Why does the inhomogenous wave equation solution change form with a different number of spatial dimensions?

Question

Assume $c = 1$ for what follows.

For the general inhomogenous wave equation in one spatial dimension $$\left(\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}\right)\phi = v(x, t),$$ the article The Wave Equation with a Source gives (where we ignore the homogenous part of the solution, e.g. by setting initial conditions to $0$): $$\phi(x, t) = \frac{1}{2} \int_0^t \int_{x - (t - s)}^{x + (t - s)} v(y, s)\ dy \ ds.$$

For the general inhomogenous wave equation in three spatial dimensions $$\left(\frac{\partial^2}{\partial t^2} - \nabla^2\right)\phi = v({\bf r}, t),$$ the article Solution of Inhomogeneous Wave Equation gives $$\phi({\bf r}, t) = \int \frac{ v({\bf r}', t - \vert{\bf r} -{\bf r}'\vert)} {4\pi \vert{\bf r} - {\bf r}'\vert}\ dV'.$$

Why does the integrand in the three-dimensional case depend inversely on $\vert{\bf r} - {\bf r}'\vert$, while the integrand in the one-dimensional case doesn't?

Answer 1

The wave equation is linear, so the inhomogeneous problem: $$ (\partial_t^2-\Delta)\phi=v $$ can be solved by a convolution with the Green’s function: $$ \phi=G*v\\ (\partial_t^2-\Delta)G=\delta(x,t) $$ Your question can therefore be reformulated as explaining the spatial dimension $D$ dependence of power law of $G$.

This can be partially tackled by dimensional analysis. Setting $c=1$, $x,t$ have dimension $1$. The delta function has dimension $-(D+1)$. A partial derivative has dimension $-1$ so the d’Alembertian has dimension $-2$. Therefore, $G$ has dimension $1-D$. You therefore expect that: $$ G\sim r^{1-D} $$ This gives the correct scaling in 1D and 2D, but not in 3D.

This is because you have an extra Dirac delta factor $\delta(r-t)$ which has dimension $-1$. Correcting this you do get for $D=3$: $$ G\sim r^{-1} $$ while still being consistent with dimensional analysis. This extra Dirac delta factor has a simple interpretation: it is Huygens’ principle.

In retrospect, the scaling could have been anticipated. However, it is best to do the mathematical derivation and then interpret the calculation. A priori hand waving arguments can easily give the wrong result.

Hope this helps.