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comment Theory behind patterns formed on Chladni plates?
Actually, Chladni plates are not described by the wave equation, i.e. by the eigenfunctions of the Laplacian, but by the eigenfunctions of the biharmonic operator, i.e. the Laplacian squared. More information and historical references can be found in a beautiful article by Gander and Wanner. Even if you try to derive the wave equation for a string, using the "masses connected by strings" model, you have to assume that it is under tension, i.e. the springs are pre-stretched, or you will get the biharmonic operator and not the wave equation.
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accepted Propagator for Dirac equation in real space
Feb
7
answered Propagator for Dirac equation in real space
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reviewed Edit Why does ice form on bridges even if the temperature is above freezing?
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revised Why does ice form on bridges even if the temperature is above freezing?
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comment Propagator for Dirac equation in real space
@Trimok: Moving the differentiation under the integral sign would mean that the propagator involves the derivative of a delta function, though, which I find odd. You make the argument that you "involving a derivative" is not an invariant property of the propagator, which I can't deny. One way to formalize it is to ask whether the solution changes when we add a constant to the function $f \to f + C$. (But this may be forbidden due to boundary conditions.)
Oct
25
comment Propagator for Dirac equation in real space
I understand that, what I'm looking for is an explicit formula for the Greens function $G_1(x,x;t,t')$. I have the impression that it involves the derivative of a $\delta$-function, and I just want to make sure that this is correct by comparing to an explicit reference. (All textbooks that I've looked at only calculate the Fourier transform $G_1(k;t,t')$).
Oct
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Oct
18
comment Propagator for Dirac equation in real space
@Trimok: We can choose $f(x',t') = g(x')\delta(t'-t_0)$ to model initial conditions at a time $t_0$. In general, the propagator for a driving force $f$ can always be used to construct the propagator for initial conditions $ψ(x,t_0)$.
Oct
16
asked Propagator for Dirac equation in real space