] [Bera 1975] [Black 1969] [Blank 1995] [Bonner 1971] [Bons et al. 2020] [Bradley et al. 1969] [Briceno et al. 2002] [Briceño and Mavris 2002] [Brooks and Powers 2004] [Brown 1989] [Bruckner 2002] [Herrera and Pak 2007] [Buonanno and Mavris 2005] [Bushnell 1990] [Bushnell 1992] [Carafoli 1969] [Caros et al. 2020] [Henne 2005] [Catalano and Sturek 2001b] [Catalano and Sturek 2001a] [Cenko 1979] [Cenko 1980] [Chang 1952] [Chang 1992] [Chapter 3. Aerodynamics 1957] [Chapter 4. Wing-Tail Plane 1957] [Chapter 5. Wing-Body Interference 1957] [Chaudhari and Takahashi 2022] [Chester 1953] [Chester 1956b] [Chester 1956a] [Chima et al. 2010] [Chin 1977] [Chin 1978] [Choi et al. 2009b] [Choudhari et al. 2003] [Chudoba et al. 2006] [Coen 2022] [Coen and Richwine 2017] [Coen et al. 2023] [Covell et al. 1990] [Cunningham 1973] [Daumas et al. 2006] [Davies 2020] [Davies 2025] [Deremaux et al. 2008] [Devan 1987] [Devan and Kania 1985] [Driver 1974] [Driver 1981] [Drougge 1990] [Dupuis 1988] [Durston and Stonum 1987] [Dutton 1967] [Ehlers 1961] [Elkin 1961] [Eminton and Lord 1956] [Eremenko et al. 2003] [Falunin and Ul’yanov 1967] [Fernández et al. 2002] [Fischer and Vemuru 1991] [Flat Wings in Inviscid 2020] [Fletcher and Burns 1979] [Follé 1983] [Fomin et al. 2009] [Fraenkel 1958] [Fuhrmann 1993] [G. the Experimental Aerodynamics 1957] [Gaglio et al. 2021] [Gall 1985] [Garcia et al. 1995] [Garcia-Fogeda and Liu 1988] [Garzon and Matisheck 2012] [Geiger 2002] [Gibson and Gerhardt 1995] [Goldsworthy 1952] [Gonzalez Gallego et al. 2018] [Gonzalez-Linero 2015] [Gopal and Maddalena 2018b] [Gopal and Maddalena 2018a] [Goto et al. 2007] [Goto et al. 2021] [Graham 1950] [Greatrex 1963] [Greene and Seebass 1996] [Gringorten 1966] [Gringorten 1967] [Gringorten and Tattelman 1970] [Guan 2013] [Guan 2014a] [Guan 2014b] [Guderley 1987] [Guderley 1988] [Guimarães et al. 2022] [Guimarães et al. 2023] [Han et al. 2026] [Hansen 2000] [Harris 1965] [Harris 1989] [Heppe and Hong 1964] [Herrmann 2008] [History of Supersonic Transport 2020] [Hui 1972] [Hutt and Howe 1988] [Volkov 2023] [Ide et al. 2012] [Theoretical Studies to Refine 1970] [Iuliano et al. 2009] [Iuliano et al. 2011] [James 1998] [Jischke 1978] [Johnson 1985] [Jones 1966] [Jones 1967b] [Jones 1971] [Jumper 1983] [Kanamori et al. 2019] [Kariappa and Smith 1972] [Kawamura and Karashima 1957] [Keenan 1961] [Kelly 1980] [Khan et al. 2025] [Kiyici and Aradag 2015] [Kraiko and Tilliaeva 1973] [Kraiko and Tkalenko 1967] [Kroo 2008] [Kubota 2008] [Kulfan 1978] [Kulfan 2002] [Kulfan 2008] [Kulfan 2009] [Kuznetsov 1981] [KÜChemann 1962] [Lance 1952] [Lapshina and Ostapenko 2009] [Сергеевна et al. 2024] [Lee 1994] [Lei 2008] [Leslie and Perry 1954] [li and Rowe 1973] [Li et al. 1995] [Li et al. 2018] [Li et al. 2026] [Lighthill 1956] [Livne 2023a] [Livne 2023b] [Lock 1957] [Lock and Bridgewater 1967] [Loftin 1967a] [Lord and Eminton 1954] [Loubeau 2022] [Loubeau et al. 2023a] [Loubeau et al. 2023b] [Lovell 1999] [Lovell 2001] [Lynde and Campbell 2016] [Maglieri 2011] [Maikapar 1959] [Maklakov 2001] [Malmuth and Shalaev 2004] [Malmuth et al. 1986] [Malmuth et al. 1987] [Mann and Carlson 1994] [Marconi and Moretti 1976] [Marshall 2000] [Martin and Gerber 1953] [Martins et al. 2002] [Martins et al. 2004] [Maruyama et al. 2006] [Maruyama et al. 2009] [Mateescu 1986] [Mavris and Hayden 1996] [Mazhul’ 2010] [McLean 1985] [Mendenhall 1980] [Mertens 1997b] [Bashkirov 2024] [Miles 1952b] [Miles 1952a] [Miles 1953] [Mills and Harding 1983] [Moga 1980] [Mora et al. 2013] [Morgenstern et al. 2015] [Morito and Sidwell 1967] [Mutz et al. 1964] [Nadim Melhem et al. 2024] [Narayan et al. 2025] [Nastase 1998] [Naval Ordnance Systems Command Washington DC 1957] [Nemec et al. 2024] [Nie et al. 2024] [Nie et al. 2025] [Nie et al. 2026] [Nikolic and Jumper 2004] [Nikolic and Jumper 2005] [Nomura and Kuroda 2002] [Ohrenberger and Baum 1972] [Osawa and Hewitt 1986] [Papamoschou 2002] [Parikh et al. 1990] [Petrescu 2020] [Pfaff 1965] [Pfenninger and Vemuru 1988] [Phan et al. 2003] [Pipko 1966] [Platzer and Hoffman 1966] [Prescott 1963] [Properties and Design 2012] [Puckett 1946] [Quagliarella and Iuliano 2017] [Quagliarella and Iuliano 2018a] [Quagliarella and Iuliano 2018b] [Spivey et al. 2007] [Hartman et al. 2001] [Wlezien and Veitch 2002] [Radkey et al. 1976] [Radkey et al. 1978] [Rallabhandi and Mavris 2003] [Rallabhandi and Mavris 2006a] [Rallabhandi and Mavris 2008] [Rasmussen 1978] [Ray and Lele 2006] [Ray et al. 2004] [Reed and Saric 2002] [Requirements of a Commercial 1967] [Richardson 1965] [Rivers et al. 2008] [Rodriguez 2007] [Rosenhead 1956] [Rothwell 1953] [Rowe et al. 1981] [Sachs et al. 2025] [Sahu 1986] [Satre 1969] [Schlette and Staudacher 2022] [Schnell et al. 2025] [Schouten and Bakker 2000] [Schuelein 2008] [Schuermann et al. 2014] [Schutte et al. 2005] [Seebass 1997] [Setoguchi and Kanazaki 2020] [Siewert and Stewart 1965] [Simmons and Freund 2005b] [Sisk and Andrews 1963] [Sklar and Rusak 2020] [Smith 2007] [Sotomayer and Weeks 1978] [Sparkes 2024] [Spearman 1979] [Rathsam et al] [Stewartson 1966] [Stratford 1973] [Sturdza 2007] [Sturdza and Rajnarayan 2014] [Subsonic and Supersonic Jets 1975] [Sun and Smith 2017] [Sun and Smith 2018b] [Supersonic Aircraft 1997] [Supersonic Cruise 1981] [Supersonic Cruise Research 1979 1979a] [Supersonic Cruise Research 1979 1979b] [Supersonic Flight 2025] [Supersonic Transport Wing Minimum 1992] [Surzhikov and Shang 2005] [Swan 1969] [Takovitskii 2020] [Technical Applications 1976] [Tetervin 1963] [Whitham 1952] [Thompson 2023] [Tice and Martin 1992] [Tkalenko 1969] [Tokugawa and Yoshida 2006] [Tokugawa et al. 2008] [Tokugawa et al. 2014] [Tolkachev et al. 2024] [Torenbeek 2020] [Tracy et al. 2002] [Transonic and Supersonic Flight 1992] [Transonic and Supersonic Flows 2012] [Traoré and Lemée 2001] [Turbojet and Turbofan Engines 1964] [Tuttle 1980] [U.S. Supersonic Commercial Aircraft 1997] [Ueda et al. 2014] [Unger et al. 1999] [Unsteady Supersonic Flow past 1981] [Utomo and Bura 2019] [van Dam et al. 2002] [Van Dyken 1995] [Vaughn 1982] [Velden 1997a] [Veresnikov and Goncharenko 2025] [Verhoff and Stookesberry 1992] [Vigneron et al. 2008] [Waechter 1969] [Waechter 1971] [Walkley and Martin 1979] [Wang et al. 2026a] [Warren 1954] [Washington and Humphrey 1969] [Wasserman 1952] [Whitham 1950] [Wing Theory in Supersonic 1969] [Wright et al. 1978] [Yahalom 1971] [Yamazaki and Kusunose 2014] [Zartarian 1956] [Zartarian and Hsu 1955]

Getting it to the ground, and whether the shape survives the trip

The question that decides whether the premise is useful. Nonlinear propagation, ray tracing, the augmented Burgers equation, rise time, molecular relaxation, turbulence, caustics and cutoff. A shape that exists in the near field and not at the ground is an aerodynamic result and not a regulatory one, and this is the shelf that tells them apart.

