\documentstyle[twoside]{article}
\input amssym.def
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\markboth{\hfil Reflectionless Boundary Propagation Formulas \hfil EJDE--1995/17}%
{EJDE--1995/17\hfil J. Navarro \& H.A. Warchall\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1995}(1995), No.\ 17, pp. 1--14. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu (147.26.103.110)\newline
telnet (login: ejde), ftp, and gopher access:
ejde.math.swt.edu or ejde.math.unt.edu}
\vspace{\bigskipamount} \\
Reflectionless Boundary Propagation Formulas
for Partial Wave Solutions to the Wave Equation
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35L05, 35L15,
35C10, 35A35, 35A22.
\newline\indent
{\em Key words and phrases:} One-sided wave propagation, Wave equation,
Reflectionless \newline\indent
boundary conditions, Partial waves, Spherical-harmonic decomposition,
Open-space \newline\indent boundary conditions.
\newline\indent
\copyright 1995 Southwest Texas State University and University of
North Texas.\newline\indent
Submitted May 20, 1994. Published Nobember 28, 1995.} }
\date{}
\author{Jaime Navarro \\ Henry A. Warchall}
\maketitle
\begin{abstract}
We consider solutions to the wave equation in 3+1 spacetime dimensions whose
data is compactly supported at some initial time. For points outside a ball
containing the initial support, we develop an outgoing wave condition, and
associated one-way propagation formula, for the partial waves in the
spherical-harmonic decomposition of the solution. The propagation formula
expresses the $l$-th partial wave at time $t$ and radius $a$ in terms of
order-$l$ radial derivatives of the partial wave at time $t-\Delta t$ and
radius $a-\Delta t$. The boundary propagation formula can be applied to any
differential equation that is well-approximated by the wave equation outside a
fixed ball.
\end{abstract}
\section{Introduction}
For hyperbolic partial differential equations with analytic coefficients,
Warchall [3-4] established the local domain of dependence for solutions that,
intuitively speaking, consist of waves that are outgoing outside a convex
region. Analytic continuation was employed to establish those results, leaving
open the question of whether there are explicit Òone-sidedÓ propagation
formulas that serve to advance solutions in time in spatial regions where waves
are outgoing. To date, the only explicit example of such a propagation formula
has been that in [3] for the wave equation in one spatial dimension. Here we
provide another example, for the wave equation in three spatial dimensions.
For the wave equation ${{\partial ^2u} \over {\partial t^2}}-\Delta u=f$ in 3+1
spacetime dimensions, the results in [3] imply the following. Suppose that for
all times $t$ the source $f$ (which could depend on $u$) has spatial
support in the ball $B$ of radius $b$ centered, say, at the origin.
Suppose $u(x,t)$ is a solution whose data at time $t_0$ is supported in $B.$
Let $t_2>t_1>t_0,$ and set $\Delta t\equiv t_2-t_1.$ Let
$x_2\in \Bbb R^3$
be such that $\left| {x_2} \right|>b+\Delta t,$ and let $A$ be the ball of
radius $a\equiv \left| {x_2} \right|$ centered at the origin. Then
$u(x_2,t_2)$ and ${{\partial u} \over {\partial t}}(x_2,t_2)$ are completely
determined by the data at time $t_1$ in the spatial region that is the
intersection of $A$ with the ball of radius $\Delta t$ centered at $x_2$.
This region is shown by the shaded area in the schematic Figure 1.
\begin{figure}
\psboxto(\hsize;0pt){fig.ps}
\caption{Spacetime Regions}
\end{figure}
In this paper, we do not quite achieve the goal of making this dependence
explicit. Instead, we exhibit an $l$-dependent one-sided propagation formula
for the $l$-th partial wave in the spherical harmonic decomposition of $u$
outside of $B$. This results in a formula for $u(x_2,t_2)$ in terms of
(radial derivatives of) the data on the sphere of radius $\left| {x_2}
\right|-\Delta t$ at the time $t_1$. While this formula is local in the
radial coordinate, it involves data on an entire sphere surrounding $B$,
shown as a heavy circle in Figure 1. Still open is the problem of
determining a formula for $u(x_2,t_2)$ in terms of data at time $t_1$ in
the intersection of $A$ with the ball of radius $\Delta t$ centered at
$x_2.$
Our construction begins with the idea of Grote and Keller ([2]) to expand $u$ in
spherical harmonics and to determine an operator that converts the partial
waves into solutions of the wave equation in one spatial dimension. Our work
differs from theirs in that we employ a differential operator instead of an
integral operator, allowing us to obtain a single-point propagation formula, in
addition to differential boundary conditions.
