Lamb Waves - The Higher-order Modes

The Higher-order Modes

As the frequency is raised, the higher-order wave modes make their appearance in addition to the zero-order modes. Each higher-order mode is “born” at a resonant frequency of the plate, and exists only above that frequency. For example, in a ¾ inch (19mm) thick steel plate at a frequency of 200 kHz, the first four Lamb wave modes are present and at 300 kHz, the first six. The first few higher-order modes can be distinctly observed under favorable experimental conditions. Under less than favorable conditions they overlap and can not be distinguished.

The higher-order Lamb modes are characterized by nodal planes within the plate, parallel to the plate surfaces. Each of these modes exists only above a certain frequency which can be called its "nascent frequency". There is no upper frequency limit for any of the modes. The nascent frequencies can be pictured as the resonant frequencies for longitudinal or shear waves propagating perpendicular to the plane of the plate, i.e.

where n is any positive integer. Here c can be either the longitudinal wave velocity or the shear wave velocity, and for each resulting set of resonances the corresponding Lamb wave modes are alternately symmetrical and antisymmetric. The interplay of these two sets results in a pattern of nascent frequencies that at first glance seems irregular. For example, in a 3/4 inch (19mm) thick steel plate having longitudinal and shear velocities of 5890 m/s and 3260 m/s respectively, the nascent frequencies of the antisymmetric modes a₁, a₂ and a₃ are 86 kHz, 257 kHz and 310 kHz respectively, while the nascent frequencies of the symmetric modes s₁, s₂ and s₃ are 155 kHz, 172 kHz and 343 kHz respectively.

At its nascent frequency, each of these modes has an infinite phase velocity and a group velocity of zero. In the high frequency limit, the phase and group velocities of all these modes converge to the shear wave velocity. Because of these convergences, the Rayleigh and shear velocities (which are very close to one another) are of major importance in thick plates. Simply stated in terms of the material of greatest engineering significance, most of the high-frequency wave energy that propagates long distances in steel plates is traveling at 3000–3300 m/s.

Particle motion in the Lamb wave modes is in general elliptical, having components both perpendicular to and parallel to the plane of the plate. These components are in quadrature, i.e. they have a 90° phase difference. The relative magnitude of the components is a function of frequency. For certain frequencies-thickness products, the amplitude of one component passes through zero so that the motion is entirely perpendicular or parallel to the plane of the plate. For particles on the plate surface, these conditions occur when the Lamb wave phase velocity is √2c_t or c_l, respectively. These directionality considerations are important when considering the radiation of acoustic energy from plates into adjacent fluids.

The particle motion is also entirely perpendicular or entirely parallel to the plane of the plate, at a mode's nascent frequency. Close to the nascent frequencies of modes corresponding to longitudinal-wave resonances of the plate, their particle motion will be almost entirely perpendicular to the plane of the plate; and near the shear-wave resonances, parallel.

J. and H. Krautkrämer have pointed out that Lamb waves can be conceived as a system of longitudinal and shear waves propagating at suitable angles across and along the plate. These waves reflect and mode-convert and combine to produce a sustained, coherent wave pattern. For this coherent wave pattern to be formed, the plate thickness has to be just right relative to the angles of propagation and wavelengths of the underlying longitudinal and shear waves; this requirement leads to the velocity dispersion relationships.