294 records. [Alberts et al. 2008] [Carr et al] [Carr et al] [Carr et al] [Pierce 1992b] [Auger and Coulouvrat 2002] [Auvermann and Goedecke 1993] [Bakhne et al. 2024] [Barlas et al. 2017] [Bass and Raspet 1992] [Bass et al. 1998] [Bass et al. 2002] [Baudoin 2006] [Baudoin et al. 2011] [Bauer 1971] [Bedard 1990] [Berci and Vigevano 2011] [Berci and Vigevano 2012] [Best 1997] [Biringen et al. 2005] [Blanc-Benon 2006] [Blanc-Benon 2021] [Blanc-Benon and Ollivier 2006] [Boulanger et al. 1995] [Bowden and Betar 2025] [Broyles et al. 2019] [Busch et al. 2017] [Béquin 2022] [Callsen et al. 2024] [Carr 2021] [Carr et al. 2020] [Carr et al. 2021] [Carr et al. 2022b] [Carr et al. 2022a] [Carrier and Coulouvrat 2022] [Carrier et al. 2021a] [Cheng et al. 1996] [Chernyshev 2011] [Chernyshev 2018] [Chernyshev et al. 2022] [Chessell 1973] [Cleveland 1995] [Cleveland and Blackstock 1994] [Cliatt et al. 2015] [Cliatt et al. 2016b] [Collmar and Salamone 2017] [Corcoran et al. 2012] [Coulouvrat 1998c] [Coulouvrat 1998a] [Coulouvrat et al. 2003] [Coulouvrat et al. 2015] [Crow 1969] [Darmofal et al. 2023] [Davis and Austin 1969] [Davis 1971] [Di et al. 1995] [Doebler and Loubeau 2023] [Doebler and Sparrow 2017b] [Dougherty 1996] [Downing and Lee 1992] [Downs et al. 2020] [Downs et al. 2022a] [Dragna et al. 2023] [Elmer et al. 2013] [Emmanuelli et al. 2023] [Ermolaev et al. 2018] [Ferri and Schwartz 1971] [Ferri and Schwartz 1972] [Ferri et al. 1977] [Haering et al. 2005b] [Franke et al. 1988] [Fujino et al. 2017] [Gainville et al. 2012] [George and Kim 1979] [Gilbert et al. 1996] [Gionfriddo 1992] [Gionfriddo 1993] [Gionfriddo and Sparrow 1992] [Gionfriddo et al. 1991] [Glorioso et al. 2025b] [Goldman 2017] [Gostev et al. 2006] [Graziani et al. 2025a] [Haefeli 1969] [Haefeli et al. 1969b] [Haering 2006] [Haering et al. 1998] [Haering et al. 2008] [Haering et al. 2013] [Haering et al. 2012] [Hammerton 2001] [Hayes and Runyan 1971] [Heath and Cobb 2021] [Hobbs and Bradley 2018] [Housman et al. 2019] [Hussain and Dutilleux 2020] [Hutchinson and Bowersox 2021] [Ignat and Pozo 2017b] [Ignat and Pozo 2017a] [Iura et al. 2023] [Ivanteeva et al. 1998] [Ivanteyeva et al. 2002] [Lonzaga] [Johnson and Hammerton 2003] [Jäschke and Liebhardt 2022] [Kanamori et al. 2017a] [Kanamori et al. 2021] [Kandil 2006] [Kandil and Khasdeo 2006] [Kapcsos and Sparrow 2023a] [Kapcsos and Sparrow 2023b] [Kapcsos and Sparrow 2025] [Kirby 2021] [Kirkpatrick Alberts et al. 2009] [Kishi et al. 2020] [Koganezawa et al. 2023] [Kornyakov et al. 2023] [Korunov 2025] [L’Espérance et al. 1992] [Lam 2009] [Larsson 1997] [Cliatt et al. 2016a] [Leconte et al. 2019] [Lee and Ribner 1973] [Lee et al. 1992] [Lefkowitz and Sparrow 2009] [Liebhardt 2021] [Liebhardt et al. 2022] [Lipkens 1995] [Lipkens 2002] [Lipkens and Blackstock 1998b] [Lipkens and Blackstock 1998a] [Lipkens et al. 1998] [Liszka 1978] [Liu and Liu 2019] [Locey 2006] [Locey and Sparrow 2007] [Locey et al. 2008] [Loftin 1967b] [Lonzaga 2019b] [Lonzaga 2019c] [Lonzaga 2019a] [Lonzaga 2020] [Lonzaga 2021] [Lonzaga 2023] [Loubeau 2021] [Loubeau and Coulouvrat 2008] [Loubeau and Coulouvrat 2009] [Loubeau and Doebler 2019] [Luquet] [Velmani and Suresh 2022] [Marchiano et al. 2003] [Marini and Riegel 2024] [McBride et al. 1990] [McBride et al. 1992] [Henderson et al. 1971] [Mishchenko and Gimon 2023] [Mishchenko et al. 2022] [Mishchenko et al. 2024] [Molin 2017] [Ménéxiadis and Varnier 2008] [Nelson et al. 1970] [Nonlinear Acoustic Behavior 1975] [Norris et al. 1995] [Nyborg et al. 2024] [Nyborg et al. 2025] [Ohira and Kwak 2014] [Onyeonwu 1975b] [Onyeonwu 1975a] [Ortega et al. 2019] [Ostashev and Goedecke 2005] [Ostashev and Norris 2022] [Lonzaga et al. 2025] [Pekeris 1946] [Pena and Diamond 1965] [Pettit and Wilson 2021] [Piacsek 1991] [Piacsek 1998] [Piacsek 2002] [Piacsek et al. 2008b] [Piacsek et al. 2008a] [Piacsek et al. 2010] [Pierce 1969] [Pierce 1970] [Pierce 1971] [Pierce 1972] [Pierce 1992a] [Pierce 1995] [Pierce 1996] [Pierce and Maglieri 1971] [Pierce and Maglieri 1972] [Pierce 1994] [Pilon 2010] [Pilon 2011] [Plotkin 1995] [Plotkin 2006b] [Plotkin 1992] [Poling 1996] [Polly 1966] [Potter and Murphy 1959] [Pridmore-Brown 1956] [Pridmore-Brown and Ingard 1955a] [Pridmore-Brown and Ingard 1955b] [Qiao et al. 2019] [Rallabhandi 2011b] [Rallabhandi 2018a] [Rallabhandi 2021] [Rallabhandi 2022a] [Rallabhandi and Fenbert 2017] [Rallabhandi and Loubeau 2017b] [Rallabhandi and Loubeau 2021] [Rallabhandi and Loubeau 2022] [Rallabhandi et al. 2023] [Salamone 2009] [Riegel and Sparrow 2012] [Riegel and Sparrow 2019] [Riegel and Sparrow 2024] [Riegel and Sparrow 2025] [Riegel et al. 2021] [Robinson 1992b] [Robinson 1993] [Robinson 1992a] [Rochat and Sparrow 1996] [Rochat and Sparrow 1997c] [Rochat and Sparrow 1997b] [Rouse 2014] [Rubin and Graber 1966] [Rusak and Cole 1993] [Rusak and Cole 1994] [Ryan et al. 2022] [Sabatini et al. 2016] [Salamone and Sparrow 2012] [Salamone et al. 2013a] [Salamone et al. 2013b] [Sanai et al. 1975] [Sasoh et al. 2016] [Schweitzer 1968] [Sekundov 1974] [Shamanaeva et al. 2018] [Song et al. 2012] [Yang et al. 2014] [Coulouvrat 2002] [Haefeli et al. 1969a] [Hayes and Runyan 1972] [Sparrow 1993] [Sparrow and Gionfriddo 1994] [Rallabhandi] [Rallabhandi and Loubeau] [Standard Values of Atmospheric] [Stout and Sparrow 2016] [Stout and Sparrow 2018] [Stout and Sparrow 2019] [Stout and Sparrow 2022] [Stout et al. 2021] [McCurdy 1994] [Sukharev 2019] [Page et al. 2015] [Sutherland and Daigle 1997] [Takahashi et al. 2017] [Takahashi et al. 2018] [Takayama 2019] [Takeno et al. 2015] [Tam and Chen 1993] [Terletskyi et al. 2023] [Terletskyy and Trushevskyy 2024] [Kim 2008] [Trono Figueras et al. 2024] [Tsuchiya and Kanamori 2025] [Turo et al. 2017] [Vecchiotti et al. 2024] [Verdier et al. 2026] [Wade 2021] [Wade and Sparrow 2019] [Wade and Sparrow 2021] [Warren 1972a] [Whitlow and Harris 1976] [Willshire et al. 1992] [Wilson 1998a] [Wilson 1998b] [Wilson et al. 1996] [Yamamoto et al. 2012] [Yamamoto et al. 2015] [Yamashita and Makino 2022] [Yamashita and Makino 2023b] [Yamashita and Makino 2023a] [Yamashita and Obayashi 2009] [Yamashita and Suzuki 2015b] [Yamashita and Suzuki 2018] [Yamashita and Suzuki 2019] [Yuldashev 2021] [Yuldashev et al. 2021] [Zelias 2021] [Zelias et al. 2021a] [Zelias et al. 2021b]

Shaping the wave, which is the premise the aeroplane existed to test

The premise. Boom minimisation, equivalent area, the F-function, nose-bluntness relaxation and the inverse design methods built on them. The X-54’s entire justification is that this shelf is correct [Seebass and George 1972] [Darden 1979].

