  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.
