Unique continuation for water waves and dispersive multiplier equations

We show that if a solution to the water wave equation, for an arbitrary short time interval, is flat on an open set and the horizontal fluid velocity at the surface is zero on the same open set, then the wave must vanish everywhere for all times. In addition, we use a result from non-harmonic Fourier analysis to show that (1 + 1d) linear dispersive PDE with Fourier multipliers also have this unique continuation property, subject to a natural asymptotic growth condition on the multiplier symbol.