Final D

raft

1

Design of Reactive Rectangular Expansion Chambers for Broadband Acoustic

Attenuation Performance based on Optimal Port Location†

Akhilesh Mimani1, ‡ and M. L. Munjal2 1School of Mechanical Engineering, The University of Adelaide, South Australia 5005, Australia

e-mail: akhilesh.mimani@adelaide.edu.au (Corresponding author) 2Facility of Research in Technical Acoustics, Department of Mechanical Engineering, Indian Institute of Science,

Bangalore 560 012, India

e-mail: munjal@mecheng.iisc.ernet.in Abstract This paper analyses the Transmission Loss (TL) performance of rectangular expansion chambers having a Single-Inlet and Single-Outlet (SISO) or Single-Inlet and Double-Outlet (SIDO) by means of a 3-D semi-analytical formulation based on the modal expansion and the Green’s function approach. To this end, the acoustic field inside the rigid-wall rectangular chamber is obtained as the orthogonal modal solution of 3-D homogeneous Helmholtz equation. The SISO/SIDO rectangular chamber system is characterised using the uniform piston-driven model in terms of the impedance [Z] matrix parameters (equivalently, the acoustic pressure response function) obtained by computing the average of the 3-D Green’s function over the surface area of the inlet/outlet ports modelled as rigid pistons oscillating with uniform velocity. The TL graphs computed using the 3-D semi-analytical formulation are found to be in an excellent agreement with that obtained from the 3-D FEA for SISO test-cases, thereby validating the technique presented here. A parametric investigation is conducted to study the effect of arbitrary location of inlet/outlet ports (on the chamber surface) on the TL graph resulting in the formulation of guidelines towards designing an axially short and long SISO/SIDO rectangular chambers exhibiting a broadband TL performance in terms of optimal angular and axial location of ports (on the appropriate acoustic pressure nodes). In addition, characteristic features of the TL spectrum of a general reciprocal and conservative Single-Inlet and Multiple-Outlet (SIMO) muffler system such as (1) the effect of interchanging the position of inlet and outlet ports and (2) analysis of peaks and troughs are proved analytically by means of the scattering [S] matrix parameters. These features are corroborated through the analysis of TL graph (obtained using the 3-D semi-analytical formulation) of SIDO rectangular chambers. Keywords Rectangular plenum chambers, Reciprocal and conservative systems, 3-D Green’s function, Uniform piston-driven model, Single inlet and single/multiple outlet mufflers, Optimal port location. 1. INTRODUCTION

Rectangular expansion chambers are popularly used as plenum chambers in Heating Ventilation and Air-Conditioning (HVAC) duct systems and other industrial air-handling applications. The evaluation and analysis of the acoustical attenuation performance of rigid-wall (unlined) and dissipative (lined) plenum chambers as well as the problem of break-out/break-in noise and sound transmission through the thin-wall rectangular chambers has been a subject matter of several investigations, see Refs. [1-8], [8-10] and [11, 12], respectively. In particular, previous papers present different analytical (modal summation approach) and numerical modelling techniques [1-6, 13, 14] as well as experimental procedures [7, 8] for the determination of the four-pole parameters and thence, the Transmission Loss (TL) performance of a Single-Inlet and Single-Outlet (SISO) and Single-Inlet and Double-Outlet (SIDO) rigid-wall rectangular plenum. The [T] matrix multiplication approach (also known as cascading [13, 15]) based on the 1-D plane wave theory is the simplest analytical technique to model wave propagation in axially long chambers; however, it is valid only in the low-frequency range, i.e., up to the cut-on frequency of the first transverse mode. Owing to the simplicity of the orthogonal modes (given by the trigonometric functions) inside a rigid-wall rectangular cavity that models its 3-D acoustic field, the evaluation of the four-pole parameters and the TL performance of rectangular chambers using a 3-D analytical/semi-analytical approach based on modal summation is popular [1-6].

Munjal [1] devised a simple semi-analytical point-collocation approach to obtain the four-pole (or the transfer [T] matrix) parameters and the TL characteristics of a simple expansion chamber type rectangular (and circular) chamber by incorporating the effect of higher-order transverse modes. This approach made use of the compatibility conditions for acoustic pressure and acoustic particle velocity at a number of equally spaced points (or nodes) on the end face on which the port (sudden area discontinuity) is located. This generated the required number of algebraic equations for evaluation of the modal

† A preliminary version of this work has been published in the proceedings of the International Congress on Sound and Vibration (ICSV- 20) held in Bangkok, Thailand during 7th to 11th July, 2013. ‡ A part of this work was carried out when the corresponding author was pursuing his doctorate at the Facility of Research in Technical Acoustics, Department of Mechanical Engineering, Indian Institute of Science, Bangalore – 560 012, India.

Final D

raft

2

amplitudes associated with the 3-D modal functions, the total number of which is proportional to the area ratio. One of the main limiting factors of this approach was the convergence of the modal solution which depends on the location of the nodal points requiring a constant track of the nodal mesh so that the collocated points do not fall on the pressure nodes. Furthermore, this approach was not suitable for fractional area ratio besides being rather inconvenient to use for chambers having a side port. Regardless of these limitations, the point-collocation approach was used by Chu et al. [4] to analyse a Single-Inlet and Single-Outlet (SISO) reverse-flow rectangular chamber configuration and more recently, by Wu et al. [5] wherein a SIDO as well as a Double-Inlet and Single-Outlet (DISO) straight-flow rectangular chamber configurations were analysed.

Ih [2] modelled the inlet/outlet ports (of a square or circular cross-section) as uniform velocity pistons and characterised a 2-port (SISO) rigid-wall rectangular chamber (in terms of the four-pole parameters) based on the orthogonal modal summation and superposing the velocity potentials due to individual pistons. It is noted that the analytical expressions for the velocity potentials were obtained by solving the homogeneous 3-D Helmholtz equation subject to inhomogeneous boundary conditions at the face on which the piston-source (port) was located. This 3-D analytical approach does not suffer from the aforementioned limitations of the point-collocation approach [1] and enabled him to readily conduct parametric studies on the effect of port location; the TL performance of different SISO configurations such as end-inlet/end-outlet chambers (straight-flow and reverse-flow configurations), end-inlet/side-outlet chambers (cross-flow configuration), Helmholtz resonators as well as 90 and 180 bends were analysed.

A conceptually same analytical approach was followed by Venkatesham et al. [3] to characterise SISO rigid-wall rectangular chambers wherein the inlet/outlet ports (of square cross-section) was modelled as uniform velocity pistons. They obtained the velocity potentials due to individual pistons in terms of the 3-D Green’s function or point-source response (for a rigid-wall rectangular cavity) which was integrated over the port area to yield the average response. The total acoustic pressure at the ports was obtained by superposing their individual average velocity potentials, following which, the transfer [T] matrix of the 2-port system was obtained. The individual velocity potentials are in essence, the different [Z] matrix parameters for a 2-port system. Similar to Ih [2], Venkatesham et al. [3] also conducted a parametric investigation to study the TL characteristics of different SISO configurations of rectangular chamber. However, it is noted that Refs. [2] and [3] do not explicitly formulate guidelines that recommends optimal location of ports towards designing rectangular chambers (of axially short or long length) for broadband attenuation pattern. It is important to note that Venkatesham et al. [3] obtain analytical expression for velocity potentials by solving the inhomogeneous 3-D Helmholtz equation subject to homogeneous boundary conditions. From a mathematically formal point-of-view, this is the main difference from the analytical formulation adopted by Ih [2] wherein the homogeneous 3-D Helmholtz equation was solved subject to inhomogeneous boundary conditions at the port-chamber interface (as indicated earlier). Although, these two modelling techniques yield identical result (TL graph), their mathematical equivalence had not yet been shown for rectangular chambers. The present work demonstrates that both these analytical methods of obtaining the acoustic pressure response functions or equivalently, the [Z] matrix parameters for a rectangular chamber are mathematically equivalent, see Appendix A.

Kadam and Kim [7] presented an experimental procedure to obtain the four-pole parameters of a rigid-wall 3-D rectangular cavity (made of hard wood) from the measured pressure response functions. Their experimental results was found to be in a good agreement with the analytical results [6], thereby validating the experimental approach for the first time. Li and Hansen [8] compared the experimental results of the TL performance of lined and unlined plenum chambers against several different and well-known prediction models such as the low- and high-frequency model proposed by Cummings [9] to analyse lined plenum chambers and the 3-D analytical model of Ih [2] to analyse unlined (rigid-wall) plenum chambers. Pan et al. [16] analysed the low-frequency acoustic response in a damped rectangular enclosure using the modified method of weighted residual. Their model was able to successfully predict the general features of the acoustic response of both a helicopter passenger cabin and a laboratory enclosure. Ali et al. [17] presented a review of the different Boundary-Element Method (BEM) techniques used for solving the acoustic eigenvalue problem in a rigid cavity and compared the results of different techniques for a rectangular cavity (for which the analytical expression for resonance frequencies is well-known).

While the Refs. [2, 3] consider the problem of characterising a 2-port (SISO) rigid-wall rectangular plenum chamber by means of the 3-D analytical uniform piston-driven model and evaluate its TL performance, this approach has not yet been used for characterising and evaluating the TL performance of a multi-port rigid-wall rectangular plenum having a single inlet and multiple (in general, M) outlet ports that may be located on the end or side face. This problem is important because a typical HVAC system may have multiple (two or more) outlet ports. Furthermore, it is well-known through the previous papers by Ih [2], Eriksson [18, 19] and the monograph by Munjal [13] that certain mode(s) of a rectangular duct may be suppressed by appropriately locating the port centre on their respective pressure node(s). Indeed, this technique of optimal port location has been exploited for the case of elliptical and circular cylindrical chambers to obtain broadband attenuation behaviour [13, 20-23]. Despite its fundamental nature and previous parametric studies on the effect of port location on TL performance [2, 3], the problem of determining optimal port location (with a view to obtain broadband attenuation) for the case of a SISO and in general, a Single-Inlet and Multiple-Outlet (SIMO) rigid-wall rectangular plenum configuration has not been considered. In fact, given the recent advancements presenting novel algorithms towards optimising a specific muffler configuration/geometry such as sub-chamber optimisation approach [24], the simulated annealing method for mufflers with perforated inlet extensions [25] and the differential evolution method for mufflers composed of multiple rectangular fin-shaped chambers [14], it becomes

Final D

raft

3

imperative to solve the outstanding problem of optimal port locations for the relatively simpler SISO/SIMO rectangular plenum; these optimised port locations are intended to form the basis of design guidelines for such chambers.