289 records. [Abraham et al. 2020] [Akatsuka 2024] [Allahverdi et al. 2016] [Alonso and Colonno 2012] [Rathsam] [Anderson 2021] [Application of sonic-boom minimization concepts in supersonic transport design] [Argrow et al. 2003] [Aronstein and Schueler 2004] [Aronstein and Schueler 2005] [Darden and Shepherd 1992] [Babcock and Salamone 2006] [Ban et al. 2018] [Bashkirov et al. 2021] [Bonavolontà et al. 2023] [Boonjaipetch et al. 2021] [Bowersox et al. 2003] [Bozeman et al. 2024] [Lowe et al] [Brezillon et al. 2011] [Bridges et al. 2018] [Buonanno et al. 2014] [Campbell et al. 2015] [Candon et al. 2016] [Carlson 1970] [Carpenter and Cizmas 2020] [Carpenter et al. 2025] [Carrier et al. 2021b] [Carter et al. 2011] [Castner 2010b] [Chattopadhyay et al. 1995] [Chen et al. 2023] [Chen et al. 2024] [Chen et al. 2026] [Chernyshev et al. 2008] [Chernyshev et al. 2016] [Cheung 1995] [Cheung et al. 1999b] [Chima 2011] [Chima et al. 2011] [Choi et al. 2004b] [Choi et al. 2004a] [Choi et al. 2008] [Chung and Alonso 2002] [Cliff et al. 2016a] [Cliff et al. 2015] [Cliff et al. 2016c] [Cliff et al. 2016b] [Coen et al. 2017] [Colonno and Alonso 2010] [Conners and Howe 2006] [Conners and Wayman 2011] [Darden 1975a] [Darden and Mack 1976] [McCurdy 1999] [Demiroglu and Nikbay 2023] [Demiroglu et al. 2021] [Howe et al. 2008] [Ding et al. 2023] [Dixon and Hunsaker 2023] [Doebler 2021] [Doebler et al. 2018] [Doebler et al. 2023c] [Doebler et al. 2023b] [Durston et al. 2011] [Durston et al. 2014] [Durston et al. 2015] [Elmiligui et al. 2011] [Morgenstern et al. 2005] [Farhat et al. 2002] [Farhat et al. 2004] [Farhat et al. 2007] [Feng et al. 2014] [Fomin et al. 2013] [Foster et al. 2025] [Furukawa et al. 2008] [Gan and Zha 2015a] [George 1968] [George 1969b] [Goel and Jawahar 2022] [Goto et al. 2005] [Graham et al. 2005a] [Graham et al. 2005b] [Graziani et al. 2025d] [Haas and Kroo 2010] [Haglund 1991] [Hague and Jones 1971] [Hao et al. 2016] [Heath et al. 2016] [Heath et al. 2017] [Higgins 1965] [Howe 2011] [Howe 2005] [Imrak et al. 2021] [Inagaki et al. 1998] [Ishikawa et al. 2011] [Ishikawa et al. 2017] [Ishikawa et al. 2019] [Ishikawa et al. 2021b] [Jim et al. 2022] [Jung et al. 2011] [Karr et al. 2003] [Kasuga et al. 2016] [Kasuga et al. 2017] [Kim et al. 2011] [Kirz 2018] [Kirz 2019] [Kirz 2022b] [Kishi et al. 2023] [Klos et al. 2008] [Knight 2022] [Kuratani et al. 2007] [Lazzara et al. 2019] [Lazzara et al. 2020] [Lazzara et al. 2021] [Leatherwood and Sullivan 1994a] [Li 2015] [Li 2023] [Li and Geiselhart 2020] [Li and Geiselhart 2021b] [Li and Geiselhart 2021a] [Li and Geiselhart 2022] [Li and Geiselhart 2023b] [Li and Geiselhart 2024a] [Li and Geiselhart 2024b] [Li and Geiselhart 2026] [Li and Rallabhandi 2011] [Li and Rallabhandi 2014] [Li and Shields 2011] [Li et al. 2008] [Li et al. 2009] [Li et al. 2010] [Li et al. 2011] [Li et al. 2016] [Li et al. 2017b] [Li et al. 2019] [Li et al. 2025b] [Li et al. 2025c] [Li et al. 2025a] [Liebhardt et al. 2020] [Loth et al. 2016] [Lux et al. 1999] [Lukaczyk et al. 2012] [Lyman and Morgenstern 2004] [Ma et al. 2024b] [Ma et al. 2024a] [Mabe et al. 2023] [Mack and Darden 1979a] [Mack and Darden 1980] [Mack and Needleman 1990] [Mack 1999] [Mack 2003] [Mack 2005] [Magee et al. 2013] [Makino and Kroo 2006] [Makino et al. 1997] [Makino et al. 1998] [Marmel et al. 2024] [Marmel et al. 2025] [Matsushima et al. 2004] [McLean and Shrout 1965] [McLean and Shrout 1966b] [McLean and Shrout 1966a] [Miller and Carlson 1971] [Miller 1971b] [Min et al. 2026] [Minelli et al. 2012] [Minelli et al. 2013] [Minelli et al. 2014] [Darden 1975b] [Morgenstern 2012c] [Morgenstern 2013] [Morgenstern et al. 2012] [Munguía et al. 2017] [Murakami 2006] [Nadarajah et al. 2002] [Nadarajah et al. 2003] [Nadarajah et al. 2005] [Naohiko et al. 2016] [Narayan et al. 1999] [Niedzwiecki and Ribner 1977] [Nikbay et al. 2023] [Ocheltree and Choi 2015] [Ordaz and Li 2011] [Ordaz and Li 2012] [Ordaz and Li 2016] [Ordaz and Rallabhandi 2010] [Ordaz et al. 2014] [Ordaz et al. 2015] [Otto et al. 2015] [Padron et al. 2025] [Padron et al. 2026] [Pagaldipti et al. 1996] [Page 2017] [Pak 2015a] [Pak 2018a] [Pak 2018b] [Pak 2015b] [Palacios et al. 2012] [Pan 1970] [Park 2011] [Park and Carter 2021] [Park and Carter 2022] [Petty 1970] [Phillips and Heath 2020] [Phillips and West 2017] [Phillips and West 2018] [Phillips and West 2019] [Piccionello 2023] [Plotkin 2009] [Plotkin and Maglieri 2013] [Plotkin et al. 2008a] [Plotkin et al. 2009] [Potapkin and Moskvichev 2014] [Potapkin and Moskvichev 2019] [Qiao et al. 2022] [Rallabhandi 2013] [Rallabhandi 2014a] [Rallabhandi 2014b] [Rallabhandi and Mavris 2005] [Rallabhandi and Mavris 2006c] [Rallabhandi and Mavris 2007a] [Rallabhandi et al. 2012] [Rallabhandi et al. 2014] [Rallabhandi et al. 2017] [Ribeiro et al. 2021] [Richiardi 2023] [Rodriguez et al. 2021] [Ryan-Simmons and Agarwal 2025] [Sasaki and Obayashi 2003] [Scarselli and Marulo 2010] [Schaefer et al. 2023] [Schrass et al. 2020] [Seebass 1971] [Seebass and Argrow 1998] [Seebass and George 1973] [Shields and Li 2011] [Siclari and Darden 1991] [Siegelman 1970] [Silva et al. 2014] [Silva et al. 2015] [George and Seebass 1971] [Howe 2003] [Seebass and George 1972] [Darden 1979] [Sun and Smith 2019b] [Sun and Smith 2019a] [Sun and Smith 2021] [Sun and Smith 2022] [Sun et al. 2020] [Tamayo] [Tyushina et al. 2022] [Ueno and Makino 2021] [Ueno et al. 2016b] [Ueno et al. 2016a] [Ueno et al. 2017] [Ueno et al. 2023] [Utsumi 2025] [Utsumi and Obayashi 2010] [Vyas et al. 2011] [Waithe 2011] [Page et al. 2014] [Wayman et al. 2011a] [Wayman et al. 2011b] [West et al. 2014] [West et al. 2017] [White et al. 2024] [Wilcox et al. 2012] [Mack and Darden] [Wintzer and Castner 2015] [Wintzer and Kroo 2012] [Wintzer and Ordaz 2015] [Li and Geiselhart] [Li and Geiselhart] [Li and Geiselhart] [Li and Geiselhart] [Li and Geiselhart 2021c] [Li and Geiselhart 2023a] [Li et al] [Xiao et al. 2026] [Yamaguchi and Nakamura 1998] [Ye et al. 2021] [Yoshida 1994] [Yoshida and Tokuyama 1992] [Yoshimoto and Uchiyama 2003] [Yuhara and Rinoie 2012] [Zhang et al. 2019b] [Zhang et al. 2019a] [Zheng et al. 2026]

The other noise a supersonic aeroplane makes, which is the one that got a standard

The noise that is not the boom, and it is worth separating. Jet noise, fan noise, flyover and takeoff noise, and the perceived-noise metrics built for them. The international noise and emissions standards for supersonic aeroplanes date from the 1970s and were written around Concorde, and rulemaking to revise them near airports and en route is under way [Rötger et al. 2023], and this cluster is the evidence base for the airport half of that work.

