Lamb Waves - The Zero-order Modes

The Zero-order Modes

The symmetrical and antisymmetric zero-order modes deserve special attention. These modes have "nascent frequencies" of zero. Thus they are the only modes that exist over the entire frequency spectrum from zero to indefinitely high frequencies. In the low frequency range (i.e. when the wavelength is greater than the plate thickness) these modes are often called the “extensional mode” and the “flexural mode" respectively, terms that describe the nature of the motion and the elastic stiffnesses that govern the velocities of propagation. The elliptical particle motion is mainly in the plane of the plate for the symmetrical, extensional mode and perpendicular to the plane of the plate for the antisymmetric, flexural mode. These characteristics change at higher frequencies.

These two modes are the most important because (a) they exist at all frequencies and (b) in most practical situations they carry more energy than the higher-order modes.

The zero-order symmetrical mode (designated s₀) travels at the "plate velocity" in the low-frequency regime where it is properly called the "extensional mode". In this regime the plate stretches in the direction of propagation and contracts correspondingly in the thickness direction. As the frequency increases and the wavelength becomes comparable with the plate thickness, curving of the plate starts to have a significant influence on its effective stiffness. The phase velocity drops smoothly while the group velocity drops somewhat precipitously towards a minimum. At higher frequencies yet, both the phase velocity and the group velocity converge towards the Rayleigh wave velocity - the phase velocity from above, and the group velocity from below.

In the low-frequency limit for the extensional mode, the z- and x-components of the surface displacement are in quadrature and the ratio of their amplitudes is given by:

where is Poisson's ratio.

The zero-order antisymmetric mode (designated a₀) is highly dispersive in the low frequency regime where it is properly called the "flexural mode". For very low frequencies (very thin plates) the phase and group velocities are both proportional to the square root of the frequency; the group velocity is twice the phase velocity. This simple relationship is a consequence of the stiffness/thickness relationship for thin plates in bending. At higher frequencies where the wavelength is no longer much greater than the plate thickness, these relationships break down. The phase velocity rises less and less quickly and converges towards the Rayleigh wave velocity in the high frequency limit. The group velocity passes through a maximum, a little faster than the shear wave velocity, when the wavelength is approximately equal to the plate thickness. It then converges, from above, to the Rayleigh wave velocity in the high frequency limit.

In experiments that allow both extensional and flexural modes to be excited and detected, the extensional mode often appears as a higher-velocity, lower-amplitude precursor to the flexural mode. The flexural mode is the more easily excited of the two, and often carries most of the energy.