Time-harmonic and transient propagation and diffraction phenomena can be described alternatively by progressing and oscillatory waves that express the wave motion in terms of direct and multiple wavefronts or rays, and in terms of resonances or modes, respectively. Each description is convenient and physically appealing when it requires few contributing elements. It is inconvenient and physically less transparent when it requires many elements, and it would then be desirable to combine many inconvenient elements into fewer convenient ones. For a variety of propagation environments, including layered or other guiding regions, this can be done by expressing a group of rays collectively in terms of modes, or a group of modes collectively in terms of rays. When this ray-mode equivalence is invoked selectively, there emerges a hybrid representation that combines ray fields and modal fields in uniquely defined proportions. The theory is based on Poisson summation and on alternative treatments of wave spectra. It has been applied to electromagnetics, underwater acoustics, and SH elastic motion, and is now being extended to general propagation in elastic media.

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