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Consider the wave equation initial value problem in $\mathbb R^3$ with spatially variable wave speed, denoted by

\begin{align*} \frac{\partial^2}{\partial t^2}u(x,t)-c(x)\Delta u(x,t)&=0\hspace{.2in}\text{in }\mathbb R^3\times(0,\infty),\\ u(x,0)&=f(x),\\\frac{\partial}{\partial t}u(x,0)&=g(x). \end{align*}

My question: Is there a way to explicitly write down a solution similar to Kirchhoff's formula for the case of constant speed $c(x)=c_0$?

Intuitivly, i would expect the solution $u(x_0,t_0)$ for fixed space and time to be dependent on values of spherical integrals of $f$ and $g$ as in the constant speed case, but in a sound-speed dependent metric.

This is related to a similar question concerning a wave equation in 1 dimension: Wave propagation with variable wave speed

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1 Answer 1

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There is an extensive literature on wave equations with variable coefficients, which are equivalent to wave equations on curved spacetime - as in the monograph by Friendlander. Depending on what you are willing to accept as 'explicit', the answer is that yes, there is an explicit generalisation of the Kirchhoff integral. However, it is very difficult to calculate the coefficients of the data $f,g$ in this integral, which are determined by the retarded Green function $G_R$ of the wave operator: $$\square = \partial_t^2 - c(x)\Delta.$$ This Green function is intimately related to the geometry of the spacetime with line element $ds^2=dt^2-c(x)d\vec{x}\cdot d\vec{x}$. In order to calculate $G_R$, you essentially need to fully solve the geodesic equations obtained by extremising the action $\int ds$.

The retarded Green's function satisfies $$ \square G_R(t,x;t',x')=-4\pi\delta_4(t,x;t',x'),$$ where $\delta_4$ is the 4-dimensional Dirac distribution. Then the generalized Kirchhoff formula allows us to explicitly write down the value of a solution $\Psi$ of the wave equation at a point $(t,x)$ to the future of an initial data hypersurface $\Sigma^\prime=\{(t',x'):x'\in\mathbb{R}^3\}$:

$$ \Psi(t,x)=-\frac{1}{4\pi}\int_{\Sigma^\prime}(G_R(t,x;t',x')g(x')-f(x')\partial_{t'}G_R(t,x;t',x'))d\Sigma^\prime,$$

where $d\Sigma^\prime=c^{3/2}(x')d_3x'$ is the volume element on $\Sigma^\prime$.

These things (Green's function, waves in curved spacetime) are of great interest in General Relativity, and the GR literature is a good place to look for more details. In particular, I recommend starting with Poisson's Living Review article which covers the geometrical background in a very readable way. You'll find details here (and in the citations) on how to calculate $G_R$, which is required for practical applications of the Kirchhoff formula.

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