Geometrical orbits in the power spectra of waves

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Abstract
We investigate the relationship between the power spectrum of a wave field and that of a spatially uncorrelated source exciting it. Temporal correlations between values of the field at different positions are also examined. It is found that interference effects can significantly alter the structure of the power spectrum, leading to oscillations in it, even when the power spectrum of the source is a smooth function of frequency. We derive a semiclassical approximation in which these oscillations are related to orbits of the geometrical limit of the wave system. We also derive a trace formula that approximates a spatial average of the wave power spectrum as a sum over periodic orbits. These calculations explain the structure of a measured power spectrum of the fluctuating height of a fluid surface, generated by the circular hydraulic jump, which provided the motivation for the study.