Laplace's Equation
The velocity potential [math]\displaystyle{ \Phi }[/math] 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
[math]\displaystyle{ \nabla^2\phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial z^2} = 0 }[/math]
and in three dimensions
[math]\displaystyle{ \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 }[/math]
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.
