Difference between revisions of "Laplace's Equation"
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| − | The velocity potential satisfies Laplace equation if we can assume that the fluid is inviscid, incompressible, and irrotational. | + | 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 | Laplace's equation is the following in two dimensions | ||
| − | <math>\ | + | <math>\nabla^2\phi = \frac{\partial^2 \phi}{\partial x^2} |
+ \frac{\partial^2 \phi}{\partial z^2} = 0</math> | + \frac{\partial^2 \phi}{\partial z^2} = 0</math> | ||
and in three dimensions | and in three dimensions | ||
| − | <math>\ | + | <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> | + \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 | 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 another. The linear water wave arises as a boundary wave which | ||
| − | decays in the vertical | + | decays in the vertical direction and has wave properties in the horizontal |
direction. | 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.
