# Green's function of Helmholtz eqn (with time)

by coffee1729
Tags: function, green, helmholtz, time
 P: 4 Hi, I have been trying to find the (causal) Green's function of $$\frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + m^2 \phi = 0.$$ 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!
 Sci Advisor PF Gold P: 1,765 Green's function of Helmholtz eqn (with time) The solution is easy if you assume time-harmonic functions. That is, if your function can be expressed as $$\phi(\rho,t) = \sum_{n=-\infty}^\infty{ \alpha_n\phi_n(\rho)e^{-i\omega_nt}}$$ Then the second order time derivative just becomes $$\frac{\partial^2}{\partial t^2} \rightarrow -\omega_n^2$$ 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, $$\partial_t^2\phi-\nabla^2\phi+m^2=0$$ 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.