Allocation and movement of roots of the Lamb wave dispersion equation in complex plane

Abstract

The classical Lamb problem is considered for steady-state harmonic waves in a free elastic layer under the applied load. An integral representation of the solution is reduced to an infinite series in terms of residues at the poles of Green's matrix. These poles coincide with the roots of the Lamb's dispersion equation and define eigenwaveforms that are normal modes of the layer. The pole arrangement on the continual dispersion curves in complex plane of wave number and their frequency dependency are studied. Two transformation mechanisms of the complex wave numbers into the real ones are observed. The first mechanism is regular. In this case, complex roots become real through the imaginary axis. Second one is irregular, when complex pole becomes real without passing imaginary axis and forms a backward wave. These Lamb waves have zero group velocity at the cutoff frequencies. Their appearances are accompanied by resonance phenomena.