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The spectral decomposition of the elastic wave operator in a layeredisotropic half-space is derived by means of standard functional analyticmethods. Particular attention is paid to the coupled $P$-$SV$ waves. Theproblem is formulated directly in terms of displacements which leads to a $2\times 2$ Sturm-Liouville system. The resolvent kernel (Green function) isexpressed in terms of simple plane-wave solutions. Application of Stone'sformula leads naturally to eigenfunction expansions in terms of generalizedeigenvectors with oscillatory behavior at infinity. The generalized normal modeexpansion is employed to define a diffuse field as a white noise process inmodal space. By means of a Wigner transform, we calculate vertical tohorizontal kinetic energy ratios in layered media, as a function of depth andfrequency. Several illustrative examples are considered including energy ratiosnear a free surface, in the presence of a soft layer. Numerical comparisonsbetween the generalized eigenfunction summation and a classical locked-modeapproximation demonstrate the validity of the approach. The impact of the localvelocity structure on the energy partitioning of a diffuse field isillustrated.

Normal modes of layered elastic media and application to diffuse fields