On the non-linear energy transfer in a gravity wave spectrum Part 2. Conservation theorems; wave-particle analogy; irrevesibility

Abstract

From the conditions of energy and momentum conservation it is shown that all four interaction coefficients of the elementary quadruple interactions discussed in Part I of this paper are equal. A further conservative quantity is then found which can be interpreted as the number density of the gravity-wave ensemble representing the random sea surface. The well-known analogy between a random linear wave field and a mass particle ensemble is found to hold also in the case of weak non-linear interactions in the wave field. The interaction conditions for energy transfer between waves to correspond to the equations of energy and momentum conservation for a particle collision, the final equation for the rate of change of the wave spectrum corresponding to Boltzmann's equation for the rate of change of the number density of an ensemble of colliding mass particles. The stationary wave spectrum corresponding to the Maxwell distribution for a mass particle ensemble is found to be degenerate. The question of the irreversibility of the transfer process for gravity waves is discussed but not completely resolved.