The purpose of this project is to study non-linear wave-wave interaction.
For this the Zakharov equation and the Hasselmann equation have been
re-derived. The singularities of the interaction coefficient and shallow
water theory which is closely related were examined in order to determine
the behaviour of these singularities. It was found that the terms with
singularities could be neglected in the vicinity of the singularities.
An efficient exact integration from Hasselmann and Hasselmann (1981) was
set up and implemented. The discrete interaction approximation from
Hasselmann et al. (1985) was justified and implemented. Exact
calculations and the discrete interaction approximation was carried out
for a number of selected spectra at infinite water depth and finite water
depth. The results were in good agreement with previous calculations
carried out by for example Hasselmann and Hasselmann (1981). The case of
an unidirectional spectrum was considered too, and it was found that
there was a rather significant transfer in the shallow water case, and a
zero transfer in the infinite depth case as the interaction coefficient
becomes zero identically for interaction configurations with a non-zero
spectral product. The transfer in the case of an unidirectional spectrum
was in general from waves with frequencies in the vicinity of the peak
frequency, to waves propagating in the same direction as the original
spectrum with frequencies in the vicinity of 3/2 times the peak
frequency, and to waves propagating in the opposite direction as the
original spectrum with frequencies in the vicinity of 1/2 times the peak
frequency.
