Lamb wave A0 and S0 velocities in steel plate

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.

fd = frequency × plate thickness  (MHz·mm)
Rayleigh–Lamb, symmetric:      tan(qh)/tan(ph) = −4k²pq/(q²−k²)²
Rayleigh–Lamb, antisymmetric:  tan(qh)/tan(ph) = −(q²−k²)²/(4k²pq)
p² = ω²/c_L² − k²,  q² = ω²/c_T² − k²,  k = ω/c_phase,  h = d/2
group velocity c_g = dω/dk
wavelength λ = c_phase / f
wedge angle: sin θ = c_wedge / c_phase
higher mode cut-offs at fd = n·c_T/2 and n·c_L/2