Wave propagation in a periodic medium with application to the ocean

Abstract

The problem of linear wave motion in a medium with sinusoidal spatial variations is considered. For properties with small enough variation (measured by the parameter q), a Mathieu equation is often obtained; a full dispersion relation is constructed, which includes resonant (linear) interaction between two waves aided by the medium. Gaps (the classic ‘stopping bands’ of Brillouin) appear in the dispersion curve, implying that no solutions, everywhere finite in an unbounded medium, exist within certain bands of frequency. Various combinations of normal modes exhibit temporal and spatial modulations; the spatial modulations represent trapping of wave energy in a band lying along contours of the medium. This feature was anticipated by O. Phillips with a qualitative argument. The simple analysis also encompasses the approaches to exact resonance. If the wave numbers are detuned by a fraction q, for example, the energy transferred between waves is reduced by 9/10. Solutions in the ‘forbidden’ band of frequencies, appropriate to a bounded medium, are given. The wave field is then confined to the neighborhood of a fixed or moving wall. Three examples relevant to the ocean are considered: internal gravity waves in a sinusoidal shear flow (for large Richardson number), long gravity waves over a sinusoidal bottom (for small fractional depth changes, δ) and Rossby waves over a sinusoidal bottom (for small δ/λ, where λ is the frequency in cycles per day, itself small).