The plate that carries two kinds of sound

The wave a surface is enough to hold found that a solid needs only one free face to trap a wave. The Rayleigh wave clings to the surface of a half-space, dies away within a wavelength of it, travels slower than any wave in the bulk and carries no dispersion at all, because a half-space has no length in it for a wavelength to be compared with. Its speed in aluminium is 2.92 km/s at every frequency.

Give the solid a second face, parallel to the first, and the length appears. A plate of thickness $d$ is the oldest and commonest waveguide in engineering — every aircraft skin, ship’s hull, pipe wall, tank, rail web and car door is one — and what it guides is not a Rayleigh wave on each side but a family of waves that belong to the whole thickness at once. Horace Lamb worked them out in 1917. The answer is more interesting than either a pipe’s or a surface’s, because in a plate the two kinds of bulk sound, compression and shear, cannot travel independently: every reflection at a free face turns some of each into the other.

The figures below compute the whole family for an aluminium plate, with a compression speed $c_L$ of 6.32 km/s and a shear speed $c_T$ of 3.13 km/s, from Lamb’s equations solved numerically, and then follow the two waves that exist at every frequency — one that stretches the plate and one that bends it — through what each does with a pulse and what each can tell about the plate it is in.

Two bulk waves that must travel as one #

Inside the metal there are only two waves. A compression wave squeezes the material along its direction of travel; a shear wave slides it sideways. In a block they travel independently at their own speeds. At a free face they cannot, because the face must carry no force — neither a push across it nor a drag along it — and one wave alone cannot make both tractions vanish. A compression wave striking a free face obliquely reflects as a compression wave and a shear wave, and so does a shear wave. This mode conversion is the whole difference between elastic waves and the sound in a gas, which has only one kind and reflects from a free surface as itself.

In a plate the reflected waves strike the other face and convert again, and a guided wave is a pattern of compression and shear that reproduces itself after a round trip. Write each as travelling along the plate as $e^{i(kx - \omega t)}$ with a standing profile across the thickness, impose zero traction on both faces, and the four conditions have a non-trivial solution only when a determinant vanishes. By symmetry the plate’s motions split into two classes that never mix: symmetric ones, in which the two faces bulge outwards and pull inwards together, mirror images across the mid-plane; and antisymmetric ones, in which both faces move the same way and the plate bends. Each class has its own equation. With $h$ the half-thickness and

$$p^2 = \frac{\omega^2}{c_L^2} - k^2, \qquad q^2 = \frac{\omega^2}{c_T^2} - k^2,$$

the symmetric condition is

$$\frac{\tan qh}{\tan ph} = -\frac{4k^2pq}{(q^2-k^2)^2},$$

and the antisymmetric one is its reciprocal. These are the Rayleigh–Lamb equations. They contain no adjustable constant, only the two bulk speeds and the product of frequency and thickness. A plate twice as thick at half the frequency is the same problem, which is why every chart of them is drawn against $fd$ in megahertz-millimetres.

The chart has the shape every waveguide’s has, and one feature none of them has. The higher modes, from 1.56 MHz·mm upwards, behave exactly like the modes of the pipe that will not carry a low note: each enters at a cut-off frequency, where the thickness holds a whole number of half-wavelengths of a bulk wave bouncing straight across, arriving with infinite phase velocity and slowing as the frequency rises. Below its cut-off it does not travel. The first antisymmetric cut-off, at $c_T/2 = 1.56$ MHz·mm, is a shear wave resonating across the thickness; the next symmetric ones sit near $c_T$ and $c_L/2$, 3.13 and 3.16.

What a pipe does not have is a pair of modes with no cut-off at all. At the lowest frequencies a plate always carries exactly two waves, the fundamental symmetric wave $S_0$ and the fundamental antisymmetric wave $A_0$, and they are as different from each other as two waves in one solid can be. One stays near 5.4 km/s; the other falls towards zero.

The stretching wave and its own speed #

At low frequency, $S_0$ is a wave of stretching. The plate is pulled and pushed along its length uniformly through its thickness, like a rod carrying a compression, and its speed is

$$c_{\text{plate}} = 2c_T\sqrt{1 - \frac{c_T^2}{c_L^2}} = 5.44 \text{ km/s}$$

for aluminium. That number is neither bulk speed. It is slower than compression in a block, 6.32, because the plate is free to thin as it is stretched: Poisson’s ratio lets the faces move in and out, and a material allowed to contract sideways is softer along its length than one held in by the surrounding metal. It is faster than a stretching wave in a thin rod, $\sqrt{E/\rho} = 5.1$ km/s, because a plate can thin in one sideways direction but not the other — it is held by its own width. Three different speeds for the “same” compression wave in the same alloy, depending on how much the material around it constrains it, is the first lesson of the chart, and the one Nothing is allowed to be rigid points towards: a sound speed is the square root of a stiffness, and stiffness depends on what the material is not allowed to do.