In parallel with the aforementioned studies, the TL performance of SIDO circular cylindrical chambers have been investigated using the 3-D analytical mode-matching approach [26] and the uniform piston-driven model via the 3-D Green’s function method [27]. In a previous work [28], the authors had presented a general algorithm to characterise a network of multi-port elements (using the impedance [Z] matrix) and developed novel expressions for determining the TL (in terms of scattering [S] matrix) for a general Multiple-Inlet and Multiple-Outlet (MIMO) muffler system. Their investigation was however, confined to validate the proposed algorithm by comparing the TL graphs of arbitrary multi-port network configurations computed using the 1-D axial plane wave theory against the 3-D Finite-Element Analysis (FEA) results. Yang and Ji [29] also investigated the problem of characterising a network of multi-port (2-port and 3-port) elements based on the [Z] matrix by using different numerical techniques such as point-collocation, BEM and numerical mode-matching to compute the individual [Z] matrices of sub-systems. Subsequently, the TL performance of the overall SIDO muffler system was obtained. In these papers, the characteristic features of a general SIMO muffler system satisfying acoustic reciprocity and energy conservation were not explicitly derived or proved, although Mimani and Munjal [23] later showed that interchanging the location of inlet and outlet ports of a SIDO elliptical cylindrical chamber completely alters its TL characteristics. In light of the background provided above, the objective of this work is therefore, to first present a 3-D analytical approach based on the modal summation technique and the uniform piston-driven model for characterising a multi-port rectangular chamber having arbitrary number of ports that may be located on the end or side face. By virtue of this analytical formulation, parametric studies are conducted with a view to examine the effect of axial and angular location of the inlet/outlet ports on the TL spectrum and subsequently, identify specific SISO/SIDO rectangular chamber configurations (of short and long lengths) that yield a broadband attenuation performance based on optimal port location. Furthermore, this paper analytically proves certain characteristic properties of a general reciprocal and conservative SIMO muffler system which facilitate a better understanding or analysis of its TL spectrum. The parametric studies enable one to computationally corroborate these properties for SIDO rectangular chambers.

This paper is organised as follows. Section 2 presents the theoretical formulation based on the uniform piston-driven model and the modal summation approach (via the 3-D Green’s function) for characterising a multi-port rigid-wall rectangular chamber having ports located arbitrarily on the chamber surface. Towards the end of this section, an expression for computing the TL performance of a general SIMO/SISO system is obtained. Section 3 analytically shows that the TL performance of a reciprocal and/or conservative SIMO muffler is significantly altered by interchanging the location of inlet and outlet ports. This section also shows that the peaks and troughs in the TL graph of a conservative SIMO and SISO system can be readily predicted or analysed in terms of the scattering [S] matrix parameters whilst for the special case of a SISO system, the [Z] matrix parameters are better suited for this purpose. Section 4 presents the TL performance of different SISO and SIDO rectangular chamber configurations obtained using the 3-D semi-analytical approach, validates the method by comparing the results with 3-D FEA prediction (and a previous result [2]) and carries out parametric studies to analyse the effect of axial and angular location of inlet/outlet ports on the broadband acoustic attenuation range. The paper is concluded in Section 5 wherein different SISO and SIDO rectangular chamber configurations exhibiting a broadband TL performance are mentioned.

2. THEORETICAL FORMULATION A multi-port chamber having N M ports denoted by P1, P2, …, PN + M is characterised by means of an impedance [Z] matrix representation shown hereunder [23, 28].

1 1 1 1x

... ... ... ... ,T TN N N M N N N MN M N Mp p p p v v v v

Z (1)

where

1 1 111 1

1 1

x1 1 1 1 1 1

1 1

.

N N MN

N NN N N N N M

N M N MN N N N N N N M

N M N M N N M N N M N M

Z ZZ Z

Z Z Z Z

Z Z Z Z

Z Z Z Z

Z

(2)

The acoustic pressure (Pa) and mass-velocity 3kg m at the ports P1, P2,…, PN+M of the N+M port system are

denoted by 1 1... ...N N N Mp p p p and 1 1... ... ,N N N Mv v v v respectively. It is noted that direction of the mass-

velocity is considered positive looking into the system and a harmonic time-dependence is assumed so that ωj ti ip p e and

Final D

raft

4

ω ,j ti iv v e i =1, 2,…, N+M. Furthermore, 1j and is the excitation angular frequency 1radian s and is given by

2 ,f where f is the frequency in Hz. In the ensuing subsection, a multi-port rectangular expansion chamber having N M ports is characterised using the

uniform piston-driven model via the 3-D Green’s function approach in terms of the [Z] matrix representation. It is noted that the walls of the plenum chamber are considered rigid (with no absorptive/dissipative linings), therefore, a conservative system is being considered. Furthermore, a stationary medium, i.e., a zero mean flow is assumed implying that the rectangular expansion chamber satisfies the principle of acoustic reciprocity [30].

2.1 Characterisation of a multi-port Rectangular chamber The acoustic field in a rigid-wall rectangular chamber is obtained by solution of the homogeneous 3-D Helmholtz equation in Cartesian coordinates , , x y z shown as follows [1-7, 16, 31].

2 2 2

2 2 20 2 2 20, ,k p

x y z

(1, 2)

where ( , , )p p x y z represents the acoustic pressure field, 0 0k c is the excitation wavenumber 1(m ) and 0c denotes the sound speed 1(m s ). The application of homogeneous rigid-wall boundary condition on the chamber faces yields the orthogonal modal solution given by [1-7, 16, 31]

= = =

0 0 = 0

, , cos cos cos ,l m n

nmll m n

n m lp x y z A x y zB H L

(3)

where B, H and L denote the breadth, height and length of the rectangular chamber, respectively, nmlA denotes the modal coefficient corresponding to the ( , , )n m l mode. It is noted that the resonance frequency of the ( , , )n m l mode of the rectangular chamber is given by [2, 3, 16]

2 2 2

0 .2nmlc n m lf

B H L

(4)

The acoustic pressure response function for characterising the rectangular chamber is obtained by modal solution of the

inhomogeneous 3-D Helmholtz equation (subject to homogeneous rigid-wall boundary conditions) shown as follows [3, 6, 31] 2 2

0 0 0δ δ δ ,S S Sk p j ρ Q x x y y z z (5)

where 0 is the ambient density -3kg m of the air, 0Q is the volume flow-rate 3 -1m s due to the source port that is

modelled as point-source, δ denotes the Dirac Delta function (in Cartesian co-ordinates) whilst , , S S Sx y z denotes the location of the centre of the source port. The orthogonal modal solution of Eq. (1) expressing the 3-D acoustic field inside the rigid-wall rectangular chamber given by Eq. (3) is inserted into Eq. (5) whereby the modal coefficients nmlA are evaluated using mode orthogonality of circular functions; these are back-substituted in the modal solution to obtain the 3-D Green’s function , , , ,R R R S S SG x y z x y z or the acoustic pressure response due to a point-source port S located arbitrarily , , S S Sx y z on the chamber surface [3, 31].

0 0 0 0

0 0 2

, ,

, , , , , , , ,

cos cos cos cos cos cos

R R R S S S R R R S S S

S S SR R R

n m l

p x y z x y z G x y z x y zρ Q ρ Q

m y n x l zm y n x l zH B L H B Ljk c

n mNB H

2 20,1,2,... 0,1,2,... 0,1,2,... 2

0

e ,n m l

j t

n m l l kL

(6)

where 2 2 2

, ,0 0 0

cos d cos d cos d ε ,y Hx B z L

n m l nmlx y z

n x m y l zN x y z BHLB H L

(7)

Final D

raft

5

are product of the integrals of the square of a particular set of mode shape functions ( , , )n m l integrated over the volume of rectangular chamber whilst

ε 1 for 0 ,

0.5 for 0, 0 , 0, 0, 0 , 0, 0, 0 ,

0.25 for 0, 0, 0 , 0, 0, 0 , 0, 0, 0 ,

0.125 for 0, 0, 0 .

nml m n l

m n l m n l m n l

m n l m n l m n l

m n l

(8a-d)

and , , R R Rx y z denotes the co-ordinates of the centre of the receiver port R. It is observed from Eq. (6) that on interchanging the location of source and receiver ports, the Green’s function remains unaltered, thereby indicating that the 3-D acoustic field inside the rectangular chamber with zero mean flow satisfies the principle of acoustic reciprocity [30]. Furthermore, it is also noted that the Green’s function response is purely imaginary which analytically shows that the rigid-wall rectangular chamber (i.e., the absence of a dissipative lining) having zero mean flow is both reciprocal and conservative [30].

The point-source modelling is the least accurate method for obtaining the acoustic pressure response of a cavity due to excitation at a port (sudden-area discontinuity) because the ports of finite cross-section area are approximated as points on the chamber surface [32, 33]. In fact, the point-source model is known to have convergence issues at the source port; while the modal summation series in the cross-impedance parameters , ,ijZ i j (i.e., at the far-field of the source port) exhibit good converge when computed using the point-source model, the modal summation series in the self-impedance parameters

iiZ converge very slowly (i.e., an extraordinarily large number of modal terms are required) when computed using the point-source model, see Zhou and Kim [6]. Therefore, in view of this limitation, the more realistic and accurate uniform piston-driven model is used to compute all the acoustic pressure response functions, i.e., both self- and cross-impedance parameters (using the 3-D Green’s function given by Eq. (6)) by modelling the source port as rigid oscillating piston having uniform velocity distribution which is equal to the normal acoustic particle velocity in the chamber over the cross-sectional area of the port. Furthermore, the normal acoustic particle velocity in the chamber over the annular cross-section of the rigid face (on which the source port is located) is set to zero whilst the acoustic pressure field in the chamber and port are taken equal over the port-chamber interface. Therefore, the uniform piston-driven model assumes planar wave propagation in the ports from the port-chamber interface, i.e., it neglects the higher-order transverse evanescent modes in the ports. This assumption is justified because the diameter of ports is significantly smaller than the dimensions of the rectangular chamber, therefore, the transverse modes propagate (or become cut-on) only at high frequencies whilst throughout the frequency region of interest (for an automotive muffler), these modes are evanescent that decay within a short distance, approximately less than two times the port diameter [13]. Incidentally, Zhou and Kim [6] termed the uniform piston-driven model as the surface-source model and showed that this method is equivalent to computing the average of the Green’s function over the source port area and it guarantees overcoming the slow convergence issue (of the iiZ parameters) arising out of the point-source representation. Nevertheless, it is mentioned here that the accuracy of modelling the acoustic fields at the sudden-area discontinuity can be further improved by also considering the higher-order transverse modes in the ports by using the 3-D analytical mode-matching approach [34, 35] which is the most accurate of all modelling techniques. However, it is algebraically much more tedious as compared to the uniform piston-driven model and yields nearly identical results throughout the frequency range of interest. Therefore, in view of its relative simplicity and comparable accuracy, use of the 3-D uniform piston-driven model is popular [2, 3, 6, 7, 20-23] and is indeed used here.

