Sound attenuation in lined bends
S. Félixa) and V. Pagneux
Laboratoire d’Acoustique de l’Université du Maine, UMR-CNRS 6613, Université du Maine,
Avenue Olivier Messiaen 72085 Le Mans Cedex 09, France
共Received 23 February 2004; revised 23 June 2004; accepted 12 July 2004兲
In the present paper we are concerned with sound propagation and attenuation in two- or
three-dimensional lined bends. First it is shown that the effect of locally reacting absorbing materials
at the walls of a waveguide can easily be taken into account in the multimodal formulation proposed
in earlier papers by the authors, and, for bends, algebraic solutions are carried out for the acoustic
field and scattering properties. Then a study of the sound attenuation in lined bends is given using
the multimodal formulation and the properties of such waveguides are shown and discussed, in
particular, the presence of a plateau of attenuation at high frequencies and a whispering gallery
effect that occurs in bends. © 2004 Acoustical Society of America. 关DOI: 10.1121/1.1788733兴
PACS numbers: 43.20.Mv, 43.20.Hq, 43.20.Dk 关LLT兴
I. INTRODUCTION
1,2
In recent papers by the authors on multimodal wave
propagation in curved ducts, perfectly reflecting rigid walls
and a lossless linear medium were assumed.
Our first aim in the present study is to introduce the
effects of a lining of the walls in the multimodal formulation
of the wave propagation in bends. One can include such
mechanisms and extend the field of applications of the multimodal approach to more realistic problems, without increasing the complexity of the formulation. It is of particular
interest, for both scientific and practical reasons, to take into
account the joint effects of liners and curvature in ducts,
since these properties are often present in practical duct systems, notably in engine noise suppressors.
In Sec. II, the multimodal equations governing the components of the pressure and velocity projected on the local
transverse modes in a bend are modified to take into account
localized impedance conditions at the walls, and algebraic
solutions for the acoustic field and reflection and transmission matrices are given. Thus, this extended formulation allows us to propose a theoretical method to predict the sound
attenuation in lined bends.
Two formulations of the sound attenuation are given in
Sec. III, one of which is defined for a known incident wave
and takes into account the mode coupling and the scattering
at each end of the duct, while the other, deduced from the
conductance, characterizes the attenuation in a more general
scope, for an arbitrary incident wave field. In Sec. IV the
calculation with the two expressions of the sound attenuation
is done for bends with various configurations of the wall
treatment, and the results, as well as the results and conclusions of the preceeding works on the subject,3– 8 are discussed.
II. FORMULATION
In the present section the formulation of the multimodal
wave propagation is developed for a bend lined with an aba兲
Electronic mail: simon.felix@univ-lemans.fr
J. Acoust. Soc. Am. 116 (4), Pt. 1, October 2004
Pages: 1921–1931
sorbing material. An adiabatic lossless linear medium is assumed. Both two-dimensional and three-dimensional bends
are considered.
A. Two-dimensional bend
Consider a circularly curved section of a twodimensional duct system of width h 共Fig. 1兲. If the complex
acoustic pressure is written as p̂⫽ ␳ 0 c 20 p exp(j␻t), with corresponding velocity v̂⫽c 0 v exp(j␻t), where ␳ 0 is the density
of air and c 0 the speed of sound, then dimensionless pressure
p and velocity v satisfy the linearized Euler equations,
⫺ jkv⫽“p
共1兲
“"v⫽⫺ jk p,
共2兲
and
with frequency k⫽ ␻ /c 0 . By eliminating the radial component v r of the velocity, these equations becomes, in (r,s)
coordinates 共see Appendix A兲,
⫺ jk 共 1⫺ ␬ r 兲v s ⫽
⫺ jk 共 1⫺ ␬ r 兲 p⫽
⳵p
,
⳵s
共3兲
冉
冊
⳵vs 1 ⳵
⳵p
,
⫺
共 1⫺ ␬ r 兲
⳵ s jk ⳵ r
⳵r
共4兲
with v s the axial velocity and ␬ ⫽1/R 0 the curvature of the
bend axis.
The boundary conditions at the walls are
冉 冊
⳵p
⳵r
⫽⫺ jk ␨ i p,
r⫽h/2
冉 冊
⳵p
⳵r
⫽ jk ␨ e p,
共5兲
r⫽⫺h/2
where ␨ i ⫽ ␳ 0 c 0 /z i and ␨ e ⫽ ␳ 0 c 0 /z e are reduced admittances, and z i ⫽z i ( ␻ ) and z e ⫽z e ( ␻ ) the surface impedances
of the internal and external wall, respectively.
The pressure and axial velocity in the bend are now
expressed using infinite series
0001-4966/2004/116(4)/1921/11/$20.00
© 2004 Acoustical Society of America
1921
FIG. 1. Geometry of the 2-D duct system with a bend and system of coordinates. The wall treatment is characterized by localized impedances z i and
ze .
p 共 r,s 兲 ⫽
兺␣ P ␣共 s 兲 ␺ ␣共 r 兲 ,
v s 共 r,s 兲 ⫽
共6兲
兺␣ U ␣共 s 兲 ␺ ␣共 r 兲 ,
共7兲
FIG. 2. Geometry of the 3-D duct system with a bend and system of coordinates. The wall treatment is characterized by the localized impedance
z( ␾ ).
where
␺ ␣ 共 r 兲 ⫽ 冑2⫺ ␦ 0 ␣ cos
冉 冉 冊冊
␣␲
h
r⫺
h
2
,
␣ 苸N,
U ␣⬘ ⫽
共8兲
are the rigid transverse eigenfunctions satisfying the homogeneous Neumann condition at the walls. Rigid eigenmodes
are chosen instead of the eigenmodes that satisfy the boundary condition 共5兲 for the simplicity of their calculation and
use, notably when joining the solutions in the lined duct and
in the rigid ducts upstream and downstream. Moreover, the
rigid eigenmodes should not be calculated for each frequency. These eigenfunctions ␺ ␣ and the pressure p do not
satisfy the same boundary conditions; one can thus expect a
lower convergence rate of the multimodal method, compared
with the rigid case.1
The projection of Eq. 共3兲 on the basis ( ␺ ␣ ) ␣ 苸N gives the
equation
P⬘ ⫽⫺ jkBU,
共9兲
independent of the conditions at the walls, and therefore unchanged compared with the rigid case.1 B is a matrix depending on the geometrical parameters 共see Appendix B兲. The
projection of Eq. 共4兲, however, gives a result modified by the
new boundary conditions:
冉冉 冊
冉 冉 冊冊 冉 冊 冉 冊
冉 冉 冊冊 冉 冊冊
1
1 1
共 C⫹KB 兲 ␣␤ P ␤ ⫹
jk
jk h
⫻ ⫺ jk ␨ i p
␬h
2
h
␬h
⫺ 1⫹
2
2
h
2
␺␣
h
2
␺␣ ⫺
⫻ jk ␨ e p ⫺
1⫺
h
2
,
共11兲
with matrices K, B, and C given in Appendix B. Thus,
U⬘ ⫽
1
共 C⫹KB⫹F 兲 P,
jk
共12兲
with
᭙ 共 ␣ , ␤ 兲 苸N2 ,
F ␣␤ ⫽⫺ jk
冑2⫺ ␦ 0 ␣ 冑2⫺ ␦ 0 ␤
冉
⫹ ␨ e 1⫹
h
冊
冉冉
冊
␨ i 1⫺
␬h
共 ⫺1 兲 ␣ ⫹ ␤ .