The $S_0$ curve is almost flat up to about 1 MHz·mm. Above that it falls steeply, through a region near 2 to 2.5 MHz·mm where its displacement changes character, and ends hugging the Rayleigh line. At high frequency the wavelength is short against the thickness, the two faces no longer feel each other, and each carries its own Rayleigh wave; $S_0$ and $A_0$ become the sum and the difference of the same two surface waves and their speeds converge on 2.92 km/s.

The bending wave that slows as it lengthens #

$A_0$ is the wave anyone who has struck a sheet of metal has heard. At low frequency it is a flexure: the whole plate bends, one face in tension and the other in compression, and the restoring force is the plate’s bending stiffness rather than the stiffness of the metal in either direction. The physics is the same as a vibrating beam’s, and thin-plate theory gives its speed in one line. A plate’s bending stiffness per unit width is $D = Ed^3/12(1-\nu^2)$, its mass per unit area is $\rho d$, and a bending wave of wavenumber $k$ oscillates at $\omega = k^2\sqrt{D/\rho d}$, so that

$$c = \frac{\omega}{k} = \sqrt{\frac{\omega\, d\, c_{\text{plate}}}{\sqrt{12}}}.$$

The speed grows as the square root of the frequency. Halve the pitch and the wave goes $1/\sqrt{2}$ as fast; take the frequency to zero and the speed goes to zero with it. A long, slow bend of a thin sheet is a very slow wave, because a long bend has very little curvature, and curvature is all that bending stiffness resists.

On logarithmic axes the thin-plate result is a straight line of slope one-half, and the exact $A_0$ lies on it to within a few per cent up to 0.1 MHz·mm: 0.313 km/s at 0.01, 0.958 at 0.1. Beyond that the two part company. Thin-plate theory assumes that cross-sections stay plane and perpendicular to the bent mid-plane, and that the plate’s own rotation and shear cost nothing. Both assumptions fail once the bending wavelength is comparable with the thickness. A wave 3 mm long in a 1 mm sheet is not bending the sheet so much as shearing it, and shear has a speed limit the bending formula knows nothing of. The formula predicts a flexural wave faster than shear, faster than compression and eventually faster than anything; the true $A_0$ bends over and settles at 2.92 km/s. Timoshenko’s correction for shear and rotary inertia captures most of the turn; Lamb’s equations capture all of it, because they never assumed the plate was thin.

The two levels of approximation are worth keeping apart. The first, thin-plate theory, is exact in a limit and wrong outside it in a way it does not announce: nothing in the formula warns that it has left its range. The second, the Rayleigh–Lamb equations, is exact within linear elasticity at every frequency and contains the first as a limit. Most of the practical use of plate waves happens in exactly the region where the first stops being right, between 0.1 and 2 MHz·mm.

Energy that runs at another speed #

Phase velocity is the speed of crests. A pulse — a burst of a few cycles, which is what anyone measuring anything actually sends — travels at the group velocity, $c_g = d\omega/dk$, which differs from the phase velocity whenever the wave is dispersive. The packet that moves at another speed made that argument in general; the plate makes it vivid, because its two fundamental waves sit on opposite sides of it.

For a bending wave at low frequency $\omega \propto k^2$, and differentiating gives $c_g = 2c$ exactly. The energy of a flexural pulse runs at twice the speed of its crests, so that watching one, the ripples appear at the back of the packet, run forwards through it, and vanish at the front. $A_0$ at 0.05 MHz·mm has crests moving at 0.69 km/s and energy moving at 1.33.

$S_0$ is the opposite case at low frequency. Its phase velocity is nearly flat, so its group velocity equals it and a pulse keeps its shape. That holds up to about 1 MHz·mm, and then $S_0$ does something no simple dispersion law produces: its group velocity falls from 5.4 to 1.79 km/s over about one MHz·mm and climbs back. The fall coincides with the steep part of the phase-velocity curve, where the wave’s motion is changing from a stretch distributed evenly through the thickness to a pair of surface waves; energy that spends that region reorganising its shape across the plate is slow to move along it. $A_0$’s group velocity does the reverse, rising past its phase speed to a broad maximum of about 3.15 km/s around 1 MHz·mm, then settling with its phase velocity on the Rayleigh speed.

