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Solve the equation of wave equation with fixed source at the origin $$\Delta p + \omega^2p=-\delta(x)$$ The question asks for solution of the form $p(r) = C\frac{e^{i\lambda r}}{r}$, where $r = |x|$.

I first plug the proposed solution $p$ into the equation when $x$ is away from 0(so that the equation becomes homogeneous) and get that $\lambda = \pm\omega$. Then I integrate the equation over a small ball around 0. $$\int_{\partial B_\epsilon(0)}\frac{\partial p}{\partial r}dS=\int_{B_\epsilon(0)}\Delta pdx = \int_{B_\epsilon(0)}(\delta(x) - \omega^2p) dx = 1-\omega^2\int_{B_\epsilon(0)}pdx$$

I follow the hints and get the above results, but I don't know how to deal with the first and last term in the last equation that I obtain to finally determine the $\lambda$ and $C$. Can someone give some ideas?

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  • $\begingroup$ I think the first term should be $\int_{\partial B_{\epsilon}(0)}\frac{\partial p}{\partial r}dS = 4\pi \epsilon^2 p_r(\epsilon)$, right? $\endgroup$ – QD666 Oct 22 '18 at 0:32

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