Introduction to Acoustics

74 Part A Propagation of Sound
Part A 3.11
the scattered wave. This term is required to be made up
of waves that propagate out from the sphere, which is
centered at the origin. With some generality, one can use
the principle of superposition and set
�
ˆpsc = ˆPinc
ℓ=0
Aℓ Pℓ(cos θ)h (1)
ℓ (kr) , (3.425)
with the factors in the terms in the sum representing the
Legendre polynomials and spherical Hankel functions.
The individual terms in the sum are sometimes referred
to as partial waves. The analytical task is to determine the
coefficients Aℓ. (The expression above applies to scattering
from any axisymmetric body, including spheroids
and penetrable objects, where the properties might vary
with r and θ. The discussion here, however, is limited to
that of the rigid sphere.)
With the plane wave expansion (3.423) inserted for
the direct wave, the sum of incident and scattered waves
takes the form
ˆp = ˆPinc
∞�
ℓ=0
� (2ℓ + 1)i ℓ jℓ(kr) + Aℓh (1)
ℓ (kr)�
× Pℓ(cos θ) . (3.426)
the boundary condition imposed by the rigidity of the
sphere is that ˆvr = 0 at r = a, or equivalently that
∂ ˆp/∂r = 0atr = a. With the aid of the radial component
of the Euler equation (3.175) and of the linear
independence of the various Legendre polynomials, this
yields
d � ℓ
(2ℓ + 1)i jℓ(kr) + Aℓh
dr
(1)
ℓ (kr)� = 0
at r = a . (3.427)
The desired coefficients are consequently
Aℓ =−(2ℓ + 1)i ℓ
� ddr
jℓ(kr) �
r=a
. (3.428)
� ddr h (1)
ℓ (kr)�
r=a
Far-Field Scattering
In the limit of large kr, the asymptotic expression (3.412)
for the spherical Hankel function can be used, and the
scattered wave takes the asymptotic form
where
ˆpsc → ˆPinc f (θ) eikr
kr
f (θ) =
, (3.429)
∞�
(−i) ℓ+1 Aℓ Pℓ(cos θ) (3.430)
ℓ=0
is a complex dimensionless function of only the angle θ.
This asymptotic form holds for scattering from any
axisymmetric object of bounded extent. In the more general
case when the object is not axisymmetric, one should
regard the function f as being also a function of the
azimuthal angle φ,sothat
f (θ) → f (θ,φ) . (3.431)
The prediction in all cases is that the far-field scattered
wave resembles an outgoing spherical wave, but with an
amplitude that depends on the direction of propagation.
The far-field intensity associated with the scattered
wave is asymptotically entirely in the radial direction,
and in accord with (3.281), its time average is given by
Isc = 1 1
2 ρc | ˆpsc| 2 , (3.432)
or,
Isc = 1
2
| ˆPinc| 2
ρc
| f (θ,φ)|
2 1
. (3.433)
2
k 2 r
(ka) 2
0.5 (ka) 2
ka