The spectrum of finite depth water waves

In this contribution we study the spectrum of periodic traveling gravity waves on a two-dimensional fluid of finite depth. We extend the stable and highly accurate method of Transformed Field Expansions to the finite depth case in the presence of both simple and repeated eigenvalues, and then numerically simulate the changes in the spectrum as the wave amplitude is increased. We also calculate explicitly the first non-zero correction to the flat-water spectrum, which we observe to accurately predict the stability (or instability) for all amplitudes within the disc of analyticity of the spectrum. In addition to computations of the spectrum, we also compute the radius of the disc of analyticity of the spectrum—the amplitude boundary beyond which neither the asymptotics nor the TFE method is applicable. We observe an instability which is analytically connected to the flat state for kh∈(0.855,1)kh∈(0.855,1).