244 records. [A Review 2011] [Abdelhamid 1970] [Action against Aircraft Noise 1989] [Nagamatsu et al. 1975] [Ahuja 1990] [Ahuja 1993] [Ahuja et al. 1993] [Aircraft Noise and Certification 2019] [Aircraft Noise 2022] [Aircraft Noise 2012a] [Aircraft Noise Level Reduction] [Aircraft Noise Metrics 2022] [Aircraft Noise 2012b] [Aircraft Noise Prediction 1989] [Aircraft Noise Propagation 2011] [Aircraft Noise Research Needs] [Aircraft Noise Sources 1989] [Akatsuka and Ishii 2020] [Akatsuka and Ishii 2021] [Akatsuka and Ishii 2022] [Akatsuka et al. 2013] [Akatsuka et al. 2016] [Akatsuka et al. 2025] [An Experimental Study 1976] [Atvars et al. 1975] [Aujogue et al. 2022] [Bailly and Bailly 1997] [Baklacioglu et al. 2025] [Balakumar 1998] [Barbarino et al. 2024] [Baysal and Idres 2001] [Berry and Harris 2024] [Berry and Speakman 1993] [Berton 2022b] [Berton 2022a] [Berton 2023] [Berton and Majjigi 2025] [Bertsch et al. 2026b] [Bertsch et al. 2026a] [Bhat and Seiner 1993] [Bousquet and Blandeau 2021] [Breen and Ahuja 2020] [Bridges 2019] [Bridges et al. 2022] [Brown et al. 2010] [Bryce and Pinker 1977] [Busch and Bridges 2025] [Calculation of Aircraft Noise 1989] [Calder and Gupta 1977] [Champagne 1991] [Chen et al. 2017] [Clauss et al. 1980] [Cole and England 1967] [Collin Dominique 2012] [Cuppoletti et al. 2013] [Cuppoletti et al. 2018] [Density Fluctuations and Radiated 1976] [Doelling 1961] [Doelling and Bolt 1961b] [Doelling and Bolt 1961a] [Dosanjh 1971] [Dosanjh 1975] [Dosanjh 1981] [Dosanjh et al. 1976] [Driver and Maglieri 1980] [Effect of Temperature 1975] [Eldred and Hubbard 1972] [Eller and Cavanagh 2000] [Envia 2018] [Erwin and Sinha 2011] [Erwin and Sinha 2013] [Evaluation of Headphones for] [Farmer 2024] [Fidell et al. 1989b] [Fidell et al. 1989a] [Fidell et al. 1998b] [Fisher et al. 1998] [FitzGerald 1971] [Gabrielian et al. 2021] [Gautam et al. 2026] [Generation of Mach Waves 2016] [Gilinsky et al. 1993] [Golberg and Gurevich 2023] [Graebert et al. 2025] [Busch and Bridges] [Ground-Plane Microphone Configuration for] [Gutmark et al. 1989] [Gély and Márki 2022] [Hall 2024] [Helicopter and V/Stol Aircraft] [Henderson and Papamoschou 2014] [Herrmann 2006] [Hines and Sabatella 1973] [Homans et al. 1978] [Horton and Busch 2024] [Horvay and Nagamatsu 1970] [House Noise-Reduction Measurements for] [Hromisin 2024a] [Hromisin 2024b] [Hubbard and Maglieri 1961] [Huet et al. 2025] [Huff et al. 2016] [Human 2000] [Human Reaction to Aircraft 1989] [Ilie and Sullivan 2020] [Investigation of Screech Tone 1983] [James and Prasad 2025] [Jaron et al. 2023] [Berton] [Berton] [Horton and Busch] [Jou 1979] [Kabaliswaran and Das 2026] [Kandula 2006] [Kandula 2008] [Khavaran et al. 1994] [Kibens and Wlezien 1983] [Kirz et al. 2023] [Kish 2019] [Knott 1974] [Knott and Benzakein 1973] [Kobayashi et al. 1994] [Kopiev et al. 2025] [Krothapalli et al. 1996] [Krothapalli et al. 1997] [Kumar et al. 2024] [Kuo 2013] [Laufer et al. 1975] [Lepe et al. 2025] [Lepe et al. 2026] [Loh and Hultgren 2005] [Long 2005] [Long 2010] [Makarewicz and Wojciechowska 2003] [Marzouk 2008] [McLaughlin 1976] [Measurements of Aircraft Exterior] [Method for Calculating Attenuation] [Method for Predicting Lateral] [Methods for Aircraft Noise 2011] [Michel 1987] [Michel 2013] [Miller and Eagan 1997] [Millet 2005] [Min et al. 2015] [Mirzoyan and Khaletskii 2021] [Mirzoyan et al. 2004] [Monitoring of Aircraft Noise 2011] [Morfey and Howell 1980] [Morris and Bhat 1993] [Morris and McLaughlin 2018] [Munday et al. 2009] [Murray 2012] [Murray et al. 2014] [Myers and Wydeven 1989] [Nagel and Papathansiou 1983] [Newell and Zwernemann 1999] [Nihart and Brown 1988] [Nonomura and Fujii 2008] [Norum 1989] [Norum 1992] [Norum and Brown 1993] [Norum and Seiner 1980] [Norum and Shearin 1986] [Nuckolls and Ng 1993] [Nuckolls and Ng 1995] [O’Keefe et al. 1981] [Oertel sen. et al 2017] [Pao 1971] [Papamoschou and Papamoschou 1997] [Papamoschou and Phong 2018] [Parthasarathy and Massier 1976] [Peterse 2024] [Piccirillo 2023] [Piccirillo et al. 2022] [Piccone 2022] [Pilon et al. 2014] [Pilon et al. 2017] [Pinker 2024] [Powers et al. 2014] [Powers et al. 2015] [Powers et al. 2018] [Preisser 1980] [Propagation of Aircraft Noise 1976] [Proskurov et al. 2023] [Ramakrishnan et al. 2018] [Rask et al. 2011] [Review of Theory 1976] [Rizzi et al. 2020] [Rogers and Cook 1952] [Rowe et al. 1979] [Sarohia 1977] [Sarohia 1978] [Sarohia et al. 1979] [Sarohia et al. 1980] [Schlinker et al. 2010] [Schlinker et al. 2011] [Seiner and Krejsa 1989] [Semrau and Frye 1974] [Shapiro and Drell 1965] [Shapiro et al. 1966] [Shivashankara and Stubbs 1984] [Shivashankara and Stubbs 1987] [Smith 1986] [Soell 1980] [Some Recent Developments 1976] [Southerland et al. 1986] [Stephens 2017] [Stern and Peracchio 1989] [Stone 1984] [Stone et al. 2000] [Supersonic Jet Noise Suppression 1976] [Tam 1973] [Tam 1983b] [Tam 1989] [Tam 1991] [Tam 2015] [Tam 2025] [Tam and Reddy 1996] [Tam and Salikuddin 1984] [Tanna and Dean 1975] [Tanna et al. 1973] [Tester et al. 1978] [The Main Sources 2011] [Tinney and Panickar 2013] [Tokashiki 2023] [Typical Aircraft Noise Levels 1989] [Venkatakrishnan and Krothapalli 2006] [Venkatakrishnan et al. 2000] [Ventland 2022] [Voet et al. 2022] [Voet et al. 2024a] [Voet et al. 2024b] [Wang et al. 2026b] [Wat and Sarohia 1981] [Wing Noise Simulation 2024] [Woodburn 2024] [Woodward et al. 1983] [Woodward et al. 1984] [Zaman and Podboy 2010] [Zhang et al. 2023] [Zilz and Wlezien 1990] [Zuckerwar and Shope 1974] [Zwernemann 1999]

Computing the wave before there is an aeroplane to make it

Where the work went once the aeroplane stopped being built. Computational fluid dynamics, mesh adaptation, near-field extraction and the prediction workshops that compare codes against each other. A prediction workshop is what a discipline does when it cannot fly, and all three of the AIAA sonic boom prediction workshops appear in this cluster, together with the NASA workshop of 2008 that preceded them.

169 records. [Aftosmis and Nemec 2014] [Akatsuka and Ishii 2018] [Alaa et al. 2022] [Anderson et al. 2017] [Anderson et al. 2019] [Barthelemy et al. 1991] [Barthelemy et al. 1992] [Biblarz et al. 1998] [Bos 1996] [Bui 2009] [Bui 2011] [Burn and Plotkin 1989] [Campbell et al. 2008] [Cao et al. 2024] [Carlson 1964] [Carpenter et al. 2022] [Carter and Deere 2008] [Carter and Park 2021] [Carter and Park 2022] [Chen and Zhang 2006] [Cheung et al. 1990] [Cheung et al. 1992] [Choi et al. 2004c] [Choi et al. 2009a] [Chung et al. 2003] [Clare and Oman 2006] [Cliff and Thomas 1991] [Cliff and Thomas 1993] [Cliff et al. 2013] [Cui et al. 2026] [Dagrau et al. 2014] [Darden 1990] [Darden 1993] [De Rosa et al. 2026] [Derlaga et al. 2019] [Djomehri and Erickson 1993] [Downing and Plotkin 1996] [Duan et al. 2026] [Duensing et al. 2022] [Emre Tekaslan et al. 2023] [Endo and Phillips 2022] [Ferri et al. 1973] [Ferri et al. 1976] [Fouladi 1993] [Friedlander et al. 2018] [Furukawa and Makino 2007] [Gan and Zha 2015b] [Glorioso et al. 2024] [Graziani 2023] [Graziani et al. 2023] [Graziani et al. 2024a] [Guo and Hafez 1995] [Haering et al. 2005a] [Hafez and Wahba 2003] [Herrmann and Orlowski 1997] [Hoch et al. 2025] [Housman et al. 2014] [Hu et al. 2011] [Hu et al. 2012] [Ishikawa et al. 2021a] [Ishikawa et al. 2022] [Jameson et al. 2005] [Lonzaga] [Jones et al. 2006] [Vanharen et al] [Kanamori et al. 2022] [Kandil et al. 2002] [Kim et al. 2001] [Kirz 2021] [Kirz 2022a] [Kirz and Rudnik 2017a] [Kirz and Rudnik 2017b] [Kirz and Rudnik 2019] [Kirz et al. 2025] [Kiseleva and Lukashevich 2021] [Kiseleva and Lukeshevich 2019] [Koegler 1966] [Kwak et al. 2013] [Laflin 2006] [Laflin et al. 2006] [Lengyan and Zhansen 2015] [Li et al. 1999] [Li et al. 2017a] [Loseille et al. 2019] [Loubeau et al. 2019] [Ma et al. 2017] [Ma et al. 2018] [Madson 1994] [Makino et al. 2002] [Endo and Phillips] [Mangano and Martins 2021] [Manning and Kroo 1999] [Maute et al. 2008] [Miller 2023] [Minelli] [Mishchenko and Kiseleva 2019] [Morgenstern 2014] [Newsome 1985] [Norgia and Norgia 1997] [Numerical Study of Noise 2025] [Ojiaku and Prakash 2026] [Ordaz and Li 2013] [Ordaz and Li 2014] [Ozcer 2007] [Page and Plotkin 1991] [Park 2017] [Park and Darmofal 2008] [Park and Darmofal 2010] [Park and Morgenstern 2014] [Park and Morgenstern 2016] [Park and Nemec 2017] [Park and Nemec 2019] [Park et al. 2014b] [Park et al. 2014a] [Pilon and Lyrintzis 1996] [Plotkin and Page 2002] [Pooja Chaubdar 2025] [Rallabhandi 2011c] [Rallabhandi 2011a] [Rallabhandi 2018b] [Rallabhandi and Loubeau 2017a] [Rallabhandi and Loubeau 2019] [Rallabhandi and Mavris 2006b] [Rallabhandi and Mavris 2007b] [Rao and Zumwalt 1970a] [Rao et al. 2023] [Reuther 2002] [Reuther and Jameson 1995] [Reuther et al. 1996a] [Reuther et al. 1996b] [Reuther et al. 1999] [Rodriguez et al. 2025] [Rodriguez et al. 2026] [Roe et al. 2020] [Saito et al. 2014] [Scarselli et al. 2007] [Schnell 2023] [Seraj and Martins 2022] [Shaw 1991] [Shen 2017] [Shen and Alonso 2025] [Shen and Lazzara 2016] [Shen and Lazzara 2024] [Siclari and Darden 1990] [Siclari and Darden 1993] [Siclari 1992] [Siclari 1999] [Smith-Velazquez and Theunissen 2018] [Sonic Boom Analysis 1975] [Spurlock et al. 2021] [Spurlock et al. 2022] [Takovitskii 2023] [Tran et al. 2018] [Vanharen et al. 2022] [Velden 1997b] [Waithe 2008] [White 1993] [Wintzer et al. 2008] [Woan et al. 1991] [Xiao et al. 2024] [Yamashita and Ishikawa 2023] [Yamashita et al. 2021] [Yamashita et al. 2022] [Yamashita et al. 2025] [Yang et al. 2018] [Yeh and Clark 2007] [Yoshida and Tani 1993] [Yoshida et al. 1993] [Zou et al. 2023]

Measuring a wave that arrives once and is gone

The instrumentation problem. Microphone arrays, in-flight probing, near-field pressure measurement, wind-tunnel signatures and the low-frequency response a boom measurement needs. The Quiet Spike’s only aerodynamic result came from this shelf [Smolka et al. 2007].

