Changes in the form of short gravity waves on long waves and tidal currents

TLDR

Short gravity waves superposed on longer waves tend to become shorter and steeper at the crests of the longer waves and longer and lower in the troughs. The paper aims to rigorously calculate how non‑linear interactions between short and long gravity waves change the wavelength and amplitude of the short waves. This is achieved by analytically accounting for the non‑linear interactions between the two wave trains to compute the resulting wavelength and amplitude changes. The calculations reveal that the short‑wave energy variation corresponds to work done by the longer waves against the short‑wave radiation stress—a factor previously overlooked—and that the results differ fundamentally from earlier estimates by Unna.

Abstract

Short gravity waves, when superposed on much longer waves of the same type, have a tendency to become both shorter and steeper at the crests of the longer waves, and correspondingly longer and lower in the troughs. In the present paper, by taking into account the non-linear interactions between the two wave trains, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated. The results differ in some essentials from previous estimates by Unna. The variation in energy of the short waves is shown to correspond to work done by the longer waves against the radiation stress of the short waves, which has previously been overlooked. The concept of the radiation stress is likely to be valuable in other problems.