The higher modes do stranger things still. The first symmetric mode above the cut-off, $S_1$, has a stretch near 2.9 MHz·mm in aluminium where its curve doubles back above the top of the chart: two branches meet at a point where the wave has a finite wavelength and a group velocity of exactly zero. Energy put into the plate at that frequency does not leave. It rings in place as a local resonance, and laser ultrasonics uses that ringing to measure the thickness of a small spot of a plate without touching it, because the frequency of the resonance is fixed by $d$ and the two bulk speeds.

One burst, two arrivals #

The practical meaning of the two group velocities appears when a burst is sent along a plate and received some distance away. The figure below synthesises it: a five-cycle tone burst at 200 kHz launched into a 1 mm aluminium sheet, its spectrum split into some hundred frequencies, each sent along the plate with the wavenumber the computed dispersion curves give it, and the whole reassembled 300 mm away. Nothing is assumed about the shape of the arrival; it is what the curves imply.

$S_0$ arrives first, at 300 mm ÷ 5.4 km/s ≈ 55 μs, as a compact copy of what was sent, because at 0.2 MHz·mm its dispersion is negligible across the burst’s bandwidth. $A_0$ arrives later and longer. Its centre comes in near 130 μs, at its group velocity of about 2.3 km/s, and the packet has been stretched into a chirp: the higher frequencies in the burst bend faster and lead, the lower ones lag, and five cycles have become a dozen of varying spacing. Another 300 mm and $S_0$ would look the same while $A_0$ was twice as smeared, because dispersion spreads a pulse in proportion to the distance it has run — the argument of the packet that will not keep its shape, here with a dispersion law that is computed rather than assumed.

Which wave a transducer launches is a matter of how it pushes. A probe pressing on one face, or a wedge that strikes the plate at an angle matched to one wave’s phase velocity, favours one mode over another; a pair of transducers on opposite faces driven together launches the symmetric family and driven opposite the antisymmetric. A hammer tap launches mostly $A_0$ — the same flexural wave that carries a ball’s energy away in the bounce an elastic plate cannot give back — which is why the sound of a struck sheet sweeps downward in pitch: the bending wave delivers its high frequencies first, the way a lightning stroke’s radio pulse arrives as the whistle that arrives sorted. Every arrival on such a trace is identifiable by its timing because the curves above say exactly when each mode should arrive at each frequency.

What the plate’s thickness does to each wave #

The two fundamental waves respond to the plate’s thickness in completely different ways, and that difference is what makes plate waves useful for inspection rather than merely interesting. Because the equations depend only on $fd$, changing the thickness at a fixed frequency is the same as moving along the frequency axis, and where a curve is flat nothing happens.

At 100 kHz $S_0$ travels at 5.44 km/s through 1 mm of aluminium and at 5.42 through 5 mm — a change of less than half a per cent. $A_0$ travels at 0.96 km/s through 1 mm, 1.31 through 2 mm and 1.89 through 5 mm: its speed goes roughly as the square root of the thickness, as the bending formula says it must. Corrosion that thins a 2 mm skin to 1.6 mm slows $A_0$ by 9 per cent and leaves $S_0$ unchanged. The bending wave is a thickness gauge spread along a line, and the stretching wave is a reference that ignores thickness and responds instead to cracks and cuts that interrupt the stretch.

This is the working principle of guided-wave inspection. Conventional ultrasonic testing sends a pulse straight through a wall and times its echo from the far side, at megahertz frequencies chosen for the reason the depth an ultrasound image pays for its sharpness sets out, which measures the thickness under the probe to a fraction of a millimetre and nowhere else; inspecting a pipeline that way means scanning every square centimetre of it, with the insulation stripped off. A guided wave launched along the pipe at low $fd$ travels tens of metres, and anywhere the wall is thinned, cracked or welded, part of it reflects. A single collar of transducers at one dig site screens a long length of buried pipe for wall loss, and only the places that echo need excavating. In aircraft the same idea runs on permanently bonded sensors that send $A_0$ and $S_0$ across a panel and compare each new trace with the one recorded when the panel was sound.

