Variational formulations, integral approaches and series expansion techniques for linear water waves

Abstract

Some general criteria are given for the classification of several numerical and semi-analytical approaches used to solve problems concerning the propagation of linear surface water waves over a bottom of variable depth and in the presence of natural or artificial obstacles. The various approaches have been subdivided in two fundamental categories, namely the direct approaches, for which the main unknown variable is the potential field or the surface wave displacement, and the indirect approaches, in which first a solution of the problem in terms of the normal velocity along the lines of a certain network drawn in the considered domain has to be found. Standard procedures have been detected in order to establish variational principles for problems in which different representations of the unknown fields are employed in the various parts of the whole domain. The different kinds of numerical difficulties that can be encountered have been studied and the links among several levels of investigation stressed. A comparison has been carried out among the various methods, regarding the possible onset of numerical instabilities and the accuracy of the results. In particular, it has been shown that the indirect methods are able to represent a wavelength by much less parameters than the direct ones.