A boom is aged in the thin air it starts in

The edge is a rumble, not a quieter bang priced how a sonic boom spreads across its carpet and ended on the half of the problem it could not price. The rays that reach the outer part of the carpet have come a long way — at Mach 1.8 from 15 km, the one that lands nine-tenths of the way to the edge has travelled 2.53 times as far as the one straight under the track. The signature that forgets the shape is the essay about what distance does to a boom: the pressure steepens into shocks, the signature becomes an N-wave, and the N-wave keeps lengthening and weakening as it travels. Put the two together and the edge should hear a much older boom.

It does not, and the reason is not in the rays. It is in the air they pass through. The distance that matters to a boom is not measured in kilometres but in how much the wave has distorted, and a pressure wave distorts thin air much faster than dense air. So a boom does most of its ageing high up, near the aeroplane, where every ray is still close to every other — and the long, low paths to the edge of the carpet are through air that barely ages it at all.

What ageing is, along a ray #

A point of a boom’s signature at an overpressure $p$ travels slightly faster than sound, because the air it is in has been compressed and warmed. Over a small distance $ds$ it gains, in arrival time, $\beta p\,ds/(\rho a^3)$ on the undisturbed wave, with $\beta = (\gamma+1)/2$. That is the whole mechanism of every compression becoming a shock: the high-pressure parts of the signature catch up with the low-pressure parts ahead of them.

The overpressure itself changes along the way, because the ray tube carrying it widens and the air changes. A weak wave keeps $p^2A/(\rho a)$ fixed, where $A$ is the tube’s cross-section, so $p = B(s)\,f$ with $B = \sqrt{\rho a/A}$ and $f$ the shape of the signature. Put the two together and every point of the signature has been advanced, by the time it reaches the ground, by $f$ times

$$\lambda = \int \frac{\beta\,B(s)}{\rho(s)\,a(s)^3}\,ds,$$

the age. It is exactly the number the earlier essay fed to the Lax formula to turn a near-field shape into an N-wave. In a uniform medium with conical spreading $B$ falls as $s^{-1/2}$ and the age grows as $\sqrt{s}$, which is where that essay’s square-root laws came from. In the real atmosphere nothing in the integrand is constant, and the ray tracer supplies all of it.

The tube’s area comes from the same tracer that found the carpet. For level flight, the tube between two neighbouring azimuths and two neighbouring moments of emission has, at any height, a cross-section normal to the ray equal to the flight speed times the lateral spacing of the two azimuths times the vertical component of the ray’s direction. In air of one temperature and density that is a cone, and the computed age then grows as the square root of the path to six parts in a million at every azimuth tried. That is the check on the geometry and on the integral together.

Where the age is gathered #

The integrand has the density in it twice, once in $B$ and once in the denominator, and the net effect is that the age gathered per kilometre goes as one over the square root of the density. Air at the tropopause is about a third as dense as air at the ground, and air at 15 km less than a fifth. So a kilometre of path near the aeroplane is worth more than twice a kilometre near the ground, before the tube’s spreading is counted — and the spreading counts the same way, since the tube is narrowest, and the amplitude largest, where the wave starts.

The result for the ray under the track is that 66 per cent of its age is gathered above the tropopause, in 26 per cent of its path. By 3 km above the ground it has 95 per cent. The edge ray is more extreme in path and less in age: 58 per cent of its age is gathered above the tropopause in the first 17 per cent of its path, and the lowest 3 km, where it spends 39 per cent of its path travelling almost horizontally, add 11 per cent.

That turns the picture of a boom’s journey upside down. The wave that reaches the ground has not been slowly aged over fifteen or thirty kilometres of air. It was aged mostly in its first few kilometres, in the thin air around the aeroplane, and then carried down through a dense atmosphere that does little more than spread it.

Two and a half times as far, half as old again #

Across the carpet the difference compounds. A quarter of the way out, 9 km from the track, the path is 1.18 times the under-track path and the age 1.08 times. Halfway, at 19 km, the path is 1.62 times and the age 1.24. Three-quarters of the way, at 28 km, the path is 2.17 times and the age 1.40. At the last ray computed, 34 km out, the path is 2.53 times and the age 1.47.

