The interaction of periodic surface gravity waves with slowly varying water currents

Abstract

The governing equations for interactions between surface gravity wavetrains and slowly-varying water currents are derived and the incorporation of Vocoidal water wave theory into this framework is discussed. The emphasis throughout is on the derivation of the general form of the governing equations plus a detailed discussion of the qualitative physical behaviour implied by the equations. Particular solutions are usually given only where they serve to clarify the general method or some physical feature of the analysis. The thesis proper is introduced by a derivation of wave kinematics on still water. A review of the kinematics and dynamics of an inviscid and irrotational fluid follows. The wave and fluid properties are then combined via the definition of wave integral properties. A derivation of the Airy and Stokes O(a2) wave theories is given and used to illustrate a number of points. Water currents (following or opposing the waves) are introduced via their influence on the wave-kinematics. The wave/current dynamics are then presented in two ways: firstly using a wave energy approach and secondly by introducing the wave action concept. Wave action is more convenient because it is a conserved quantity unlike wave energy. The general equations for two dimensional wave/current interactions are derived and discussed. At this point three topics are reconsidered: group velocity, momentum density in wave motion and Lagrangian mean forms of averaging. The general equations for wave/current interaction are shown to be compatible with the Vocoidal water wave theory and applications of the theory to wave/current problems are discussed.