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It was never built, and the regulation is being repealed without it.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 The prohibition it was designated to help lift was ordered repealed by the President of the United States on 6 June 2025, with a deadline that falls four days after the date on this article [Executive Order 14304, Leading the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]

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

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

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

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 1969]

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

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]

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

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 a positive-phase 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],

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

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

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

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]

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

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]

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

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

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

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

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

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],

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

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 1969]

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

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],

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

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 1975] [Darden 1975].

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 2005] [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 2005].

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, Final Rule, 38 FR 8051]. The rule was published on 28 March 1973 and took effect on 27 April 1973 [Civil Aircraft Sonic Boom, Final Rule, 38 FR 8051].

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],

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

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],

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

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],

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

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,

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

so the stronger overtakes the weaker after a travel distance

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

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,

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

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],

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

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],

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

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],

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

whose width varies inversely with the amplitude,

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

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],

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

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,

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

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],

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

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 1992].

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

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

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 1992] [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 2013].

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 1994], 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 1992].

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,

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

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],

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

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 2016] [Cliatt et al 2016]. What arrives is not silence but an evanescent rumble, and the programme measured it and named a metric for it [Haering et al 2005].

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 1969],

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

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

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

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 2016].

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 2005], 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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al 2023] [The Design of the 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 2010] [Castner 2012] [Castner 2012] [Castner 2011] [Chen and Suzuki 2021] [Cheng 2005] [Cheng and Lee 1996] [Cheng and Lee 1998] [Cheng and Lee 1998] [Cheng and Lee 2004] [Cheung et al 1999] [Cho 2009] [Cho 2009] [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 1998] [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 2020] [Doebler 2020] [Doebler and Rathsam 2019] [Dogan 2008] [Downing 2017] [Downs 2021] [Downs et al 2022] [Dragna 2021] [Drobzheva et al 2021] [Dubois 1970] [Durrant et al 2021] [Durrant et al 2022] [Durrant et al 2022] [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] [Formation of the layout 2025] [Frampton et al 1993] [Ganjehi et al 2008] [Gao and Sun 2025] [George 1969] [Gottlieb and Ritzel 1988] [Gottlieb et al 1978] [Graziani et al 2024] [Graziani et al 2025] [Graziani et al 2025] [Ha et al 2021] [Haber 1993] [Haglund and Kane 1972] [Haglund and Kane 1974] [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 1974] [Henderson and Hilton 1974] [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 2017] [Kandil et al 2005] [Kandil et al 2006] [Kane 1973] [Kane 1973] [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 2024] [Korunov et al 2024] [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 1996] [Lee and Downing 1996] [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 1967] [Ludlow and Morgan 1972] [Lympany and Page 2022] [Lynch and Speake 1978] [Lysenkov 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 2017] [Morgenstern 2021] [Simmons and Freund 2005] [Moskvichev and Potapkin 2019] [Naka et al 2008] [Needleman and Mack 1990] [Newberry 1967] [Nikbay et al 2017] [Nixon 1970] [Nixon and Borsky 1965] [Novakovich et al 2019] [Nozzle Plume Effects on 2013] [Numata et al 2015] [Plotkin 1994] [On the propagation of weak shock waves] [Oswatitsch and Sun 1972] [Ou and Wang 2025] [Ozcer and Kandil 2009] [Page 1975] [Palmer 1970] [Palmer and Sparrow 2015] [Pan and Varner 1972] [Pan 1968] [Panel Discussion Sonic Boom 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 2006] [Pilon 2007] [Plotkin 1989] [Plotkin 1990] [Plotkin 1993] [Plotkin 2006] [Plotkin and Cantril 1976] [Plotkin et al 1989] [Plotkin et al 1990] [Plotkin et al 1992] [Plotkin et al 2005] [Plotkin et al 2005] [Plotkin et al 2008] [Porter 1968] [Potapkin and Moskvichev 2017] [Potapkin and Moskvichev 2018] [Potapkin and Moskvichev 2020] [Potapkin and Moskvichev 2021] [Potapkin and Moskvichev 2021] [Power 1963] [Power 1965] [Pretlove and Bowler 1972] [Simmons et al 2007] [Rachami and Page 2010] [Rallabhandi 2019] [Rallabhandi 2022] [Rallabhandi and Mavris 2004] [Rallabhandi et al 2010] [Rao and Zumwalt 1970] [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 1997] [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 2020] [Scarselli and Castorini 2013] [Schultz et al 2016] [Schultz et al 2017] [Sears 1969] [Seebass 1969] [Seebass 1969] [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] [Sonic boom investigation Computation 2017] [Plotkin and Maglieri 2003] [Sonic boom results for 1975] [Soroka 1963] [Sparrow 1995] [Sparrow 1995] [Sparrow 1997] [Sparrow 1998] [Sparrow 1998] [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 2018] [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 1972] [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 2015] [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 2001] [Catalano and Sturek 2001] [Cenko 1979] [Cenko 1980] [Chang 1952] [Chang 1992] [Chapter 3. Aerodynamics of 1957] [Chapter 4. Wing-Tail Plane 1957] [Chapter 5. Wing-Body Interference 1957] [Chaudhari and Takahashi 2022] [Chester 1953] [Chester 1956] [Chester 1956] [Chima et al 2010] [Chin 1977] [Chin 1978] [Choi et al 2009] [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 2018] [Gopal and Maddalena 2018] [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 2014] [Guan 2014] [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] [Hypersonic and