Lamb waves

Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plate normal (the direction perpendicular to the plate). In 1917, the English mathematician Horace Lamb published his classic analysis and description of acoustic waves of this type. Their properties turned out to be quite complex. An infinite medium supports just two wave modes traveling at unique velocities; but plates support two infinite sets of Lamb wave modes, whose velocities depend on the relationship between wavelength and plate thickness.

Since the 1990s, the understanding and utilization of Lamb waves has advanced greatly, thanks to the rapid increase in the availability of computing power. Lamb's theoretical formulations have found substantial practical application, especially in the field of nondestructive testing.

The term Rayleigh–Lamb waves embraces the Rayleigh wave, a type of wave that propagates along a single surface. Both Rayleigh and Lamb waves are constrained by the elastic properties of the surface(s) that guide them.

In general, elastic waves in solid materials are guided by the boundaries of the media in which they propagate. An approach to guided wave propagation, widely used in physical acoustics, is to seek sinusoidal solutions to the wave equation for linear elastic waves subject to boundary conditions representing the structural geometry. This is a classic eigenvalue problem.

Waves in plates were among the first guided waves to be analyzed in this way. The analysis was developed and published in 1917 by Horace Lamb, a leader in the mathematical physics of his day.

Lamb's equations were derived by setting up formalism for a solid plate having infinite extent in the x and y directions, and thickness d in the z direction. Sinusoidal solutions to the wave equation were postulated, having x- and z-displacements of the form

This form represents sinusoidal waves propagating in the x direction with wavelength 2π/k and frequency ω/2π. Displacement is a function of x, z, t only; there is no displacement in the y direction and no variation of any physical quantities in the y direction.

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