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We describe a computational framework for computing hybrid traveling-standing waves that return to a spatial translation of their initial conditions at a later time. We introduce two parameters to describe these waves, and explore bifurcations from pure traveling or pure standing waves to these more general solutions of the free-surface Euler equations.

Next, we combine Floquet theory in time and Bloch theory in space to study the stability of traveling-standing waves to harmonic and subharmonic perturbations. For the latter, we have developed new boundary integral methods for the spatially quasi-periodic Dirichlet-Neumann operator. While much is known about the spectral stability of pure traveling waves, this is the first study of general subharmonic perturbations of pure standing waves. Our unified approach for traveling-standing waves simplifies the eigenvalue problem that arises in the pure traveling case as well.