The mathematical rendition of the uniform piston-driven model assuming the source port S as rigid oscillating piston that is located on the X-Y face is shown hereunder [3, 6, 7, 31].

2 20 0 0 1, , , ,Sk p x y z j ρ U f x y z z (9)

where 1 , 1, Sf x y S X-Y 0, ,SS S (10a, b)

In Eq. (10a) and (b), SS and X-YS denote the cross-sectional area of the source port S and the X-Y face of the rectangular chamber, respectively, whilst the symbol X-Y SS S denotes the annular area (excluding the port). On making use of Eqs. (5), (6), (9) and (10), it can be shown that the acoustic pressure response based on the uniform piston-driven is given by

S

1, , , , , , , , d d .R R R S S S R R R S S S S SS S

p x y z x y z G x y z x y z x yS

(11)

Final D

raft

6

Equation (11) is further integrated over the cross-sectional area of the receiver port R and divided by it to obtain the average response whereby the Z matrix parameter RSZ is obtained. The RSZ parameter for different cases of location of receiver port on the X-Y, Y-Z or Z-X face is given by

0 0 0 0

0 0

, , , , , , , ,1 1 d d d d , Port R located on X-Y plane ,

, , , ,1 1 d

R S

R R R S S S R R R S S SRS S S R R

R SS S

R R R S S S

R S

p x y z x y z G x y z x y zZ x y x y

ρ Q S S ρ Q

G x y z x y zx

S S ρ Q

0 0

d d d , Port R located on Y-Z plane ,

, , , ,1 1 d d d d , Port R located on Z-X plan

R S

R S

S S R RS S

R R R S S SS S R R

R SS S

y y z

G x y z x y zx y z x

S S ρ Q

e ,

(12a-c)

where RS denotes the cross-sectional area of the receiver port. The self-impedance parameter SSZ (for source port located on the X-Y face) is obtained by replacing the suffix R by S in Eq. (12) to obtain

0 0 0 0

, , , , , , , ,1 1 d d d d .S S

S S S S S S S S S S S SSS S S S S

S SS S

p x y z x y z G x y z x y zZ x y x y

ρ Q S S ρ Q

(13)

The [Z] matrix parameters characterising a rectangular chamber having N M ports are obtained by using Eqs. (12) and (13). It is noted that Venkatesham et al. [3] characterised a 2-port rectangular chamber following the same approach. 2.2 Integration of the Green’s function over the port cross-sectional area

The integration of the 3-D Green’s function over the cross-sectional area of the source/receiver ports shown in Eq. (12) is carried out using numerically using the Simpson’s three-eighths rule [36] for ports having a circular cross-section, although Ih [2] had previously obtained analytical expressions for these integrals in terms of the first-order ordinary Bessel function of the first kind. It is for this reason that the present modelling approach is termed as semi-analytical. For a port of diameter Sd or equivalently, radius Sr centred at ( , )S Sx y on the X-Y face, the integral expression is given by

22

0 00 0

220 00 0

220cos cos d d 2 cos d , 0,

S SS S

S SS S

y r x xx r x rS S S

S S S S Sx r x ry r x x

n x m y n xy x r x x x mB H B

0

0

22002 cos sin cos d , 0.

S

S

x rS S S

Sx r

m r x xm y n xH x mm H H B

(14)

It is noted for the plane wave mode, i.e., 0,m n Eq. (14) is evaluates to 2 ,Sr the port cross-sectional area. Similarly, for ports located on the Y-Z and X-Z planes, the integral expressions are given by

2 22 2

0 0 0 00 0

2 22 20 00 0 0 0

cos cos d d and cos cos d d ,S S S S

S S

S SS S S S

y r z z x r z zz r z rS S S S

S S S Sz r z ry r z z x r z z

l z m y l z n xy z x zL H L B

(15, 16)

respectively. The integrals for ports of square cross-section can be readily evaluated through closed-form analytical expressions, see Refs. [2, 3, 31]. 2.3 Influence of the port location on the mode propagation/suppression The location of ports can significantly influence the excitation or suppression of the higher-order transverse modes (along the x and y directions) or axial modes (along the z direction) of the rectangular chamber [18, 19] which directly influences the [Z] matrix parameters. It is studied analytically in this subsection using the point-source model or the Green’s function given by Eq. (6). To this end, the test-case of a port S centred at 0.5 , 0.5S Sx B y H on the X-Y end face is considered wherein it is observed that

Final D

raft

7

cos 0, 1, 3, 5,.... cos 0, 1, 3, 5,....2 2

m nm n

(17, 18)

implying that a centred location of the port on the end face results in the modal coefficient associated with the corresponding odd mode amount to zero, thence not allowing the odd modes to propagate although the frequency may be greater than the cut-on frequency of that mode. It is for this reason that for a concentric rectangular expansion chamber, only the even modes propagate which explains the repetitive nature of the domes and troughs for a frequency range beyond the cut-on frequency of the first order higher mode.

The test-case of a port S located at 0.25 or 0.75 , 0.25 or 0.75S Sx B B y H H on the X-Y end face is now considered wherein it is observed that

3 3cos cos 0, 2, 6, 10,... cos cos 0, 2, 6, 10,...4 4 4 4

m m n nm n

(19, 20)

Equations (19) and (20) signify that even modes given by m = n = 2, 6, 10… do not propagate if the port S is centred at

the aforementioned co-ordinates. Similar comments on the suppression of even or odd axial modes also hold for a side port located on the Y-Z or X-Z faces. These results/comments on the influence of port location on propagation/suppression of certain rigid-wall modes will be used in the ensuing section to explain the nature of TL graph, in particular, the broadband attenuation characteristics of different 2-port (SISO) rectangular chamber configurations. 2.4 Computation of TL of a SIMO/SISO muffler system An expression for the TL performance for a muffler system having a single inlet (say, port P1) and multiple outlet (ports P2, P3,…, PM + 1) is obtained in terms of the S matrix or Z matrix parameters. To this end, the S matrix is expressed in terms

of the Z matrix of a 1M port muffler system shown as follows [22, 23, 28].

1 11 1 1

1 1 2 .M M

S Z Y Ι Z Y I I Z Y Ι (21)

In Eq. (21), the number of inlet ports N = 1 (port P1), I is the identity matrix and [Y] is a diagonal matrix consisting of

the characteristic impedances of the ports. It is assumed that the ports are acoustically compact, i.e., their diameter is much smaller compared to the shortest wavelength of interest (to automotive muffler designer) so that only planar waves propagate. Anechoic termination is imposed at M outlet ports (P2, P3,…, PM+1) to yield

1 1 11 2 1 21 3 1 31 1 1 1 1, , ,..., ,M MB A S B A S B A S B A S (22)

following which the TL expression is obtained for a SIMO muffler system given by

11 10 2

221 13121

2 3 1

1

TL 10log .

... M

M

Y

SSSY Y Y

(23)

In Eq. (23), 1A is the incident wave amplitude at port P1 whilst 1 2 1 MB B B is the vector of reflected wave

amplitudes at ports P1, P2, P3,…, PM+1, respectively, propagating downstream. It is noted that at ports P2, P3,…, PM+1, the incident wave amplitudes 2 3 1 0,MA A A respectively, due to the imposition of anechoic conditions. The TL for the SIMO system for the different cases of inlet port located at port P2, P3,…, PM + 1 is similarly given by

2 12 10 1012 2 2 2

221 2 1 1 2 1 13212

1 3 1 1 2

1 1

TL 10log ,..., TL 10log ,

... ...

MM

M M M M M

M M

Y Y

S S S SSSY Y Y Y Y Y

(24, 25)

Final D

raft

8

respectively. The TL for a SISO muffler system (with ports P1 and P2 taken as the inlet and outlet, respectively) is obtained by substituting M = 1 in Eq. (23). On expressing the 21S parameter in terms of the Z matrix parameters followed by further algebraic manipulation, one obtains [22]

211 1 22 2 21 122

1 10 1021 1 2 2121

1 1TL 10log 10log .4

Z Y Z Y Z ZYY Y Y ZS

(26)

It is noted that the Abom [37] obtained an identical expression for TL for a SISO muffler system in terms of the [S]

matrix parameter 21,S wherein the convective effects of mean flow at the ports were also taken into account. 3. TRANSMISSION LOSS PROPERTIES OF A RECIPROCAL AND CONSERVATIVE SIMO MUFFLER SYSTEM 3.1 Interchanging the positions of inlet and outlet ports of a reciprocal SIMO system An analytical proof is outlined in this subsection to show that the TL performance of a reciprocal SIMO muffler system is significantly altered by interchanging the position of inlet and outlet ports. To this end, the Z matrix of a 1M port

muffler system is expressed in terms of the S matrix shown as follows [28].

1 1

1 1 2 . M M

Z Ι S I S Y I S I Y (27)

For a muffler system satisfying the principle of acoustic reciprocity [30], T Z Z and on use of this reciprocity

condition in Eq. (27), one obtains the following relation between the S matrix parameters.

1T S Y S Y or , , 1,2, ... , 1 .iij ji

j

YS S i j MY

(28)

From Eq. (28), one obtains the relation 12 21 1 2S S Y Y that is substituted in TL2 to yield

12 10 2

2221 2 232 1 2

2 3 1 1 1

1

TL 10log .... M

M

YS Y YS SY Y Y Y Y

(29)

It is noted that in general, since the S parameters S32 ≠ S31,…, S(M+1)2 ≠ S(M+1)1, the TL2 ≠ TL1 for a general reciprocal

SIMO muffler system. Equation (28) is similarly used to simplify the expression for TLj, where 1 < j ≤ (M+1), to obtain

110 2

2 2 2212 1 1 1

2 1 1 1 1 1 1 1

1

TL 10 log .

