Difference between revisions of "Laplace's Equation"

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decays in the vertical condition and has wave properties in the horizontal
 
decays in the vertical condition and has wave properties in the horizontal
 
direction.
 
direction.
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 +
More details can be found on the [http://en.wikipedia.org/wiki/Laplace%27s_equation: Wikipedia entry]

Revision as of 22:05, 8 June 2006

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.

More details can be found on the Wikipedia entry