On finite amplitude water waves

The modulation theory for finite amplitude water waves is developed using the variational technique. It is shown how Levi-Civita’s relation, Starr’s relation and the conservation equations all follow very simply and naturally from this approach. The present paper is limited to deep water waves, but the results can be extended to arbitrary depth. For deep water, the appropriate Lagrangian can be reduced to a single function, which can be taken from recent numerical calculations on periodic waves. This is used to discuss the stability of wavetrains to long modulations.