Ray theory is used to investigate the interaction of a short high-frequency progressive internal wave of infinitesimal amplitude with a long progressive near-inertial wave of arbitrary amplitude. Weak-interaction theory would, if applicable, predict that the largest changes in short-wave properties occur when the resonance condition c = cg is satisfied, where c is the phase velocity of the long wave and cg is the group velocity of the short wave. The present calculation confirms this prediction only when the long wave has exceedingly small amplitude (peak velocities of order 0.1 cm/s).
However, when the background velocity has a realistic amplitude (e.g. oceanic values are of order 20 cm/s) the resonance condition fails to be relevant. For example, waves which initially have c = cg become trapped in low-shear regions and consequently experience very small changes in wavenumber. Other short waves, which initially have cg [Lt ] c and hence violate the resonance condition, exhibit large and permanent changes in vertical wavenumber.
Remarkably, it is found that these permanent changes are much more likely to be decreases, rather than increases, in wavenumber. This can be explained as follows. Short waves which enter an inertial-wave packet experience both increases and decreases in wavenumber. However, at times when the wavenumber is relatively large, the group velocity is relatively small and the short wave is unlikely to escape from the inertial packet, whereas small wavenumber and large group velocity assist the escape of the short-wave group. Consequently the short waves that leave the inertial packet tend to have a smaller average wavenumber than those that enter. Thus the net effect of a near-inertial packet on a collection of short waves appears to be an increase in vertical wavelength and frequency.