Multipole Expansion of Scalar Wave Equation

(1409) |

can be Fourier analyzed in time,

with each Fourier harmonic satisfying the homogeneous Helmholtz wave equation,

where . We can write the Helmholtz equation in terms of spherical coordinates , , :

(1412) |

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

(1413) |

Here, we restrict our attention to physical solutions that are well-behaved in the angular variables and . The spherical harmonics satisfy the following equations:

(1414) | ||

(1415) |

where is a non-negative integer, and is an integer that satisfies the inequality . The radial functions satisfy

With the substitution

(1417) |

Equation (1418) is transformed into

(1418) |

which is a type of Bessel equation of half-integer order, . Thus, we can write the solution for as

where and are arbitrary constants. The half-integer order Bessel functions and have analogous properties to the integer order Bessel functions and . In particular, the are well behaved in the limit , whereas the are badly behaved.

It is convenient to define the *spherical Bessel functions*,
and
, where

(1420) | ||

(1421) |

It is also convenient to define the

(1422) | ||

(1423) |

Assuming that is real, is the complex conjugate of . It turns out that the spherical Bessel functions can be expressed in the closed form

(1424) | ||

(1425) |

In the limit of small argument,

where . In the limit of large argument,

(1428) | ||

(1429) |

which implies that

It follows, from the above discussion, that the radial functions , specified in Equation (1421), can also be written

(1432) |

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

(1433) |

Moreover, it is clear from Equations (1412) and (1432)-(1433) that, at large , the terms involving the Hankel functions correspond to outgoing radial waves, whereas those involving the functions correspond to incoming radial waves.