A Note on Counting the Multiplicities of Elastic Surface Waves Using Weyl’s Law

Surface wave dispersion curves are widely used to constrain earth velocity structures and are important to compute theoretical synthetic seismograms with a mode-summation approach. While the computation of dispersion curves requires searching roots of nonlinear functions, some high-mode may be missed with improper choice of searching steps. The asymptotic distribution of eigenvalues of the elastic wave equation can be used as auxiliary information to design a sophisticated scheme to compute the surface wave dispersion curves. In this study, we show the Weyl’s law, combined with the Liouville transformation, can be exploited to derive asymptotic eigenvalue counting functions of elastic surface waves in a horizontally stratified or radially heterogeneous medium. We also show the derived formulation according to the Weyl’s law, in its simple case, agrees with previous studies. The derived asymptotic eigenvalue counting functions are validated by comparison with numerical results. This study demonstrates the Weyl’s law can be used to derive eigenvalue counting functions of surface waves in elastic media, and it is also possible to be applied to more complex media.