On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves

Abstract

In this note we describe the underlying principles - and pitfalls - of the process of non-dimensionalising and scaling the equations that model the classical problem in water waves. In particular, we introduce the two fundamental parameters (associated with amplitude and with wave length) and show how they are used, independently, to represent different approximations (with corresponding different interpretations and applications). In addition, and most importantly, we analyse how these two parameters play a role in the derivation of the Korteweg-de Vries (KdV) equation, which then lead to predictions for the regions of physical space where solitons might be expected to appear. In particular, we address the issue of whether KdV theory can be used effectively to predict tsunamis. We argue that for tsunamis the propagation distances are much too short for KdV dynamics to develop.