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The wave equation $$\nabla^2 u(r,t)-\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}(r,t)=0$$ can be Fourier transformed with respect to time, using $\frac{\partial}{\partial t}=i\omega$, to obtain the Helmholtz equation: $$\nabla^2 u(r,\omega)+\frac{\omega^2}{c^2}u(r,\omega)=0\,.$$ What I don't understand is what happens if we Fourier transform is subsequently with respect to space: $$-k^2 u(k,\omega)+\frac{\omega^2}{c^2}u(k,\omega)=0\,,$$ since we now get $$\left(-k^2+\frac{\omega^2}{c^2}\right)u(k,\omega)=0\,.$$ If this should hold for all $k$ and all $\omega$, the solution is the zero function, right?

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When solving PDE we usually allow solutions to be distributions (aka generalised functions). I'll spare you the details; you'll have to read a book on PDE by a mathematician to understand that there is a deep connection between PDE and distributions. Long story short, the distributional solution of $$ (k^2-\omega^2)u(k,\omega)=0 $$ is $$ u(k,\omega)=f(k,\omega)\delta(k^2-\omega^2) $$ for an arbitrary function $f$. This function satisfies $(k^2-\omega^2)u=0$ but $u\neq 0$. (Note that I set $c=1$ to simplify the notation).

You can find some more details about this approach in this answer of mine, where I solve $(\partial^2+m^2)\phi=0$. Note that in that post, $\partial^2=\nabla^2-\partial_t^2$; to get the standard wave equation, you can take $m=0$.

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  • $\begingroup$ Thank you! The solution of $u(k,\omega)$ is thus an arbitrary function $f(k,\omega)$ on the line $k=\frac{\omega}{c}$ and zero elsewhere? Could you perhaps recommend me a mathematical PDE book to help me explain exactly the step between your 1st and 2nd equation? $\endgroup$ – Carucel Feb 1 '17 at 19:14
  • $\begingroup$ @Carucel 1) yes, exactly. 2) I've asked a friend of mine who's a mathematician, and he told me that the best reference for PDE is Evans' Partial Differential Equations, but depending on your background it may be slightly too advanced and technical. In this math.SE post there are more recommendations on the topic. $\endgroup$ – AccidentalFourierTransform Feb 1 '17 at 22:11
  • $\begingroup$ Wow thank you very much! The book seems very interesting of appropiate level. Thank you a lot! $\endgroup$ – Carucel Feb 1 '17 at 23:27

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