Multipole Expansion of Scalar Wave Equation

Before considering the vector wave equation, let us consider the somewhat simpler scalar wave equation. A scalar field $\psi({\bf r},t)$ satisfying the homogeneous wave equation,

$$\nabla^{\,2}\psi - \frac{1}{c^{\,2}}\frac{\partial^{\,2}\psi}{\partial t^{\,2}}=0,$$ (1409)

can be Fourier analyzed in time,

$$\psi({\bf r}, t) = \int_{-\infty}^{\infty} \psi({\bf r}, \omega)\, {\rm e}^{-{\rm i}\,\omega\, t} \,d\omega,$$ (1410)

with each Fourier harmonic satisfying the homogeneous Helmholtz wave equation,

$$(\nabla^{\,2} + k^{\,2})\,\psi({\bf r}, \omega) = 0,$$ (1411)

where $k^{\,2}=\omega^{\,2}/c^{\,2}$ . We can write the Helmholtz equation in terms of spherical coordinates $r$ , $\theta$ , $\varphi$ :

$$\left(\frac{1}{r^{\,2}}\frac{\partial}{\partial r}\,r^{\,2} \frac... ...^2\theta}\frac{\partial^{\,2} }{\partial \varphi^{\,2}}+k^{\,2}\right)\psi = 0.$$ (1412)

As is well known, it is possible to solve this equation via separation of variables to give

$$\psi({\bf r}, \omega) = \sum_{l=0,\infty}\sum_{m=-l,+l} f_{lm}(r)\, Y_{lm}(\theta,\varphi).$$ (1413)

Here, we restrict our attention to physical solutions that are well-behaved in the angular variables $\theta$ and $\varphi$ . The spherical harmonics $Y_{lm}(\theta,\varphi)$ satisfy the following equations:

$$-\frac{\partial^{\,2} Y_{lm}}{\partial\varphi^{\,2}} = m^{\,2}\, Y_{lm},$$ (1414)

$$-\left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \,\sin... ...rac{1}{\sin^2\theta}\frac{\partial^{\,2} }{\partial \varphi^{\,2}}\right)Y_{lm} = l\,(l+1)\,Y_{lm},$$ (1415)

where $l$ is a non-negative integer, and $m$ is an integer that satisfies the inequality $\vert m\vert\leq l$ . The radial functions $f_{lm}(r)$ satisfy

$$\left[\frac{d^{\,2}}{dr^{\,2}} + \frac{2}{r}\frac{d}{dr} + k^{\,2} - \frac{l\,(l+1)}{r^{\,2}} \right]f_{lm}(r) =0.$$ (1416)

With the substitution

$$f_{lm}(r) = \frac{u_{lm}(r)}{r^{\,1/2}},$$ (1417)

Equation (1418) is transformed into

$$\left[\frac{d^{\,2}}{dr^{\,2}} + \frac{1}{r}\frac{d}{dr} +k^{\,2} - \frac{(l+1/2)^{\,2}}{r^{\,2}}\right] u_{lm}(r) =0,$$ (1418)

which is a type of Bessel equation of half-integer order, $l+1/2$ . Thus, we can write the solution for $f_{lm}(r)$ as

$$f_{lm}(r) = \frac{A_{lm}}{r^{\,1/2}}\,J_{l+1/2}(k\,r) + \frac{B_{lm}}{r^{\,1/2}} \,Y_{l+1/2}(k\,r),$$ (1419)

where $A_{lm}$ and $B_{lm}$ are arbitrary constants. The half-integer order Bessel functions $J_{l+1/2}(z)$ and $Y_{l+1/2}(z)$ have analogous properties to the integer order Bessel functions $J_m(z)$ and $Y_m(z)$ . In particular, the $J_{l+1/2}(z)$ are well behaved in the limit $\vert z\vert\rightarrow 0$ , whereas the $Y_{l+1/2}(z)$ are badly behaved.

It is convenient to define the spherical Bessel functions, $j_l(r)$ and $y_l(r)$ , where

$$j_l(z) = \left(\frac{\pi}{2 \,z}\right)^{1/2} J_{l+1/2}(z),$$ (1420)

$$y_l(z) = \left(\frac{\pi}{2 \,z}\right)^{1/2} Y_{l+1/2}(z).$$ (1421)

It is also convenient to define the spherical Hankel functions, $h_l^{(1)}(r)$ and $h_l^{(2)}(r)$ , where

$$h_l^{(1)} (z) = j_l(z)+ {\rm i}\,y_l(z),$$ (1422)

$$h_l^{(2)} (z) = j_l(z)- {\rm i}\,y_l(z).$$ (1423)

Assuming that $z$ is real, $h^{(2)}_l(z)$ is the complex conjugate of $h^{(1)}_l(z)$ . It turns out that the spherical Bessel functions can be expressed in the closed form

$$j_l(z) = (-z)^{\,l} \left(\frac{1}{z}\frac{d}{dz}\right)^l \left(\frac{\sin z}{z}\right),$$ (1424)

$$y_l(z) = -(-z)^{\,l} \left(\frac{1}{z}\frac{d}{dz}\right)^l \left(\frac{\cos z} {z}\right).$$ (1425)

In the limit of small argument,

$$j_l(z) \rightarrow \frac{z^{\,l}}{(2\,l+1)!!}\left[1+ {\cal O}(z^2)\right],$$ (1426)

$$y_l(z) \rightarrow -\frac{(2\,l-1)!!}{z^{\,l+1}}\left[1+ {\cal O}(z^2)\right],$$ (1427)

where $(2l+1)!! = (2l+1)\,(2l-1)\, (2l-3)\cdots 5\cdot 3\cdot 1$ . In the limit of large argument,

$$j_l(z) \rightarrow \frac{\sin(z-l\,\pi/2)}{z},$$ (1428)

$$y_l(z) \rightarrow -\frac{\cos(z-l\,\pi/2)}{z},$$ (1429)

which implies that

$$h_l^{(1)} (z) \rightarrow (-{\rm i})^{\,l+1}\, \frac{ {\rm e}^{+{\rm i}\,z} }{z},$$ (1430)

$$h_l^{(2)} (z) \rightarrow (+{\rm i})^{\,l+1}\, \frac{ {\rm e}^{-{\rm i}\,z} }{z}.$$ (1431)

It follows, from the above discussion, that the radial functions $f_{lm}(r)$ , specified in Equation (1421), can also be written

$$f_{lm}(r) = A_{lm}\,h_l^{(1)}(k\,r) + B_{lm}\,h_l^{(2)}(k\,r).$$ (1432)

Hence, the general solution of the homogeneous Helmholtz equation, (1413), takes the form

$$\psi({\bf r},\omega) = \sum_{l=0,\infty}\sum_{m=-l,+l}\left[A_{lm}\,h_l^{(1)}(k\,r) + B_{lm}\,h_l^{(2)}(k\,r)\right]Y_{lm}(\theta,\varphi).$$ (1433)

Moreover, it is clear from Equations (1412) and (1432)-(1433) that, at large $r$ , the terms involving the $h_l^{(1)}(k\,r)$ Hankel functions correspond to outgoing radial waves, whereas those involving the $h_l^{(2)}(k\,r)$ functions correspond to incoming radial waves.