Water waves with moving boundaries

Abstract

The unified transform, also known as the Fokas method, provides a powerful methodology for studying boundary value problems. Employing this methodology, we analyse inviscid, irrotational, two-dimensional water waves in a bounded domain, and in particular we study the generation of waves by a moving piecewise horizontal bottom, as it occurs in tsunamis. We show that this problem is characterised by two equations which involve only first-order derivatives. It is argued that under the assumptions of ‘small amplitude waves’ but not of ‘long waves’, the above two equations can be treated numerically via a recently introduced numerical technique for elliptic partial differential equations in a polygonal domain. In the particular case that the moving bottom is horizontal and under the assumption of ‘small amplitude waves’, but not of ‘long waves’, these equations yield a non-local generalisation of the Boussinesq system. Furthermore, under the additional assumption of ‘long waves’ the above system yields a Boussinesq-type system, which however includes the effect of the moving boundary.