Hi, I have been trying to find the (causal) Green's function of [tex] \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + m^2 \phi = 0. [/tex] What would be a good way to approach this? I have initial values for t=0, so I use Laplace transforms on t and Fourier transforms for the spatial variables. However, I end up with a divergent integral involving the Bessel function, and I haven't been able to find a contour on which the integral converges. Has anyone seen this before, and what would be a good way to approach this equation? Note that we could simply take the Green's function for the 3-d Helmholtz to solve this equation, but it wouldn't be causal. Is there something conceptual, that would allow me to go directly from the 3-d solution to the 2-d causal Green's function? Thanks!
I seem to recall that Byron and Fuller's book did this one. http://books.google.ca/books?id=D2X...resnum=1&ved=0CAwQ6AEwAA#v=onepage&q=&f=false but I could be wrong (and can't check now).
Thank you for your post - there is a section on solving for equations of the form H + (d/dt)^2 where H is self-adjoint, however, I don't think the spatial part here + m^2 is? Or do you mean another section in the book?
The solution is easy if you assume time-harmonic functions. That is, if your function can be expressed as [tex]\phi(\rho,t) = \sum_{n=-\infty}^\infty{ \alpha_n\phi_n(\rho)e^{-i\omega_nt}} [/tex] Then the second order time derivative just becomes [tex] \frac{\partial^2}{\partial t^2} \rightarrow -\omega_n^2 [/tex] You could combine that term with the m^2 term and get an expression that has been solved in many texts. I think it comes out to be a Hankel function. But I have a feeling this isn't what you want as implied by your "causal" qualifier. EDIT: Ahh... I remember why this looks familiar. I this looks like the Klein-Gordon equation, [tex] \partial_t^2\phi-\nabla^2\phi+m^2=0[/tex] Unfortunately the signs are not consistent with yours. However, perhaps if you look into the derivation of the Green's function of the Klein-Gordon wave equation (many quantum field texts should have this) then maybe it will give you some insight into how to do it yourself. My recollection though is that it is just a four-space Fourier transform. But if you understand how to do the 3D spatial version, then you may want to just look at how we derive 2D Green's functions. A classical electromagnetic text usually tackles the 2D and 3D Green's function, the former coming out to be a Hankel function as I mentioned earlier. But they usually deal with the time-harmonic case.
Thank you for your post! I think I tried this approach before - isn't this is very similar to taking the FT of the time domain? I ended up with the Helmholtz equation in 2-d again (this time with the term m^2 - w^2). The Green's function for this is the Hankel function, however, when reverting back into the time domain, I would have to integrate over its singularities at w = +m and w = -m (since m is real). One way of avoiding this is to take the LT of time, leaving us with the Hankel function and then doing the inverse Laplacian. This would require us to determine residues of the Hankel function (in particular H( sqrt(w^2 + m^2) * r)), which I haven't been able to do yet. Have you seen this before? The flipped sign makes time indistinguishable from space and makes determining causality hard. I have heard of wick rotation which are used in General Relativity to transform between Minkowski and Euclidean space-time - would this approach work or is it just another way of getting the singularities off the path of integration, which would mean that I would still have to evaluate the residues of the previously mentioned Hankel function?
Yeah, I think this is where your causality is causing trouble. I have a feeling that a time-harmonic signal is not really causal, it is always on and always has been. If we expand out into a Fourier Series, we move the time dependence onto the exponential and model the signal by describing the frequency content using constant coefficients. But even then the signal is on for all time. If you were to turn on and off the signal, and thus have a causal signal, it would require that these coefficients be time dependent (they ramp up and down as we turn on and off the signal). In which case the time-derivative of the signal would not be the simple \omega_n dependence since the time dependence is now not exclusively in the exponential. One idea to perserve causality might be to introduce a time loss to the signal. That is, the signal must go to zero at infinities. This is similar to the "radiation condition" used in electromagnetics. While the radiation condition is often only used to guarantee uniqueness of a solution (a separate problem), we often introduce an artificial loss in the Sommerfeld integration to avoid singularities along the frequency integration. A Sommerfeld integration is the integration of a Bessel function and an exponential function, conceptually a cylindrical wave scaled by an orthogonal plane wave. The result of the Sommerfeld integral is a spherical wave. However, the integration contour can have poles on the integration path if we have a lossless medium. Just like how we needed to have an "artificial loss" to free-space in the radiation condition to preserve uniqueness, we can add an artificial loss to the medium by moving our integration path down into the complex plane. This moves the contour off of the poles and adds a slight loss. The result is not going to be exact, but careful control of the detour will get you numerically there. I wonder then, this is a very similar situation. We have a pole when \omega^2 = m^2. What if we moved the integration path from \omega = x to say \omega = x-i\delta? We avoid the pole since we are now in the complex plane. Now conceptually we are adding a slight loss when we integrate along the positive omega axis, since the time dependence is exp(-i\omega t). So, for the negative omega axis, we use the contour of \omega = x+i\delta so that it is lossy there too. Essentually, this would make the signal disappear at large times, dependent on how large you make \delta. Thus, I think this should allow for a causal signal. Also note, you might be able to fold the integration, instead of going from -\infty to +\infty, you could go from 0 to +\infty. I do that a lot with my Hankel integrations but you have the m offset... There are time-domain Green's functions in electromagnetics. I'm pretty sure they exist since we have time-domain method of moments solvers. I have never used them but they may be something to look into since the wave equation in electromagnetics is a very similar Helmholtz equation. So they may treat a problem very similar to yours.