Supersonic Flight 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 1967] [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] [Lavrishcheva 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 2023] [Livne 2023] [Lock 1957] [Lock and Bridgewater 1967] [Loftin 1967] [Lord and Eminton 1954] [Loubeau 2022] [Loubeau et al 2023] [Loubeau et al 2023] [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 1997] [Methodic for three-dimensional parametric 2024] [Miles 1952] [Miles 1952] [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 of 2012] [Puckett 1946] [Quagliarella and Iuliano 2017] [Quagliarella and Iuliano 2018] [Quagliarella and Iuliano 2018] [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 2006] [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 2005] [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 2018] [Supersonic Aircraft High-Speed Civil 1997] [Supersonic Cruise 1981] [Supersonic Cruise Research 1979 1980] [Supersonic Cruise Research 1979 1980] [Supersonic Flight 2025] [Supersonic transport wing minimum 1992] [Surzhikov and Shang 2005] [Swan 1969] [Takovitskii 2020] [Technical applications for an 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 1997] [Veresnikov and Goncharenko 2025] [Verhoff and Stookesberry 1992] [Vigneron et al 2008] [Waechter 1969] [Waechter 1971] [Walkley and Martin 1979] [Wang et al 2026] [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 1992] [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 2022] [Carr et al 2022] [Carrier and Coulouvrat 2022] [Carrier et al 2021] [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 2016] [Collmar and Salamone 2017] [Corcoran et al 2012] [Coulouvrat 1998] [Coulouvrat 1998] [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 2017] [Dougherty 1996] [Downing and Lee 1992] [Downs et al 2020] [Downs et al 2022] [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 2005] [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 2025] [Goldman 2017] [Gostev et al 2006] [Graziani et al 2025] [Haefeli 1969] [Haefeli et al 1969] [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 2017] [Ignat and Pozo 2017] [Iura et al 2023] [Ivanteeva et al 1998] [Ivanteyeva et al 2002] [Lonzaga] [Johnson and Hammerton 2003] [Jäschke and Liebhardt 2022] [Kanamori et al 2017] [Kanamori et al 2021] [Kandil 2006] [Kandil and Khasdeo 2006] [Kapcsos and Sparrow 2023] [Kapcsos and Sparrow 2023] [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 2016] [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 1998] [Lipkens and Blackstock 1998] [Lipkens et al 1998] [Liszka 1978] [Liu and Liu 2019] [Locey 2006] [Locey and Sparrow 2007] [Locey et al 2008] [Loftin 1967] [Lonzaga 2019] [Lonzaga 2019] [Lonzaga 2019] [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 at 1975] [Norris et al 1995] [Nyborg et al 2024] [Nyborg et al 2025] [Ohira and Kwak 2014] [Onyeonwu 1975] [Onyeonwu 1975] [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 2008] [Piacsek et al 2008] [Piacsek et al 2010] [Pierce 1969] [Pierce 1970] [Pierce 1971] [Pierce 1972] [Pierce 1992] [Pierce 1995] [Pierce 1996] [Pierce and Maglieri 1971] [Pierce and Maglieri 1972] [Pierce 1994] [Pilon 2010] [Pilon 2011] [Plotkin 1995] [Plotkin 2006] [Plotkin 1992] [Poling 1996] [Polly 1966] [Potter and Murphy 1959] [Pridmore-Brown 1956] [Pridmore-Brown and Ingard 1955] [Pridmore-Brown and Ingard 1955] [Qiao et al 2019] [Rallabhandi 2011] [Rallabhandi 2018] [Rallabhandi 2021] [Rallabhandi 2022] [Rallabhandi and Fenbert 2017] [Rallabhandi and Loubeau 2017] [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 1992] [Robinson 1993] [Robinson 1992] [Rochat and Sparrow 1996] [Rochat and Sparrow 1997] [Rochat and Sparrow 1997] [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 2013] [Salamone et al 2013] [Sanai et al 1975] [Sasoh et al 2016] [Schweitzer 1968] [Sekundov 1974] [Shamanaeva et al 2018] [Song et al 2012] [Song et al 2014] [Coulouvrat 2002] [Haefeli et al 1969] [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] [The Subjective Response for 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 1972] [Whitlow and Harris 1976] [Willshire et al 1992] [Wilson 1998] [Wilson 1998] [Wilson et al 1996] [Yamamoto et al 2012] [Yamamoto et al 2015] [Yamashita and Makino 2022] [Yamashita and Makino 2023] [Yamashita and Makino 2023] [Yamashita and Obayashi 2009] [Yamashita and Suzuki 2015] [Yamashita and Suzuki 2018] [Yamashita and Suzuki 2019] [Yuldashev 2021] [Yuldashev et al 2021] [Zelias 2021] [Zelias et al 2021] [Zelias et al 2021]

