#!/usr/bin/env python
# coding: utf-8

# ## 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 $\epsilon >0$.