Deriving the Equations of Water Waves
Water waves obey simple differential equations derived using simplifying assumptions of incompressibility and irrotationality.
If the flow is irrotational we can define a velocity potentialsatisfyingIf the fluid is incompressible thenfor waves moving in thedirection withlabelling the depth of water, so Laplace's equation is satisfied.
If we assume that particles initially in the surface stay in the surface as the wave progresses, then the motion of a particle in the surface indicates the motion of the water surface. The vertical component of the velocity isAt the ocean bottom –for finite depth or for infinite depth - this must be zero, because water cannot move perpendicular to the surface.
A third equation can be derived by further considering the velocity potential. We know already that the velocity potential is unique only to within an additive constant, but is here time dependent , the constant here is actually a function of time and Bernoulli's equation becomes
Atwe can set(atmospheric pressure) and atThus the constant isand we may write Bernoulli's equation as
Subsequently the pressure at any time t the pressure at the surface isso
We can ignore the second order termand setto get
Differentiating this equation with respect to t gives