... ...j

j j j j jj j j j j M j

j j j M

Y

S Y Y Y YS S S S

Y Y Y Y Y Y Y Y Y

(30)

Based on a similar reasoning, it can be readily shown that TL1 ≠ TLj. Therefore, it is concluded that TLi for a reciprocal

SIMO muffler system corresponding to excitation at the ith port (port Pi being the inlet port) would, in general, be different from TL j of the same SIMO system corresponding to excitation at the jth port (port Pj being the inlet port), i.e., TL TL ,i j thereby proving that interchanging the inlet and outlet port locations may significantly alter the TL characteristics of a reciprocal SIMO system. As a corollary of the foregoing derivation, it is noted that for a 2-port muffler system satisfying acoustic reciprocity (not necessarily being conservative), the relation 21 12 2 1S S Y Y is used to obtain

2 11 10 10 22 2

1 221 12

1 1TL 10log 10log TL ,Y YY YS S

(31)

thereby signifying that the TL graph of a reciprocal SISO muffler remains unaltered on interchanging the position of inlet and

Final D

raft

9

outlet ports, see Refs. [13] and [27]. 3.2 Analysis/prediction of the observed peaks and troughs in the TL spectrum 3.2.1 A reciprocal and conservative SISO muffler system

The advantage of expressing the TL of a reciprocal and conservative SISO muffler system [30] explicitly in terms of the Z matrix parameters (given by Eq. (26)) is that it enables one to analyse/predict the characteristic features of the TL spectrum such as the frequency of occurrence of the attenuation peaks and the troughs in a rather straightforward manner. A peak in the TL graph of a general SISO muffler occurs at a frequency f when either of the following conditions is satisfied by the Z matrix parameters [38]: (a) 11 ,Z whilst 22 21, Z Z (and 12Z ) are finite, (b) 22 ,Z whilst 11 21, Z Z (and 12Z ) are finite, and (c) 11Z and 22 ,Z whilst 21Z (and 12Z ) are finite, (d) 21 0Z (or equivalently, 12 0Z ), regardless of the order of magnitudes of the self-impedance parameters 11Z and 22.Z

It is noted that since a reciprocal system is considered, the following conditions will hold good at a given frequency :f (1) if 21Z is finite it implies that 12Z is also finite and (2) 21 120 0.Z Z When conditions (a-c) are satisfied, the numerator of the RHS of Eq. (26) tends to infinity whilst when condition (d) is satisfied, its denominator tends to zero, implying a large attenuation or peak at f in the TL spectrum of a reciprocal and conservative system. However, it is worth mentioning that the TL graph of a SISO system which is not necessarily conservative or reciprocal, exhibits an attenuation peak at a frequency f when conditions (a-d) are satisfied.

A trough in the TL graph of a SISO muffler occurs at a given frequency f when all the Z matrix parameters (that

are purely imaginary for a reciprocal and conservative muffler system [30]) tend to infinity, i.e., ijZ f for , 1, 2i j at

the same rate (precisely as 2O , 0 ) which implies 11 22 12 21 0.Z Z Z Z Hence, under this condition, the TL expression given by Eq. (26) simplifies to

2 21 211 2 22 1

10 101 2 21 1 2

1TL 10log 10log ,4 4

Y YZ Y Z YY Y Z Y Y

(32)

and for ports with equal diameters, i.e., 1 2 ,Y Y the TL is exactly equal to zero, thereby indicating the occurrence of a trough. 3.2.2 A conservative SIMO muffler system The principle of energy conservation is used to formulate general conditions which when satisfied, the TL graph of a conservative SIMO muffler system (having M outlet ports) exhibits a peak or a trough at a given frequency .f To this end, the inlet port P1 is excited with a time-harmonic piston velocity and anechoic termination is implemented at the remaining M outlet ports P2, P3,…, PM + 1. The energy conservation statement is invoked, according to which the incident acoustic power at the inlet port P1 is equated to the sum total of acoustic power transmitted downstream of ports P2, P3,…, PM +1, thereby yielding

22 2 2 2

1 11 1 11 21

0 1 0 1 2 1

... ,2 2

M

M

SA A S Sρ Y ρ Y Y Y

(33)

which then is substituted in Eq. (23) to yield the following canonical form of TL1 expression

1 10 211

1TL 10log .1 S

(34)

Equation (34) may readily be used to analyse/predict the location of attenuation peaks and troughs in the TL1 spectrum

of a conservative SIMO system as may be understood from the following discussion.

Final D

raft

10

1. The physical implication of 11 1S f is that almost all the acoustic power incident due to the inlet piston at the port P1 is reflected back into the system and negligible acoustic power is transmitted downstream to the anechoic termination at the outlet ports P2, P3,…, PM +1. This signifies that the TL graph exhibits a resonance peak at the frequency f.

2. On the other hand, 11 0S f signifies that almost all the acoustic power incident due to the inlet piston at port P1 is transmitted to the anechoic outlet ports P2, P3,…, PM +1 whilst negligible fraction of acoustic power is reflected back into the system through the inlet port P1. This signifies that the SIMO muffler is acoustically transparent at the frequency f, thereby resulting in a trough in the TL graph.

On the basis of this discussion, it is evident that in the iiS versus f graph of conservative SIMO system with arbitrary

M outlet ports, the frequency locations corresponding to 0iiS f would indicate the occurrence of a trough, whereas

1iiS f would indicate the occurrence of a resonance peak when port Pi is the inlet port. Here, 1, 2,..., 1.i M The attenuation peak and trough in the TL graph of a reciprocal and conservative SISO system may also be predicted by

outlining different conditions satisfied by the [Z] matrix parameters that occur in the expression of 11S or 22S parameters that are given by

11 1 22 2 12 2111

11 1 22 2 12 21

,Z Y Z Y Z Z

SZ Y Z Y Z Z

11 1 22 2 12 21

2211 1 22 2 12 21

.Z Y Z Y Z Z

SZ Y Z Y Z Z

(35, 36)

From Eqs. (35) and (36), it may be readily shown that an attenuation peak occurs at a frequency f if any one of the

conditions (a-d) indicated in section 3.2.1 are satisfied. Similarly, a trough occurs at a frequency f when all the Z matrix

parameters tend to infinity, i.e., ijZ f for , 1, 2.i j It is worth mentioning that due to algebraically simple forms of S11 and S22 parameters, the different conditions for occurrence of attenuation peak and troughs in the TL graph can be explained in terms of the [Z] matrix parameters in a rather straightforward manner. However, for a conservative SIDO system (i.e., 2M ), outlining such conditions in terms of the [Z] matrix parameters is difficult as may be appreciated from the algebraic tediousness of the S11 parameter for a 3-port system given by

11 1 22 2 33 3 23 32 11 1 13 31 22 2 12 21 33 3 31 12 23 13 21 3211

11 1 22 2 33 3 23 32 11 1 13 31 22 2 12 21 33 3 31 12 23 13 21 32

.Z Y Z Y Z Y Z Z Z Y Z Z Z Y Z Z Z Y Z Z Z Z Z Z

SZ Y Z Y Z Y Z Z Z Y Z Z Z Y Z Z Z Y Z Z Z Z Z Z

(37)

For SIMO muffler systems with more than 2 outlet ports, i.e., 3,M outlining different such conditions (when an attenuation peak or trough occurs a frequency f ) in terms of the [Z] matrix parameters is still more formidable due to the increasing algebraic tediousness of the expression for iiS parameter. In view of the foregoing discussion, it is concluded that analyzing iiS versus f graph offers a convenient and a rather powerful method for analysing/explaining the salient features of TL spectrum of a conservative SIMO system.

The TL expressions for a conservative SIMO system with ports P2, P3,…, PM +1 taken as the inlet ports are similarly obtained and are shown as follows.

2 10 3 10 1 102 2 222 33 1 1

1 1 1TL 10log ,TL 10log ,..., TL 10log .1 1 1

M

M MS S S

(38)

It is noted from Eqs. (34) and (38) that in general, for a conservative (not necessarily a reciprocal) SIMO system, 1 2 3 1TL TL TL TLM inasmuch as 11 22 33 1 1... ,M MS S S S in general. Therefore, in general, interchanging the

inlet and outlet port locations would also significantly alter the TL characteristics of a conservative SIMO system. 4. RESULTS, ANALYSIS AND FORMULATION OF DESIGN GUIDELINES This section presents the TL performance of different configurations of the SISO and SIDO rectangular chambers computed using the 3-D semi-analytical model (discussed in section 2) and validates the method by comparing the results with those obtained using the 3-D Finite Element Analysis (FEA) and also against a previous result (from literature [2]). Unless otherwise noted, the breadth and height of the rectangular chamber are both taken equal and given by 200 mm,B H the length L of the axially short and long chamber is taken as 50 mm and 300 mm, respectively, ports of circular cross-section are considered and their diameters are taken equal given by d0 = 40 mm. Furthermore, the ambient temperature field in the rectangular

Final D

raft

11

chamber and ports is assumed to be uniform and is taken as 0 20 CT implying that the sound speed 10 343.14 m sc and a

uniform ambient density 30 1.20 kg m . It is noted that for the different SISO and SIDO configurations analysed in this

work, the infinite modal summation in the Green’s function response (given by Eq. (6)) is truncated to the first 20 modes in each of the x, y and z directions, i.e., 0, 1, 2,..., 20.n m l The truncated Green’s function response coupled with the uniform piston-driven model exhibits good convergence throughout the frequency range of interest. 4.1 Chambers having a single end-inlet and single/double end-outlet: Straight-flow/Reverse-flow configuration Figure 1(a) shows the three orthogonal views of a 2-port rectangular expansion chamber configuration having an end-inlet port 1 and end-outlet port 2 located on the opposite B-H end faces whilst Fig. 1(b) shows its 3-D view. This configuration is referred to as the SISO straight-flow configuration.

Figure 1 A 2-port straight-flow configuration of a rectangular expansion chamber having an end-inlet port (marked as 1) and end-outlet port (marked as 2) located on the opposite B-H end faces: (a) Orthogonal projections and (b) 3-D view.

Figure 2(a) and (b) shows the orthogonal views and the 3-D view, respectively, of a 2-port rectangular expansion chamber configuration having an end-inlet port 1 and end-outlet port 2 located on the same B-H end face. This configuration is referred to as the SISO reverse-flow configuration.