84 records. [Abel 1971] [Ahlefeldt 2017] [Air Force Test Pilot School Edwards Afb Ca 1962] [Air Force Test Pilot School Edwards Afb Ca 1991] [Akatsuka et al. 2023] [Alberts 2012] [Altman 1965] [Cliatt] [Aoki et al. 2019] [Balchanos et al. 2024] [Bolander and Hunsaker 2020] [Bravo et al. 2012] [Carlson and Morris 1966] [Carlson and Morris 1967] [Castner 2009] [Castner et al. 2013] [Castner et al. 2015] [Castner et al. 2018] [Comparison of Aircraft Noise 1975] [Doebler and Sparrow 2017a] [Durston et al. 2017] [Durston et al. 2023] [Durston et al. 2018] [Brenner and Kukreja 2007] [Flight test measurements and analysis of sonic boom phenomena near the shock wave extremity] [Moua et al. 2007] [Smolka et al. 2007] [Frederick et al. 2015] [Glorioso et al. 2025a] [Ha 2025] [Haering et al. 1999] [Haglund and Kane 1974a] [Haglund and Kane 1974c] [Haglund and Kane 1973] [Heineck et al. 2016] [Hicks and Mendoza 1971] [Ikuta et al. 2023] [Stansbery et al. 1999] [Ishikawa et al. 2008] [Ito et al. 2008] [Jung et al. 2012] [Kamali and Pierce 1972] [Kanamori et al. 2018] [Kanamori et al. 2025] [Kendall 1952] [Kozelkov et al. 2022] [Downing 1992] [Levier 1957] [Ling et al. 2022] [Liu et al. 2023] [Maeno and Ota 2015] [Marcolini and Brooks 1987] [Marcolini and Brooks 1992] [Miller 1971a] [Mizobata et al. 2024] [Morgenstern 2004] [Morgenstern 2012b] [Morgenstern 2012a] [Morris and Miller 1972] [Murovec et al. 2018] [Pan 1971] [Pan and Wang 1972] [Plotkin et al. 2004] [Haering et al. 2007] [Freund et al. 2008] [Freund et al. 2007] [Knight et al. 2008] [Castner and Simerly 2022] [Sakata 2002] [Schairer et al. 2015] [Shantz 1953] [Sheffield 2000] [Moua et al. 2008] [Kukreja 2008] [Su] [Sugioka et al. 2019] [Vermeersch et al. 2015] [Waggoner et al. 2018] [Wei] [Wilcox and Elmiligui 2013] [Winski et al. 2018] [Yamazaki et al. 2019] [Zhang and Li 2025] [Zhang et al. 2025]

The rule, and the standards written to replace it

The second smallest cluster in the survey and the one the aeroplane was designated to serve. Noise certification, aviation noise regulation and policy, and the international standards machinery. An aeroplane whose registry entry names a regulatory change process as its purpose has the second thinnest literature here for that purpose, after the atmosphere.

50 records. [Abeyratne 2001] [Albert et al. 2017] [Altree and Baxter 1968] [Baeva et al. 2023] [Batson 1970] [Bentley 2024] [Berton et al. 2017] [Bolt Beranek and Newman INC LOS Angeles CA 1965] [Bradley 2024] [Brentner et al. 2013] [Callaway and Peart 1976] [Christopher 1971] [Connor et al. 2024] [Covrig and Heyes 2022] [Cowart 2013] [Electroacoustics. Instruments for measurement] [Fidell and Mestre 2020] [Fish and Haight 1976] [Galloway 1971a] [Gardner 1978] [Hardesty et al. 1996] [Harris 1978] [Heller 1984] [Torija 2018] [Jurin and Pavlovic 2005] [Khorrami and Fares 2019] [Kütahya 2026] [Lappas and Pergamalis 2019] [Larsen et al. 2012] [McPike 1976] [Mertens 1997a] [Metzger and Foley 1970] [Milde 1978] [Noeding et al. 2023] [Nöding and Bertsch 2021] [Paullin 1973] [Petree 2018] [Rolf 2000] [Rötger et al. 2023] [Scata et al. 2024] [Scheeper 2024] [Smith 1982] [Specification for electro-acoustical measuring] [Sutherland 1980] [Truxal 2016] [Ulfvengren 2024] [Veenman and Liefferink 2013] [Von Glahn and Groesbeck 1981] [Yun,Jong-Seol 2010] [Zhang et al. 2024]

The atmosphere the wave travels through

Small, and it should not be. Standard atmosphere tables, temperature and wind profiles, and the stratification that refracts a boom. The medium is half the physics of this subject and it is one of the smallest clusters in the survey, which is a fact about how the literature is indexed rather than about how the work is done, since atmospheric variability is treated inside the propagation papers rather than beside them.

41 records. [Appendix a Standard Atmosphere 2021] [Appendix a. the Standard 2011] [Appendix B 2003] [Arntzen 2017] [Bedard 1989] [Brown 2012] [Chapter 3. Sound Propagation 2016] [Crighton and Lee-Bapty 1981] [Dean 1979] [Essenhigh 2006] [Groves 1955] [Groves 1956] [Herbert] [Kang et al. 2023] [Kaushik 2018] [Kosyakov et al. 2024] [Kurzke and Halliwell 2018] [Kurzke et al. 2025] [Li et al. 1994] [Mungiole and Wilson 2003] [Nyborg and Mintzer 1955] [Pierce and Kinney 1976] [Properties of the U.S. 2014] [Properties of the U.S. 2024] [Soilleux 2024] [Space environment natural and] [Standard Atmosphere 1997] [Standard Atmosphere 2005] [Standard Atmosphere 2024] [National Bureau of Standards 1927a] [National Bureau of Standards 1927b] [Takeya and Okumoto 1959] [The Flight Environment 2021] [The International Standard Atmosphere 2017] [The International Standard Atmosphere 2026] [U.S.Committee on Extension to the Standard Atmosphere 1966] [U.S. Standard Atmosphere 1976 1976] [1976 US Standard Atmosphere 2014] [Vaughan 2003] [Wilson and Liu 2004] [Yager 2013]

Where the Framing Breaks Down

This series reads an X-designation as a promise that something will be flown, and the X-54 breaks that reading in three places.

The first is that the aeroplane was never built. Its question was posed by a registry entry and assigned a decade later to a different aeroplane under a different number, which at this article’s date had not yet flown supersonically. No record this article can reach explains why the X-54 was not built, so the framing has no decision to examine.

The second is that its question was regulatory, and the regulation moved without it. The prohibition was ordered repealed by Executive Order 14304 on 6 June 2025 [Executive Order 14304, Leading the World in Supersonic Flight]. The order specifies no technique [Executive Order 14304, Leading the World in Supersonic Flight], and the article argues that the technique able to meet a noise-based rule without a new airframe is Mach cutoff rather than shaping. Signatures at cutoff Mach number were measured in flight and published in 1971 [Henderson et al. 1971]. The subsection headed The Alternative That Never Needed the Aeroplane sets out the argument. An aeroplane justified by a rulemaking was overtaken by an older technique that needed no airframe. Even the evidence a rulemaking consumes is human-response evidence, and the aeroplane would have supplied the stimulus for it rather than the evidence itself, as Thin Subjects records.

The third is that the case for the aeroplane is narrower than the report that originated it. The Quiet Spike report states that the aeroplane’s own shocks would overtake the spike’s within a short distance. With geometric spreading the coalescence does not complete at a strength difference of 0.01, so the case holds for a host the size of an F-15 only if its strength difference is 0.02 or more, which this article assumes, and the result cannot be extended to spikes in general. The calculation is in The Arithmetic the Report Set Out and Did Not Perform.

The Source Base

The eighty-six curated sources that carry the argument and the 2,357 that map the field are different things, and this article keeps them apart on purpose.

The evidentiary base is unusually documentary for an article about an aeroplane. Eighty-six curated research records, seven books and thirty-two reference sources carry every claim made here, and the ones that matter most are a designation registry, an executive order, the final rule of 1973 and a flight test report that states in plain words why the experiment it describes could not produce the result its successor was designated to produce. Those are not commentary on the evidence. They are the evidence.

The research survey maps the field, and none of it is cited in support of any claim about the X-54. It holds 2,357 records in ten clusters, drawn from the Crossref scholarly registry, NASA’s Technical Reports Server and the Defense Technical Information Center, of which 279 are report primaries, being 11.8 percent, by the identifier rule set out under The Definition of Primary. There is nothing in it about the aeroplane, because the aeroplane was never built and no title in the survey names it.

The Age of the Literature Is Itself a Finding

The literature of this subject has a shape, and the shape is the funding history. 2,286 of the 2,357 research records carry a year in their reference anchor or in their entry in the References list, and their median is 2008. Every rate and share in this section is counted on that year, so the article’s own source is enough to recount it.

