Modulational instabilities of periodic traveling waves in deep water

•The spectrum of traveling water waves is asymptotically approximated, considering two and three-dimensional perturbations.•A multiple scale expansion is employed, coupling wave slope to Bloch parameter.•The radius of the disc of analyticity of the spectrum is predicted, and compared to numerical simulations.•Modulational instabilities are computed.

The spectrum of periodic traveling waves in deep water is discussed. A multi-scale method is used, expanding the spectral data and the Bloch parameter in wave amplitude, to compute the size and location of modulated instabilities. The role of these instabilities in limiting the spectrum’s analyticity is explained. Both two-dimensional and three-dimensional instabilities are calculated. The asymptotic predictions are compared to numerical simulations.