Figure 2 A 2-port reverse-flow configuration of a rectangular expansion chamber having an end-inlet port (marked as 1) and end-outlet port (marked as 2) located on the same B-H end face: (a) Orthogonal projections and (b) 3-D view.

For both straight-flow and reverse-flow configurations, the end ports 1 and 2 are centred at 1 1( , )x y and 2 2( , ),x y respectively, on the B-H end face. It is noted that in this subsection, only the axially short rectangular chamber configurations are analysed because the TL graph of axially long chambers are well-known to have either a simple expansion chamber type

Final D

raft

12

behaviour for straight-flow configuration (characterised by the regular occurrence of domes and troughs in the low-frequency range [1, 13]) or a quarter-wave type behaviour for reverse-flow configuration [35, 38, 39].

Figures 3(a) and (b) presents the TL performance obtained using the 3-D semi-analytical approach and 3-D FEA for

axially short straight-flow and reverse-flow configurations, respectively, when the port 1 is centred at 1 1 100 mmx y (the pressure node of odd transverse modes) and port 2 is centred at 2 2 150 mmx y (the pressure node of even transverse modes). It is noted that the vertical lines in Fig. 3 denote the resonance frequency of , , n m l mode and the same convention is followed henceforth. An excellent agreement is observed between the 3-D semi-analytical approach and 3-D FEA in Figs. 3(a) and (b) which validates the present method. It is important to mention here that the 10-noded tetrahedral elements were used during 3-D FEA which are especially attractive to use because of the availability of automatic tetrahedral meshing programs, exhibit good numerical convergence and thus, are a good general purpose element. The element size Δl of the tetrahedron is so chosen that there is a minimum of eight nodes per wavelength and for the sound speed considered and the maximum frequency of interest max 3000 Hz,f this amount to minΔ 8 14.3 mm,l where min 114.38 mm.

A broadband attenuation performance is obtained throughout the frequency range for both straight-flow and reverse-flow configurations as shown in Figs. 3(a) and (b), respectively. Indeed, both these configurations yield an attenuation over 20 dB in the frequency range 858, 2713 Hzf where the lower and upper limits correspond to the resonance frequency of the (1, 0, 0) or (0, 1, 0) mode and (3, 1, 0) or (1, 3, 0) mode, respectively. The broadband attenuation pattern is due to the occurrence of peak at the resonance frequencies of the transverse modes which is explained in terms of the Z matrix parameters as follows. At the resonance frequencies of (1, 0, 0), (0, 1, 0), (1, 1, 0), (3, 0, 0), (0, 3, 0), (3, 1, 0) and (1, 3, 0) modes, i.e., the odd transverse modes, the 11 21,Z Z (and 12Z ) parameters are finite whilst 22Z which signifies that the TL graph exhibits a peak at these frequencies (see section 3.2.1). Similarly, at the resonance frequencies of (2, 0, 0), (0, 2, 0), and (2, 2, 0) modes, the even transverse modes, the 22 21,Z Z (and 12Z ) parameters are finite whilst 11Z which implies that the TL graph exhibits a peak at these frequencies. It is noted that both straight-flow/reverse-flow configurations exhibit a trough at the resonance frequency of the (4, 0, 0) or (0, 4, 0) mode (coincident in this case because B H ), i.e., at 3431.4f Hz (not shown in Fig. 3), thereby leading to the breakdown of the broadband attenuation pattern. This frequency corresponds to the maximum frequency limit up to which an axially short straight-flow/reverse-flow SISO rectangular chamber can exhibit a broadband attenuation and is given by 400 upper 02f f c B or 040 upper 02 ,f f c H whichever occurs earlier, provided that the resonance frequency of the first axial mode given by 001 00.5f c L occurs after or is coincident with 400f or 040f implying that maximum length L for the short chamber must satisfy either 0.25L B or 0.25.L H Incidentally, for the chamber dimensions considered, 0.25L B L H implying maximum attenuation obtained whilst having the largest possible broadband frequency range of attenuation.

Figures 3(a) and (b) demonstrate that one may easily exploit the influence of higher-order transverse modes in an axially short rectangular chamber by optimally locating the end-inlet port 1 at x1 = 0.5B, y1 = 0.5H and end-outlet port 2 at x2 = 0.75B, y2 = 0.75H on the same/opposite end face to obtain a broadband attenuation pattern. The optimal location of end ports is indeed similar to the double-tuning of the length of concentric inlet and outlet pipes in an extended-inlet and extended-outlet chamber muffler [40]. In order to mathematically demonstrate the dominance of higher-order transverse modes over the axial plane wave modes, a model based on the 2-D transverse modes is considered by ignoring all those modes in Eq. (6) that has axial variation/dependence to obtain the impedance matrix parameter RSZ .

0 0 2 20,1,2,... 0,1,2,...0 0 2

0 0

1 1cos cos d d cos cos d d ( , , , , )

ε

S R

S S R RS S R R

m n S RS SR R R S S SRS

m nnm

m y n x m y n xx y x yS H B S H Bp x y z x y z

Z jk cρ Q n mHBL k

B H

.

(39)

It is important to mention that the 2-D transverse model considered in this work (Eq. (39)) is in essence, the same as

that developed in previous papers [41, 42] for thin (or axially short) muffler elements of uniform rectangular or circular cross-section and is valid up to the frequency when the corresponding wavelength is sufficiently larger than axial length.

Figures 3(a) and (b) indicate that the 2-D transverse model is nearly coincident with the 3-D predictions for almost throughout the frequency range, in particular, for straight-flow configuration. This proves that for axially short rectangular chambers, the transverse modes completely dominate the acoustic field and in fact, the axial modes may be altogether neglected up to the resonance frequency of the first axial mode, i.e., 001 00.5f c L without significantly affecting the accuracy. Furthermore, from the point-of-view of 2-D transverse model, the axially short straight-flow and reverse-flow configurations are acoustically identical or equivalent inasmuch as this model does not distinguish between the relative axial

Final D

raft

13

location of end ports. In fact, in previous papers by the authors, a similar conclusion was also arrived at wherein axially short elliptical and circular end-chambers were analysed using the 1-D transverse plane wave approach [43].

Figure 3 Broadband TL performance of axially short SISO rectangular chamber in (a) straight-flow configuration shown in Fig. 1 and (b) reverse-flow configuration shown in Fig. 2. The transverse location of the end ports 1 and 2 (on the X-Y end faces) are identical and is given by x1 = 0.5B, y1 = 0.5H, x2 = 0.75B, y2 = 0.75H. The TL graph obtained using the 3-D semi-analytical approach is compared with those obtained using the 3-D FEA and 2-D transverse model. (c) Broadband TL performance of axially short SISO straight-flow and reverse flow configurations (computed using the 3-D semi-analytical method) having the same transverse location of end ports 1 and 2 given by x1 = 0.5B, y1 = 0.75H and x2 = 0.75B, y2 = 0.5H, respectively. The chamber and port dimensions in parts (a) to (c) are same and given by B = H = 200 mm, L = 50 mm and d0 = 40 mm.

Figure 3(c) presents the TL performance (obtained using the 3-D semi-analytical approach) for axially short straight-flow and reverse-flow configurations when the end port 1 is centred at 1 1100 mm, 150 mmx y and end port 2 centred at

2 2150 mm, 100 mm.x y It is observed from Fig. 3(c) that a broadband attenuation performance is obtained throughout the frequency range for both straight-flow and reverse-flow configurations. The TL graphs in Fig. 3(c) exhibit peak at the resonance frequencies of transverse modes and are nearly identical with those shown in Figs. 3(a) and (b), except the absence of peaks at 110 220 310, ,f f f or 130f (because all [Z] parameters are finite and non-zero at these frequencies) and the occurrence of an additional peak at 210f or 120.f Similar to the configurations analysed in Figs. 3(a) and (b), the TL graphs shown in Fig. 3(c) also exhibit a trough at 400 040 3431.4 Hz,f f thereby indicating the broadband attenuation range for this configuration.

Another set of parametric study was carried out to investigate the effect of angular location of the ports 1 and 2 on the broadband attenuation range of the axially short end-inlet and end-outlet straight-flow/reverse-flow configurations. To this end, the following test-configurations were considered: (1) x1 = 0.5B, y1 = 0.5H and x2 = 0.5B, y2 = 0.75H and (2) x1 = 0.5B, y1 = 0.75H and x2 = 0.5B, y2 = 0.25H. (The end ports may be located on the same B-H face or on opposite faces.) It was found that

Final D

raft

14

the TL graph (not shown here) for the test-configurations (1) and (2) exhibit a trough at 200f f and 010 ,f f respectively, which results in a breakdown of the broadband attenuation pattern significantly before 400f or 040.f

In view of the outcome of the foregoing parametric studies, it is concluded that for an axially short straight-flow/reverse-flow SISO chamber, the transverse location of the end ports 1 and 2 should be taken either as (1) x1 = 0.5B, y1 = 0.5H, x2 = 0.75B, y2 = 0.75H or (2) x1 = 0.5B, y1 = 0.75H, x2 = 0.75B, y2 = 0.5H to nullify the troughs at the resonance frequencies of the first few transverse modes and maximise the broadband attenuation range up to 400f f or 040.f f Indeed, use of these high-performance short chamber configurations are recommended for engineering application such as hermetically sealed refrigeration compressors [41, 42] and for medical application such as Continuous Positive Airway Pressure (CPAP) devices [44] where severe space/volume constraint is often a limiting requirement.

Figure 4(a) shows the three orthogonal views of a SIDO (3-port) rectangular expansion chamber configuration having

end ports 1 and 2 that are located on the same B-H end face and centred at 1 1( , )x y and 2 2( , ),x y respectively, whilst the end port 3 is located on the opposite B-H end face and is centred at 3 3( , ).x y Figure 4(b) shows its 3-D view.

Figure 4 A 3-port rectangular expansion chamber having two end ports (marked as 1 and 2) located on a B-H end face and the third end port (marked as 3) located on the opposite B-H end face: (a) Orthogonal projections and (b) 3-D view.