2
␬h
2
冊
共13兲
The elements F ␣␤ being simply expressed as functions of k
and ␨ i,e , the matrix F can be calculated at each frequency
without difficulty.
᭙ 共 ␣ , ␤ 兲 苸N2 ,
1
U ␣⬘ ⫽
jk
兺␤
⫺␬r兲
冋
B. Three-dimensional bend of circular cross section
1 1
共 C⫹KB 兲 ␣␤ P ␤ ⫹
共1
jk h
⳵p
␺
⳵r ␣
册
h/2
.
共10兲
⫺h/2
Hence, considering Eq. 共5兲, this equation becomes
1922
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
Consider now a three-dimensional bend, being part of a
duct system of circular cross section of radius r 0 共Fig. 2兲.
Equations 共1兲 and 共2兲 are expressed in toroidal coordinates
(r, ␾ ,s) and p and v s then satisfy2
⳵p
⫽⫺ jk 共 1⫺ ␬ r cos ␾ 兲v s ,
⳵s
共14兲
Felix and Pagneux: Sound attenuation in lined bends
冋
tween the symetrical and antisymetrical modes, which can
therefore be considered separately兲, and F becomes
⳵vs 1
⳵2p
⫽
共 1⫺ ␬ r cos ␾ 兲 k 2 p⫹ 共 1⫺ ␬ r cos ␾ 兲 2
⳵s
jk
⳵r
1 ⳵p
1 ⳵2p
⫹ 共 1⫺2 ␬ r cos ␾ 兲
⫹ 共 1⫺ ␬ r cos ␾ 兲 2
r ⳵r
r ⳵␾2
册
⳵p
1
⫹ ␬ sin ␾
.
r
⳵␾
共15兲
Assuming an azimuthally varying wall treatment, given
by an impedance z⫽z(k, ␾ ), the boundary condition is
冉 冊
⳵p
⳵r
⫽⫺ jk ␨ 共 k, ␾ 兲 p,
共16兲
r⫽r 0
where ␨ ⫽ ␳ 0 c 0 /z.
The pressure and axial velocity are projected on the orthonormal basis ( ␺ ␣ ) of the transverse modes of a rigid
straight duct of radius r 0 . As detailed in Ref. 2, these modes
are sorted in increasing order of their cutoff frequency, such
that ␣ is a simple integer index, referring to the triplet
(m,n, ␴ ) of the usual circumferential, radial, and symmetry
indexes. However, due to the azimuthally varying wall impedance, the symmetry invoked in Ref. 2 to set apart symmetrical 共␴⫽1兲 and antisymmetrical 共␴⫽0兲 modes is no
longer valid. All modes must be taken into account when
sorted.
Again, the equation obtained by projection of the momentum conservation equation 共14兲 on the modes ␺ ␣ remains unchanged:
P⬘ ⫽⫺ jkBU,
共17兲
while equation of mass conservation 共15兲 gives
U⬘ ⫽
1
jk
⫻
1
1
兺␤ 共 C⫹KB 兲 ␣␤ P␤ ⫹ jk ␲ r 2
冕 冋
2␲
0
0
r 共 1⫺ ␬ r cos ␾ 兲
⳵p
␺
⳵r ␣
册
r0
d␾ ,
共18兲
0
with K, B, and C given in Appendix C. Considering Eq. 共16兲,
this equation becomes
共19兲
the matrix F is given by
⫻
冕
2␲
0
冉
A ␣A ␤
␲ r 20
J m 共 ␥ mn 兲 J ␮ 共 ␥ ␮ ␯ 兲
共 1⫺ ␬ r cos ␾ 兲 ␨ 共 k, ␾ 兲
⫻sin m ␾ ⫹
冊 冉
A ␣A ␤
␲ r 20
J m 共 ␥ mn 兲 J ␮ 共 ␥ ␮ ␯ 兲共 Im ␮ ⫺ ␬ r 0 Jm ␮ 兲 ,
共21兲
where Im ␮ and Jm ␮ are given in Appendix B. As in the
two-dimensional case, the elements of F are simply expressed as functions of k and ␨.
C. Acoustic field, reflection and transmission
matrices
The two infinite first-order differential equations 共9兲
关resp., 共17兲兴 and 共12兲 关resp., 共19兲兴 in a two-dimensional
共resp., three-dimensional兲 bend have been constructed, assuming a duct of constant curvature and cross section and a
wall lining invariant with s. Under these assumptions the
coefficients in the differential equations are invariant along
the axis of the bend, and algebraic solutions of these equations can be carried out to calculate the acoustic field in the
bend and its reflection and transmission matrices.
Indeed, P obeys the equation P⬙ ⫹B(C⫹KB⫹F)P⫽0,
of which a solution can be constructed in terms of the eigenvalues ␯ ␣2 ( ␣ 苸N) and eigenvectors x␣ of the matrix B(C
⫹KB⫹F):
P⫽XD 共 s 兲 C1 ⫹XD ⫺1 共 s 兲 C2 ,
冊
␴␲
␵␲
sin ␮␾ ⫹
d␾ ,
2
2
共20兲
where J m is the mth-order Bessel function of the first kind
⬘ . ␤ refers to the triplet
and ␥ mn the (n⫹1)th zero of J m
共␮,␯,␵兲 and A ␣ is given in Appendix B. In case of a uniform
wall treatment, one can take ␴⫽␵ 共there is no coupling beJ. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
共22兲
where X⫽ 关 x0 ,x1 ,... 兴 , D(s) is diagonal and given by
D ␣ (s)⫽exp(⫺j␯␣s). C1 and C2 are constant column vectors,
functions of the conditions at each end of the bend.