\section{Outgoing Wave Condition}
Suppose $u$ is a classical solution to the homogeneous wave equation in 3+1
spacetime dimensions. Let $(r,\theta ,\phi )$ be spherical coordinates for
$\Bbb R^3$. Let
$$Y_{lm}(\theta ,\phi )=\sqrt {{{(2l+1)(l-m)!} \over {4\pi (l+m)!}}}
P_l^m(\cos \theta )e^{im\phi }$$
be the normalized spherical harmonic function, where
$$P_l^m(z)={{(-1)^m} \over {2^ll!}}(1-z^2)^{m / 2}{{d^{l+m}} \over
{dz^{l+m}}}\left[ {(z^2-1)^l} \right]$$
is the associated Legendre function. We expand $u$ in spherical
harmonics:
$u(x,t)=\sum\limits_{l=0}^\infty {\,\,\sum\limits_{m=-l}^l
{\,\,\,u_{lm}(x,t)}}$ , where
$u_{lm}(x,t)\equiv v_{lm}(r,t)Y_{lm}(\theta ,\phi )$ and
$$v_{lm}(r,t)\equiv \int_0^{2\pi } {\int_0^\pi {\overline {Y_{lm}(\theta ,\phi )
\,\,}u(r,\theta ,\phi,t)\,\,\sin \theta \,\,d\theta \,\,d\phi \,\,}}.$$
Then $v_{lm}$ satisfies
${{\partial ^2v_{lm}} \over {\partial t^2}}=v''_{lm}+{2 \over r}v'_{lm}-
{{l(l+1)} \over {r^2}}v_{lm},$
where a prime denotes partial differentiation with respect to $r$. We may
transform this equation to remove one term by setting
$y_{lm}(r,t)\equiv r^{-l}v_{lm}(r,t).$
Then $y_{lm}$ satisfies
${{\partial ^2y_{lm}} \over {\partial t^2}}=y''_{lm}+{{2(l+1)} \over r}y'_{lm}.$
This is the Euler-Poisson-Darboux equation in odd ÒspatialÓ dimension, which may
be transformed to the one-dimensional wave equation. Assuming
$y_{lm}$ is $(l+2)$ times continuously differentiable in $r$, we set
$z_{lm}(r,t)\equiv \left( {{1 \over r}{\partial \over {\partial r}}}
\right)^l\left[ {r^{2l+1}y_{lm}(r,t)} \right].$
Then, by virtue of the identity ([1])
$$
{{\partial ^2} \over {\partial r^2}}\left( {{1 \over r}{\partial \over
{\partial r}}} \right)^{l- 1}\left[ {r^{2l-1}\psi } \right]=\left(
{{1 \over r}{\partial \over {\partial r}}} \right)^l\left[
{r^{2l}{{\partial \psi } \over {\partial r}}} \right]
\eqno{(1)}
$$
for $\psi \in C^{l+1},$ the function $z_{lm}$ satisfies
${{\partial ^2z_{lm}} \over {\partial t^2}}={{\partial ^2z_{lm}} \over {\partial
r^2}}$.