Figure 5(a) depicts the TL graphs for the axially short SIDO configuration (shown in Fig. 4) for different cases when (a) end port 1 is the inlet and ports 2 and 3 are outlet, (b) end port 2 is the inlet and ports 1 and 3 are outlet and (c) end port 3 is the inlet and ports 1 and 2 are outlet. It is noted that end port 1 is centred at 1 1 100 mmx y whilst the end ports 2 and 3 are centred at 2 2 150 mmx y and 3 3 150 mm,x y respectively. Figure 5(a) indicates that when port 1 is the inlet port, a broadband attenuation pattern is observed throughout the frequency range which is qualitatively similar to the TL graphs shown in Figs. 3(a) and (b) for axially short SISO chambers. Incidentally, the peaks in this TL graph at the resonance frequencies can be readily explained by analysing the 11S versus frequency f graph shown in Fig. 6(a). It is observed from

Fig. 6(a) that 11 1S for the frequency range 700, 2800 Hzf and in particular, at resonance frequencies, 11 1S which signifies that no acoustic power is transmitted downstream of the anechoic ports 2 and 3, thereby explaining the occurrence of peaks at these frequencies, thence the broadband attenuation characteristics.

Figure 5(a) indicates that the TL characteristics for the axially short SIDO configuration when the port 2 or port 3 is taken as inlet is significantly different in comparison to the test-case when the port 1 is the inlet; in this case, the TL graphs exhibits a trough at the resonance frequency of (1, 0, 0) or (0, 1, 0) mode leading to the breakdown of the broadband attenuation pattern at this frequency. This demonstrates that interchanging the location of the inlet and outlet ports of a SIDO muffler completely alters the nature of TL graph, thereby corroborating the theoretical developments in section 3.1. The troughs in the TL graph when port 2 or 3 is the inlet may be explained by analysing the spectral variation of 22S or

33S parameters shown in Fig. 6(b). It is noted that near the resonance frequencies of (1, 0, 0) or (0, 1, 0), (1, 1, 0), (3, 0, 0) or

(0, 3, 0) and (3, 1, 0) or (1, 3, 0) modes, both 22 0S and 33 0,S thereby signifying that almost all the acoustic power incident at the inlet port is transmitted downstream into the anechoic outlet ports which renders the SIDO muffler acoustically transparent at these frequencies, thereby explaining the breakdown of the broadband attenuation characteristics. Incidentally, it is also observed from Fig. 6(b) that the spectrum of 22S and 33S parameters is nearly co-incident, thereby explaining the nearly identical nature of the TL graphs in Fig. 5(a) for the test-case when ports 2 and 3 are taken as inlet.

Final D

raft

15

Figure 5 (a) TL performance of axially short SIDO rectangular chamber shown in Fig. 4 – Effect of interchanging the location of the inlet and outlet ports. The chamber and port dimensions for the SIDO configuration are identical with the axially short SISO configuration analysed in Fig. 3 whilst the location of end ports 1, 2 and 3 is given by 1 1 1( , , ) 0.5 ,0.5 ,0 ,x y z B H

2 2 2( , , ) 0.75 ,0.75 ,0x y z B H and 3 3 3( , , ) 0.75 ,0.75 , ,x y z B H L respectively. (b) Comparison of the TL performance of axially short SIDO configuration (with port 1 as the inlet) with its SISO counterparts analyzed in Fig. 3 – Effect of an additional outlet port.

Figure 6 Variation of the iiS parameter ( 1, 2, 3)i versus frequency f for the axially short SIDO rectangular expansion chamber configuration analysed in Fig. 5(a).

Figure 5(b) compares the TL performance of the axially short SIDO chamber configuration analysed in Fig. 5(a) (with port 1 taken as inlet) with its SISO counterpart analysed in Fig. 3 with a view to study the effect of considering an additional outlet port on attenuation performance. The following points are observed from the TL graphs shown in Fig. 5(b):

Final D

raft

16

1. The TL performance of the straight-flow/ reverse-flow type axially short SISO rectangular chamber is qualitatively similar to its equivalent SIDO counterpart wherein the TL graphs of the SISO and SIDO configurations are almost parallel to each other.

2. The SIDO configuration yields slightly lower acoustic attenuation performance than the SISO configuration as the presence of an additional outlet port results in a smaller effective expansion ratio [23].

4.2 Chambers having a single end-inlet and single side-outlet: Cross-flow configuration

Figure 7(a) shows the three orthogonal views of a 2-port rectangular expansion chamber configuration having an end-inlet port 1 located on the B-H face and a side-outlet port 2 located on the B-L face whilst Fig. 7(b) shows its 3-D view. This configuration is referred to as the SISO cross-flow configuration. It is noted that end and side ports of this configuration are of a square cross-section.

Figure 7 A 2-port cross-flow configuration of a rectangular expansion chamber having an end-inlet port (marked as 1) located on the B-H end face and a side-outlet port (marked as 2) located on the B-L face: (a) Orthogonal projections and (b) 3-D view.

An axially long chamber is considered and the chamber dimensions are given by B = H = 150 mm, L = 225 mm. Furthermore, the breadth and height of both ports are taken to be equal and given by b0 = h0 = 50 mm. The end-inlet port 1 and the side-outlet port 2 is centred at x1 = 0.5B, y1 = 0.5H, z1 = 0 and x2 = 0.5B, y2 = H, z2 = 0.5L, respectively. It is noted that Ih [2] and Venkatesham et al. [3] had previously analysed this configuration with the same chamber and port dimensions and port locations as noted above.

Figure 8 demonstrates that the TL graph computed using the 3-D semi-analytical model presented in this work, the 3-D analytical prediction by Ih [2] and the 3-D FEA results are in excellent agreement throughout the frequency range of interest, thereby validating the present model. In particular, it is observed that the TL graph exhibits peak at the resonance frequencies of the (0, 0, 1), (0, 1, 0) or (1, 0, 0) modes resulting in a broadband attenuation pattern up to the resonance frequency of the (0, 0, 2) or the second axial mode. Indeed, the TL graph of this cross-flow configuration is similar to that of an axially long elliptical/circular chamber having an end-inlet/outlet and a side-outlet/inlet [21, 23, 45].

The peaks at the resonance frequencies in Fig. 10 are explained in terms of the Z matrix parameters as follows. At the resonance frequency of (0, 0, 1) or the first axial mode, the 22 21,Z Z (and 12Z ) parameters are finite whilst 11Z which signifies that the TL graph exhibits a peak. Incidentally, the occurrence of this peak may also be explained by noting that the resonance frequency of the (0, 0, 1) chamber mode is coincident with the first resonance frequency of the quarter-wave resonator of axial length 2L formed due to the cavity between the side port 2 and the rigid B-H end face opposite to the B-H end face on which the end port 1 is located. On the other hand, at resonance frequency of the (0, 1, 0) transverse mode, the

11 21,Z Z (and 12Z ) parameters are finite whilst 22 ,Z thereby explaining the occurrence of peak at this frequency. Similar to the peak at resonance frequency of the (0, 0, 1) mode, this peak may also be explained by the noting the first resonance frequency of a quarter-wave resonator of height 2H formed along the y direction of the rectangular chamber.

At the resonance frequency of the (0, 0, 2) or the second axial mode, all the Z matrix parameters tend to infinity (due to the location of port 1 on the B-H end face) which by the virtue of Eq. (32) implies that TL 0, thereby leading to the breakdown of broadband attenuation pattern at this frequency.

Final D

raft

17

Figure 8 TL performance of the cross-flow configuration shown in Fig. 7 having the following chamber and port dimensions: B = H = 150 mm, L = 225 mm, b0 = h0 = 50 mm and port location given by x1 = 0.5B, y1 = 0.5H, z1 = 0, x2 = 0.5B, y2 = H, z2 = 0.5L - Comparison of the 3-D semi-analytical approach used in this work, the 3-D analytical approach used by Ih [1992], 3-D FEA and the 1-D axial plane wave theory.

The 1-D axial plane wave model [13, 28] is also used to obtain the TL performance of this configuration and the 1-D results are compared against the 3-D predictions. To this end, the [Z] matrix based on the 1-D axial plane wave is first obtained (given by Eq. (40)) that characterises this cross-flow configuration [28].

0 20

01 10

2 20 2 0 2 0 2

0 0

coscot

sin,

cos cos cossin sin

k L zk L

k Lp vjY

p vk L z k z k L zk L k L

(40)

where 0 0Y c BH is the characteristic impedance of the rectangular chamber along the z direction. It is observed from Fig. 10 that the 1-D axial plane wave model is in a satisfactory agreement with the 3-D approaches in the low-frequency range, i.e., up to a frequency slightly less than the resonance frequency of the (1, 0, 0) or (0, 1, 0) mode beyond which significant deviation is observed between the 1-D model and the more accurate 3-D predictions, thereby indicating the breakdown of 1-D axial plane wave theory. 4.3 Chambers having a single side-inlet and single/double side-outlet: Cross-flow configuration

Figure 9(a) shows the three orthogonal views of a 2-port rectangular expansion chamber configuration having a side-inlet port 1 that is centred at x1, y1 = H, z1 and located on the B-L face and a side-outlet port 2 centred at x2 = B, y2, z2 and located on the H-L face whilst Fig. 9(b) shows its 3-D view. This configuration is also referred to as the SISO cross-flow configuration.

Final D

raft

18

Figure 9 A 2-port cross-flow configuration of a rectangular expansion chamber having a side-inlet port (marked as 1) located on the B-L face and a side-outlet port (marked as 2) located on the H-L face: (a) Orthogonal projections and (b) 3-D view.

Figure 10 presents the TL performance of the axially long SISO cross-flow configuration (shown in Fig. 9) obtained using the 3-D semi-analytical approach and 3-D FEA when the side port 1 is centred at x1 = 0.5B, y1 = H, z1 = 0.5L (the pressure node of the first axial mode) and the side port 2 is centred at x2 = B, y2 = 0.5H, z2 = 0.75L (one of the pressure nodes of the second axial mode). It is noted that this location of side ports 1 and 2 corresponds to an equivalent relative angle

12 2 between their respective centres. An excellent agreement is observed between the results of 3-D semi-analytical approach and 3-D FEA as may be observed from Fig. 10 which validates the present method.

Figure 10 Broadband TL performance of the cross-flow configuration shown in Fig. 9 having the following chamber and port dimensions: B = H = 200 mm, L = 300 mm, d0 = 40 mm and side port location given by x1 = 0.5B, y1 = H, z1 = 0.5L, x2 = B, y2 = 0.5H, z2 = 0.75L - Comparison of the 3-D semi-analytical approach used in this work, 3-D FEA and the 1-D axial plane wave theory.

