Lamb Waves - Velocity Dispersion Inherent in The Characteristic Equations

Velocity Dispersion Inherent in The Characteristic Equations

Lamb waves exhibit velocity dispersion; that is, their velocity of propagation c depends on the frequency (or wavelength), as well as on the elastic constants and density of the material. This phenomenon is central to the study and understanding of wave behavior in plates. Physically, the key parameter is the ratio of plate thickness d to wavelength . This ratio determines the effective stiffness of the plate and hence the velocity of the wave. In technological applications, a more practical parameter readily derived from this is used, namely the product of thickness and frequency:

The relationship between velocity and frequency (or wavelength) is inherent in the characteristic equations. In the case of the plate, these equations are not simple and their solution requires numerical methods. This was an intractable problem until the advent of the digital computer forty years after Lamb's original work. The publication of computer-generated "dispersion curves" by Viktorov in the former Soviet Union, Firestone followed by Worlton in the United States, and eventually many others brought Lamb wave theory into the realm of practical applicability. Experimental waveforms observed in plates can be understood by interpretation with reference to the dispersion curves.

Dispersion curves - graphs that show relationships between wave velocity, wavelength and frequency in dispersive systems - can be presented in various forms. The form that gives the greatest insight into the underlying physics has (angular frequency) on the y-axis and k (wave number) on the x-axis. The form used by Viktorov, that brought Lamb waves into practical use, has wave velocity on the y-axis and, the thickness/wavelength ratio, on the x-axis. The most practical form of all, for which credit is due to J. and H. Krautkrämer as well as to Floyd Firestone (who, incidentally, coined the phrase "Lamb waves") has wave velocity on the y-axis and fd, the frequency-thickness product, on the x-axis.

Lamb's characteristic equations indicate the existence of two entire families of sinusoidal wave modes in infinite plates of width . This stands in contrast with the situation in unbounded media where there are just two wave modes, the longitudinal wave and the transverse or shear wave. As in Rayleigh waves which propagate along single free surfaces, the particle motion in Lamb waves is elliptical with its x and z components depending on the depth within the plate. In one family of modes, the motion is symmetrical about the midthickness plane. In the other family it is antisymmetric. The phenomenon of velocity dispersion leads to a rich variety of experimentally observable waveforms when acoustic waves propagate in plates. It is the group velocity c_g, not the above-mentioned phase velocity c or c_p, that determines the modulations seen in the observed waveform. The appearance of the waveforms depends critically on the frequency range selected for observation. The flexural and extensional modes are relatively easy to recognize and this has been advocated as a technique of nondestructive testing.