We denote with a dot partial differentiation with respect to time $t$. Under
the hypothesis that both $u$ and $\dot u$ have spatial support in the ball
$B$ at time $t_0,$ it follows that the supports of $z_{lm}(\,\,\cdot ,t_0)$
and $\dot z_{lm}(\,\,\cdot ,t_0)$ are contained in $[0,b],$ since
$z_{lm}(r,t)=\left( {{1 \over r}{\partial \over {\partial r}}}
\right)^l\left[ {r^{l+1}v_{lm}(r,t)} \right].$
Because $z_{lm}$ is a solution to the one-dimensional wave equation, it follows
that, for all $t>t_0$ and $r>b,$
${{\partial z_{lm}} \over {\partial t}}(r,t)+{{\partial z_{lm}} \over
{\partial r}}(r,t)=0.$
Consequently, $v_{lm}$ satisfies the outgoing wave condition
$$
{\partial \over {\partial r}}\left( {{1 \over r}{\partial \over {\partial
r}}} \right)^l\left[ {r^{l+1}v_{lm}(r,t)} \right]+\left( {{1 \over r}
{\partial \over {\partial r}}} \right)^l\left[
{r^{l+1}{{\partial v_{lm}} \over {\partial t}}(r,t)} \right]=0
\eqno{(2)}
$$
for $t>t_0$ and $r>b.$ This, evaluated at radius $r=a,$
is the boundary condition of [2].
It is convenient to rewrite the outgoing wave condition (2) in a more compact
form. Define the differential operator
$L_l\equiv s^l\left( {-{\partial \over {\partial s}}{1 \over s}} \right)^l,$
and denote by $L_l^*$ its formal adjoint
$L_l^*\equiv \left( {{1 \over s}{\partial \over {\partial s}}} \right)^l
\left[{s^l\cdot } \right]$.
With this notation, the outgoing wave condition for $v = v_{lm}(r,t)$
can be written as
$$
{\partial \over {\partial s}}L_l^*\left( {s\,v} \right)+
L_l^*\left( {s\,\dot v} \right)=0
\eqno{(3)}
$$
\section{One-Sided Propagation Formula}
We may use (3) to advance the solution $u$ in time at locations with $r>b$ as
follows. Each partial wave $u_{lm}$ satisfies the (3+1)-dimensional wave
equation, so we may apply the usual propagation formula to advance $u_{lm}$
in time:
$$
\begin{array}{rl}
4\pi u_{lm}(x,t_2)=\oint \{ & u_{lm}(x+(\Delta t)\omega ,t_1)+
(\Delta t)\left[ \dot u_{lm}(x+(\Delta t)\omega ,t_1) \right. +
\\
&+ \left. (\omega \cdot \nabla u_{lm})(x+(\Delta t)\omega,t_1) \right]\ \}\,d^2\omega,
\end{array}
$$
where the integration is over the unit sphere.
Consider a location $x_2$ with $\left| {x_2} \right|>b+\Delta t,$ and set
$a\equiv \left| {x_2} \right|.$ We will temporarily assume that the direction
of $x_2$ is along the north pole of the spherical coordinate system. Because
$Y_{lm}(\theta \,=\,0,\phi )=0$ for $m\ne 0,$ we see that, with this
assumption, $u_{lm}(x_2,t)=0$ for $m\ne 0,$ for all $t$. Thus
$u(x_2,t)=\sum\limits_{l=0}^\infty {u_{l0}(x_2,t)},$ and we consider only the
time development of
$$u_{l0}(x,t)=\sqrt {{\textstyle{{2l+1} \over {4\pi }}}}P_l(\cos \theta
)v_{l0}(r,t),$$ where $P_l$ is the $l$th-order Legendre polynomial.
