Laplace's Equation: Difference between revisions
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The velocity potential satisfies | The velocity potential <math>\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>\nabla^2\phi = \frac{\partial^2 \phi}{\partial x^2} | |||
+ \frac{\partial^2 \phi}{\partial z^2} = 0</math> | |||
and in three dimensions | |||
<math>\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. | |||
==See Also== | |||
* [http://en.wikipedia.org/wiki/Laplace%27s_equation Laplace's equation] | |||
Latest revision as of 03:19, 23 September 2008
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.
