Notes on shallow water extension of Miles Theory

We present results from a straightforward extension of Miles’ theory of wind-wave generation towards
shallow water. Both numerical results and an approximate analytical expression for the
growthrate are obtained. The analytical expression forms the basis for a parametrization of wind
wave growth in numerical wave prediction models.

The present wind input term in third generation models is based on the assumption that the windwave
generation depends, regarding ocean wave properties, only on the phase speed of the waves. In
this work it is shown that for the short waves (kD > 1, with k the wavenumber and D the local depth)
this is indeed a good assumption, but for the long waves, with kD<<1, there will be an overestimate
of wind input. Nevertheless, simulations with the energy balance equation for the duration limited
case show that the present wind input term and the newly proposed wind input term give almost identical
evolution of wave height and spectrum. This may be understood by noting that in equilibrium
conditions kpD ≃ 1 (with kp the peak wavenumber), in other words in practical circumstances waves
are short enough so that the difference between present and new wind input source function does not
matter.