Parametric Excitation of Coupled Waves I. General Formulation

The coupling of two waves due to the presence of a third wave with large amplitude is studied. On the basis of simple model equations, the conditions for excitation of the first two waves are discussed for the following three cases: i) \(\omega_{1}+\omega_{2}{\risingdotseq}\omega_{0}\) and ω 1 , ω 2 are large compared with their frequency shift, ii) \(\omega_{1}{\ll}\omega_{2}{\lesssim}\omega_{0}\) and iii) \(\omega_{1}{\ll}\omega_{0}{\lesssim}\omega_{2}\), where ω 1 , ω 2 are the unperturbed frequencies of the two waves under consideration and ω 0 is the frequency of the incident large amplitude wave. In the first two cases, the excited wave is found oscillatory, while in the third it is found non-oscillatory. The threshold power of the incident wave for the onset of excitation, the frequency shift at the threshold and the growth rate above threshold are calculated in each case.