Introduction to Acoustics

68 Part A Propagation of Sound
Part A 3.11
the desire to use a linear approximation, one approximates
the unit normal vector for a given point on the
surface of the sphere to be the same as when the sphere
is centered at the origin, so that n ≈ er, wherethelatter
is the unit vector in the radial direction. The normal
component of the fluid velocity is then approximately
vn = vc (ez·er) cos(ωt) . (3.348)
The dot product is cos θ,whereθ is the polar angle.
One also makes the approximation that the boundary
condition is to be imposed, not at the actual (moving)
location of the point on the surface, but at the place in
space where that point is when the sphere is centered
at the origin. All these considerations lead to the linear
acoustics boundary condition
ˆvr = vc cos θ at r = a (3.349)
for the complex amplitude of the fluid velocity.
The feature distinguishing this boundary condition
from that for the radially oscillating sphere is the factor
cos θ. The plausible conjecture that both ˆvr and ˆp continue
to have the same θ dependence for all values of r
is correct in this case, and one can look for a solution
of the Helmholtz equation that has such a dependence,
such as
ˆp = B ∂
�
eikr �
= B cos θ
∂z r
d
�
eikr �
. (3.350)
dr r
The first part of this relation follows because a derivative
of a solution with respect to any Cartesian coordinate
is also a solution and because, as demonstrated
in a previous part of this section, eikr /r is a solution.
The second part follows because r2 = z2 + x2 + y2 ,
so ∂r/∂z = z/r = cos θ. The quantity B is a complex
numerical constant that remains to be determined.
The radial component of Euler’s equation (3.175)
for the constant-frequency case requires that
∂ ˆp
−iωρˆvr =− , (3.351)
∂r
where on the right side the differentiation is to be carried
out at constant θ. Given the expression (3.350), the
corresponding relation for the radial component of the
fluid velocity is consequently
ˆvr = B d2
cos θ
iωρ dr2 �
eikr �
. (3.352)
r
The boundary condition at r = a is satisfied if one takes
B to have a value such that
vc = B
�
d2 �
eikr ��
. (3.353)
iωρ r
dr 2
r=a
The indicated algebra yields
�
iωρa
B =−
3vc 2 + 2ika + k2a2 �
e −ika , (3.354)
so the complex amplitude of the acoustic part of the
pressure becomes
k
ˆp =ρcvc
2a2 2 + 2ika + k2a2 �
1 − 1
� �a �
ikr r
×e −ik[r−a] cos θ, (3.355)
while the radial component of the fluid velocity is
�
2 + 2ikr + k2r 2
ˆvr = vc
2 + 2ika + k2a2 ��
a3 r3 �
e −ik[r−a] cos θ.
(3.356)
The radiation impedance is
�
k2a2 + ika
Zrad = ρc
2 + 2ika + k2a2 �
. (3.357)
Various simplifications result when considers limiting
cases for the values of kr and ka. A case of common
interest is when ka ≪ 1 (small sphere) and kr ≫ 1(far
field), so that
�
ω2ρvca ˆp =
3 �
eikr cos θ. (3.358)
c r
3.11.4 Axially Symmetric Solutions
The example discussed in the previous subsection of
radiation from a transversely oscillating sphere is one of
a class of solutions of the linear acoustic equations where
the field quantities depend on the spherical coordinates
r and θ but not on the azimuthal angle φ. The Helmholtz
equation for such circumstances has the form
1
r2 � �
∂ 2 ∂ ˆp
r +
∂r ∂r
1
r2 � �
∂ ∂ ˆp
sin θ + k
sin θ ∂θ ∂θ
2 ˆp = 0 .
(3.359)
A common technique is to build up solutions of this
equation using the principle of superposition, with the
individual terms being factored solutions of the form
ˆpℓ = Pℓ(cos θ)Êℓ(kr) , (3.360)
where ℓ is an integer that distinguishes the various particular
separated solutions. Insertion of this product into
the Helmholtz equation leads to the conclusion that each
factor must satisfy an appropriate ordinary differential