Helmholtz

Helmholtz - Physics 318 Electromagnetism R.G. Palmer...

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Physics 318 R.G. Palmer Electromagnetism 3/26/10 The Helmholtz equation is ( 2 + k 2 ) ψ = 0 If k 6 = 0, then is called the Helmholtz equation. k 2 is usually positive in most applications. If k = 0, then is called the Laplace equation. The wave equation and heat/diffusion equations give rise to the Helmholtz equation when time is separated out, with time dependence e ± ikvt for the wave equation (with speed v ) and e - α 2 k 2 t for the heat/diffusion equation (with thermal diffusivity or diffusion constant α 2 ). In the following summary, any linear combination of the functions listed together in braces {} is allowed. Whenever I write cos x and sin x , you could equally well have e ± ix or any linear combination thereof. Similarly cosh x and sinh x should be taken to include any linear combination of e ± x . If there is a zero separation constant ( i.e. α = 0) for cosh αx /sinh αx /cos αx /sin αx , then the functions change to
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Helmholtz - Physics 318 Electromagnetism R.G. Palmer...

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