Consider now a bend with a given radiation condition
U(s f )⫽Y (s f )P(s f ) at the outlet 共the relation U⫽Y P defines
a generalized admittance Y within the context of the multimodal formalism兲 and a given pressure P共0兲 at the inlet. The
continuity conditions at these two interfaces give the following system of four equations:
P共 0 兲 ⫽XC1 ⫹XC2 ,
共23兲
Y 共 0 兲 P共 0 兲 ⫽⫺HY 共 C1 ⫺C2 兲 ,
P共 s f 兲 ⫽X 共 DC1 ⫹D
⫺1
C2 兲 ,
Y 共 s f 兲 P共 s f 兲 ⫽⫺HY 共 DC1 ⫺D ⫺1 C2 兲 ,
1
U⬘ ⫽ 共 C⫹KB⫹F 兲 P;
jk
F ␣␤ ⫽⫺ jkr 0 ␨
F ␣␤ ⫽⫺ jkr 0 ␨
共24兲
共25兲
共26兲
whose unknowns are the constant vectors C1 and C2 , the
admittance Y (0) at the inlet of the bend, and the pressure
P(s f ) at the outlet. D⫽D(s f ), H and Y are functions of B, C,
F, K, and X—the reader may refer to Ref. 1 for details.
Finding the solutions Y (0) and P(s f ) in case where
Y (s f ) is the characteristic admittance Y c , which is diagonal
and given by Y c ␣ ⫽k ␣ /k, where k ␣2 ⫽k 2 ⫺( ␣ ␲ /h) 2 共2D兲 or
k ␣2 ⫽k 2 ⫺( ␥ mn /r 0 ) 2 共3D兲, allows us to define and calculate
the reflection and transmission matrices of the bend. The
algebraic calculation of these matrices have been developed
in Ref. 1 for a rigid bend; with the appropriate matrices D, H,
X, and Y taking into account the wall treatment as above, the
same process leads to the reflection and transmission matrices of a lined bend.
With the solutions C1 and C2 one determines the pressure 共22兲 in the bend. However, as dicussed in Ref. 1, the
Felix and Pagneux: Sound attenuation in lined bends
1923
matrix D ⫺1 (s) in 共22兲 is a source of numerical problems of
convergence. One thus defines C̃2 ⫽D ⫺1 C2 and the pressure
in the bend is then written as
P⫽X„D 共 s 兲 C1 ⫹D 共 s f ⫺s 兲 C̃2 …,
共27兲
and this formulation depends on D only, with positive arguments s and s f ⫺s, and does not depend on D ⫺1 . The solutions C1 and C̃2 are
C1 ⫽ 共 1⫺ ␦ 兲 ⫺1 X ⫺1 P0 ,
共28兲
C̃2 ⫽⫺„Y 共 s f 兲 X⫺HY …⫺1 „Y 共 s f 兲 X⫹HY …
共29兲
⫺1
where ␦ ⫽D„Y (s f )X⫺HY … „Y (s f )X⫹HY …D.
This algebraic formulation, established for a duct lined
with a longitudinally constant wall impedance, is also suitable for a piecewise constant impedance.9
When the coefficients in the matricial differential equations governing P and U are not invariant along the bend,
i.e., if the curvature, cross section, or wall admittance vary,
algebraic solutions cannot be carried out for the acoustic
field or the reflection and transmission matrices. However, as
developed in Refs. 1–2 and 10–11, the acoustic field and
reflection matrix can be obtained numerically by first calculating the impedance matrix along the bend. The impedance
obeys a Riccati equation, which can be integrated numerically after truncation at a sufficient number of modes 共see
Appendix D兲.
A. Energy flux in the bend
The attenuation of an acoustic wave propagating in a
waveguide can be defined as the decrease of the energy flux:
冕
1
1
Re共 pv* 兲 dSជ ⫽ Re共 t PU* 兲
2
S2
共30兲
between the entrance and the exit of the waveguide. S is the
cross-sectional area.
Since a lossless medium is assumed, only the wall treatment may cause an attenuation of the wave. Indeed, we
verify that the multimodal equations lead to a constant energy flux W(s) in a rigid bend, when the wall admittance is
equal to zero: the variations of W(s) are then given by
1
dW 共 s 兲
⫽W ⬘ 共 s 兲 ⫽ Re共 t P⬘ U* ⫹ t PU⬘ * 兲 ,
ds
2
共31兲
with P and U satisfying1,2 P⬘ ⫽⫺ jkBU and U⬘ ⫽(1/jk)(C
⫹KB)P. Thus,
冉
冊
1
1
W ⬘ 共 s 兲 ⫽ Re ⫺ jk t UBU* ⫺ t P共 C⫹KB 兲 P* .
2
jk
共32兲
Since B and C⫹KB are real symmetrical matrices, the terms
⫺ jk t UBU* and (⫺1/jk) t P(C⫹KB)P* are purely imaginary numbers and W ⬘ (s)⫽0: the energy flux is constant.
However, if the wall admittance is finite—not equal to
zero—the previous calculus leads to
1924
共33兲
B. Attenuation of a given incident wave P„ i …
The sound attenuation is given in decibels by
冉 冊
W trans.
,
W inc.
共34兲
where W inc. and W trans. are the incident and transmitted energy fluxes, respectively.
Considering an incident wave P(i) , emitted by a source
upstream from the bend 关region 共I兲 in Figs. 1 and 2兴, with
corresponding incident velocity U(i) , the incident energy flux
is W inc.⫽1/2 Re(tP(i) U(i) * ). Since for the incident wave
U(i) ⫽Y c P(i) , this equation becomes
1
W inc.⫽ Re共 P共 i 兲 † Y c P共 i 兲 兲 ,
2
共35兲
where † denotes the adjoint operator.
If T is the transmission matrix of the lined bend, the
transmitted wave is P(t) ⫽TP(i) , with corresponding velocity
U(t) ⫽Y c TP(i) . The transmitted energy flux is then
1
W trans.⫽ Re共 P共 i 兲 † T † Y c TP共 i 兲 兲 .
2
共36兲
Then, as the transmission matrix T is known 共see Sec. II C兲,
the attenuation 共34兲 can be calculated for any given incident
wave P(i) by using Eqs. 共35兲 and 共36兲:
III. SOUND ATTENUATION IN A LINED BEND
1
S
冊
F being a complex symmetrical matrix, W ⬘ (s) is not equal to
zero, and consequently the energy flux may vary along the
lined bend.
A dB⫽⫺10 log10
⫻D 共 1⫺ ␦ 兲 ⫺1 X ⫺1 P0 ,
W共 s 兲⫽
冉
1
1
W ⬘ 共 s 兲 ⫽ Re ⫺ t PFP* ;
2
jk
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
A dB⫽⫺10 log10
冉
Re共 P共 i 兲 † T † Y c TP共 i 兲 兲
Re共 P共 i 兲 † Y c P共 i 兲 兲
冊
.
