Number and location of zero-group-velocity modes.

The frequency-wavenumber spectra of laminated media often exhibit anomalous modes with descending branches whose group velocity is negative, and these terminate at a minimum point at which the group velocity transitions from negative to positive. These minima are associated with resonant conditions in the medium, which have been clearly observed in experiments and in the nondestructive testing of laminated plates. Starting from first principles, this paper provides a theoretical analysis on the number and location of such zero-group-velocity (ZGV) modes for horizontally layered media bounded by any arbitrary combination of external boundaries. It is found that these ZGV points are few in number and show up mostly in low-order modes which are characterized by a negative second derivative at the cutoff frequencies, a condition that can readily be tested. It is also shown that the effective number of ZGVs is small even when the ratio of the dilatational and shear wave velocity is a rational number and there exist coincidences in cutoff frequencies, a condition that at first would suggest that the number of ZGVs is infinite. Finally, it is shown that the number of ZGVs decreases with the Poisson's ratio.