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 2021] [Carter et al 2011] [Castner 2010] [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 1999] [Chima 2011] [Chima et al 2011] [Choi et al 2004] [Choi et al 2004] [Choi et al 2008] [Chung and Alonso 2002] [Cliff et al 2016] [Cliff et al 2015] [Cliff et al 2016] [Cliff et al 2016] [Coen et al 2017] [Colonno and Alonso 2010] [Conners and Howe 2006] [Conners and Wayman 2011] [Darden 1975] [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 2023] [Doebler et al 2023] [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 2015] [George 1968] [George 1969] [Goel and Jawahar 2022] [Goto et al 2005] [Graham et al 2005] [Graham et al 2005] [Graziani et al 2025] [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 2021] [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 2022] [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 1994] [Li 2015] [Li 2023] [Li and Geiselhart 2020] [Li and Geiselhart 2021] [Li and Geiselhart 2021] [Li and Geiselhart 2022] [Li and Geiselhart 2023] [Li and Geiselhart 2024] [Li and Geiselhart 2024] [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 2017] [Li et al 2019] [Li et al 2025] [Li et al 2025] [Li et al 2025] [Liebhardt et al 2020] [Loth et al 2016] [Lux et al 1999] [Lukaczyk et al 2012] [Lyman and Morgenstern 2004] [Ma et al 2024] [Ma et al 2024] [Mabe et al 2023] [Mack and Darden 1979] [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 1966] [McLean and Shrout 1966] [Miller and Carlson 1971] [Miller 1971] [Min et al 2026] [Minelli et al 2012] [Minelli et al 2013] [Minelli et al 2014] [Darden 1975] [Morgenstern 2012] [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 2015] [Pak 2018] [Pak 2018] [Pak 2015] [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 2008] [Plotkin et al 2009] [Potapkin and Moskvichev 2014] [Potapkin and Moskvichev 2019] [Qiao et al 2022] [Rallabhandi 2013] [Rallabhandi 2014] [Rallabhandi 2014] [Rallabhandi and Mavris 2005] [Rallabhandi and Mavris 2006] [Rallabhandi and Mavris 2007] [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 2019] [Sun and Smith 2019] [Sun and Smith 2021] [Sun and Smith 2022] [Sun et al 2020] [Tamayo] [Tyushina et al 2022] [Ueno and Makino 2021] [Ueno et al 2016] [Ueno et al 2016] [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 2011] [Wayman et al 2011] [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 2021] [Li and Geiselhart 2023] [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 2019] [Zhang et al 2019] [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 of the 2011] [Abdelhamid 1970] [Action against aircraft noise 1989] [Aeroacoustics Fan, STOL, and 1975] [Ahuja 1990] [Ahuja 1993] [Ahuja et al 1993] [Aircraft Noise and Certification 2019] [Aircraft Noise Flight Operations 2022] [Aircraft Noise Flight Trajectories 2012] [Aircraft Noise Level Reduction] [Aircraft Noise Metrics and 2022] [Aircraft Noise Noise Sources 2012] [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 of 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 2022] [Berton 2022] [Berton 2023] [Berton and Majjigi 2025] [Bertsch et al 2026] [Bertsch et al 2026] [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 1961] [Doelling and Bolt 1961] [Dosanjh 1971] [Dosanjh 1975] [Dosanjh 1981] [Dosanjh et al 1976] [Driver and Maglieri 1980] [Effect of Temperature on 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 1989] [Fidell et al 1989] [Fidell et al 1998] [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 2024] [Hromisin 2024] [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 and 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 et al 1974] [Shapiro and Drell 1965] [Shapiro et al 1966] [Shivashankara and Stubbs 1984] [Shivashankara and Stubbs 1987] [Smith 1986] [Soell 1980] [Some Recent Developments in 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 1983] [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 of 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 2024] [Voet et al 2024] [Wang et al 2026] [Wat and Sarohia 1981] [Wing Noise Simulation of 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 2004] [Choi et al 2009] [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 2015] [Glorioso et al 2024] [Graziani 2023] [Graziani et al 2023] [Graziani et al 2024] [Guo and Hafez 1995] [Haering et al 2005] [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 2021] [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 2022] [Kirz and Rudnik 2017] [Kirz and Rudnik 2017] [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 2017] [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 2014] [Park et al 2014] [Pilon and Lyrintzis 1996] [Plotkin and Page 2002] [Pooja Chaubdar 2025] [Rallabhandi 2011] [Rallabhandi 2011] [Rallabhandi 2018] [Rallabhandi and Loubeau 2017] [Rallabhandi and Loubeau 2019] [Rallabhandi and Mavris 2006] [Rallabhandi and Mavris 2007] [Rao and Zumwalt 1970] [Rao et al 2023] [Reuther 2002] [Reuther and Jameson 1995] [Reuther et al 1996] [Reuther et al 1996] [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 for 1975] [Spurlock et al 2021] [Spurlock et al 2022] [Takovitskii 2023] [Tran et al 2018] [Vanharen et al 2022] [Velden 1997] [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 2017] [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 2025] [Ha 2025] [Haering et al 1999] [Haglund and Kane 1974] [Haglund and Kane 1974] [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 1971] [Mizobata et al 2024] [Morgenstern 2004] [Morgenstern 2012] [Morgenstern 2012] [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 1971] [Gardner 1978] [Hardesty et al 1996] [Harris 1978] [Heller 1984] [J.Torija 2018] [Jurin and Pavlovic 2005] [Khorrami and Fares 2019] [Kutahya 2026] [Lappas and Pergamalis 2019] [Larsen et al 2012] [McPike 1976] [Mertens 1997] [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 Jr. 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 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 Properties of 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] [Standard atmosphere chart 1927] [Standard atmosphere chart supersedes 1927] [Takeya and Okumoto 1959] [The Flight Environment Standard 2021] [The International Standard Atmosphere 2017] [The international standard atmosphere 2026] [U.S. standard atmosphere supplements 1966] [U.S. Standard Atmosphere, 1976] [US Standard Atmosphere Model 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 1,420 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.

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