Difference between revisions of "Helmholtz's Equation"
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<math>\nabla^2 \phi - k^2 \phi = 0 </math>. | <math>\nabla^2 \phi - k^2 \phi = 0 </math>. | ||
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It applies to a wide variety of situations such as electromagnetics and acoustics. | It applies to a wide variety of situations such as electromagnetics and acoustics. | ||
In water waves it arises when we [[Removing The Depth Dependence|Remove The Depth Dependence]]. Often there is then a cross | In water waves it arises when we [[Removing The Depth Dependence|Remove The Depth Dependence]]. Often there is then a cross | ||
Revision as of 09:09, 23 August 2008
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.
