Laplace's Equation

The velocity potential $\Phi$ satisfies Laplace's equation if we can assume that the fluid is inviscid, incompressible, and irrotational.

Laplace's equation is the following in two dimensions

$\nabla^2\phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial z^2} = 0$

and in three dimensions

$\nabla^2\phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2}+ \frac{\partial^2 \phi}{\partial z^2} = 0$

The typical solution to Laplace's equation oscillates in one direction and decays in another. The linear water wave arises as a boundary wave which decays in the vertical direction and has wave properties in the horizontal direction.