Substituting
$w_l(x,t)\equiv \sqrt {{\textstyle{{4\pi } \over {2l+1}}}}u_{l0}(x,t)=
v_{l0}(r,t)P_l(\cos \theta )$
into the propagation formula, changing to integration variable
$s\equiv {\textstyle{1 \over a}}\left| {x_2+(\Delta t)\omega } \right|,$
and integrating by parts the term involving
${{\partial } \over {\partial r}}v_{l0},$
we arrive at the formula
$$
2w_l(x_2,t_2)={{s(s-\mu )} \over \tau }P_l(\mu )\,v(s)\left| {_{s=1-\tau
}^{s=1+\tau }} \right.+\int_{1-\tau }^{1+\tau } {\left\{ {sP_l(\mu )\dot
v(s)-\tau \,P'_l(\mu )v(s)} \right\}\,ds},
\eqno{(4)}
$$
where for brevity we set
$\mu \equiv \mu (s)\equiv {{s^2+1-\tau ^2} \over {2s}}$
and $\tau \equiv {{\Delta t} \over a}$ and
$v(s)\equiv v_{l0}(as,\,\,t_1)$ and
$\dot v(s)\equiv a\,{{\partial v_{l0}} \over {\partial t}}(as,\,\,t_1).$
To make use of the outgoing wave condition (3), we will rewrite the term in (4)
involving $\dot v.$ We will manufacture the expression
$L_l^*\left( {s\,\dot v} \right)$ from the term $sP_l(\mu )\dot v$
in the integrand of (4) by determining a function $Q_l(s)$ such that
$P_l(\mu )=L_l(Q_l).$
Then integration by parts will convert the integrand term
$sP_l(\mu )\dot v=L_l(Q_l)\,\,s\dot v$ to the term $Q_l\,L_l^*(s\dot v),$
which according to the outgoing wave condition is equal to
$-\,Q_l\,{\partial \over {\partial s}}L_l^*(sv).$
Additional integration by parts converts this term to
$s\,v\,L_l\left( {{\partial \over {\partial s}}Q_l} \right),$
which turns out to be equal to the opposite of the only other integrand term,
$-\,\tau \,P'_l(\mu )v,$
thus converting the integrand in (4) to zero and reducing the integral to
boundary terms. We will furthermore choose our function $Q_l(s)$ such that
all boundary terms at $s=1+\tau $ vanish.
The required function $Q_l(s)$ could be obtained by direct $l$-fold
integration of the criterion $P_l(\mu )=L_l(Q_l)$ with judicious choice of
constants of integration. We prefer, however, to define $Q_l(s)$ as the
result of that process.
\paragraph{Definition:} For fixed $\tau $ and nonnegative integer $l$, we set
$$Q_l(s)\equiv {{(-1)^l} \over {2^ll!}}\left( {(s-\tau )^2-1} \right)^l.$$
In terms of the given definitions of $L_l$ and $\mu,$ we have the following
facts.
\paragraph{Lemma 1:}
$s\,L_l\left( {{\partial \over {\partial s}}Q_l} \right)=\tau \,P'_l(\mu )$
for all nonnegative integer $l$.
\paragraph{Lemma 2:}
$L_l(Q_l)=P_l(\mu )$ for all nonnegative integer $l$.
\paragraph{Lemma 3:} For integer $l\ge 1$ and functions $f$ and $g$ in
$C^l(\Bbb R)$,
$$
\int_\alpha ^\beta {(L_lf)(s)\,\,g(s)\,\,ds}=\left. {\Gamma _l(f,g)(s)}
\right|_{s=\alpha }^{s=\beta }+\int_\alpha ^\beta {f(s)\,\,(L_l^*g)(s)\,\,ds},
$$
where
$$
\Gamma _l(f,g)(s)\equiv -\sum\limits_{j=1}^l {\left[ {\left( {-{\partial \over
{\partial s}}{1 \over s}} \right)^{l-j}f(s)} \right]\,\,\left[ {{1 \over s}
\left( {{1 \over s}{\partial \over {\partial s}}}
\right)^{j-1}\left( {s^lg(s)} \right)} \right]}.
$$
The proofs of Lemma 1 and Lemma 2 are in Section 6. Lemma 3 is established by
straightforward integration by parts.
