Laplace's Equation

The velocity potential satisfies Laplace 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 condition and has wave properties in the horizontal direction.