共37兲
C. The case of an incoherent source
The result given above allows us to calculate the attenuation of a given incident wave in a lined bend. Thus some
particular cases for which the incident wave is known can be
studied, with the mode coupling taken into account. However, in many cases the sound source upstream from the lined
duct system is not known—it may be a number of noise
sources—and such an approach is then unadapted since the
possibly statistical aspect of the distribution of incident
modes is not taken into account. It could thus be useful to
define a quantity characterizing the attenuation in a more
general scope, for an arbitrary source.
For a given frequency k, we assume that the N p propagative modes ␺ ␣ upstream from the bend are excited by as
many incoherent unit fluxes. The transmission of such a
mode distribution in a waveguide can be characterized with
the conductance12
G⫽Tr共 T̂ † Ŷ c T̂ 兲 ,
共38兲
where Tr is the trace and T̂ and Ŷ c are the transmission
matrix and characteristic admittance matrix, respectively, restricted to the N p propagative modes.
Felix and Pagneux: Sound attenuation in lined bends
Indeed, the transmitted energy
⫽1/2 Re(P(i)† T † Y c TP(i) ), that is
1
W trans.⫽ Re
2
冉兺
flux
is
冊
共 T † Y c T 兲 ␣␤ P 共␣i 兲 * P 共␤i 兲 .
␣,␤
W trans.
共39兲
Assuming N p incoherent unit fluxes implies 具 P ␣(i) * P ␤(i) 典
⫽ ␦ ␣␤ if ␣ ⬍N p and ␤ ⬍N p , and 具 P ␣(i) * P ␤(i) 典 ⫽0 if ␣ ⭓N p or
␤ ⭓N p . Hence
1
W trans.⫽ Re
2
冉兺
冊
N p ⫺1
␣ ⫽0
共40兲
共 T † Y c T 兲 ␣␣ ,
that is
1
W trans.⫽ Tr共 T̂ † Ŷ c T̂ 兲 .
2
共41兲
The attenuation is then
A dB⫽⫺10 log10
冉 冊
冉 冊
W trans.
G
⫽⫺10 log10
.
W inc.
Tr共 Ŷ c 兲
FIG. 3. Attenuation in a 2-D bend 共—兲 and in a straight duct having the
same axis length s f 共–•–兲, both lined with a porous material, for a plane
incident wave.
共42兲
1. General results
IV. RESULTS
Expressions 共37兲 and 共42兲 are valid for both twodimensional and three-dimensional ducts, as long as the
transmission matrix of the lined duct can be calculated.
However, for simplicity, a two-dimensional bend is considered in all of the results to be described in the following,
except where otherwise stated.
For comparison, results on the attenuation in straight
lined ducts are given in the present section, together with the
results for bends. It can be easily shown that the algebraic
formulation of the reflection and transmission matrices that
has been formulated for bends 共see Ref. 1 and Sec. II C兲 can
be used for straight ducts by simply taking ␬⫽0. Therefore,
the same expressions 共37兲 and 共42兲 of the attenuation are
used in the following for both bent and straight ducts, with
the appropriate transmission matrix.
The dimensions of the bend are h⫽0.2 m, R 0 ⫽0.9 m,
and the overall angle ␪ f ⫽s f /R 0 ⫽60° 共Fig. 1兲. The walls are
lined with a rigid porous material. Assuming a locally reacting material, the surface impedance of the wall treatment is
taken as the impedance at normal incidence of a plane porous plate having the same thickness:
z 共 ␻ 兲 ⫽⫺
冉冉冊 冊
␳
j
共 ␳ K 兲 1/2 cot ␻
␾
K
1/2
d ,
共43兲
where d is the thickness of the porous layer, ␾ the porosity, ␳
the effective density, and K the effective bulk modulus of air
in the material.13 The porous material is characterized with
the following quantitites: porosity ␾⫽0.98, tortuosity ␣ ⬁
⫽1.2, flow resistivity ␴⫽40000 N m⫺4 s, characteristic dimensions ⌳⫽2⫻10⫺4 m and ⌳ ⬘ ⫽4⫻10⫺4 m, thickness d
⫽0.05 m.
A. Incident plane wave
The following results are obtained assuming an incident
plane wave, emitted by a source upstream from the bend.
Thus one use the expression 共37兲 to calculate the attenuation,
with an incident mode 0, P ␣(i) ⫽ ␦ ␣ 0 .
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
Figure 3 shows the attenuation as function of the frequency in the lined bend described above and in a straight
lined duct having the same axis length and the same wall
treatment. The attenuation in a straight lined duct is known
to follow the typical curve shown in this figure: the attenuation is weak at low frequencies; it reaches a maximum when
the wavelength is of the order of the duct width (kh/ ␲
⬃2), and then decreases. The attenuation at high frequencies
is poor, due to a ‘‘ray effect:’’3 the incident acoustic wave
being regarded as rays propagates through the duct without
‘‘colliding’’ the walls.
A way, first pointed out by empirical means,3 to avoid
this ray effect and increase the efficiency of the system is to
use a curved duct. A plateau appears then, keeping the attenuation at a high level 共for kh/ ␲ ⫽100, the attenuation is
still greater than 9 dB兲. Thus, although the curvature is weak
( ␬ h⬃0.2) in our example, it yields a substantial increase in
the attenuation at high frequencies.
At low frequencies, the effects of the curvature vanish
and the attenuation curves for the bend and the straight duct
converge to tend toward zero. Rostafinski7 concludes his
study of lined bends at very low frequencies 共the wavelength
is at least two orders of magnitude larger than the width h of
the bend, that is, kh/ ␲ ⬍10⫺2 ) by stating that the attenuation
in a bend is generally lower 共by 2% to 7%兲 than in a straight
duct. None of the configurations that have been considered in
the present study have led to this conclusions; in all cases,
the difference between the results for a bend and a straight
duct in this range of frequency was much lower than 1%.
More generally, for all frequencies, the attenuation level
is globally higher when the duct is curved than when it is
straight. However it can be locally, that is, in a small frequency range 共see, e.g., for kh/ ␲ ⬃3), lower. If the behavior
at low and high frequencies can be easily interpreted, it is not
simple to describe the effect of the curvature at medium frequencies, and therefore to explain this latter behavior.
When considering a three-dimensional bend and straight
duct with a circular cross section 共Fig. 4兲, similar results as
discussed above are obtained.
Felix and Pagneux: Sound attenuation in lined bends
1925
FIG. 4. Attenuation in a 3-D bend of circular cross section 共—, r 0
⫽0.1 m, R 0 ⫽0.9 m, s f /R 0 ⫽60°), and in straight duct having the same axis
length s f 共–•–兲, both lined with a porous material, for a plane incident wave.
