Variational Water-Wave Modeling: From Deep Water to Beaches

Abstract

The mathematical and numerical modelling of free-surface water waves is considered from the viewpoint of variational principles combined with finite-element discretisations. Luke’s classical variational principle (VP) is derived first, as opposed to Luke’s (ingenious) positing of his VP, and forms the basis for three geometric or compatible finite-element water-wave models, two of which are validated against laboratory measurements and compared. Potential-flow wave dynamics in intermediate-depth water is coupled variationally to shallow-water beach dynamics, the latter modelling breaking waves, and illustrative numerics is shown to highlight the interactions between deeper water and shallow-water waves. Throughout, photographs of intricate wave interactions are used as illustrations.

Appendix 1: Variations in Pre-Luke’s VP

While analysing the variations in (5.6), note furthermore that

$$\begin{aligned}&\int _0^T \int _{\varOmega _H}\int _0^h D\partial _t \delta \phi \,{\mathrm d}{z}\,{\mathrm d}{x}\,{\mathrm d}{y}\,{\mathrm d}{t} = \int _0^T \int _{\varOmega _H}\int _0^h -\delta \phi \partial _tD+\partial _t(D\delta \phi )\,{\mathrm d}{z}\,{\mathrm d}{x}\,{\mathrm d}{y}\,{\mathrm d}{t} \end{aligned}$$

(5.35a)

$$\begin{aligned}&= -\int _0^T \int _{\varOmega _H}\int _0^h \delta \phi \partial _tD\,{\mathrm d}{z}\,{\mathrm d}{x}\,{\mathrm d}{y}\,{\mathrm d}{t} + \int _0^T \frac{\textrm{d} }{{\mathrm d}{t}} \int _{\varOmega _h} \int _0^h D\delta \phi \,{\mathrm d}{z}\,{\mathrm d}{x}\,{\mathrm d}{y}\,{\mathrm d}{t}\nonumber \\&\qquad \qquad \qquad \quad -\int _0^T \int _{\varOmega _H} D\partial _t h\delta \phi \,{\mathrm d}{x}\,{\mathrm d}{y}\,{\mathrm d}{t}\end{aligned}$$

(5.35b)

$$\begin{aligned}&= -\int _0^T \int _{\varOmega _H}\int _0^h \delta \phi \partial _tD\,{\mathrm d}{z}\,{\mathrm d}{x}\,{\mathrm d}{y}\,{\mathrm d}{t} -\int _0^T \int _{\varOmega _H} D\partial _t h\delta \phi \,{\mathrm d}{x}\,{\mathrm d}{y}\,{\mathrm d}{t}, \end{aligned}$$

(5.35c)

in which the volumetric contribution at $t=0$ and $t=T$ cancels, since $\delta \phi (x,y,z,0)=\delta \phi (x,y,z,T)=0$, and by assuming for the moment that $\varOmega _h$ is time-independent. If $\varOmega _h$ is time-dependent, for example in the presence of a wavemaker, then an extra term will emerge.