Publication ran at 25.3 records a year through the 1970s, fell to 13.9 through the 1980s, and has grown every decade since to 79.6 in the 2020s, a decade whose count is divided by seven because the survey reaches seven years into it. The trough is 54.9 percent of the peak that preceded it, and the present rate is 5.7 times the trough. The American supersonic transport was cancelled in 1971 and the prohibition took effect in 1973, and the decade that follows those two dates is the only one in this corpus in which the yearly rate fell. Counted in records rather than records a year, the unfinished 2020s already hold 557 against 540 for the 2010s.

The corpus is split almost evenly between the recent and the old. 36.7 percent of the dated records were published in 2015 or later and 37.9 percent before 2000, which is a field with a long memory and an active present rather than either alone.

The report primaries are older than the survey and their dates follow the agency’s research programmes rather than the journals. Of the 246 report primaries that carry a year, the median is 1992 against 2008 for the corpus, a gap of 16 years, and 84 date from before 1980. The 1990s hold more of them than any other decade, at 66. Of the 2,310 dated records published in 2000 or later, 74 are report primaries, so the recent literature is mostly journal and conference work. A report-primary fraction of 11.8 percent in a corpus whose median is 2008 is therefore reporting something about when the subject was funded.

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 the NASA Technical Reports Server or it carries a digital object identifier under the 10.21236 prefix, which Crossref registers for Defense Technical Information Center reports. By that rule 279 of the 2,357 research records are report primaries, being 11.8 percent, among them the Quiet Spike flight test report, the airborne data report of the shaped-boom demonstration, the cutoff measurements of 1971 and the cutoff investigation of 2012.

The programmes’ own bibliographies lead to the older documents, and those documents are cited directly. The Quiet Spike flight test report, the shaped-boom demonstration paper and the cutoff investigation each name the documents they were built on. The curated references include the Gulfstream design chain from 2003 to 2009, the classical minimisation and propagation literature from 1956 to 1979, the human-response experiments the loudness procedure was calibrated against, and the standard atmosphere the article computes in.

The measure sees most of this article’s primary documents and misses the five that decide it. A designation registry, an executive order, a final rule in the Federal Register, a part of the code of federal regulations and a patent carry no identifier the measure can read, and those five are what the argument turns on. The rule does not reach the journal papers among the primary sources either, such as Whitham on weak shocks and George and Seebass on minimisation, so 11.8 percent is a lower bound on how much of the survey is first-hand. The count of named primary documents is reported beside the fraction rather than instead of it, and the measure is not adjusted to flatter the number.

Thin Subjects

Six subjects that this article’s derivations and conclusions 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, with a hyphen read as a space. Where a row joins two groups of terms with the word together, a title must carry a term from each group.

Subject Title terms counted Records
the X-54 itself X-54, X-54A, X54 0
the coalescence and aging of a signature coalescence, coalescing, merging, overtaking, aging, ageing together with shock, shocks, boom, booms, signature, signatures, N-wave 1
the extendable nose spike Quiet Spike, QuietSpike, extendable nose, telescoping nose 17
Mach cutoff flight Mach cutoff, Mach cut-off, cutoff Mach, no-boom thresholds, threshold Mach number 15
the reflection of a boom at the ground reflection factor, reflection factors, ground reflection, boom reflection, reflected sonic booms, reflected sonic boom 4
the overland prohibition overland, over land, prohibition, 91.817 9

No title names the X-54, which is the fact the opening of this section states.

The coalescence on which the case for the aeroplane rests is named in one title, a study of the aging of sonic booms [Plotkin 1994], so the coalescence distances computed above rest on weak-shock theory rather than on a literature that measures them.

The extendable nose spike is named in 17 titles, and 15 of them are the design, test and analysis reports of Gulfstream and the flight programme. The other two, from 2019 and 2024, study the concept on a supersonic transport. The spike’s literature is its own programme’s record, which is why this article reads that record directly.

Mach cutoff is named in 15 titles, and 2 of them date from 1971. The other 13 date from 2015 or later, so for the 43 years from 1972 to 2014 no title in this survey names the technique that needs no new airframe to keep a boom off the ground.

The reflection of a boom at the ground is named in 4 titles, so the factor of about 1.9 that turns an incident wave into a measured one rests on the in-flight technique cited where the factor is derived rather than on a body of measurement.

The overland prohibition is named in 9 titles, and one of them names the prohibition itself, a reassessment of the merits of the FAA prohibition on civil supersonic flight published in 2018 [Petree 2018]. The rest concern overland flight itself, how it might be certified and what it would be worth.

Fifty records of 2,357, or 2.1 percent, are about the rule. That is the second smallest cluster in the survey, after the atmosphere. The aeroplane’s registry entry names a regulatory change process as its purpose, and the regulatory literature is the second thinnest shelf in a survey of its subject, while 447 records, or 19.0 percent, are about what the noise does to the people who hear it. The evidence a rulemaking actually consumes is human-response evidence, and this article’s aeroplane was designated to produce the stimulus for it.

Two conclusions therefore rest on documents rather than on a literature. The first is that a shaped nose on an otherwise ordinary aeroplane cannot deliver a shaped ground signature. It rests on a primary document rather than on the survey, being the flight test report of the programme that decided not to attempt it, and on an arithmetic that report did not perform. The second is that the rule is the deliverable and the aeroplane the evidence for it. It rests on the registry entry, which names a regulatory change process as the aeroplane’s purpose, and on the executive order, and the regulatory shelf of the survey is its thinnest, at fifty records.

What the Survey Admits

A record is admitted only when a person reading its title finds that it bears on a subject this article’s argument uses, read against the article and not against a shared word. Those subjects are the sonic boom as a physical object and its minimisation, its propagation through the atmosphere and into water, the supersonic aerodynamics and configurations of the aircraft that make it, its computation and its measurement, the standard atmosphere, the response of people, buildings and animals to it, the other noise of aircraft, and the regulation and certification of aircraft noise. No residual cluster is needed, because every admitted title belongs to one of the ten named clusters.

Many words this subject depends on also name something else, and a record that shares only such a word is excluded. Impulsive noise also names an outlier in power-line, radio, image and speech data and in the algorithms of active noise control. Shadow zone and sound-speed profile also belong to ocean acoustics, sonar and marine seismic surveying, and shadow zone also to geotechnics and astrobiology. Near field and far field also belong to antenna and radar measurement, a supersonic jet is also a tool of laser spectroscopy and molecular beams, a boom is also a barrier for an oil spill and the tail of a helicopter, and Sonic Boom is also the title of a book about the music industry, an essay, an advertising study and a technique of ultrasound imaging. A tunnel sonic boom is the pressure wave a high-speed train radiates from a tunnel portal, a quiet tunnel is a low-disturbance wind tunnel, a standard atmosphere is also the conditioning climate for testing textiles and paper, and regulation also names a control law. Microphone arrays that listen for coughs, speech, gearbox faults or the modes of a structure are excluded for the same reason. A record whose only link to this article is a distant application domain is excluded as well, among them the internal flows of inlets, diffusers, nozzles and engines, which the article places out of scope, shock and boundary-layer interaction, combustion and aerodynamic heating, missile and projectile hardware, spikes and energy deposition ahead of blunt bodies where the title does not name the boom, laboratory shock physics, the subsystems and flight control of supersonic aircraft, planetary decelerators, gravity waves of the upper atmosphere, aircraft emissions and contrails, occupational and interior noise exposure and hearing protection, product sound quality, and laboratory studies of animal hearing. Correction, erratum, corrigendum, retraction and withdrawal notices are excluded as well, because a survey counts research works and a notice is an editorial event rather than a work. A review of a book, an unsigned news item, a reference or definition entry of two pages or fewer, a table of standard values of two pages or fewer printed as a textbook appendix, a journal’s listing of a newly issued publication, the resolution of a standards conference and a second deposit of a document already in the survey, such as a report-server copy of a journal or conference paper, are not admitted either, while an authored encyclopedia article of more than two pages and a textbook chapter are research works and are kept.

Some records of doubtful relevance are kept. Community response to road, rail, wind turbine, industrial and other environmental noise, and general studies of annoyance, low-frequency noise and noise sensitivity, are kept because exposure-response methodology is one field and aircraft are one of its cases. The penetration of a sonic boom into water is kept as distinct from ocean acoustics, and so are the active control of a physical impulsive sound, laser beams distorted by an aircraft’s shock, the drag of projectiles and missiles where the subject is wave drag, generic studies of shock propagation and acoustic ray tracing whose medium is not stated, and market and economic studies of supersonic aircraft and of aviation noise. Each shares the physics of the wave, its measurement or its human response with this subject, and a reader may weigh it accordingly.

Every research title has been read for relevance, so the off-topic share that remains is a matter of reading judgement and not of sampling. A reading of titles cannot show that a work is relevant in substance, so this is a statement about titles only.

Epistemic State

What is documentary fact. The X-54A designation was allocated on 5 May 2008 to Gulfstream Aerospace with NASA as the sponsoring agency, with two Rolls-Royce Tay 651 engines and the mission statement quoted in full above. The Shaped Sonic Boom Demonstration flew on 27 August 2003 and measured a shaped signature on the ground. The Quiet Spike flew 32 flights on a NASA F-15B between August 2006 and February 2007, reached Mach 1.8 at 685 pounds per square foot, probed its near field at Mach 1.4, and did not attempt a ground measurement for the reason its report states. The prohibition on civil supersonic flight over the United States was published on 28 March 1973 and took effect on 27 April 1973, and Executive Order 14304 of 6 June 2025 directs its repeal within 180 days without naming any technique. The X-59 first flew on 28 October 2025, below the speed of sound.

What is computed here and not taken from a source. The coalescence distances both with and without geometric spreading, the ray path length, the cutoff Mach number and its temperature sensitivity, the turning altitude, the lateral cutoff azimuth, the Mach angles, the true airspeeds, the standard-atmosphere quantities, Concorde’s cruise boom expressed as a level and as a pressure ratio, the consistency of 685 pounds per square foot with Mach 1.8 at 45,000 feet, and every count in the Source Base including the publication rates by decade.

The coalescence calculation rests on an assumed range of shock-strength differences that no source in this article supplies, and the article reports the range rather than a value for that reason. It also depends on a reference distance at which that strength is quoted, taken here as one aeroplane length, and both choices move the outcome. The spreading-corrected result is the one to read, and it does not support the conclusion at the weakest end of the range.

