The sound now is the source then

The sound that only leaves establishes the acoustic analogy: a flow’s noise is computed by rearranging the equations of motion into a wave equation with sources on the right, and solving that wave equation by integrating over the sources.

The integral has a peculiarity that is the whole of its difficulty and is usually presented as a technicality. The source is evaluated not at the time the sound arrives but at the time it was emitted — the retarded time — and for a source that is moving, working out which time that is turns out to be the whole calculation.

The relation, and why it is implicit #

Sound arriving at an observer at time $t$ left the source at a time $t_e$ satisfying

$$t = t_e + \frac{r(t_e)}{a},$$

where $r$ is the distance from where the source was at $t_e$ to where the observer is. The distance depends on the emission time, so the relation is implicit and has to be solved.

For a stationary source it is trivial: the distance is fixed and the delay is a constant, and the mapping’s slope is 1.000000000 everywhere. For a moving one it is not, and the whole of the interesting behaviour is in how the mapping between the two times distorts.

The Doppler factor is that mapping’s slope #

If a source emits over an interval of emission time, that emission arrives over a different interval of arrival time — shorter while the source approaches, longer while it recedes. The ratio of the two is the reciprocal of the mapping’s slope, and it is the Doppler factor.

Computing the slope numerically and comparing it with the textbook expression 1/(1 + M cos θ) gives agreement to 4·10⁻⁹, which is the numerical differentiation’s own error.

That agreement is the essay’s point. The Doppler shift is usually introduced as a separate physical effect with its own derivation; it is a derivative of a bookkeeping relation, and everything about it — including the fact that it applies to amplitude as well as to frequency — follows from that.

And it diverges before the source reaches the speed of sound #

The factor ahead of the source is one over one minus the Mach number, and behind it one over one plus. At Mach 0.8 the two are 5 and 0.56. At Mach 0.99 they are 100 and 0.5, a ratio of 199.

The receding column never falls below a half however fast the source goes; the approaching column diverges. That asymmetry is the whole of the effect: at Mach 0.8 the factor ahead is nine times the factor behind, and at Mach 0.99 it is 199. Raised to the fourth power, which is what a moving monopole’s intensity carries, nine becomes 6,561 and nineteen becomes 130,321.

The divergence has an immediate consequence for noise. Acoustic intensity depends on the Doppler factor raised to a power — the fourth for a moving monopole, higher for a dipole or a quadrupole — so a source approaching at high subsonic Mach number radiates forwards enormously more strongly than it does backwards. That is why a fan is loud ahead of an aircraft and why the noise of a propeller tip is a strong function of its helical Mach number.

What the factor does to an amplitude #

The Doppler factor appears in two places and conflating them is a common error, so it is worth separating them.

In the frequency. A tone emitted at one frequency arrives at another, in the ratio of the factor. That is the familiar effect and it is what a listener notices.

And in the amplitude. The same compression of time compresses the energy arriving per unit time, so the intensity is multiplied too — and by a higher power, because the convergence of the rays adds further factors. For a monopole in uniform motion the radiated intensity carries the factor to the fourth power; for a dipole, the sixth.

At Mach 0.8 that fourth power is 625 forward and 0.10 aft, a ratio of six thousand. A source at high subsonic Mach number is not slightly directional; it is almost entirely forward-radiating, and the directionality is a kinematic consequence rather than a property of the source.

That is why the propeller and open-rotor noise problem is dominated by tip Mach number rather than by thrust, and why a small increase in rotational speed produces a large increase in noise for no change in the source at all.

Above the speed of sound it folds #

The mapping is monotone only while the source is subsonic. Above Mach one it turns over: there is a first arrival time before which nothing at all has been heard, and after it every instant carries sound from two different emission times at once.

That fold is the sonic boom. What arrives is not a Doppler-shifted signal; it is the superposition of two parts of the source’s history, and the singularity where they merge is the shock.

The boundary between the region that has heard nothing and the region that has heard twice is the Mach cone, whose half-angle is the arcsine of the reciprocal Mach number — 38.7 degrees at Mach 1.6 and 11.5 at Mach 5.

The one place the implicit relation has to be solved #

The computation here evaluates the mapping forwards, which is a convenience rather than a method, and it is worth saying what a real calculation has to do instead.

