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Thus, the final form of our wave is given below.

y_x(x,y,t) =	rcosh(k(h+y)) sinhkh	sin(wt-kx)+x
y_y(x,y,t) =	rsinh(k(h+y)) sinhkh	cos(wt-kx)

These equations give the motion of water waves in a sea of constant depth h. One may still ask - which of these possible motions are truly allowed? Can one choose any w, any k, any r, etc.? Indeed, there are restrictions and these restrictions can be found by employing Bernoulli's Theorem.

To use Bernoulli's theorem we need a flow that is a dynamical system - that is, we need the velocity vectors to be time independent. It is clear that our model does not meet this condition. This problem can be overcome by making a change of coordinates. We use, as a new reference frame, that of a boat travelling in the direction of the waves at exactly the wave speed. From the boat's point of view the waves have a fixed shape and the water flows backwards along these waves undulating up and down. Specifically, we take X = x-wt/k as a change of coordinates. As a result, the velocities become

v_X = wR_x(y)coskX-w/k

v_y = wR_y(y)sinkX

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