Figure 5 shows the attenuation in a two-dimensional
bend and a straight duct with walls that are lined with a
fibrous sheet mounted on a locally reacting core with impervious backing, as studied by Ko and Ho5 and Lafarge.8 The
reduced admittance of the lining is given by ␨ (k)⫽ 关 R(1
⫹ jk/k 0 )⫺ j cot(kd)兴⫺1, where R⫽1.5 is the reduced resistance, k 0 ⫽70.12 m⫺1 the characteristic wave number, and
d⫽0.1 m the lining core depth. The attenuation curves describe a succession of absorption lobes and vanish each time
the transverse velocity is null on the fibrous sheet, that is,
since the thickness of the cores is d⫽h/2, for frequencies
kh/ ␲ ⫽4n, n苸N. For this configuration one can also consider that a plateau appears at high frequencies for the bend,
since the decrease in the amplitude of the absorption lobes
observed for the straight duct is no longer observed for the
bend.
This particular behavior of the attenuation in bends at
high frequencies—the appearence of a plateau—is thus not
due to some properties of one particular kind of lining and
any absorbing boundary condition would produce the same
effect. As a matter of fact, Fig. 6 shows that the plateau is
observed when considering an arbitrary constant admittance.
FIG. 5. Attenuation in a 2-D bend 共—兲 and in a straight duct having the
same axis length s f 共–•–兲, for a plane incident wave. The walls are lined
with a fibrous sheet mounted on a locally reacting core.
1926
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
FIG. 6. Attenuation in a 2-D bend 共—兲 and in a straight duct having the
same axis length s f 共–•–兲, for a plane incident wave. The reduced wall
admittance is constant: ␨ i ⫽ ␨ e ⫽0.5⫹ j0.25.
2. One-wall-lined bends
Figure 7 shows attenuation curves of one-wall-lined
bends and a straight duct. The attenuation in the outer-walllined bend is much greater than in the inner-wall-lined duct.
Furthermore, no plateau appears if the inner wall only is
lined, and then attenuation in this case is lower than in a
one-wall-lined straight duct. Thus, as remarked by
Grigor’yan,3 the curvature does not always yield an increase
in the attenuation. Ko and Ho5 suggest that the difference
between the attenuations for inner-wall-lined and outer-wall
lined bends may be due to the difference between the lengths
of the walls in a bend. There is no doubt that an increase in
the length of wall treatment generally increases the attenuation in a duct, but the difference between the length of the
walls in a bend cannot explain on its own this significant
difference. The attenuations for two bends with the same
parameters h and R 0 , one of which is inner-wall-lined while
the other is outer-wall-lined, with axis lengths such that the
length of the wall treatment is the same in both cases, are
compared in Fig. 8. The results are clearly different, although
the length that is lined is the same.
FIG. 7. Attenuation in one-wall-lined bends, for a plane incident wave; 共—兲
outer wall lined, 共¯兲 inner wall lined, 共–•–兲 straight duct with only one wall
lined.
Felix and Pagneux: Sound attenuation in lined bends
FIG. 8. Attenuation in one-wall-lined 2-D bends with various overall angles,
for a plane incident wave; 共—兲 outer wall lined, ␪ f ⫽54°, 共– – –兲 inner wall
lined, ␪ f ⫽54°, 共–•–兲 outer wall lined, ␪ f ⫽67.5°, 共¯兲 inner wall lined,
␪ f ⫽67.5°. The lining length in the configurations 共—兲 and 共¯兲 is the same.
3. Whispering gallery effect
More than a difference between the lengths of lining, the
appearance of a whispering gallery effect may explain the
difference of efficiency between the inner-wall-lined and
outer-wall-lined bends. For sufficiently high frequencies, the
incident wave ‘‘traverses the duct by glancing along the concave wall, the convex wall being left in the shadow.’’3 The
axial component j s of the acoustic intensity j
⫽1/2 Re(pv* ), calculated in a two-dimensional rigid bend
for a plane incident wave, shows this whispering gallery effect 共Fig. 9兲: the intensity, uniformly distributed on the cross
section upstream from the bend, is progressively confined
near the concave 共outer兲 wall as the frequency is increased.
Since the intensity is at its minimum—and even locally
null, see Fig. 9—near the convex 共inner兲 wall, it is clear that
the contribution of the latter to sound attenuation is poor. In
contrary, lining the outer wall, where the intensity is at its
maximum, may yield a substantial attenuation, even at high
frequencies. Besides, the relatively uniform distribution of
the intensity on the cross section of a straight duct may explain the intermediate position of the attenuation curve for a
one-wall-lined straight duct in Fig. 7.
When the frequency is increased, the zone of the maximum of intensity near the outer wall becomes more narrow.
Considering what is written above, a partial lining of the
outer wall near this narrow zone should yield an attenuation
similar to what is obtained when the two walls are wholly
lined. Consider the attenuation in a two-dimensional bend
with parameters h⫽0.2 m, R 0 ⫽0.9 m, and ␪ f ⫽60°, as before, lined on 30° in the middle of its outer wall, for a plane
incident wave 共Fig. 10兲. The result at high frequencies is
close to the one obtained for the wholly lined bend—the
difference is of the order of 2 dB—and the attenuation remains much greater than in the wholly lined straight duct. If
the lined zone is reduced to 20°, the attenuation plateau level
is close to 7 dB. Besides, we verify that the position on the
wall of a given length of wall treatment is not indifferent;
indeed, the attenuation is at its maximum when the wall is
lined where the maximum of energy flux is expected.
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
FIG. 9. Axial component j s of the acoustic intensity j⫽1/2 Re(pv* ) in a
2-D rigid bend (R 0 /h⫽4.5, ␪ f ⫽60°), for six frequencies. A plane wave
( j s ⫽1, arbitrary unit兲 is emitted upstream from the bend. For all of the
figures, except the first one (kh/ ␲ ⫽0.7), the gray scaling gives the values
of j s varying between 0 共white兲 and 5 共black兲. Since the variations of j s in
the first figure are lower than 7%, the gray scaling is different, varying
between 0.93 共white兲 and 1.07 共black兲.
B. Incoherent sources
We now assume that the incident wave field upstream
from the bend is produced by incoherent sources. Therefore
the attenuation in the following results is calculated with the
expression 共42兲, deduced from the conductance 共38兲.
Using Eq. 共42兲 to calculate the attenuation in a twodimensional bend lined on its two walls with a porous layer
共Fig. 11兲 gives a result that is qualitatively close to the one
obtained for an incident plane wave 共Fig. 3兲, with notably the
appearance of the plateau at high frequencies, after an absorption lobe at kh/ ␲ ⬃2. All of the propagative modes being taken into account and these modes being more attenuated than the plane wave mode, the level of the attenuation
plateau if naturally higher.
