## Spherically symmetric waves

Spherically symmetric three-dimensional
waves propagate in the radial direction $r$ only so that
$u = u(r,t)$. The fully three-dimensional wave equation

$$
\frac{\partial^2u}{\partial t^2}=\nabla\cdot (c^2\nabla u) + f
$$

then reduces to the spherically symmetric wave equation

<!-- Equation labels as ordinary links -->
<div id="_auto1"></div>

$$
\begin{equation}
\frac{\partial^2u}{\partial t^2}=\frac{1}{r^2}\frac{\partial}{\partial r}
\left(c^2(r)r^2\frac{\partial u}{\partial t}\right)
+ f(r),\quad r\in (0,R),\ t>0
\thinspace . 
\label{_auto1} \tag{1}
\end{equation}
$$

Assume that the wave velocity $c$ is constant. One can easily show
that the function $v(r,t) = ru(r,t)$ fulfills a standard wave equation
in Cartesian coordinates. To this end, insert $u=v/r$ in

$$
\frac{1}{r^2}\frac{\partial}{\partial r}
\left(c^2(r)r^2\frac{\partial u}{\partial t}\right)
$$

to obtain

$$
r\left(\frac{d c^2}{dr}\frac{\partial v}{\partial r} +
c^2\frac{\partial^2 v}{\partial r^2}\right) - \frac{d c^2}{dr}v
\thinspace .
$$

The two terms in the parenthesis can be combined to

$$
r\frac{\partial}{\partial r}\left( c^2\frac{\partial v}{\partial r}\right),
$$

which is recognized as the variable-coefficient Laplace operator in
one Cartesian coordinate. The spherically symmetric wave equation in
terms of $v(r,t)$ now becomes

<!-- Equation labels as ordinary links -->
<div id="_auto2"></div>

$$
\begin{equation}
\frac{\partial^2u}{\partial t^2}=
\frac{\partial}{\partial r}
\left(c^2(r)\frac{\partial v}{\partial t}
-\frac{1}{r}\frac{d c^2}{dr}v\right) + rf(r),\quad r\in (0,R),\ t>0
\thinspace . 
\label{_auto2} \tag{2}
\end{equation}
$$

In the case of constant wave velocity $c$, this equation reduces to
the wave equation in a single Cartesian coordinate:

<!-- Equation labels as ordinary links -->
<div id="wave:app:rsymm:Cart"></div>

$$
\begin{equation}
\frac{\partial^2u}{\partial t^2}=
\frac{\partial}{\partial r}
\left(c^2(r)\frac{\partial v}{\partial t}\right)
+ rf(r),\quad r\in (0,R),\ t>0
\thinspace . 
\label{wave:app:rsymm:Cart} \tag{3}
\end{equation}
$$

That is, any program for solving the one-dimensional wave equation
in a Cartesian coordinate system can be used to
solve ([3](#wave:app:rsymm:Cart)), provided the source term is
multiplied by the coordinate. Moreover, if $r=0$ is included in the
domain, spherical symmetry demands that $\partial u/\partial r=0$ at
$r=0$, which means that

$$
\frac{\partial u}{\partial r} = \frac{1}{r^2}\left(
r\frac{\partial v}{\partial r} - v\right) = 0,\quad r=0,
$$

implying $v(0,t)=0$ as a necessary condition. For practical applications,
we exclude $r=0$ from the domain and assume that some boundary
condition is assigned at $r=\epsilon$, for some $\epsilon >0$.

f446d923e8dd1d22e804564153886b98e0ccea48
ipynb
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1. YES
2. YES