Difference between revisions of "Laplace's Equation"
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| − | The velocity potential satisfies | + | The velocity potential satisfies Laplace equation if we can assume that the fluid is inviscid, incompressible, and irrotational. |
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| + | Laplace's equation is the following in two dimensions | ||
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| + | <math>\Nabla^2\phi = \frac{\partial^2 \phi}{\partial x^2} | ||
| + | + \frac{\partial^2 \phi}{\partial z^2} = 0</math> | ||
| + | |||
| + | and in three dimensions | ||
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| + | <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 condition and has wave properties in the horizontal | ||
| + | direction. | ||
Revision as of 11:13, 24 May 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.
