Exact linearisation of the nonlinear shallow-water equations over a moving bottom: classification, wave generation and run-up

Abstract

The Carrier--Greenspan hodograph transformation linearises the nonlinear shallow-water equations on a beach of constant slope. We extend the transformation to a moving bottom and identify the family of bottom motions that preserves the Carrier--Greenspan invariants. The bottom gradient must be uniform in space, that is, $h(x,t)=ρ(t)x+\mathcal{B}(t)$, and constant $ρ$ returns the classical slope. For this family a horizontally accelerating frame removes the forcing. The bottom motion then disappears from the hodograph equation, and two quadratures recover the physical variables. If the gradient varies in time, the bed displacement grows without bound offshore and the far field cannot remain at rest. The domain must therefore be finite, although the pivot may be placed anywhere. A bottom-tilting wave maker realises this family. Closed forms follow for the hinge response, for the crest-to-trough steepness of the generated wave, and for the leading nonlinear correction. The correction depends on the point to which the steepness is referred, and so differs between a leading-elevation and a leading-depression wave. Because the hinge is at the beach toe, these closed forms provide the incident wave required by the classical plane-beach solution. The run-up can therefore be predicted from the plate motion alone. The steepness is predicted without a fitted constant and agrees with the reference computations. Sixty-one runs over a level bed, reported here for the first time, carry the test to deeper water and longer plate motions. The measured run-up follows the predicted run-up, with an offset that the inviscid theory does not account for.