What is secondary and marked as such. The X-54’s cruise Mach number, altitude and perceived-level target, and the account that the agency regarded the designation as a placeholder. None of the argument depends on any of them.

What is cited from a document rather than through one. The Gulfstream design chain, the classical minimisation and propagation papers, the human-response experiments and the standard atmosphere are all cited directly. One source is absent. The standard monograph of this subject is named by the cutoff investigation’s bibliography, and its report-server identifier is not verifiable, so it is not cited here.

What is an approximation rather than a result. The spreading correction assumes both shocks weaken as the inverse square root of distance and that they weaken proportionally, neither of which is exact once the wave begins to age. It is a bound rather than a prediction, and a propagation code of the kind the discipline actually uses would answer the question properly [Lonzaga et al. 2025]. The Mach cutoff relation likewise assumes a horizontally stratified atmosphere and still air, and the article states the wind term without using it.

What is not known. Why the aeroplane was not built. No primary document reviewed here records a decision, and the absence of a cancellation record is not evidence of a cancellation. Also unknown is whether the interim standard the executive order directs was in place on the deadline, which falls four days after this article’s date and therefore outside what it can report.

What this article does not claim. It does not claim that low-boom shaping is unnecessary, and the shaping literature is the fifth largest cluster in the article’s survey for good reason. It claims that the regulatory change the X-54 was designated to support was ordered before the evidence it was designated to produce existed, by an order silent on technique, and that a demonstrator justified by a rulemaking is exposed to the rulemaking moving without it. That Mach cutoff is the technique able to meet a noise-based rule without a new airframe is this article’s inference from the cutoff relation and the 1971 and 2012 measurements, and the order does not say so.

Out of Scope

The economics of supersonic business travel. Whether a market exists for an aeroplane of this kind is the question that decided whether anybody would pay for it, and this article makes no claim about it.

The landing and take-off noise of a supersonic aeroplane, which is a separate regulatory track with a standard, a metric and a literature of its own. The survey above reports that cluster as part of the field it maps, and the article does not argue from it.

The propulsion. Two Rolls-Royce Tay 651 engines are what the registry records, and what a low-boom airframe demands of an inlet and a nozzle is a large subject this article touches only where the equivalent area distribution requires it.

Everything after this article’s date. The executive order sets a deadline four days ahead of it, and what the agency did on that deadline is not something this article can report.

Conclusion

An X-designation is usually a promise that something will be flown, and this series has already documented several that were not.

The X-54A’s registry entry made a narrower promise than most. It promised ground sonic boom signatures, in support of a regulatory change process, and it made that promise in 2008 because a flight test programme had established, in a report published the year before, that no existing aeroplane could produce them. The gap was real and the entry names it precisely.

The argument for filling it survives arithmetic, though not quite in the form the report states it. Working the overtaking through weak-shock theory shows that the spike’s shocklets are consumed before the ground, about halfway down the ray path at the weakest, at the strength differences this article assumes for an aeroplane the size of the one they were bolted to, and that the same calculation would come out differently on a weaker or longer host. The programme’s reason was right about its own aeroplane, though not about spikes in general, a smaller claim than the report makes and a sufficient one for what followed.

Nothing was built. The evidence the entry describes is assigned to a different aeroplane under a different number a decade later, which had not flown supersonically by this article’s date, and in the meantime the prohibition was ordered repealed by an order that leaves the technique open, while a technique that makes no ground boom at all, rather than a quiet one, needed no new airframe to meet the noise-based rule the order calls for.

That technique is not new and this article’s most surprising finding is how old it is. Signatures at cutoff Mach number were measured in flight and published in 1971, 2 years before the prohibition took effect and 54 years before the order to repeal it [Henderson et al. 1971]. The aeroplane was overtaken not by a newer idea but by an older one, which had been sitting in the same agency’s report server throughout and needed no airframe to make its case.

The lesson is not that the aeroplane was unnecessary. The X-59 exists, which is the strongest possible argument that somebody still needs what the X-54 was for. The lesson is that an aeroplane whose justification is a rulemaking has tied itself to a clock it does not control, and this one was overtaken by its own regulation in much the way its predecessor’s shocklets were overtaken by the aeroplane carrying them.

The number was allocated 17.6 years before the date on this article. The rule is being repealed, its replacement flew last month, and X-54A has still never been built.

Equations

This section collects the 31 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 31, 30 are established and 1 are bespoke. Atmospheric relations are in units of the International System, with altitudes in metres, temperatures in kelvin and pressures in pascals, while overpressures and dynamic pressures quoted from flight reports are in pounds per square foot and levels are in decibels referred to 20 micropascals, and several letters are kept apart deliberately, so $T$ is the temperature and $T_{N}$ the N-wave duration, $L$ is the lapse rate and $\mathcal{L}$ the coalescence distance, $R$ is the gas constant and $\mathcal{R}$ the ground reflection coefficient.

Equation 1, Mach Angle. The disturbances of a supersonic body lie on a cone whose half-angle has a sine equal to the reciprocal of the Mach number. Mach Number Liepmann and Roshko, Elements of gasdynamics

\[\mu \;=\; \arcsin\!\left(\frac{1}{M}\right), \tag{1}\]

Equation 2, Speed of Sound in a Perfect Gas. The speed of sound in a perfect gas is the square root of the ratio of specific heats times the gas constant times the absolute temperature. Anderson, Modern compressible flow

\[a \;=\; \sqrt{\gamma R T}. \tag{2}\]

Equation 3, Straight Ray Path From Cruise Altitude to the Ground. A ray leaving normal to the Mach cone makes the Mach angle with the vertical, so its straight path to the ground is the altitude divided by the cosine of that angle. Haefeli et al. 1969a

\[r \;=\; \frac{h}{\cos\mu} \;=\; \frac{h}{\sqrt{1 - M^{-2}}}. \tag{3}\]

Equation 4, Propagation Speed of a Weak Shock. A weak shock outruns the speed of sound by a fraction proportional to its strength, with coefficient the ratio of one more than the ratio of specific heats to four times that ratio. Whitham, Linear and nonlinear waves Hamilton and Blackstock, Nonlinear acoustics

\[\frac{U}{a} \;=\; 1 \;+\; \frac{\gamma+1}{4\gamma}\,\frac{\Delta p}{p} \;+\; \mathcal{O}\!\left(\frac{\Delta p}{p}\right)^{2}. \tag{4}\]

Equation 5, Linear Pressure Profile of an N-Wave. The N-wave is a linear fall in overpressure from its peak to its negative peak between two jumps over the duration $T_{N}$. Darden 1976 Application of sonic-boom minimization concepts in supersonic transport design

\[\Delta p(t) \;=\; \Delta p_{\max}\left(1 - \frac{2t}{T_{N}}\right), \qquad 0 \le t \le T_{N}, \tag{5}\]

Equation 6, Positive-Phase Impulse of an N-Wave. Integrating the overpressure over the positive half of the N-wave gives an impulse of one quarter of the peak overpressure times the duration $T_{N}$. McCurdy 1994

\[I \;=\; \int_{0}^{T_{N}/2}\Delta p(t)\,\mathrm{d}t \;=\; \frac{\Delta p_{\max} T_{N}}{4}. \tag{6}\]

Equation 7, Whitham F-Function. Whitham’s F-function gives the far-field disturbance of a slender supersonic body as an Abel-type integral of the second derivative of its equivalent area distribution. Whitham 1952 Ashley and Landahl, Aerodynamics of wings and bodies

\[F(y) \;=\; \frac{1}{2\pi}\int_{0}^{y} \frac{A_e''(\xi)}{\sqrt{y-\xi}}\;\mathrm{d}\xi . \tag{7}\]

Equation 8, Equivalent Area Including Lift. The equivalent area adds to the physical cross-section a term proportional to the lift carried ahead of each station, so a lifting surface displaces the air whether or not it is thick. Darden 1976

\[A_e(x) \;=\; A(x) \;+\; \frac{\beta}{2q}\int_{0}^{x} \ell(\xi)\,\mathrm{d}\xi , \qquad \beta = \sqrt{M^{2}-1}. \tag{8}\]

Equation 9, Lift Contribution to the Equivalent Area at the Tail. By the tail all of the lift has entered the equivalent area, contributing $\beta W / 2q$ whatever the shaping.

\[A_{L}(l) \;=\; \frac{\beta W}{2q}, \tag{9}\]

Equation 10, Square-Root Dependence of Overpressure on Weight. A fully aged N-wave amplitude goes as the square root of the F-function amplitude, which is linear in the lift, so overpressure scales as the square root of weight. McLean 1965 Application of sonic-boom minimization concepts in supersonic transport design

\[\Delta p \;\propto\; W^{1/2}. \tag{10}\]

Equation 11, Near-Field Overpressure From the F-Function. Before nonlinear steepening, the overpressure ratio is proportional to the F-function and falls as the inverse square root of distance. On the propagation of weak shock waves

\[\frac{\Delta p}{p_\infty} \;=\; \frac{\gamma M^{2}}{\sqrt{2\beta}}\,\frac{F(y)}{\sqrt{r}} , \tag{11}\]

Equation 12, Far-Field Decay of N-Wave Amplitude and Duration. Once aged into an N-wave, the overpressure falls as the minus three-quarters power of distance and the duration grows as its one-quarter power. On the propagation of weak shock waves Haefeli et al. 1969a

\[\Delta p \;\propto\; r^{-3/4}, \qquad \Delta t \;\propto\; r^{1/4}. \tag{12}\]

Equation 13, Slender-Body Wave Drag Functional. In linearised supersonic flow the wave drag of a slender body is a double integral of the second derivative of the same equivalent area distribution with a logarithmic kernel. Ashley and Landahl, Aerodynamics of wings and bodies

\[D \;=\; -\frac{q}{2\pi}\int_{0}^{l}\!\!\int_{0}^{l} A_e''(x_{1})\,A_e''(x_{2})\, \ln\lvert x_{1}-x_{2}\rvert \;\mathrm{d}x_{1}\,\mathrm{d}x_{2}. \tag{13}\]