However, in the straight duct, a significant change can
be noticed. Since the plane wave mode is weakly attenuated
at high frequencies in a straight duct, the curve shown in Fig.
3 is rapidly close to zero. The propagative modes being taken
into account in the expression 共42兲 of the attenuation, a plateau appears in Fig. 11, which is, however, still under the
Felix and Pagneux: Sound attenuation in lined bends
1927
FIG. 10. Attenuation in a 2-D bend locally lined on the outer wall, for a
plane incident wave. For comparison, the curves of Fig. 3 are also plotted.
plateau for the bend. The curvature still yields an increase in
the attenuation.
Using Eq. 共42兲 to calculate the attenuation in bends lined
as in Fig. 5 and 6 or with other kinds of wall treatments leads
to the same conclusions. The attenuation is globally increased when the duct is curved, and a plateau of attenuation
appears in straight ducts.
As in the incident plane wave case, the contribution of
the outer wall to the global attenuation is still more important
than the inner wall contribution. However, the whispering
gallery effect that has been observed in the previous paragraph is proper to the case of very low-order incident modes,
and the distribution of the energy in the bend is not limited to
the neighborhood of the outer wall when the incident wave
field is complex. The ray method to be used in the following
shows this effect of the higher-order modes on the energy
distribution. One consequence of this spread distribution of
the energy in the bend is the appearance of a plateau at high
frequencies in inner-wall-lined bends.
A classical expression of the attenuation in waveguides
and, in particular, in bend is3,5,8
A dB⫽⫺10 log10
冉
N ⫺1
p
兺 n⫽0
e 2 Im共 ␯ n 兲 ␪ f
Np
冊
,
共44兲
FIG. 12. Attenuation in a 2-D bend, calculated with 共—兲 expression 共37兲
with a plane incident wave, 共– – –兲 expression 共42兲, 共¯兲 expression 共44兲,
共–•–兲 expression 共44兲 with N p ⫽1 共fundamental mode only兲.
where the angular wave numbers ␯ n (n苸N) are the solutions
of the implicit dispersion relation,
冋 冉
冋
冋
冋
冉
冊冊 冉
冉 冉 冊冊
冉 冉 冊冊
冉 冉 冊冊
j ␨ 1 N ␯ kR 0 1⫺ ␬
h
2
冉 冊冊 册
冉 冉 冊冊 册
冉 冉 冊冊 册
冉 冉 冊冊 册
⫹N ␯⬘ kR 0 1⫺ ␬
h
2
⫻ j ␨ 2 J ␯ kR 0 1⫹ ␬
h
2
⫹J ␯⬘ kR 0 1⫹ ␬
h
2
⫺ j ␨ 1 J ␯ kR 0 1⫺ ␬
h
2
⫹J ␯⬘ kR 0 1⫺ ␬
h
2
⫻ j ␨ 2 N ␯ kR 0 1⫹ ␬
h
2
⫹N ␯⬘ kR 0 1⫹ ␬
h
2
⫽0.
共45兲
The calculation of the angular wave numbers is often limited
to the first mode,3,7,8 since the fundamental mode in a
straight or bent duct is generally the least attenuated.3,5,8,14
The result that is obtained with the expression 共44兲 of
the attenuation for the two-dimensional bend lined with porous material is close to the result obtained with Eq. 共42兲
共Fig. 12兲. This figure also shows a comparison between the
calculations of the attenuation with Eq. 共44兲 with the first
bend mode only and with Eq. 共37兲. Again, both methods give
similar results. In our example the bend is weakly curved,
and therefore the energy of the incident plane wave mode is
predominantly transferred to the first bend mode, which is
the least attenuated. But if the bend is more strongly curved
or if the wall impedance is high, Eqs. 共44兲 and 共42兲 no longer
give similar results. In such cases the simple expression 共44兲
of the attenuation in not a suitable measurement of the attenuation.
C. Rays to predict the attenuation
FIG. 11. Attenuation in a 2-D bend 共—兲 and in a straight duct having the
same axis length s f 共–•–兲, calculated with the expression 共42兲.
1928
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
A mode ␺ ␣ propagating in a two-dimensional straight
waveguide can be regarded as two interfering sets of rays at
angles ␪ ␣ ⫽sin⫺1(␣␲/kh) and ⫺ ␪ ␣ . The aim of the present
part is to use this interpretation in a simpler form—regarding
Felix and Pagneux: Sound attenuation in lined bends
FIG. 13. Propagation in a 2-D bend of a plane incident wave using a ray
method.
the mode propagating in the duct as two sets of rays with no
phase information—to predict the zones of absorption of a
given incident wave on the walls of a duct.
Consider first the propagation of an incident plane wave
in a rigid bend joining two straight ducts 共Fig. 13兲. As shown
before with the calculations of attenuation and acoustic intensity 共Sec. IV A 3兲, the inner wall, where none of the rays
collide, is ‘‘left in the shadow.’’
Consider now the propagation of a higher order mode
共␣⬎0兲 for a given frequency, as, e.g., the mode ␺ 12 with
kh/ ␲ ⫽17.5 共Fig. 14兲. Due to the finite angle of incidence of
the rays ( ␪ 12⫽43.3° at this frequency兲, the inner wall of the
bend is not in the shadow anymore, even if the ‘‘density’’ of
rays colliding this wall is still lower than the density on the
outer wall. Figure 15 shows the mean attenuation in the bend
drawn in Fig. 14, with h⫽0.2 m and the same porous lining
as in the preceeding, in the frequency range kh/ ␲
苸]17,18关 , for the incident mode ␺ 12 , and for five lining
schemes. The attenuation in the outer-wall-lined case is
greater than in the inner-wall-lined case, as expected with the
ray approach.
Two zones can be distinguished on the outer wall in Fig.
14: the first 35°, characterized by a high density of rays
colliding the wall, and the next 50°, where the density is
much lower. Figure 15 shows that the attenuation when the
FIG. 14. Propagation in a 2-D bend of the incident mode ␺ 12 at the frequency kh/ ␲ ⫽17.5 using a ray method (R 0 /h⫽1, ␪ f ⫽90°).
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
FIG. 15. Mean attenuation level in the frequency range kh/ ␲ 苸]17,18关 for
the bend shown in Fig. 14, with h⫽0.2 m, for five lining schemes 共the lined
zones on the walls are shown with bold strokes兲, for the incident mode ␺ 12 .
first 35° of the outer wall are lined is greater that when the
next 50° are lined, altough the length of lining is greater in
the latter case.
Thus, the simple observation of rays propagating in the
duct allows us to expect the relative levels of attenuation for
the different lining schemes. If one considers the practical
problem of a noise source with a frequency emerging on a
particular mode, a wall treatment localized on the efficient
absorption zones can thus be proposed with this very simple
approach.