We now use these results to carry out the computation outlined above for the
term involving $\dot v$ in our integral formula (4):
For $l=0$ we have
$$\int_{1-\tau }^{1+\tau } {\,sP_0(\mu )\dot v(s)\,ds}=
\int_{1-\tau }^{1+\tau }
{\,\,s\,\dot v(s)\,ds}=-\int_{1-\tau }^{1+\tau } {\,{\partial \over
{\partial s}}(\,s\,v)\,ds}$$
from direct application of the outgoing wave condition. Since
$P'_0(z)=0$ for all $z$, formula (4) reduces in this case to
$2w_0(x_2,t_2)=\left. {{{s(s-\mu -\tau )} \over \tau }v(s)}
\right|_{s=1-\tau}^{s=1+\tau }.$ Since $\mu (1+\tau )=\mu (1-\tau )=1,$
we have $w_0(x_2,t_2)=(1-\tau )\,\,v(1-\tau ).$
For $l\ge 1$ we have
\begin{eqnarray*}
\lefteqn{\int_{1-\tau }^{1+\tau } {\,sP_l(\mu )\dot v(s)\,ds}} \\
&=&\int_{1-\tau}^{1+\tau } {\,\,L_l(Q_l)\,s\,\dot v(s)\,ds}
\hspace{1cm}\hbox{(by Lemma 2)}
\\
&=&\left. {\Gamma _l(Q_l,\,s\dot v)} \right|_{1-\tau }^{1+\tau
}\,\,+\,\,\int_{1-\tau }^{1+\tau } {\,\,Q_l(s)\,\,L_l^*\,(s\,\dot v)\,\,ds}
\hspace{1cm}\hbox{(by Lemma 3)}
\\
&=&\left. {\Gamma _l(Q_l,\,s\dot v)} \right|_{1-\tau }^{1+\tau}
\,\,+\,\,\int_{1-\tau }^{1+\tau } {\,\,Q_l(s)\,\left( {-{\partial
\over {\partial s}}L_l^*\,(s\,v)} \right)\,\,\,ds}
\hspace{1cm}\hbox{(by (3))}
\\
&=&\left. {\left[ {\Gamma _l(Q_l,\,s\dot v)-Q_l(s)L_l^*\,(s\,v)\,} \right]}
\right|_{1-\tau }^{1+\tau }\,\,+
\\
& &\;+\int_{1-\tau }^{1+\tau } {\,\,\left(
{{\partial \over {\partial s}}Q_l(s)} \right)\,\,L_l^*\,(s\,v)\,\,\,ds}
\hspace{1cm}\hbox{(by I.B.P)}
\\
&=&\left. {\left[ {\Gamma _l(Q_l,\,s\dot v)-\Gamma _l({\textstyle{\partial
\over {\partial s}}}Q_l,\,sv)- Q_l(s)L_l^*\,(s\,v)\,} \right]}
\right|_{1-\tau }^{1+\tau }\,\,+
\\
& &\;+\int_{1-\tau }^{1+\tau } {\,\,L_l\left(
{{\textstyle{\partial \over {\partial s}}}Q_l} \right)\,s\,v(s)\,\,\,ds}
\hspace{1cm}\hbox{(by Lemma 3)}
\\
&=&\left. {\left[ {\Gamma _l(Q_l,\,s\dot v)-\Gamma _l({\textstyle{\partial
\over {\partial s}}}Q_l,\,sv)- Q_l(s)L_l^*\,(s\,v)\,} \right]}
\right|_{1-\tau }^{1+\tau }\,\,+
\\
&&\;+\int_{1-\tau }^{1+\tau } {\,\,\tau
\,P'_l(\mu )\,v(s)\,\,\,ds}.
\hspace{1cm}\hbox{(by Lemma 1)}
\end{eqnarray*}
Thus our integral formula for the solution becomes
$$
\begin{array}{rcl}
2w_l(x_2,t_2)&=&\left[
{1 \over \tau }s(s-\mu)P_l(\mu )\,v(s)+\Gamma _l(Q_l,\,s\dot v) \right.
\\
&&-\left.\left. {\Gamma _l({\textstyle{\partial \over
{\partial s}}}Q_l,\,sv)-Q_l(s)L_l^*\,(s\,v)\,} \right]
\right|_{s=1-\tau }^{s=1+\tau }
\end{array}
\eqno{(5)}
$$
We claim that the expression in square brackets in (5) vanishes for
$s=1+\tau .$ The reasons are the following.
