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We generalize the periodic Evans function approach recently used to study thespectral stability of Stokes wave and gravity-capillary (including Wiltonripples) in water of finite depth to study spectral stability of Stokes wavesin water of infinite depth. We prove waves of sufficiently small amplitude arealways low-frequency unstable regardless of the wave number and gravity, givingan alternative proof for the Benjamin-Feir modulational instability in theinfinite depth case. Here, the first proof for the infinite depth case isrecently obtained by Nguyen and Strauss. We also study the spectral stabilityat non-zero resonant frequencies and find no additional instability.

An alternative proof of modulation instability of Stokes waves in deep water