The price is the dispersion itself. The pulse figure showed $A_0$ smearing over 300 mm of travel; over 30 m it becomes a long chirp in which echoes overlap, and the higher modes, once excited, add arrivals of their own. Guided-wave systems therefore work at low $fd$ where only the fundamentals exist, choose frequencies where the chosen mode’s curve is flat, and use the computed dispersion to compress the received chirps back into pulses before reading them — a step that is only possible because the curves are known exactly from the two bulk speeds.

A plate in a fluid, and a plate on a frame #

The free plate is the cleanest case and the one the figures compute, but almost no real plate is free. Immerse it in water and every mode that moves the faces normally — $A_0$ above all — pushes on the water and radiates sound into it wherever its phase velocity exceeds water’s 1.48 km/s, leaking energy and becoming a leaky wave that dies away along the plate. Below that speed $A_0$ cannot radiate, because no wave in the water can keep pace with its crests, and it travels on with water loading it: the fluid’s inertia slows it. This is the same condition that decides whether a channel with no walls holds light, applied to a plate in a liquid: a guided wave leaks into its surroundings wherever its phase velocity is faster than the surroundings can carry.

The same arithmetic sets what a struck panel sounds like in air. $A_0$’s speed rises with frequency, and at the coincidence frequency it equals the speed of sound in air, 343 m/s. Below it a bending plate barely radiates; above it every bending wave launches sound efficiently at an angle. For a 1 mm aluminium sheet the coincidence frequency is near 12 kHz; for a 10 mm one near 1.2 kHz — which is why thick panels let speech through more easily than their mass alone predicts, and why sound insulation is not simply a matter of weight.

A plate fixed at its edges carries the same waves, reflected back and forth between the edges into standing patterns, and the drum that has no harmonics is the membrane version of that. A plate’s resonances are spaced even more irregularly than a drum’s, because the bending wave’s frequency goes as the square of its wavenumber rather than in proportion to it, and the result is the clang of a struck sheet rather than a note.

What linear elasticity leaves out #

Every number above comes from a plate that is uniform, isotropic, perfectly elastic and free on both faces, with displacements small enough for the equations to be linear. Aluminium alloy is close to all of those. Other plates are not. A fibre composite, the material of most modern aircraft skins, is strongly anisotropic, so its plate waves travel at different speeds in different directions along the panel, and their energy need not travel in the direction of their crests; the dispersion curves must be computed for each direction from the composite’s stiffness matrix. Damping is small in metal and large in polymers and composites, where the bending wave loses a few per cent of its amplitude per wavelength and the plate’s pulse figure would show a weaker, earlier-cut-off $A_0$.

Real inspection also meets features that are not a plate at all — stiffeners, rivets, curvature, coatings and welds — each of which reflects, scatters and converts one mode into others. The dispersion curves are the alphabet of the subject, not its prose. Reading a real trace from a real structure means knowing which arrivals the curves predict and accounting for the rest as scattering, and a good part of the engineering is choosing a frequency where the alphabet is short.

Still open: reading a plate from its own noise #

The figures computed waves sent on purpose. A plate in service is never silent: engines, flow, traffic and wind keep its plate waves excited all the time, at random. The same trick that recovers the Earth’s structure from its background seismic hum works here. Cross-correlating the noise recorded at two points on a panel rebuilds, over enough time, the wave that would have travelled from one point to the other — with its dispersion intact — so that a structure could in principle be inspected using only listening sensors and no transmitter. How long the averaging must run, how uneven the noise field may be before the recovered wave is biased, and whether a change in the plate can be told from a change in the noise that excites it, are being worked out on real aircraft structures and pipelines.

A cleaner question is the higher modes. Most inspection avoids them because they complicate the trace, but each carries its own sensitivity: some concentrate their energy near the faces, others in the middle of the thickness, and a crack that has not yet reached the surface is invisible to one family and conspicuous to the other. Exciting single higher modes cleanly, and reading the plate’s condition from several of them at once, is a measurement problem the dispersion curves set out completely and nobody has yet made routine.

The habit worth carrying away is to ask what the boundaries force a wave to become. A plate is two free faces, and between them compression and shear cannot travel separately: every guided wave is the two bulk waves locked together, and the only number that decides which waves exist and how fast they go is frequency times thickness. At low frequency that leaves a stretching wave at the plate velocity, 5.44 km/s in aluminium, which ignores the thickness, and a bending wave whose speed grows as the square root of both frequency and thickness — 0.96 km/s at 100 kHz in a millimetre of metal. One is the reference and the other is the gauge, and between them a single burst sent along a metre of metal reads the wall it travelled through.