Source Term Inhomogenous Helmholtz Equation

Question

So if we look at the inhomogenous Helmholtz equation
begin{equation}
nabla^2 u+k^2u=-f
end{equation}
and include the sommerfield boundary condition
begin{equation}
lim_{r rightarrow infty } r^frac{n-1}{2}(frac{partial}{partial r}-ik) u(rx)=0
end{equation}
the solution should be
begin{equation}
u(x)=(G*f)(x)
end{equation}
with the symbolic convolution operator $ * $ and the 3 dimensional Greens function
begin{equation}
G(x)=frac{e^{ik|x|}}{4 pi |x|}
end{equation}
What I am having trouble to understand now is: What is the physical interpretation of $f$ in this context? If we had a simple acoustic monopole source given by
begin{equation}
underline p=frac{underline A}{r} e^{-ikr}
end{equation}
and I want to introduce a phase shift $varphi$ to the acoustic monopole, how would I include this in those equations? $underline A$ would be the complex amplitude and $underline p$ the complex sound pressure.

lnmaurer · Accepted Answer

This isn't a subject that I know much about, but no one else has jumped in, so I'll take a crack at it. In particular, I've never done anything with acoustics; I'm just trying to work by analogy with electricity and magnetism.
My understanding of the Helmholtz equation is that its application starts with the understanding that the solution to your problem can be separated into a spatial and temporal part. I.e. $Pleft(vec{x}, tright) = uleft(vec{x}right)Tleft(tright)$. The Helmholtz equation is used to find the spatial part $u$, and after you've done that, you can just multiply in the temporal part $T$. E.g. if your source is generating sound at a constant frequency and with a given phase, $Tleft(tright) = e^{-ileft(omega t + phiright)}$.
What is the interpretation of the source $f$? A source is whatever is generating your field. In electrostatics the Helmholtz equation doesn't apply, but Poisson's equation does; a source could be a point particle (i.e. an electric monopole) like an electron, and it generates an electric field. Mathematically, an electric monopole with charge $q$ at $vec{x}=0$ looks like $fleft(vec{x}right) = q deltaleft(vec{x}right)$, where $deltaleft(vec{x}right)$ is the Dirac delta function. For acoustics, the source could be a speaker that generates a pressure field. My interpretation of an "acoustic monopole" is that it is an infinitely small (point) speaker that radiates sound equally in all directions.  Mathematically, I think that an acoustic monopole radiating a sound with amplitude $A$ at $vec{x}=0$ would look like $fleft(vec{x}right) = A deltaleft(vec{x}right)$.
So, what's $uleft(vec{x}right)$ in your case? I think it should be
$$uleft(vec{x}right)=left(G * fright)left(vec{x}right) = A frac{e^{ikleft|vec{x}right|}}{4pileft|vec{x}right|},$$
which looks a lot like your definition for the complex pressure. (Except for the factor of $4pi$. Maybe that gets grouped in with the complex amplitude somehow?)
If you then want to add time dependence, I think that you would just multiply in your $T$. E.g.
$$Pleft(vec{x}, tright) = uleft(vec{x}right)Tleft(tright) = A frac{e^{ikleft|vec{x}right|}}{4pileft|vec{x}right|} e^{-ileft(omega t + phiright)}.$$
You could group that in various ways, e.g.
$$Pleft(vec{x}, tright) = frac{A e^{-iphi}}{4pi} frac{e^{ileft(kleft|vec{x}right| - omega t right)}}{left|vec{x}right|}.$$
If $frac{A e^{-iphi}}{4pi}$ is defined as the complex amplitude (I don't know if this is the right definition), then its complex part is due to the phase shift in your source.
Hopefully what I've said is on the right track. If I'm off base, hopefully someone will correct me.

Source Term Inhomogenous Helmholtz Equation

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