Laplace's Equation: Difference between revisions
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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> | ||
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.