V. CONCLUSION
The effects of a lining of the walls on wave propagation
in curved ducts have been introduced in the multimodal formulation established in earlier papers by the authors. New
equations giving the multimodal acoustic field and the scattering and impedance matrices have been carried out, extending the field of applications of the multimodal method to
ducts lined with locally reacting absorbing materials.
Besides, two calculations have been proposed to evaluate the sound attenuation in a lined duct, for a given incident
wave or a distribution of incoherent sources, and results with
both calculations have been presented and discussed.
The known effects of the curvature on the sound
attenuation—the increase of the global attenuation level and
the appearance of a plateau of attenuation at high
frequencies—have been shown systematically for various
configurations of the wall treatment and incident wave field.
For a two-dimensional bend, the whispering gallery effect of
the fundamental mode that appears along the outer wall has
been shown, which explains the great difference between
sound attenuations induced by the inner and outer walls respectively. For a three-dimensional bend whose cross-section
is such that one cannot consider disctinct inner and outer
wall 共circular or elliptical cross-section,...兲, one can expect
that the whispering gallery effect still occurs, with the wave
glancing along the external part of the wall. This effect of a
larger attenuation on the external part of the duct wall is
observed for any kind of source, even though the whispering
gallery effect is less pronounced.
Finally, a simple study of ray propagation in a bend has
been shown to allow a prediction of the zones of the wall
where the sound absorption mainly occurs and further studies are planned on the characterization of waveguides with
ray methods.15
Felix and Pagneux: Sound attenuation in lined bends
1929
APPENDIX A: 2D COORDINATE SYSTEM
Since (r,s) coordinates are orthogonal, we have dr
⫽h r drûr ⫹h s dsûs , where the scale factors are h r ⫽1 and
h s ⫽1⫺ ␬ r. Thus, the gradient and divergence operators in
this coordinate system are
“f⫽
1
⳵f
⳵f
ûr ⫹
û ,
⳵r
1⫺ ␬ r ⳵ s s
“"F⫽
共A1兲
⳵
1 ⳵Fs
1
„共 1⫺ ␬ r 兲 F r …⫹
.
1⫺ ␬ r ⳵ r
1⫺ ␬ r ⳵ s
共A2兲
APPENDIX B: MATRICES IN THE 2D MULTIMODAL
EQUATIONS
FIG. 16. Duct with varying curvature, cross section, and wall admittance.
The matrices K, B, and C are defined and calculated in
Ref. 1. We give in the following their expression with the
notations of the present paper:
᭙( ␣ , ␤ )苸N2 ,
冉 冉 冊冊
␣␲
h
K ␣␤ ⫽ k 2 ⫺
B ␣␤ ⫽
C ␣␤ ⫽
冦
冦
where
2
␦ ␣␤ ,
A ␣⫽
if ␣ ⫽ ␤ ,
1
冑2⫺ ␦ 0 ␣ 冑2⫺ ␦ 0 ␤
␬h
␲2
„共 ⫺1 兲 ␣ ⫹ ␤ ⫺1…
␣ 2⫹ ␤ 2
共 ␣ 2⫺ ␤ 2 兲2
if ␣ ⫽ ␤
Tm ␮ ⫽
⫻
冕
0
C ␣␤ ⫽A ␣ A ␤
冊
r 2J m
␬
␣
, . 共B3兲
„共 ⫺1 兲 ␣ ⫹ ␤ ⫺1… 2
h
␣ ⫺␤2
␦ ␣␤ ,
0
Jm
mn r
r0
␮␯r
J␮
r0
dr,
␬
␲ r 20
冊
冉
cos ␾ sin m ␾ ⫹
共C5兲
if m⬎0;
冊 冉
冊
␴␲
␴␲
sin ␮␾ ⫹
d␾
2
2
共C6兲
共C7兲
and
Tm ␮
r0
0
冕
2␲
0
冉
sin ␾ cos m ␾ ⫹
冊 冉
冊
␴␲
␴␲
sin ␮␾ ⫹
d␾
2
2
共C8兲
␲
„␦
共 1⫺ 共 ␦ ␴ 1 ⫺ ␦ ␴ 0 兲 ␦ m0 兲
2 ␮ ⫺m,1
⫺ ␦ m⫺ ␮ ,1„1⫹ 共 ␦ ␴ 1 ⫺ ␦ ␴ 0 兲 ␦ ␮ 0 …….
The term Im ␮ in Eq. 共21兲 is
Im ␮ ⫽
冕 冉
2␲
0
sin m ␾ ⫹
冊 冉
共C9兲
冊
␴␲
␴␲
sin ␮␾ ⫹
d␾
2
2
⫽ ␲ ␦ m ␮ „1⫹ 共 ␦ ␴ 1 ⫺ ␦ ␴ 0 兲 ␦ m0 ….
共C10兲
APPENDIX D: DUCT WITH VARYING CURVATURE,
CROSS SECTION AND WALL IMPEDANCE
␥ mn r
␥ ␮␯r
J␮
dr,
r0
r0
Tm ␮
共C4兲
if m⫽0,
␲
␦
共 1⫹ 共 ␦ ␴ 1 ⫺ ␦ ␴ 0 兲共 ␦ m0 ⫹ ␦ ␮ 0 兲兲 ,
2 兩 m⫺ ␮ 兩 ,1
⫽
共C1兲
冉 冊 冉 冊
冕 冉 冊 冉 冊
␥ mn ␬
␲ r 30
0
Hm ␮ ⫽
B ␣␤ ⫽ ␦ ␣␤ ⫺A ␣ A ␤
r0
2␲
r0
1
m2
1⫺ 2 J 20 共 ␥ mn 兲 ,
2
␥ mn
1/
冕
冕 冉␥ 冊 冉␥ 冊
2
The integer index ␣ refers to circumferential and radial
indexes m and n, respectively, and ␤ refers to ␮ and ␯. For
details about the following expressions, the reader may refer
to Ref. 2:
᭙( ␣ , ␤ )苸N2 ,
冉
再 冑冉
⫽
if ␣ ⫽ ␤
K ␣␤ ⫽ k 2 ⫺
共 Tm ␮ ⫺Hm ␮ 兲
共B2兲
冑2⫺ ␦ 0 ␣ 冑2⫺ ␦ 0 ␤
2
␥ mn
r 20
␲ r 20
,
if ␣ ⫽ ␤ ,
0
␬
1/冑J 20 共 ␥ 0n 兲 ,
共B1兲
APPENDIX C: MATRICES IN THE 3D MULTIMODAL
EQUATIONS
1930
⫺A ␣ A ␤ m
rJ m⫹1
共C2兲
␥ mn r
␥ ␮␯r
J␮
dr,
r0
r0
共C3兲
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
In this appendix, the multimodal formulation is given for
a duct with varying curvature, cross-section and locally reacting wall treatment. For convenience, a two-dimensional
duct is considered, but this formulation can be carried out in
a three-dimensional bend without difficulty.
