New integral relations for gravity waves of finite amplitude

Abstract

Some new exact relations are derived between integral properties of a progressive irrotational gravity wave of finite amplitude in deep water. In particular it is shown that the Eulerian-mean angular momentum is directly proportional to the Lagrangian T−V, through the relation \[ \overline{A}_{\rm E} = 2c(T-V)/g, \] where c is the phase speed and g denotes the acceleration due to gravity. Moreover, for waves of constant length, the differential relation \[ {\rm d}\overline{A}_{\rm E} = 2(3T-V)\,{\rm d}c/g \] also holds.In a wave of limiting steepness it was shown previously that the level of action ya is very nearly equal to the crest level ymax. This is further discussed, and is shown to be probably a numerical coincidence.