Difference between revisions of "Helmholtz's Equation"

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{{complete pages}}
 
{{complete pages}}
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== Indroduction ==
  
 
This is a very well known equation given by
 
This is a very well known equation given by
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satisfy Helmholtz's equation. This means that many asymptotic results in linear water waves can be
 
satisfy Helmholtz's equation. This means that many asymptotic results in linear water waves can be
 
derived from results in acoustic or electromagnetic scattering.
 
derived from results in acoustic or electromagnetic scattering.
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== Solution for a Circle ==
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We can solve for the scattering by a circle using separation of variables. This is the basis
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of the method used in [[
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{{Separation of variables for the r and theta coordinates}
  
 
[http://en.wikipedia.org/wiki/Helmholtz_equation External link]
 
[http://en.wikipedia.org/wiki/Helmholtz_equation External link]
  
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Revision as of 00:07, 9 June 2010


Indroduction

This is a very well known equation given by

[math]\displaystyle{ \nabla^2 \phi + k^2 \phi = 0 }[/math].

It applies to a wide variety of situations such as electromagnetics and acoustics. In water waves it arises when we Remove The Depth Dependence. Often there is then a cross over from the study of water waves to the study of scattering problems more generally. Also, if we perform a Cylindrical Eigenfunction Expansion we find that the modes all decay rapidly as distance goes to infinity except the solutions which satisfy Helmholtz's equation. This means that many asymptotic results in linear water waves can be derived from results in acoustic or electromagnetic scattering.

Solution for a Circle

We can solve for the scattering by a circle using separation of variables. This is the basis of the method used in [[

{{Separation of variables for the r and theta coordinates}

External link