Normal modes of layered elastic media and application to diffuse fields

The spectral decomposition of the elastic wave operator in a layered isotropic half-space is derived by means of standard functional analytic methods. Particular attention is paid to the coupled $P$-$SV$ waves. The problem is formulated directly in terms of displacements which leads to a $2 \times 2$ Sturm-Liouville system. The resolvent kernel (Green function) is expressed in terms of simple plane-wave solutions. Application of Stone's formula leads naturally to eigenfunction expansions in terms of generalized eigenvectors with oscillatory behavior at infinity. The generalized normal mode expansion is employed to define a diffuse field as a white noise process in modal space. By means of a Wigner transform, we calculate vertical to horizontal kinetic energy ratios in layered media, as a function of depth and frequency. Several illustrative examples are considered including energy ratios near a free surface, in the presence of a soft layer. Numerical comparisons between the generalized eigenfunction summation and a classical locked-mode approximation demonstrate the validity of the approach. The impact of the local velocity structure on the energy partitioning of a diffuse field is illustrated.