It is observed from Fig. 10 that a broadband attenuation performance is obtained up to resonance frequency of the (2, 0, 0) or (0, 2, 0) transverse mode given by 1715.7 Hz,f in particular, this configuration yields an attenuation over 20 dB in the frequency range 100, 1500 Hz.f The broadband attenuation pattern is due to the occurrence of peak at the resonance

Final D

raft

19

frequencies of the (0, 0, 1), (1, 0, 0) or (0, 1, 0), (0, 1, 1) or (1, 0, 1), (0, 0, 2), (0, 1, 2) or (1, 0, 2) modes which is explained in terms of the Z matrix parameters as follows. At the resonance frequencies of the first axial mode denoted by (0, 0, 1) and the axial-transverse modes given by (1, 0, 1) or (0, 1, 1), the 11 21,Z Z (and 12Z ) parameters are finite whilst 22Z which signifies that the TL graph exhibits a peak. On the other hand, at the resonance frequencies of the second axial mode denoted by (0, 0, 2) and the axial-transverse modes given by (1, 0, 2) or (0, 1, 2), the 22 21,Z Z (and 12Z ) parameters are finite whilst

11Z that again explains the occurrence of peak in the TL graph. Furthermore, at resonance frequencies of (a) the (1, 0, 0) transverse mode, the 11 21,Z Z (and 12Z ) parameters are finite whilst 22 ,Z and (b) the (0, 1, 0) transverse mode, the 22 21,Z Z (and 12Z ) parameters are finite whilst 11 ,Z thereby explaining the occurrence of peak in the TL spectrum. It is noted that at the resonance frequency of the (1, 1, 0) transverse mode, all Z matrix parameters are finite and non-zero, hence, the TL graphs does not exhibit either a peak or trough.

The TL graph in Fig. 10 exhibits a trough at the resonance frequency of the (2, 0, 0) or (0, 2, 0) transverse modes leading to a breakdown of the TL performance (noted earlier). The occurrence of trough at this frequency can be readily explained by noting that at the resonance frequency of the (2, 0, 0) or (0, 2, 0) modes, all Z matrix parameters tend to infinity which implies (by virtue of Eq. (32)) that TL 0. However, it is important to mention here that based on the chamber dimensions considered, the resonance frequency of (0, 0, 3) or the third axial mode given by 003 01.5f c L is coincident with the resonance frequencies of the (2, 0, 0) or (0, 2, 0) mode given by 200 0 020 0 .f c B f c H Although, the TL graph of the SISO configuration considered exhibits a small peak at 003f (that may be explained on the basis of axial plane wave modes), this peak is not able to cancel or nullify the dominant trough due to the (2, 0, 0) or (0, 2, 0) modes, thereby indicating that the broadband attenuation range cannot be increased by fine-tuning the chamber length as 1.5L H or 1.5 .L B In fact, based on this observation, guidelines can also be formulated for specifying the minimum chamber length for axially long chambers having a side-inlet and side-outlet and is briefly discussed as follows. For chamber length 1.5 or 1.5 ,L B L H the peak due to the third axial mode occurs before 200 020 or f f because 003 200f f or 003 020 f f which signifies a maximum broadband attenuation range equal to 200 020 or ,f f whichever occurs earlier. Therefore, the minimum length of the axially long chamber for obtaining broadband attenuation over the maximum frequency range is equal to 1.5L H or 1.5 .L B

The maximum chamber length which yields a broadband attenuation over the maximum possible frequency range can also be determined by first noting that a trough always occurs at the resonance frequency of the (0, 0, 4) or the fourth axial mode given by 004 02f c L (because all Z matrix parameters tend to infinity). On increasing the chamber length, the trough at 004f tends to shift towards the left or lower side of the frequency spectrum and when 2 ,L H 004 200 020= ,f f f thereby signifying the coincidence of troughs due to the onset of these modes. A further increase in the chamber length will result in the breakdown of the broadband attenuation pattern earlier than the resonance frequency of the (2, 0, 0) or (0, 0, 2) modes as the trough due to the fourth axial mode will occur before the trough due to these transverse modes. Therefore, the maximum chamber length is taken as 2 .L H In view of the foregoing discussion, the recommended optimal length range for axially long chambers is given by 1.5 2H L H which yields a high attenuation in the low-frequency region and simultaneously, also has the maximum broadband attenuation range.

This axially long SISO rectangular chamber configuration with side-inlet port 1 and side-outlet port 2 located at x1 = 0.5B, y1 = H, z1 = 0.5L and x2 = B, y2 = 0.5H, z2 = 0.75L, respectively, is indeed the counterpart of axially long elliptical/circular chamber configurations having a side-inlet and side-outlet port that are located at axial distances given by one-half and three-quarters of chamber length with relative angular location between their centres equal to 2, see Refs. [20, 23, 45].

The 1-D axial plane wave model [13, 28] is also used to compute the TL performance of the axially long cross-flow SISO rectangular chamber and the 1-D results are compared against the 3-D predictions. To this end, the [Z] matrix based on the 1-D axial plane wave theory is first obtained (given by Eq. (41)) that characterises this configuration [28].

0 1 0 1 0 2 0 11 10

2 20 0 2 0 1 0 2 0 2

cos cos cos cos,

sin cos cos cos cos

k z k L z k z k L zp vjYp vk L k z k L z k z k L z

(41)

where 0 0 .Y c BH Figure 10 demonstrates that the 1-D axial is in a good agreement with the 3-D prediction in the low-frequency range, i.e., up to a frequency slightly smaller than the resonance frequency of the (1, 0, 0) or (0, 1, 0) modes beyond which significant deviation is observed between the 1-D and 3-D approaches, thereby indicating the breakdown of the 1-D model.

A parametric study was carried out to investigate the effect of angular location of the side-outlet port 2 on the broadband attenuation range of the axially long side-inlet and side-outlet configuration. To this end, the location of side-inlet port 1 was fixed at x1 = 0.5B, y1 = H, z1 = 0.5L, the axial location of the side-port 2 was also kept constant at z2 = 0.75L (same

Final D

raft

20

parameters as those considered in Fig. 10) whilst its angular location was varied: (1) x2 = 0.75B, y2 = H, (2) x2 = B, y2 = 0.75H, (3) x2 = B, y2 = 0.25H and (4) x2 = 0.75B, y2 = 0. It is noted that the side port 2 is located on the same and opposite B-L faces for test-configurations (1) and (4), respectively, whilst for the test-configurations (2) and (3), the side-port 2 is located on the H-L face. Indeed, the test-configurations (1)-(4) are the counterparts of axially long elliptical/circular chamber muffler having a side-inlet port (z1 = 0.5L) and a side-outlet port (z2 = 0.75L) with relative angular location between their centres equal to 0, 0.25 , 0.75 , , respectively. It was found that the TL graphs (not shown here) for each of the test-configurations (1)-(4) exhibits a peak at 001f f and 002f f due to the axial location of side ports 1 and 2 on the pressure nodes of the first and second axial modes, respectively. However, a trough was observed at 100f f or 010 ,f f thereby leading to a breakdown in the broadband attenuation pattern significantly before 002.f In particular, the TL graph for test-configurations (1) and (2) exhibits a peak in the low-frequency region (near 001f f ) and resembles that of a side-branch resonator. In view of the outcome of this parametric study, it is concluded that angular location of the side ports 1 and 2 given by x1 = 0.5B, y1 = H and x2 = B, y2 = 0.5H, respectively, is crucial to nullify the trough at 100f or 010f and thus, significantly enhance the frequency range of broadband attenuation up to 200f f or 020.f f

Figure 11(a) and (b) shows the three orthogonal views and the 3-D view, respectively, of a SIDO (3-port) rectangular

expansion chamber configuration having side port 1 located on B-L face and side ports 2 and 3 located on opposite H-L face.

Figure 11 A 3-port cross-flow configuration of a rectangular expansion chamber having a side-outlet port (marked as 1) located on the B-L face and two side ports (marked as 2 and 3) located on opposite H-L faces: (a) Orthogonal projections and (b) 3-D view.

Figure 12 depicts the TL graphs for the axially long cross-flow SIDO configuration (shown in Fig. 11) for different cases when (a) side port 1 is the inlet and side ports 2 and 3 are outlet and (b) side port 2 is the inlet and side ports 1 and 3 are outlet. It is noted that the side port 1 is centred at x1 = 0.5B, y1 = H, z1 = 0.5L whilst the side ports 2 and 3 are centred at x2 = 0, x3 = B, y2 = y3 = 0.5H, z2 = z3 = 0.75L, respectively. Figure 12 indicates that when port 1 is the inlet port, a broadband attenuation pattern is observed throughout the frequency range which is qualitatively similar to the TL graphs shown in Fig. 10 for axially long SISO chamber. Incidentally, the peaks in this TL graph at the resonance frequencies can again be readily explained by analysing the 11S versus frequency f graph shown in Fig. 13(a). It is observed from Fig. 13(a) that 11 1S for

the frequency range 200, 1500 Hzf and in particular, at resonance frequencies, 11 1S which signifies that no acoustic power is transmitted downstream of the anechoic ports 2 and 3, thereby explaining the occurrence of peaks at these frequencies, thence the broadband attenuation characteristics.

Figure 12 indicates that the TL characteristics when the side port 2 is taken as inlet is significantly different in comparison to the test-case when the side port 1 is the inlet; in this case, the TL graphs exhibits a trough at the resonance frequency of (0, 0, 1) or the first axial mode leading to the breakdown of the broadband attenuation pattern at this frequency. This demonstrates that interchanging the location of the inlet and outlet ports of a SIDO muffler completely alters the nature of TL graph, thereby again corroborating the theoretical developments in section 3.1. The troughs in the TL graph when the side port 2 is the inlet may also be explained by analysing the spectral variation of 22S parameters shown in Fig. 13(b). It is noted

Final D

raft

21

that near the resonance frequencies of (0, 0, 1), (1, 0, 0), (0, 1, 0), (0, 1, 1), (1, 0, 1), (0, 2, 0), (2, 0, 0) and (0, 0, 3) modes, 22 0,S thereby signifying that almost all the acoustic power incident at the inlet port is transmitted downstream into the

anechoic outlet ports which renders the SIDO muffler acoustically transparent at these frequencies, thereby explaining the breakdown of the broadband attenuation characteristics.

