Once a plate becomes thin compared with the wavelength, bulk longitudinal and shear waves stop existing independently. They reflect and mode-convert at both surfaces until only combinations that satisfy the traction-free boundary conditions on both faces survive. Those combinations are Lamb waves — guided plate modes, neither compression nor shear, which is why this calculator has no single wave-mode velocity behind it.

The modes split by symmetry about the mid-plane. The symmetric family stretches the plate along its length; the antisymmetric family bends it. The two fundamental modes, S0 and A0, exist at every frequency. At low frequency-thickness product S0 tends to the plate velocity, 2c_T√(1 − (c_T/c_L)²), about 5415 m/s in steel, and behaves almost non-dispersively; A0 tends to zero as a flexural wave and is strongly dispersive, its velocity rising steeply with fd. Both converge on the Rayleigh velocity, near 2998 m/s in steel, once the plate is many wavelengths thick.

Dispersion is the practical problem. Because velocity depends on the product of frequency and thickness, a broadband pulse spreads as it travels — the wavepacket travels at the group velocity while the phase fronts move at the phase velocity, and where the two differ sharply the pulse smears out and both amplitude and timing become unreliable. Inspections are therefore set up in the flattest region of the curve available, and a change of plate thickness within a scan is a change of operating point.

Excitation follows Snell’s law: a wedge generates the mode whose phase velocity satisfies sin θ = c_wedge / c_phase, which is why a mode with a phase velocity below the 2730 m/s wedge velocity cannot be launched by a wedge at all and needs a comb transducer, an EMAT or a magnetostrictive collar instead. Above fd ≈ 1.62 MHz·mm in steel the A1 mode appears and above 2.95 MHz·mm S1 appears, so a real signal at high fd contains more than the two modes tabulated here.

Worked example

Lamb mode	s0
Frequency	2 MHz
Plate thickness	1 mm
Wedge velocity	2730 m/s
Propagation distance	500 mm
Frequency-thickness product	2 MHz·mm
Phase velocity	4992.3 m/s
Group velocity	3647.3 m/s
Wavelength	2.496 mm
Wedge angle to excite this mode	33.15 deg
Arrival time over the propagation distance	137.09 µs
A1 cut-off	1.62 MHz·mm
S1 cut-off	2.95 MHz·mm

2 MHz on 1 mm steel plate is fd = 2.00 MHz·mm. Solving the symmetric Rayleigh–Lamb equation there for cL 5900 and cT 3240 gives a phase velocity of 4992.3 m/s, well down from the 5415 m/s low-frequency plate velocity, and a group velocity of 3647.3 m/s — the mode is already strongly dispersive. The wavelength is 4992.3/2000 = 2.496 mm, a 2730 m/s wedge must be cut to asin(2730/4992.3) = 33.15°, and an echo from 500 mm away arrives 500/3.6473 = 137.09 µs after the pulse. A1 has already cut on at 1.62 MHz·mm, so a third mode is present.

Use at your own risk — verify before you act

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