Abstract
The two‐dimensional initial‐value problem is examined for an inviscid incompressible fluid on a plane beach within the framework of a linearized but non‐hydrostatic theory. Solutions generated by initial water surface distributions (or surface velocities) are obtained in integral form and are both computed and studied under asymptotic limits of small and large times for a number of special cases. The theory is extended to cover forced motion by fluctuating surface pressure or by forcing on the (otherwise) solid bed. Comparisons are made with the classical results for the alternative geometry of uniform depth. An extension to three dimensions is given without rigorous proofs and forced surface‐motion results indicate that outgoing waves are produced by a disturbance which is time oscillatory and spatially oscillatory along the shoreline direction. One of the chief results of the present paper is the observation that, in general, this disturbance will lead to unbounded wave heights for large time when its longshore wavelength is equal to the wavelength of an Ursell edge‐wave mode.
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