Introduction to Acoustics

is the Wronskian for the Bessel and Neumann functions.
Here the primes denote differentiation with respect to the
argument η = kw of the indicated function. One can derive
from the differential equation (3.453) that these two
functions independently satisfy the Wronskian relation
ηW(J0, Y0) = constant , (3.495)
and this is so regardless of the specialized definitions of
the two functions. The constant can be evaluated from
values and derivatives at any given point. In particular,
one can use the asymptotic expressions, and the derived
constant is 2/π. Thus one has
Iw,av = 1 |C|
π
2
� �
1 1 �
= �
d ˆm �2
ρcw ρcw 16π �ω2 �
dℓ � . (3.496)
3.12.3 Radially Oscillating Cylinder
A classic example for the radiation of cylindrical waves
is that where a cylinder has nominal radius a, andan
instantaneous radius
wcyl(t) = a + vo
sin(ωt) . (3.497)
ω
Here vo/ω is the amplitude of the deviations of the actual
radius from the nominal value a. For the linear acoustics
idealization to be valid, it is required that this deviation
be substantially less than a,sothat
vo ≪ ωa . (3.498)
The boundary condition on the fluid dynamic equations
should ideally be
vw = vo cos(ωt) at w = wcyl(t) , (3.499)
but (also in keeping with the linear acoustics idealization)
it is replaced by
vw = vo cos(ωt) at w = a . (3.500)
The corresponding boundary condition on the complex
amplitude is
ˆvw = vo at w = a . (3.501)
If one takes the complex amplitude of the acoustic
pressure to be of the form
ˆp = AH (1)
0 (kw) , (3.502)
Basic Linear Acoustics 3.12 Cylindrical Waves 81
where A is a complex number to be determined, then
the radial component of the fluid velocity is
ˆvw = A d
iωρ dw H(1)
0 (kw) . (3.503)
The imposing of the boundary condition determines the
value of A and one obtains
ˆp = iωρvo
H (1)
0 (kw)
� d
dw H (1)
0 (kw)�
w=a
. (3.504)
3.12.4 Transversely Oscillating Cylinder
If the cylinder is rigid and oscillating back and forth
in the x-direction, the analysis is similar to that for
the transversely oscillating sphere. The complex amplitude
of the acoustic part of the pressure should have
the general form
ˆp = B d
dx H(1)
d
0 (kw) = B cos φ
dw H(1)
0 (kw) ,
(3.505)
where B is to be determined from the boundary condition.
The latter can also be written
ˆp =−kBcos φH (1)
1 (kw) , (3.506)
in accord with the recursion relation (3.471). The latter
version is of the standard form for a factored solution of
the Helmholtz equation in cylindrical coordinates.
Euler’s equation gives the radial component of the
fluid velocity as
ˆvw = 1
iωρ
B cos φ d2
dw
H(1)
2 0
(kw) . (3.507)
The appropriate approximate boundary condition is that
ˆvw = vc cos φ at w = a . (3.508)
This allows one to identify
B =
iωρvc
� d 2
dw 2 H (1)
0 (kw)�
w=a
. (3.509)
Part A 3.12