For $l\ge 1,$ it follows immediately from the definition of $Q_l$ that
$Q_l(1+\tau )=0.$ Thus the last term $-Q_l(s)L_l^*\,(s\,v)$
vanishes for $s=1+\tau .$
It also follows from the definition that derivatives of $Q_l(s)$
of orders less than $l$ vanish at $s=1+\tau ,$
because every such derivative contains at least one factor of
$\left( {(s-\tau )^2-1} \right).$ Because the quantity
$\Gamma _l(f,g)$ involves derivatives of $f$ and $g$ only of orders
$0,\,\,1,\,\,\ldots ,\,\,(l-1),$ the term $\Gamma _l(Q_l,\,s\dot v)$
vanishes for $s=1+\tau .$
In contrast, the summation in the third term
$-\Gamma _l({\textstyle{\partial \over {\partial s}}}Q_l,\,sv)$
contains exactly one term with an order-$l$ derivative of
$Q_l$. To evaluate this term at $s=1+\tau$, let V.T. stand for terms that
are polynomial in $s$ and contain at least one factor of
$\left( {(s-\tau )^2-1} \right)$; such terms vanish when evaluated with
$s=1+\tau$. Bearing in mind that derivatives of $Q_l(s)$ of orders less
than $l$ vanish at $s=1+\tau$, for the third term in (5) we have
\begin{eqnarray*}
-\Gamma_l({\textstyle{\partial \over {\partial s}}}Q_l,\,sv)
& =&\sum\limits_{j=1}^l {\left[ {\left( {-{\partial \over
{\partial s}}{1 \over s}} \right)^{l-j}
{\textstyle{\partial \over {\partial s}}}Q_l } \right]
\,\,\left[ {{1 \over s}
\left( {{1 \over s}{\partial \over {\partial s}}}
\right)^{j-1}\left( {s^l s v} \right)} \right]} \\
& =& {\left[ {\left( {-{\partial \over
{\partial s}}{1 \over s}} \right)^{l-1}
{\textstyle{\partial \over {\partial s}}}Q_l } \right]
\,\,\left[ {s^l v} \right]} + {\rm V.T.} \\
& =& -s v (-1)^l {\textstyle{\partial^l \over {\partial s^l}}}Q_l + {\rm V.T.} \\
& =& -s v {\textstyle{1 \over {2^l l!}}}
{\textstyle{\partial^l \over {\partial s^l}}}
\left( (s-\tau)^2 -1 \right) ^l + {\rm V.T.} \\
& =& -s v (s-\tau)^l + {\rm V.T.}
\end{eqnarray*}
Thus
$$
\left. {-\Gamma_l({\textstyle{\partial \over {\partial s}}}Q_l,\,sv)}
\right| _{s=1+\tau } = -(1+\tau )\,\,v(1+\tau ).
$$
On the other hand, because $\mu (1+\tau )=1$ and $P_l(1)=0$
for all $l$, the first term in (5) yields
$\left. {{\textstyle{1 \over \tau }}s(s-\mu )P_l(\mu )\,v(s)}
\right|_{s=1+\tau }=(1+\tau )\,v(1+\tau ),$
which cancels the contribution from the third term.
Since $\mu (1-\tau )=1,$ our propagation formula becomes finally
$$
\begin{array}{rl}
2w_l(x_2,t_2)= & (1-\tau )\,v(1-\tau )+
\\
&+ \left. {\left({Q_l(s)L_l^*\,(s\,v)+
\Gamma _l({\textstyle{\partial \over {\partial s}}}Q_l,\,sv)-
\Gamma _l(Q_l,\,s\dot v)\,} \right)} \right|_{s=1-\tau },
\end{array}
\eqno{(6)}
$$
where $\tau \equiv {{\Delta t} \over a}$ and $v(s)\equiv v_{l0}(as,\,\,t_1)$
and $\dot v(s)\equiv a\,{{\partial v_{l0}} \over {\partial t}}(as,\,\,t_1).$
This formula expresses the value of $w_l(x_2,t_2)$ in terms of derivatives of
$v_{l0}$ at the single point $(r,t)=(a-\Delta t,\,\,t_1).$ Specifically,
radial derivatives of $v_{l0}(r,\,\,t_1)$ to order $l$ and of $\dot
v_{l0}(r,\,\,t_1)$ to order $(l-1)$ are required, only at $r=a-\Delta t.$ We
note that formula (6) is also valid for $l=0,$ provided $\Gamma _0(f,g)$
is defined to be zero.