Even a uniform medium would have given the edge ray only $\sqrt{2.53} = 1.59$ times the age, because age grows as the square root of distance there. The real atmosphere gives less than that, because every extra kilometre of the edge ray’s path is added at the bottom, in the densest air. Path length is the wrong measure of how far a boom has travelled. The age is the right one, and across a whole carpet it varies by less than half.

What the age does to the edge’s boom #

An N-wave that has aged by $\lambda$ has an overpressure proportional to $B/\sqrt{\lambda}$ and a duration proportional to $\sqrt{\lambda}$, so both follow directly. At Mach 1.8, where spreading alone left the last ray with 0.68 of the under-track overpressure, the extra age takes it to 0.56. At Mach 1.3 the corresponding pair is 0.87 and 0.77, 21 km out; at Mach 2.5, 0.63 and 0.51, 38 km out. The boom falls, and it still does not halve anywhere on any of the three carpets. Against distance, the Mach 1.8 and Mach 2.5 curves nearly coincide once again: a faster aeroplane’s carpet is wider, but a place a given distance from the track hears much the same fraction of the boom from either.

The duration is the more surprising number. The edge’s N-wave is 21 per cent longer than the one under the track at Mach 1.8, and 24 per cent at Mach 2.5 — not the factor of 1.6 a uniform medium would give, and nothing like the factor of two and a half in the path. A boom’s duration is what sets its lowest frequencies, and if the edge’s rumble were a matter of the signature having been stretched over a long path, this is where it would show. It does not show. Whatever makes the edge sound different is mostly not the ageing, which leaves the other candidate the edge essay named, diffraction into the shadow beyond the last ray, as the stronger one.

What this does to a measurement #

The distribution of the age has a consequence for how booms are measured, which is worth drawing out before turning to design.

Flight tests put their microphones under the track, because that is where the boom is strongest and where a flight’s path can be relied on to deliver it. The boom that turns back before the ground showed that the carpet’s width is set by the temperature profile near the ground, which varies from day to day; the age says something different about the signature itself. Because two-thirds of the ageing is done in the stratosphere, the shape of the N-wave under the track depends mostly on air that is steady — the isothermal layer above the tropopause changes far less than the boundary layer does — and on the aeroplane. A measurement under the track is therefore a fairly clean measurement of the aeroplane’s signature, aged through a nearly standard stratosphere.

The same reasoning says why an accelerating aeroplane’s focus is harder to predict than its ordinary carpet. A focus is made where rays converge, and rays converge near the ground, where the air varies most; the ageing that shapes the signature arriving at a focus has been done high up, in a region the focusing calculation does not need to know much about, but the focusing itself happens in the region it needs to know everything about.

And the edge of the carpet is, by the same token, the worst place to infer an aeroplane’s signature from. Its rays have been aged by nearly the same stratosphere, but they have spent the rest of their journey in the part of the atmosphere that is least standard, and every effect this calculation leaves out — relaxation, turbulence, the ground, the shadow — acts on them more.

A boom shaped not to shock #

The practical question is the one quiet-supersonic design asks. An aeroplane can be shaped — along the lines the least-drag body begins — so that its near-field signature rises gradually rather than all at once. Ageing steepens that ramp, and the ramp breaks into a shock when the age reaches one over its steepest slope. For the inviscid Burgers equation that moment is exact: marched by the Lax formula, a ramped front is still a steep but continuous ramp at nine-tenths of its break age, and a jump of its full height — a shock — at 1.15 times it.

A design is judged under the track, so the natural description of it is a margin: the age at which its front would break, divided by the age it actually gathers under the track. A margin of 1.25 means the shaping would have survived a quarter more ageing than it received.

Under the track the shaped front arrives as a ramp, steepened to a fifth of its launch width but still a ramp. Out at 34 km, with 1.47 times the age, it arrives as a shock like any other boom. The question is where between the two it changes, and the age curve answers it directly: the shaping survives out to the distance at which the age ratio reaches the margin.