Equation 14, Troposphere of the Standard Atmosphere. Below the tropopause the temperature falls linearly with altitude and hydrostatic equilibrium in a perfect gas makes the pressure a power of the temperature ratio. U.S. Standard Atmosphere

\[T(z) \;=\; T_{0} - Lz, \qquad p(z) \;=\; p_{0}\left(\frac{T(z)}{T_{0}}\right)^{g/LR}, \tag{14}\]

Equation 15, Isothermal Layer of the Standard Atmosphere. Above the tropopause the temperature is constant and the pressure falls exponentially with altitude. U.S. Standard Atmosphere 1976 1976

\[p(z) \;=\; p_{11}\exp\!\left[-\frac{g\,(z - z_{11})}{R\,T_{11}}\right]. \tag{15}\]

Equation 16, Dynamic Pressure in Terms of Static Pressure and Mach Number. The dynamic pressure of a perfect gas is half the ratio of specific heats times the static pressure times the square of the Mach number. Liepmann and Roshko, Elements of gasdynamics

\[q \;=\; \tfrac{1}{2}\rho V^{2} \;=\; \frac{\gamma}{2}\,p\,M^{2}. \tag{16}\]

Equation 17, Excess Propagation Speed of a Weak Shock. The speed by which a weak shock exceeds the speed of sound is the speed of sound times the strength coefficient times the pressure ratio. On the propagation of weak shock waves

\[\Delta U \;=\; a\,\frac{\gamma+1}{4\gamma}\,\frac{\Delta p}{p}, \tag{17}\]

Equation 18, Coalescence Distance of Two Shocks. Two shocks a distance $s$ apart merge after a travel distance equal to $s$ divided by the difference in their excess speeds, in which the speed of sound cancels. On the propagation of weak shock waves

\[\mathcal{L} \;=\; \frac{a\,s}{\Delta U_{1}-\Delta U_{2}} \;=\; s\,\frac{4\gamma}{(\gamma+1)}\left[\frac{\Delta p_{1}-\Delta p_{2}}{p}\right]^{-1}. \tag{18}\]

Equation 19, Aging Length of a Weak Wave. The aging length of nonlinear acoustics is the distance over which a wave of length $s$ and relative strength $\Delta p / p$ distorts into its asymptotic form. Whitham, Linear and nonlinear waves Hamilton and Blackstock, Nonlinear acoustics

\[r_{\text{age}} \;\sim\; s\,\frac{4\gamma}{(\gamma+1)}\,\frac{p}{\Delta p}. \tag{19}\]

Equation 20, Bespoke Coalescence Distance With Cylindrical Spreading. The article lets the strength difference between the two shocks decay as the inverse square root of distance from a reference distance it takes as one aeroplane length, and integrates the separation to zero. Darden et al. 1989

\[\frac{\mathrm{d}s}{\mathrm{d}r} \;=\; -\frac{\gamma+1}{4\gamma}\,\delta_{0}\sqrt{\frac{r_{0}}{r}} \qquad\Longrightarrow\qquad \sqrt{r_{c}} \;=\; \sqrt{r_{0}} \;+\; \frac{\mathcal{L}}{2\sqrt{r_{0}}}, \tag{20}\]

Equation 21, Ray Invariant in a Stratified Atmosphere. In a horizontally stratified medium the local speed of sound divided by the cosine of the ray angle below the horizontal is conserved along a ray as the trace velocity. Pierce, Acoustics, an introduction to its physical principles and applications

\[\frac{a(z)}{\cos\theta(z)} \;=\; V_{t} \;=\; \text{constant}, \tag{21}\]

Equation 22, Hyperbolic Tangent Shock Profile. A steady balance between nonlinear steepening and diffusion gives a shock front with a hyperbolic tangent profile of characteristic time $\tau$. Blackstock, Fundamentals of physical acoustics

\[\Delta p(t) \;=\; \frac{\Delta p_{\max}}{2}\left[1 - \tanh\!\left(\frac{t}{\tau}\right)\right], \tag{22}\]

Equation 23, Inverse Dependence of Shock Thickness on Amplitude. The rise time of the steady shock varies inversely with its amplitude, so a weaker boom also rises more slowly. Blackstock, Fundamentals of physical acoustics

\[\tau \;\propto\; \frac{1}{\Delta p}. \tag{23}\]

Equation 24, Sound Pressure Level. An overpressure becomes a level as twenty times the base-ten logarithm of its ratio to the reference pressure of 20 micropascals. A loudness calculation procedure applied to shaped sonic booms

\[L_{p} \;=\; 20\log_{10}\!\left(\frac{\Delta p}{p_{\mathrm{ref}}}\right), \qquad p_{\mathrm{ref}} = 20\ \mu\mathrm{Pa}, \tag{24}\]

Equation 25, Concorde Overpressure as a Fraction of Ambient Pressure. The measured Concorde overpressure of 1.94 pounds per square foot at Mach 2 and 52,000 feet is about one part in 1,090 of the ambient pressure. NASA Facts FS-2003-11-016, Sonic Booms

\[\frac{\Delta p}{p_{\infty}} \;=\; 9.17 \times 10^{-4}, \tag{25}\]

Equation 26, Ground Reflection of the Incident Wave. A gauge on a hard surface reads the incident wave plus its reflection, about 1.9 times the incident overpressure for a reflection coefficient near 0.9. Stansbery et al. 1999

\[\Delta p_{\text{measured}} \;=\; (1 + \mathcal{R})\,\Delta p_{\text{incident}} \;\approx\; 1.9\,\Delta p_{\text{incident}}, \qquad \mathcal{R} \approx 0.9, \tag{26}\]

Equation 27, Sound Exposure Level. The sound exposure level is ten times the base-ten logarithm of the time integral of the squared pressure ratio, referred to one second.

\[\mathrm{SEL} \;=\; 10\log_{10}\!\left[\frac{1}{t_{\mathrm{ref}}}\int \frac{p^{2}(t)}{p_{\mathrm{ref}}^{2}}\,\mathrm{d}t\right], \qquad t_{\mathrm{ref}} = 1\ \mathrm{s}. \tag{27}\]

Equation 28, Mach Cutoff Number in Still Air. A boom fails to reach the ground when the shock’s ground intersection is slower than sound at the surface, so the cutoff Mach number is the square root of the ratio of ground to cruise temperature. U.S. Standard Atmosphere Anderson, Modern compressible flow

\[M_{\mathrm{co}} \;=\; \frac{a_{\text{ground}}}{a_{\text{cruise}}} \;=\; \sqrt{\frac{T_{\text{ground}}}{T_{\text{cruise}}}}. \tag{28}\]

Equation 29, Turning Altitude in the Standard Troposphere. Setting the linear tropospheric speed of sound equal to the trace velocity gives a turning altitude linear in the square of the Mach number. Coulouvrat 2002

\[z_{\text{turn}} \;=\; \frac{T_{0} - M^{2}\,T_{\text{cruise}}}{L}. \tag{29}\]

Equation 30, Effective Sound Speed and Cutoff Mach Number With Wind. Wind adds its component along the ray to the thermodynamic speed of sound, so the cutoff Mach number is the ratio of the effective ground sound speed to the sound speed at cruise altitude. Haefeli et al. 1969a

\[c_{\mathrm{eff}}(z) \;=\; a(z) + \mathbf{u}(z)\cdot\hat{\mathbf{n}}, \qquad M_{\mathrm{co}} \;=\; \frac{c_{\mathrm{eff}}(0)}{a(z_{\text{cruise}})} . \tag{30}\]

Equation 31, Lateral Extent of the Boom Carpet. A ray leaving at azimuth $\phi$ reaches the ground only while the Mach number times the cosine of the azimuth exceeds the cutoff Mach number, which bounds the half-width of the carpet. Cliatt et al. 2016a

\[M\cos\phi \;\ge\; M_{\mathrm{co}} \qquad\Longrightarrow\qquad \phi_{\max} \;=\; \arccos\!\left(\frac{M_{\mathrm{co}}}{M}\right). \tag{31}\]

Notation

  • $\mu$, the Mach angle
  • $M$, the flight Mach number
  • $a$, the speed of sound
  • $\gamma$, the ratio of specific heats, taken as 1.4 for air
  • $R$, the specific gas constant of air
  • $h$, the cruise altitude
  • $U$, the propagation speed of a weak shock
  • $\Delta p$, the overpressure, with $\Delta p_{\max}$ its peak
  • $p_\infty$, the ambient pressure at cruise altitude
  • $T_{N}$, the duration of the N-wave
  • $F(y)$, Whitham’s F-function
  • $A_e$, the equivalent area distribution, with $A$ the geometric cross-section
  • $\ell$, the lift per unit length
  • $\beta$, the Prandtl-Glauert factor $\sqrt{M^{2}-1}$
  • $A_{L}$, the lift contribution to the equivalent area at the tail
  • $W$, the aircraft weight
  • $q$, the dynamic pressure
  • $D$, the wave drag
  • $L$, the temperature lapse rate of the troposphere
  • $T_{11}$, the temperature of the isothermal layer, with $p_{11}$ and $z_{11}$ the pressure and altitude at its base
  • $\mathcal{L}$, the coalescence distance of two shocks
  • $r_{\text{age}}$, the aging length
  • $\delta_{0}$, the strength difference between two shocks at the reference distance $r_{0}$
  • $r_{c}$, the coalescence distance corrected for spreading
  • $V_{t}$, the horizontal trace velocity of the wavefront
  • $\theta$, the ray angle below the horizontal
  • $\tau$, the characteristic rise time of the shock
  • $L_{p}$, the sound pressure level
  • $p_{\mathrm{ref}}$, the reference pressure of 20 micropascals
  • $\mathcal{R}$, the ground reflection coefficient
  • $\mathrm{SEL}$, the sound exposure level
  • $M_{\mathrm{co}}$, the cutoff Mach number
  • $z_{\text{turn}}$, the altitude at which a ray turns horizontal
  • $c_{\mathrm{eff}}$, the effective sound speed including the wind component along the ray
  • $\phi$, the azimuth of a ray from the ground track

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