Introduction to Acoustics

Rodrigues relation, one has
�1
−1
=
[Pℓ(ξ)] 2 dξ
�
1
2ℓ �2 ℓ!
(−1) ℓ
�
1
−1
(ξ 2 − 1) ℓ
× d2ℓ
dξ2ℓ (ξ2 − 1) ℓ dξ (3.389)
�
1
=
2ℓ �2 ℓ!
�1
(2ℓ)! (1 − ξ 2 ) ℓ dξ
−1
�
1
=
2ℓ �2 �π/2
(2ℓ)!2
ℓ!
0
sin θ 2ℓ+1 dθ. (3.390)
The trigonometric integral Iℓ in the latter expression is
evaluated using the trigonometric identity
d
dθ (sin2ℓ θ cos θ) =−sin 2ℓ+1 θ
+ 2ℓ(sin 2ℓ−2 θ)(1 − sin 2 θ) ,
(3.391)
the integral of which yields the recursion relation
Iℓ = 2ℓ
(2ℓ + 1) Iℓ−1 , (3.392)
and from this one infers
Iℓ = (2ℓ ℓ!) 2
(2ℓ + 1)! I0 . (3.393)
The integral for ℓ = 0 is unity, so one has
�1
−1
[Pℓ(ξ)] 2 �
1
dξ =
2ℓ �2 �
(2ℓℓ!) 2 �
(2ℓ)!2
ℓ! (2ℓ + 1)!
= 2
. (3.394)
2ℓ + 1
Spherical Bessel Functions
The ordinary differential equation (3.362) for the factor
Êℓ(η) takes the form
d
dη
�
η
2 dÊℓ
dη
�
− ℓ(ℓ + 1)Êℓ + η 2 Êℓ = 0 , (3.395)
with the identification for the separation constant that
results from the requirement that the θ-dependent factor
Basic Linear Acoustics 3.11 Spherical Waves 71
be finite at θ = 0andθ = π.Forℓ = 0, a possible solution
is
Ê0 = A0
e iη
η
, (3.396)
as can be verified by direct substitution, with A0 being
any constant. Since there is no corresponding θ dependence
this is the same as the solution (3.327) foran
outgoing spherical wave.
For arbitrary positive integer ℓ, a possible solution
is
h (1)
�
ℓ (η) =−iηℓ − 1
�ℓ d eiη , (3.397)
η dη η
so that, in particular,
h (1) eiη
0 (η) =−i
η ;
h (1) d
1 (η) = i
dη
e iη
η =−
�
1 + i
�
eiη η η
. (3.398)
Alternately, both the real and imaginary parts should be
solutions, so if one writes
h (1)
ℓ (η) = jℓ(η) + iyℓ(η) , (3.399)
then
jℓ(η) = η ℓ
yℓ(η) =−η ℓ
jl (η)
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.1
–0.2
l=0
�
− 1
η
�
− 1
η
l=1
�ℓ d sin η
dη η
l=2
d
dη
l=3
, (3.400)
�ℓ cos η
, (3.401)
η
–0.3
2 4 6 8 10 12 14
Fig. 3.26 Spherical Bessel functions for various orders
η
Part A 3.11