Survival is a distance, not a fraction #

The result has a simple shape that was not obvious in advance. How far out a shaped boom stays shaped depends on its margin and hardly at all on the Mach number. A 5 per cent margin keeps the shaping to about 7 km from the track, a 10 per cent margin to 10.5 km, a 25 per cent margin to 19 km, for all three flights to within a few hundred metres. What changes with the Mach number is only where the carpet ends: at Mach 1.3 a 25 per cent margin covers 85 per cent of the half-width, at Mach 1.8 half of it, and at Mach 2.5 43 per cent. A 50 per cent margin covers the whole of the slower two carpets, as far out as rays were computed.

That is the answer to the question the edge essay closed on, and it reads both ways. For a slow flight, a design with a modest margin is quiet across most of its carpet, because the carpet is narrow. For a fast one, the same design is quiet over a strip about 40 km wide in the middle of a carpet twice that — and the width that describes its quietness is the strip’s, not the carpet’s. And because the strip’s width is set by the atmosphere and the flight height rather than by the Mach number, it is a number a designer can know before choosing the speed.

The checks #

Three checks hold the argument up. In air of one temperature and density both the age and the amplitude follow the cone’s square-root laws to six parts in a million, which tests the tube geometry, the density, and the treatment of the aeroplane end of the integral, where the integrand is infinite and the first interval is integrated as an inverse square root rather than by trapezoid. A shaped front stays unbroken below its break age and shocks above it. And the edge ray’s age ratio is below the square root of its path ratio at every Mach number, which is the claim that the extra path is cheap, tested in the form it is made.

What the age does not include #

The rise time. The shocks here are discontinuities. A real boom’s shock has a thickness, set far more by the molecular relaxation of oxygen and nitrogen than by viscosity, and the rise time that results depends on the humidity along the path. That is a property of the lowest, wettest air, which is exactly where the edge ray spends its time — so the rise time may well vary across the carpet much more than the age does. This calculation cannot say.

Absorption of the whole signature. Relaxation also takes energy from the low frequencies over long paths, and the edge’s path is long in the air where that happens. It would lower the edge’s amplitude further; it is not here.

The shadow beyond the last ray. Every number here stops at the last ray computed, nine-tenths of the way out. Beyond it the field is diffracted, and the signature that arrives there is not an aged N-wave of any age.

Turbulence. The lowest kilometre or two of the atmosphere is turbulent, and turbulence scatters the steep parts of a signature — spiking some and rounding others — in a way measured booms show clearly. Again that is low air, and again it acts more on the edge rays than on the one under the track.

The atmosphere and the flight. A standard atmosphere with no wind, a flat ground, level flight at constant speed, and the speed of every signal the local sound speed. A real day changes the numbers; it should not change the distribution of the age between thin and dense air, which is set by the density ratio alone.

Who worked it out #

Wallace Hayes and colleagues set out the age variable for a stratified atmosphere in the 1960s and 1970s, in the form that NASA and industry sonic-boom propagation codes still use; the Blokhintzev invariant for a moving inhomogeneous medium dates from the 1940s, and Whitham’s F-function from 1952. Richard Seebass and Albert George showed around 1970 how a signature could be shaped to reach the ground without the full N-wave, which is the design idea every quiet-supersonic programme since has pursued. What is added here is the decomposition of the age by height, and the observation that a shaped boom’s quiet strip is a distance set by its margin rather than a share of its carpet.

Still open: the humid air at the bottom #

The age turned out to be decided at the top of the atmosphere. The rise time is very likely decided at the bottom, where the edge rays spend their time and where the humidity sets how fast oxygen and nitrogen relax behind a shock.

The next calculation carries an augmented Burgers equation — the same nonlinear term, plus thermoviscous diffusion and two relaxation processes with the humidity-dependent relaxation frequencies of a standard absorption model — along the same rays, from the ground-level age upwards, and asks for the rise time at every station. If the rise time grows across the carpet much faster than the age does, the edge’s rumble is a property of the wet air near the ground and not of the distance, and a quiet design’s strip would be wider on a humid day than a dry one. Whether it does is a matter of computing it.