\section{Propagation at General Locations}
We know how to advance $u(x_2,t)$ in time, using the one-sided propagation
formulas for
$w_l(x_2,t)=\sqrt {{\textstyle{{4\pi } \over {2l+1}}}}\,\,u_{l0}(x_2,t),$
in the case when the direction of $x_2$ is along the north pole ($\theta =0$)
of the spherical coordinate system $(r,\theta ,\phi ).$ For other directions of
$x_2$ with, say, $(\theta ,\phi )=(\theta _1,\phi_1),$ we may rotate the
coordinate system to place $x_2$ along the new polar axis, compute the
coefficients $v_{lm}^{(1)}(r,t)$ in the expansion of $u$ with respect to
spherical harmonics in the new coordinate system, and apply our propagation
formula (6) to $v_{l0}^{(1)}(r,t)$ in place of $v_{l0}(r,t)$ to determine
$w_l^{(1)}(x_2,t_2),$ and hence determine
$u(x_2,t_2)=\sum\limits_{l=0}^\infty {\,\,}\sqrt {{\textstyle{{2l+1} \over
{4\pi }}}}\,\,w_l^{(1)}(x_2,t_2).$
If we envisioned a numerical algorithm based on a truncated spherical-harmonic
expansion combined with these results, we would find the recomputation of
coefficients for different orientations of coordinate system to be wasteful.
Such recomputation is in fact unnecessary, because the coefficients are related
by the addition formula for spherical harmonics:
\begin{eqnarray*}
v_{l0}^{(1)}(r,t) &=&\sqrt {{{2l+1} \over {4\pi }}}\int_0^{2\pi }
{\int_0^\pi {P_l(\cos \theta ')\,\,u(r,\theta ,\phi ,t)\,\,\sin \theta
\,\,d\theta \,\,d\phi }}
\\
&=&\sqrt {{{4\pi } \over {2l+1}}}\int_0^{2\pi } {\int_0^\pi
{\sum\limits_{m=-l}^l {Y_{lm}(\theta _1,\phi _1)\,\overline {Y_{lm}
(\theta
,\phi )}}\,u(r,\theta ,\phi ,t)\sin \theta \,d\theta \,d\phi }}
\\
&=&\sqrt {{{4\pi } \over {2l+1}}}\sum\limits_{m=-l}^l {Y_{lm}(\theta _1,\phi
_1)\,\,v_{lm}(r,t)}
\end{eqnarray*}
This gives the coefficients $v_{l0}^{(1)}(r,t)$ in terms of the coefficients
$v_{lm}(r,t)$ that can be computed once and for all in a fixed coordinate
system.
We now insert this last expression for $v_{l0}^{(1)}$ into (6) to compute
$w_l^{(1)}(x_2,t_2).$ Because (6) is linear in $v$ and $\dot v$, and
involves only radial derivatives, we may rearrange the sums to obtain
\begin{eqnarray*}
2w_l^{(1)}(x_2,t_2)&= & \sqrt {{\textstyle{{4\pi } \over
{2l+1}}}}\sum\limits_{m=-l}^l
\, Y_{lm}(\theta _1,\phi _1) \biggl[ (1-\tau )\,v(1-\tau )+
\\
&&+ \left. \left. \left( Q_l(s)L_l^*\,(s\,v)+
\Gamma_l({\textstyle{\partial \over {\partial s}}}Q_l,\,sv)-
\Gamma _l(Q_l,\,s\dot v)\, \right) \right|_{s=1-\tau } \right]
\end{eqnarray*}
where now $v(s)\equiv v_{lm}(as,\,\,t_1)$
and $\dot v(s)\equiv a\,{{\partial v_{lm}} \over {\partial t}}(as,\,\,t_1).$
Thus, for general $x_2=(a,\theta _1,\phi _1),$ we have
$$
u(x_2,t_2)=\sum\limits_{l=0}^\infty {\,\,}\sqrt {{\textstyle{{2l+1} \over
{4\pi }}}}\,\,w_l^{(1)}(x_2,t_2)=\sum\limits_{l=0}^\infty {\,\,}
\sum\limits_{m=-l}^l {\,\,}Y_{lm}(\theta _1,\phi _1)v_{lm}(a,t_2)
