Trapping and resonance of long shelf waves due to groups of short waves

Abstract

Groups of short waves within a narrow frequency band are known to be accompanied by second-order long waves travelling at the group velocity of the predominant short waves. When the short waves are refracted by bottom topography, new long waves can be further radiated and propagated away from the topography at the shallow-water speed. Since over a long submarine ridge there can be trapped modes of long-period waves, incident groups of short waves can excite the trapped waves through a second-order mechanism. In this paper we study such excitations over a rectangular shelf which scatters the first-order short waves. By employing asymptotic methods we examine the transient excitation of the trapped long wave by both sinusoidal wave groups and wave packets. The effects of a small angular spread of the incident waves are also included.