On wave generation by a submerged oscillating source over Roseau's beach profile with outline application to a scattering problem

Abstract

The exact solution for linearised water wave motion over a specific curved bottom beach profile, developed by Roseau in his book Asymptotic Wave Theory (North-Holland 1976) is here generalised to account also for the insertion of a submerged oscillating line source randomly placed above the beach. An explicit expression is obtained for the associated velocity potential in various scenarios. In particular, the one where the wave field at infinity is devoid of incoming waves allows Sretenski's (Prikl. Mat. Meh. 27 (1963) 1012–1025) established discovery on how the source is made invisible at long distances by its strategic placement, to be verified by the asymptotic theory. Meanwhile, the opportunity is taken to construct the solution in the form of a Green's function and its use is exemplified with a detailed guide to solving a problem of waves attacking a finite floating dock above the Roseau profile. The symmetry property of the Green's function is verified numerically thereby providing a robust check on the validity of the procedures used in computation and leaving the reader with a tool to explore a number of different problems over this curved beach such as might be required from time to time to validate and calibrate more detailed computational models ultimately designed for application with a wide variety of bottom profiles.