$$
where $v_{lm}(a,t_2)$ is obtained from the explicit one-sided propagation
formula
$$
\begin{array}{rcl}
v_{lm}(a,t_2) &= &{\textstyle{1 \over 2}} (1-\tau )\,v(1-\tau )+
\\
&&+ \left. {\textstyle{1 \over 2}}{\left( {Q_l(s)L_l^*\,(s\,v)+
\Gamma _l({\textstyle{\partial \over {\partial s}}}Q_l,\,sv)-
\Gamma _l(Q_l,\,s\dot v)\,} \right)} \right|_{s=1-\tau }
\end{array}
\eqno{(7)}
$$
where $\tau \equiv {{\Delta t} \over a}$ and $v(s)\equiv v_{lm}(as,\,\,t_1)$
and $\dot v(s)\equiv a\,{{\partial v_{lm}} \over {\partial t}}(as,\,\,t_1).$
We note that, because ${{\partial u} \over {\partial t}}(x,t)$ also satisfies
the wave equation, we may obtain analogous propagation formulas for
${{\partial v_{lm}} \over {\partial t}}(a,t_2)$ by applying (7) with $v(s)$
on the right-hand side replaced by
${{\partial v_{lm}} \over {\partial t}}(as,t_1),$
and with $\dot v(s)$ on the right-hand side replaced by
$$a{{\partial ^2v_{lm}} \over {\partial t^2}}(as,t_1)=av''_{lm}(as,t_1)+
{2 \over s}v'_{lm}(as,t_1)-{{l(l+1)} \over {as^2}}v_{lm}(as,t_1).$$
In detail, our algorithm for advancing $u$ on the boundary sphere
$\left| x \right|=a>b+\Delta t$ from time $t_1$ to time $t_2=t_1+\Delta t$
is the following:
\ \\
\hangindent=1cm \hangafter=2
(I) \ \ Given initial data $u(x,t_1)$ and $\dot u(x,t_1)$ in a spatial
neighborhood of the sphere of radius $(a-\Delta t),$ compute \\
$v_{lm}(r,t_1)\equiv \int_0^{2\pi } {\int_0^\pi {\overline {Y_{lm}
(\theta,\phi ) \,\,}u(r,\theta ,\phi ,t_1)\,\sin \theta \,\,d\theta \,d\phi}}$
and \\ $\dot v_{lm}(r,t_1)\equiv \int_0^{2\pi }{\int_0^\pi {\overline
{Y_{lm}(\theta ,\phi) \,\,}\dot u(r,\theta,\phi ,t_1)\,\sin \theta
\,\,d\theta \,d\phi }}$ for $r$ near $(a-\Delta t),$ for $0\le l\le N$
and $-l\le m\le l,$ where $N$ is the highest order of spherical harmonic
to be used.
\ \\
\hangindent=1cm \hangafter=2
(II) \ \ Tabulate the numbers \\
$v_{lm}(a-\Delta t,t_1),$ $v'_{lm}(a-\Delta t,t_1),$
$v''_{lm}(a-\Delta t,t_1),$ . . . , $v_{lm}^{(N+1)}(a-\Delta t,t_1),$
\\
$\dot v_{lm}(a-\Delta t,t_1),$ $\dot v'_{lm}(a-\Delta t,t_1),$
$\dot v''_{lm}(a-\Delta t,t_1),$ . . . , $\dot v_{lm}^{(N)}(a-\Delta t,t_1),$
\\
for $0\le l\le N$ and $-l\le m\le l.$
\ \\
\hangindent=1cm \hangafter=2
(III) \ \ Apply formula (7) to compute $v_{lm}(a,t_2)$ for $0\le l\le N$
and $-l\le m\le l.$ Apply the indicated modification of (7) to compute
$\dot v_{lm}(a,t_2).$ Then
$u(a,\theta ,\phi ,t_2)=\sum\limits_{l=0}^\infty {\,\,}\sum\limits_{m=-l}^l
{\,\,}Y_{lm}(\theta ,\phi )\,\,v_{lm}(a,t_2),$
with an analogous formula for $\dot u$ in terms of $\dot v_{lm}.$
\ \\
\hangindent=1cm \hangafter=2
(IV) \ \ Use these values for $u$ and $\dot u$ on the boundary sphere
$\left| x \right|=a$ together with the numerical algorithm of choice to update
$u$ and $\dot u$ for $\left| x \right|