New integral relations for gravity waves of finite amplitude

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 ĀE is directly proportional to the Lagrangian T−V, through the relation \[ ĀE = 2c(T-V)/g, \] where c is the phase speed and g denotes the acceleration due to gravity. Moreover, for waves of constant length, the differential relation \[ { d}ĀE = 2(3T-V)\,{ d}c/g \] also holds. In a wave of limiting steepness it was shown previously that the level of action y a is very nearly equal to the crest level ymax. This is further discussed, and is shown to be probably a numerical coincidence.