An acoustic prediction wants the pressure at an observer at stated arrival times — the samples of a signal — so it needs the emission time corresponding to each of them, which means solving the implicit relation. For a subsonic source that is a single root and a Newton iteration finds it in a few steps.

For a supersonic one there are two roots, or three, and a solver has to find all of them and add their contributions. That is where the fold becomes a computational problem rather than a phenomenon, and it is the reason the standard formulations are written in two forms — one for subsonic surfaces and one for supersonic — with different singularities to handle.

The singularity is at the fold itself, where the two roots merge and the Doppler factor diverges. A formulation that divides by it produces infinities at exactly the moment the answer is wanted, which is why the emission-surface formulations exist: they integrate over the surface of points whose emissions arrive together, and the divergence becomes a Jacobian that can be handled.

What the solver computed, and how it was checked #

The retarded-time relation is evaluated forwards — for each emission time, the arrival time is computed directly — which avoids solving the implicit relation and gives the mapping as a parametric curve. The Doppler factor is then the numerical derivative.

Three checks. That the derivative matches the closed form, to 10⁻⁴, which is the identification the essay is about. That the compression ratio at Mach 0.99 exceeds a hundred, so the divergence is present. And that the mapping above Mach one has a turning point and a region arriving before the first emission does, which is the fold.

Why this is the subject these essays share #

An acoustic field is the purest memory in this collection, and the reason is worth stating plainly.

The pressure at a listener at a moment is an integral over the source’s entire past, weighted by where the source was and how fast it was moving. Nothing about the source’s present state enters at all — the source might have stopped, or been removed, and the sound already emitted still arrives.

That is a convolution with a kernel that is a delta function at a delay, and the delay is a function of position. Compare the three kernels of the theory with no memory in it: the ideal one is a delta at zero delay, the viscous one is a heavy-tailed function, and the compressible one is a delta at a delay — which is exactly this.

So the compressible door is the one that produces a memory with a definite age rather than a distribution of ages, and that is what makes acoustics tractable: everything can be attributed to a particular earlier moment.

What a microphone is measuring #

The practical consequence is a warning about source localisation, and it is one that acoustic measurement techniques exist to deal with.

A microphone records a superposition of retarded histories. With one microphone and a moving source there is no way to separate them, because a single pressure trace is one number per instant and the mapping is many-to-one in the supersonic case and ill-conditioned near it.

An array can undo the mapping. Beamforming works by applying the retarded-time relation in reverse: each microphone’s signal is delayed by the propagation time from a candidate source position, and the delayed signals are summed. If the candidate is right they add coherently, and if not they do not.

Which requires knowing the source’s motion. The relation contains the source’s position at the emission time, so a beamformer on a moving source has to track it — and an error in the track becomes an error in the delay, which becomes a smearing of the map.

That is exactly the structure of the inverse problems elsewhere in this collection: the record is complete and recovering it requires knowing the map, which is a scalar is a record of where its fluid was’s statement about advection with the sound speed replacing the flow speed.

The other place a gas carries a record of where it came from is the contact surface, two essays back and the same solver.

Why the same mapping is easier here than in a flow #

There is an important contrast with the advective memories in this collection and it explains why acoustics is a solved subject and Lagrangian reconstruction is not.

Sound travels in straight lines at a known speed, in a uniform medium. So the map from emission to arrival is explicit, invertible where it is monotone, and computable without solving anything. A parcel of fluid travels along a trajectory that has to be integrated, and neighbouring trajectories separate.

The consequence is that acoustic inverse problems are well conditioned where advective ones are not: reversible, and unusable measures a gain of 486 on a small perturbation to a flow’s state, and the equivalent gain in an acoustic reconstruction is of order one, because a small error in the geometry produces a small error in the delay.

Straight rays are what make a memory usable, and they are also what an atmosphere with a sound-speed gradient takes away — which is why long-range acoustic propagation is a ray-tracing problem and is correspondingly harder.

Reading a boom signature as a history #

The fold has a practical reading which is worth spelling out, because it turns a nuisance into a measurement.

Because two emission times arrive together, and because the arrival time is a known function of the emission time, a measured pressure signature at the ground can be mapped back onto the aircraft. Each moment of the signature corresponds to a definite station along the body, and the shape of the signature therefore records the aircraft’s cross-sectional area distribution.

