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

See Also