  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.