The region in plane to be analyzed here is shown in Fig.
16, delimited by two lined duct walls, Ci and Ce . Consider a
curve C represented by a vector function R(s), where s is the
Felix and Pagneux: Sound attenuation in lined bends
arclength measured from some reference point. ûs and ûr
being the tangent and normal vectors of the curve C, the
inner wall Ci and outer wall Ce of the duct are represented by
the vector functions
Ri 共 s 兲 ⫽R共 s 兲 ⫹r i 共 s 兲 ûr ,
共D1兲
Re 共 s 兲 ⫽R共 s 兲 ⫹r e 共 s 兲 ûr .
共D2兲
As in Sec. II, pressure p and axial velocity v s in the duct
are expressed using the infinite series
p 共 r,s 兲 ⫽
兺␣ P ␣共 s 兲 ␺ ␣共 r,s 兲 ,
v s 共 r,s 兲 ⫽
1
h共 s 兲
共D3兲
兺␣ U ␣共 s 兲 ␺ ␣共 r,s 兲 ,
共D4兲
where h(s)⫽r i (s)⫺r e (s). Note that a different development, more convenient in varying cross-section
waveguides,10 has been chosen for v s compared to Eq. 共7兲.
The rigid transverse eigenfunctions ␺ ␣ are
冉
␺ ␣ 共 r,s 兲 ⫽ 冑2⫺ ␦ 0 ␣ cos ␣ ␲
冊
r i ⫺r
,
r i ⫺r e
␣ 苸N.
共D5兲
Therefore, by projecting Eqs. 共1兲 and 共2兲, expressed in
(r,s) coordinates, the following equations are obtained for P
and U:
P⬘ ⫽⫺ jkB
U⬘ ⫽
h⬘
1
U⫹ t ⌸P,
h
h
共D6兲
1
h⬘
h 共 C⫹KB⫹F 兲 P⫺ ⌸U,
jk
h
共D7兲
where matrices B, C, K, and F are as given in previous sections and apendices, but with the width h, the curvature ␬ of
C, and the wall impedances z i and z e depending on s. The
matrix ⌸ is given by
⌸ ␣␤ ⫽
冦
共 1⫺ ␦ 0 ␣ 兲 /2,
if ␣ ⫽ ␤ ,
冑2⫺ ␦ 0 ␣ 冑2⫺ ␦ 0 ␤
r ⬘i ⫺ 共 ⫺1 兲 ␣ ⫹ ␤ r ⬘e
␣2
r ⬘i ⫺r e⬘
␣ 2⫺ ␤ 2
if ␣ ⫽ ␤
J. Acoust. Soc. Am., Vol. 116, No. 4, Pt. 1, October 2004
,.
共D8兲
As Eqs. 共D6兲 and 共D7兲 cannot be integrated directly, the
impedance matrix Z is defined, fulfilling P⫽ZU. Substituting this relation in Eqs. 共D6兲 and 共D7兲 leads to the Riccati
equation
Z ⬘ ⫽⫺ jk
1
h⬘
h⬘
1
B⫹ t ⌸Z⫹Z ⌸⫺ hZ
h
h
h
jk
⫻ 共 C⫹KB⫹F 兲 Z,
共D9兲
which is numerically workable. Hence, by first calculating
the impedance in the duct, the acoustic field or reflection
matrix can then be obtained.
1
S. Félix and V. Pagneux, ‘‘Sound propagation in rigid bends: A multimodal approach,’’ J. Acoust. Soc. Am. 110, 1329–1337 共2001兲.
2
S. Félix and V. Pagneux, ‘‘Multimodal analysis of acoustic propagation in
three-dimensional bends,’’ Wave Motion 36, 157–168 共2002兲.
3
F. E. Grigor’yan, ‘‘Soundproofing by means of ducts with curved porous
walls,’’ Akust. Zh. 16, 229–235 共1970兲 关English translation: Sov. Phys.
Acoust. 16, 192–196 共1970兲兴.
4
M. K. Meyer and P. Mungur, ‘‘Sound propagation in curved ducts,’’ Prog.
Astronaut. Aeronaut. 44, 347–362 共1976兲.
5
S. H. Ko and L. T. Ho, ‘‘Sound attenuation in acoustically lined curved
ducts in the absence of fluid flow,’’ J. Sound Vib. 53, 189–201 共1977兲.
6
S. H. Ko, ‘‘Three-dimensionnal acoustic waves propagating in acoustically lined cylindrically curved ducts without fluid flow,’’ J. Sound Vib.
66, 165–179 共1979兲.
7
W. Rostafinski, ‘‘Propagation of long waves in acoustically treated, curved
ducts,’’ J. Acoust. Soc. Am. 71, 36 – 41 共1982兲.
8
D. Lafarge, ‘‘Etude 2D de l’atténuation acoustique dans un guide semicirculaire traité en parois par un matériau poreux,’’ XXe Journée Technologique, C. T. T. M. Le Mans, France, 1998.
9
W. Bi, V. Pagneux, D. Lafarge, and Y. Aurégan, ‘‘Sound propagation in
nonuniform lined duct by the multimodal method,’’ submitted to J. Sound.
Vib.
10
V. Pagneux, N. Amir, and J. Kergomard, ‘‘A study of wave propagation in
varying cross-section waveguides by modal decomposition. I. Theory and
validation,’’ J. Acoust. Soc. Am. 100, 2034 –2048 共1996兲.
11
N. Amir, V. Pagneux, and J. Kergomard, ‘‘A study of wave propagation in
varying cross-section waveguides by modal decomposition. II. Results,’’ J.
Acoust. Soc. Am. 101, 2504 –2517 共1997兲.
12
S. Datta, Electronic Transport in Mesoscopic Systems 共Cambridge University Press, Cambridge, 1998兲.
13
J.-F. Allard, Propagation of Sound in Prous Media: Modelling Sound Absorbing Materials 共Chapman & Hall, London, 1993兲.
14
P. E. Doak and P. G. Vaidya, ‘‘Attenuation of plane wave and higher order
mode sound propagation in lined ducts,’’ J. Sound Vib. 12, 201–224
共1970兲.
15
S. Félix and V. Pagneux, ‘‘Ray-wave correspondence in bent
waveguides,’’ to be published in Wave Motion 共2004兲.
Felix and Pagneux: Sound attenuation in lined bends
1931