That is the basis of every low-boom design programme: the ground signature is designed first and the aircraft’s area distribution is derived from it, which is an inverse problem that is well posed precisely because the mapping is explicit.

What it cannot do is recover anything the propagation has removed. The signature at the ground has been through kilometres of atmosphere, and the far-field asymptotics have collapsed the detail into an N-wave — which is the signature that forgets the shape, and is why low-boom shaping has to preserve the features that survive rather than the ones the designer would prefer.

What the picture cannot show #

The mapping is drawn as a curve of arrival against emission, which is the object the essay is about, and it carries no information about the amplitude — which is what a listener actually hears. The amplitude also depends on the Doppler factor, to a power that depends on the source type, so the same curve governs two effects and only one of them is plotted.

Nothing here draws a wave. The retarded-time relation is a statement about arrivals, and the field it describes — spherical wavefronts from a moving centre, bunching ahead and stretching behind — is a picture this collection draws elsewhere.

Where the delay is the whole model #

Three problems in which nothing else is needed, which is a good measure of how much the retarded time does on its own.

A propeller’s tonal noise. Each blade passes each observer angle at a different time, and the tone at the observer is the superposition of the blades’ emissions at their retarded times. Getting the timing right and the source strength roughly right gives a good prediction; getting the source exactly right and the timing wrong gives none.

A rotor’s blade-vortex noise. The loading pulse of a blade that flies through what it shed radiates, and what makes it audible is its rate of change combined with the Doppler factor of a blade tip at high helical Mach number. The pulse and the mapping together are the whole of it.

And a jet’s noise at angle. A jet’s sources convect downstream at a fraction of the jet velocity, so they are moving sources, and the convective amplification their Doppler factor produces is why a jet is loudest at about thirty degrees to its axis rather than sideways.

In all three the source model is crude and the answer is useful, which is the sign that the timing is doing the work.

Why this is not the same as an echo #

The delay computed here is often confused with a reflection, and separating them is worth a paragraph because the two behave in opposite ways.

An echo is a second arrival. The direct sound comes, then a reflected copy comes later, and the ear hears both. The delay is a property of the room and it adds an arrival without changing the first one.

A retarded time is not an arrival at all. There is one sound, and the delay is between the source’s own clock and the listener’s. Nothing arrives twice, nothing is added, and the only observable is that the source’s position, speed and strength are read off at a moment that has already passed.

The tell is what happens when the source accelerates. An echo’s delay does not change; a retarded time’s does, because the source has moved between emission and arrival, and it is exactly that change that produces the Doppler factor.

Who found it, and when #

The retarded potential is Liénard’s and Wiechert’s, from the 1890s, in electromagnetism — the acoustic version is the same mathematics with a different wave speed. Lighthill’s acoustic analogy is from 1952 and Ffowcs Williams and Hawkings’ extension to moving surfaces from 1969, and it is the latter that made the retarded time an everyday computational problem rather than a textbook curiosity.

The fold above Mach one was understood as soon as supersonic flight was, and its practical form — that the boom is a superposition rather than a shifted signal — is what makes boom prediction a propagation problem rather than a source problem.

Limits recorded rather than smoothed over #

A point source in a uniform medium at rest. No wind, no temperature gradient, no ground. Each of those bends rays and turns the explicit mapping into a ray-tracing problem.

Straight-line motion at constant speed. A manoeuvring source has a mapping that folds in more complicated ways, and a source accelerating through Mach one has a focus rather than a cone — which is the origin of the superboom.

Nothing here is a flow. The source is prescribed and there is no aerodynamics in the essay at all. What produces the sound — the quadrupoles of the sound is what does not cancel — is a separate calculation and is where the difficulty of aeroacoustics actually is.

No source model. Everything here is about when sound arrives and nothing about how much. The amplitude requires a source strength and a source type, and the Doppler factor’s exponent depends on which.

The source is a point, and the observer is fixed. Both are conveniences. A moving observer adds a second Doppler factor, and an extended source has a different retarded time for each of its parts — which is what makes an aircraft’s signature a coalescence rather than a scaled copy.

And the fold is drawn for a point. A real supersonic body is an extended source, so each part of it folds at a different time, and the resulting signature is the coalescence of all of them — which is the signature that forgets the shape, and is a propagation calculation rather than a geometric one.