Introduction to Acoustics

equation, the two differential equations being (with η
replacing kr)
�
1 d
sin θ
sin θ dθ
dPℓ
�
+ λℓ Pℓ = 0 , (3.361)
dθ
� �
d 2 dÊℓ
η − λℓÊℓ + η
dη dη
2 Êℓ = 0 . (3.362)
Here λℓ is a constant, termed the separation constant.
Equivalently, with Pℓ regarded as a function of ξ = cos θ,
the first of these two differential equations can be written
�
d
�
1 − ξ
dξ
2� �
dPℓ
+ λℓ Pℓ = 0 . (3.363)
dξ
Legendre Polynomials
Usually, one desires solutions that are finite at both θ = 0
and θ = π,oratξ = 1andξ =−1, but the solutions for
the θ-dependent factor are usually singular at one of
the other of these two end points. However, for special
values (eigenvalues) of the separation constant λℓ,
there exist particular solutions (eigenfunctions) that are
finite at both points. To determine these functions, one
postulates a series solution of the form
∞�
Pℓ(ξ) = aℓ,nξ n , (3.364)
n=0
and derives the recursion relation
n(n − 1)aℓ,n = [(n − 1)(n − 2) − λℓ] aℓ,n−2 .
(3.365)
The series diverges as ξ →±1 unless it only has a finite
number of terms, and such may be so if for some n the
quantity in brackets on the right side is zero. The general
choice of the separation constant that allows this is
λℓ = ℓ(ℓ + 1) , (3.366)
where ℓ is an integer, so that the recursion relation
becomes
n(n − 1)aℓ,n = [(n − ℓ − 2)(n + ℓ − 1)] aℓ,n−2 .
(3.367)
However, aℓ,0 and aℓ,1 can be chosen independently and
the recursion relation can only terminate one of the two
possible infinite series. Consequently, one must choose
aℓ,1 = 0 if ℓ even ;
aℓ,0 = 0 if ℓ odd . (3.368)
If ℓ is even the terms correspond to n = 0, n = 2, n = 4,
up to n = ℓ, while if ℓ is odd the terms correspond to
Basic Linear Acoustics 3.11 Spherical Waves 69
n = 1, n = 3, up to n = ℓ. The customary normalization
is that Pℓ(1) = 1, and the polynomials that are derived are
termed the Legendre polynomials. A general expression
that results from examination of the recursion relation
for the coefficients is
�
Pℓ(ξ) = aℓ,ℓ ξ ℓ ℓ(ℓ − 1)
−
2(2ℓ − 1) ξℓ−2
+ ℓ(ℓ − 2)(ℓ − 1)(ℓ − 3)
(2)(4)(2ℓ − 1)(2ℓ − 3) ξℓ−4 �
+ ... ,
(3.369)
where the last term has ξ raised to either the power of 1 or
0, depending on whether ℓ is odd or even. Equivalently,
if one sets,
aℓ,ℓ = Kℓ
(2ℓ)!
2ℓ , (3.370)
(ℓ!) 2
where Kℓ is to be selected, the series has the relatively
simple form
M(ℓ) �
Pℓ(ξ) = Kℓ (−1) m (2ℓ − 2m)!
2ℓm!(ℓ − m)!(ℓ − 2m)! ξℓ−2m
m=0
M(ℓ) �
= Kℓ bℓ,mξ ℓ−2m . (3.371)
m=0
Here M(ℓ) = ℓ/2 ifℓ is even, and M(ℓ) = (ℓ − 1)/2 if
ℓ is odd, so M(0) = 0, M(1) = 0, M(2) = 1, M(3) = 1,
M(4) = 2, etc.
The coefficients bℓ,m as defined here satisfy the
relation
(ℓ + 1)bℓ+1,m = (2ℓ + 1)bℓ,m − ℓbℓ−1,m−1 ,
(3.372)
as can be verified by algebraic manipulation. A consequence
of this relation is
(ℓ + 1) Pℓ+1(ξ)
Kℓ+1
= (2ℓ + 1)ξ Pℓ(ξ)
Kℓ
− ℓ Pℓ−1(ξ)
Kℓ−1
(3.373)
when ℓ ≥ 1.
The customary normalization is to take Pℓ(1) = 1.
The series for ℓ = 0andℓ = 1 are each of only one
term, and the normalization requirement leads to K0 = 1
and K1 = 1. The relation (3.373) indicates that the normalization
requirement will result, via induction, for all
successive ℓ if one takes Kℓ = 1forallℓ. With this definition,
the relation (3.373) yields the recursion relation
among polynomials of different orders
(ℓ + 1)Pℓ+1(ξ) = (2ℓ + 1)ξ Pℓ(ξ) − ℓPℓ−1(ξ) .
(3.374)
Part A 3.11