Difference between revisions of "Laplace's Equation"
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− | More details can be found on the [http://en.wikipedia.org/wiki/Laplace%27s_equation: Wikipedia entry] | + | 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