Introduction to Acoustics

90 Part A Propagation of Sound
Part A 3.15
y
z0
y0
x
Fig. 3.41 Generic sketch of a duct that carries guided sound
waves
rigid wall boundary condition imposed, are such that the
cross-sectional eigenfunctions are orthogonal, so that
�
ΨnΨm dA = 0 (3.574)
if n and m correspond to different guided modes. The
eigenvalues α2 n , moreover, are all real and nonnegative.
However, for cross sections that have some type of symmetry,
it may be that more than one linearly independent
eigenfunction Ψn (modes characterized by different values
of the index n) correspond to the same numerical
value of α2 n . In such cases the eigenvalue is said to be
degenerate.
The variation of guided mode amplitudes with
source excitation is ordinarily incorporated into the
axial wave functions Xn(x), which satisfy the onedimensional
Helmholtz equation
z
d 2 Xn
dx 2 + (k2 − α 2 )Xn = 0 . (3.575)
Here k = ω/s is the free-space wavenumber. The form
of the solution depends on whether α2 n is greater or less
than k2 .Ifα2 n < k2 , the mode is said to be a propagating
mode and the solution for Xn is given by
Xn = An e iknx + Bn e −iknx , (3.576)
where the kn,definedby
kn = (k 2 − α 2 n )1/2 , (3.577)
are the modal wavenumbers. However, if the value
of α 2 n is greater than k2 , the mode is evanescent (not
propagating), and one has
Xn = An e −βnx + Bn e βnx , (3.578)
where βn is given by
βn = � α 2 n − k2� 1/2 . (3.579)
Unless the termination of the duct is relatively close to
the source, waves that grow exponentially with distance
from the source are not meaningful, so only the term that
corresponds to exponentially dying waves is ordinarily
kept in the description of sound fields in ducts.
3.15.2 Cylindrical Ducts
For a duct with circular cross section and radius a, the
index n is replaced by an index set (q, m, s), and the
eigenfunctions Ψn are described by
�ηqmw Ψn = Kqm Jm
a
� �
cos mφ
sin mφ
�
. (3.580)
Here either the cosine (s = 1) or the sine (s =−1) corresponds
to an eigenfunction. The quantities Kqm are
normalization constants, and the Jm are Bessel functions
of order m. The corresponding eigenvalues are given by
αn = ηqm/a , (3.581)
where the ηqm are the zeros of ηJ ′ m (η), arranged in ascending
order with the index q ranging upwards from 1.
The smaller roots [3.85] for the axisymmetric modes
are η1,0 = 0.00, η2,0 = 3.83171, and η3,0 = 7.01559,
while those corresponding to m = 1areη1,1 = 1.84118,
η2,1 = 5.33144, and η3,1 = 8.53632.
w a vw
Fig. 3.42 Cylindrical duct
3.15.3 Low-Frequency Model for Ducts
In many situations of interest, the frequency of the acoustic
disturbance is so low that only one guided mode can
propagate, and all other modes are evanescent. Given
that the walls can be idealized as rigid, there is always
one mode that can propagate, this being the plane wave
vx
x