Anomalous Dispersion of the Lamb Mode

The mode of the Lamb spectrum of an isotropic plate exhibits negative group velocity in a narrow frequency domain. This anomalous behavior is explained analytically by examining the slope of each mode first in its initial state and then near its turning points. 1. Introduction The dispersion relation for symmetric Lamb modes propagating in an infinite isotropic plate of thickness is given by the well known Rayleigh-Lamb equation [1]: where In (1), and , respectively, denote the phase speeds of the transverse and longitudinal bulk waves in the material. Also, and , respectively, denote the frequency and the wave number of the mode. The phase velocity, , of a mode is given by If is plotted as a function of the frequency, the spectrum appears as in Figure 1, which depicts the spectrum for a steel plate with ？km/s and ？km/s. Figure 1: Symmetric Lamb modes on a steel plate ( ) showing phase velocity as a function of normalized frequency. The most striking feature in Figure 1 is the shape of the mode which has a turning point at , and the phase velocity becomes double valued for in . This phenomenon of negative group velocity is of great technical significance and has been observed in a large number of experiments [2–8]. The afore mentioned feature of mode was first noticed by Tolstoy and Usdin [9] in . In all isotropic materials with , only the mode has this “anomalous behavior” and other modes behave normally. We will call it the anomaly. An explanation of this peculiar shape of mode has posed a challenge since its discovery in . For the special case of a material with that is , each of the modes , , exhibits anomalous behavior [10]. Anomalous pairs of modes may also occur for certain special values of the Poisson ratio. We will call it the pair anomaly. Although (1) governs the behavior of all modes, anomalous or otherwise, no simple theory seems to exist which should provide a satisfactory explanation of why certain modes in the spectrum should possess a bulge while others proceed in a normal manner. However, certain physical explanations of this phenomenon exist. In , Whitaker and Haus [11] noted the fact that “propagation of waves with dispersion of this sort has been experimentally verified [2] but the reason for their appearance is not well understood.” They used the coupled mode theory to argue that, when the fundamental mode and the first harmonic mode are nearly degenerate at cutoff, a coupling effect can occur at the boundaries. Uberall et al. [12] hypothesize that “one observes a repulsion phenomenon between neighboring dispersion curves similar