For the axially long SIDO rectangular chamber considered, the TL graph (and 33S spectrum) when side port 3 is taken

as the inlet port was found to be coincident with the corresponding TL graph (and 22S spectrum) when the side port 2 was taken as the inlet. It is for this reason that the test-case of side port 3 as the inlet port is not presented in Figs. 12 and 13(b).

Figure 12 TL performance of the axially long cross-flow SIDO configuration shown in Fig. 11 having the following chamber and port dimensions: B = H = 200 mm, L = 300 mm, d0 = 40 mm and location of side ports given by x1 = 0.5B, y1 = H, z1 = 0.5L, x2 = 0, x3 = B, y2 = y3 = 0.5H, z2 = z3 = 0.75L - Effect of interchanging the location of the inlet and outlet ports.

Figure 13 Variation of the iiS parameter ( 1, 2)i versus frequency f for the axially long SIDO rectangular expansion chamber configuration analysed in Fig. 12.

Final D

raft

22

Figure 14 compares the TL performance of the axially long SIDO chamber configuration analysed in Fig. 12 (with side port 1 taken as inlet) with its SISO counterpart analysed in Fig. 10 with a view to study the effect of considering an additional side outlet port on the attenuation performance. To this end, the following two configurations of the SIDO chamber are considered

1. Side-inlet port 1 located at x1 = 0.5B, y1 = H, z1 = 0.5L and side-outlet ports 2 and 3 located at x2 = 0, y2 = 0.5H, z2 = 0.75L and x3 = B, y3 = 0.5H, z3 = 0.75L, respectively.

2. Side-inlet port 1 located at x1 = 0.5B, y1 = H, z1 = 0.75L and side-outlet ports 2 and 3 being located at x2 = 0, y2 = 0.5H, z2 = 0.5L and x3 = B, y3 = 0.5H, z3 = 0.5L, respectively.

Figure 14 Comparison of the TL performance of the SISO cross-flow rectangular chamber configuration shown in Fig. 9 with that of the SIDO configuration shown in Fig. 11 (when port 1 is the inlet). The chamber and port dimensions for the SISO and SIDO configurations are identical and location of the centre of side ports is annotated in the legend.

It is observed from Fig. 14 that there is no appreciable difference in the TL performance of the two different SIDO configurations, thereby signifying that location of the center of the single end-inlet port 1 at either z1 = 0.5L or 0.75L is inconsequential from a practical point-of-view provided that the side-outlet ports 2 and 3 are located at an axial distances given by z2 = z3 = 0.75L in the former case and at z2 = z3 = 0.5L in the latter. Furthermore, similar to Fig. 5(b) for axially short chambers, it is also observed from Fig. 14 for axially long chambers that the TL performance of SISO and SIDO chambers are qualitatively same; however, the SIDO configuration yields a slightly decreased attenuation in comparison to its SISO counterpart due to a smaller effective expansion volume because of the presence of an additional side outlet port [23].

5. CONCLUSIONS

This paper has analysed the TL performance of SISO and SIDO rectangular expansion chambers by means of a 3-D semi-analytical uniform piston-driven model based on the modal expansion technique and the Green’s function approach. The TL performance predicted by the 3-D semi-analytical approach is shown to be in an excellent agreement with that obtained by 3-D FEA simulations and the 3-D analytical method followed in previous studies, thereby validating the approach presented here. The 3-D semi-analytical technique enables one to (a) model the effect of arbitrary location of ports on computation of the Z matrix parameters and thence, explain the TL characteristics (such as peaks and troughs) on the basis of excitation/suppression and interaction of rigid-wall modes of the rectangular chamber as well as (b) analyse both axially short and long chambers alike. In fact, the 3-D semi-analytical model implemented as a programme on a personal computer makes it computationally more efficient (in comparison to 3-D FEA) to conduct an extensive parametric investigation on the effect of location of inlet/outlet ports resulting in the formulation of guidelines in terms of optimal port location for designing SISO and SIDO rectangular chambers exhibiting a broadband TL performance some of which are indicated as follows. (a) Axially short reverse-flow SISO configuration with an end-inlet port 1 centred at x1 = 0.5B, y1 = 0.5H, z1 = 0 and end-

outlet port 2 located on the same end face with its centre at x2 = 0.75B, y2 = 0.75H and z2 = 0.

Final D

raft

23

(b) Axially short straight-flow SISO configuration with an end-inlet port 1 centred at x1 = 0.5B, y1 = 0.5H, z1 = 0 and end-outlet port 2 located on the same end face with its centre at x2 = 0.75B, y2 = 0.75H and z2 = L.

(c) Axially short SIDO configuration with an end-inlet port 1 centred at x1 = 0.5B, y1 = 0.5H, z1 = 0 and end-outlet ports 2 and 3 centred at x2 = 0.75B, y2 = 0.75H, z2 = 0 and x3 = 0.75B, y3 = 0.75H, z3 = L, respectively.

(d) Axially short SIDO configuration with an end-inlet port 1 centred at x1 = 0.75B, y1 = 0.75H, z1 = 0 and end-outlet ports 2 and 3 centred at x2 = 0.5B, y2 = 0.5H, z2 = 0 and x3 = 0.5B, y3 = 0.5H, z3 = L, respectively.

(e) Axially short reverse-flow SISO configuration with an end-inlet port 1 centred at x1 = 0.5B, y1 = 0.75H, z1 = 0 and end-outlet port 2 located on the same end face with its centre at x2 = 0.75B, y2 = 0. 5H and z2 = 0.

(f) Axially short straight-flow SISO configuration with an end-inlet port 1 centred at x1 = 0.5B, y1 = 0.75H, z1 = 0 and end-outlet port 2 located on the same end face with its centre at x2 = 0.75B, y2 = 0.5H and z2 = L.

(g) Axially short SIDO configuration with an end-inlet port 1 centred at x1 = 0.5B, y1 = 0.75H, z1 = 0 and end-outlet ports 2 and 3 centred at x2 = x3 = 0.75B, y2 = y3 = 0.5H, z2 = 0 and z3 = L, respectively.

(h) Axially short SIDO configuration with an end-inlet port 1 centred at x1 = 0.75B, y1 = 0.5H, z1 = 0 and end-outlet ports 2 and 3 centred at x2 = x3 = 0.5B, y2 = y3 = 0.75H, z2 = 0 and z3 = L, respectively.

The optimal location of end ports in aforementioned configurations (a-h) yields a broadband attenuation performance approximately up to 400f f provided that 4L B or 040f f provided that 4L H depending on whichever is smaller. In particular, for configurations (a), (b), (e) and (f), the optimal port location demonstrates double-tuning of axially short rectangular chamber similar to that of a concentric extended-inlet and outlet expansion chamber [40]. (It is noted that the TL graphs for configurations (d), (g) and (h) are not presented here.) (i) Axially long cross-flow SISO configuration with a side-outlet port S1 located on the x-z plane and centred at x1 = 0.5B,

y1 = H, z1 = 0.5L and a side-outlet port S2 located on the y-z plane centred at x2 = 0, y2 = 0.5H, z2 = 0.75L. (j) Axially long cross-flow SIDO configuration with a side-outlet port S1 located on the x-z plane and centred at x1 = 0.5B,

y1 = H, z1 = 0.5L and side-outlet ports S2 and S3 located on the opposite y-z planes and centred at x2 = 0, y2 = 0.5H, z2 = 0.75L and x3 = B, y3= 0.5H, z3 = 0.75L.

(k) Axially long cross-flow SIDO configuration with a side-outlet port S1 located on the x-z plane and centred at x1 = 0.5B, y1 = H, z1 = 0.75L and side-outlet ports S2 and S3 located on the opposite y-z planes and centred at x2 = 0, y2 = 0.5H, z2 = 0.5L and x3 = B, y3= 0.5H, z3 = 0.5L.

The configurations (i-k) exhibit a broadband attenuation performance up to 200f or 020f depending on whichever occurs earlier.

The broadband TL performance of SISO configurations (a), (b), (e) and (f) are found to be qualitatively similar to the SIDO configurations (c), (d), (g) and (h) whilst the TL graphs of SISO configuration (i) is qualitatively similar to the SIDO configurations (j) and (k); the only noticeable difference between the SISO and SIDO systems is that the acoustic attenuation of a SISO muffler is slightly higher than its SIDO counterpart due to a larger effective expansion ratio. Nonetheless, as discussed in previous studies [26], the SIDO configurations have a great potential to substantially reduce the backpressure as compared to its SISO counterpart without compromising too dearly on the TL performance.

It is analytically proved that (1) interchanging the position of inlet and outlet ports for a general reciprocal and/or conservative Single-Inlet and Multiple-Outlet (SIMO) muffler system significantly alters the TL characteristics whilst (2) for a general conservative SIMO system with port i taken as the inlet port, the frequency f corresponding to 0iiS f indicates

the occurrence of a trough whereas 1iiS f indicates the occurrence of a peak in the TL graph, thereby offering a useful analysis tool. These characteristic features (1) and (2) of the TL spectrum of a general SIMO system has been corroborated by analysing the TL graphs of SIDO rectangular expansion chambers computed using the 3-D semi-analytical formulation. ACKNOWLEDGEMENTS The authors are grateful to Dr. Nils Wagner from INTES GmbH, Stuttgart, Germany for providing the results (TL graphs) of 3-D FEA that was used to validate the 3-D semi-analytical model used in this work.

APPENDIX A: MATHEMATICAL EQUIVALENCE OF THE TWO ANALYTICAL METHODS FOR CHARACTERISING RECTANGULAR CHAMBERS

The mathematical equivalence of the two analytical methods, namely, (1) the solution of inhomogeneous 3-D Helmholtz equation subject to homogeneous rigid-wall boundary conditions [3] and (2) the solution of homogeneous 3-D Helmholtz equation subject to inhomogeneous rigid-wall boundary conditions [2] (at the face on which the source port is located) towards obtaining the acoustic pressure response function of a rectangular chamber due to uniform piston source excitation is demonstrated here. To this end, a 2-port reverse-flow rectangular expansion chamber configuration (shown in Fig. 2) is reconsidered and an analytical expression for the cross-impedance matrix parameter 21Z is derived through the solution of homogeneous 3-D Helmholtz equation (see Eq. (1)) subject to inhomogeneous rigid-wall conditions (at z = 0) given by

Final D

raft

24

X-Y0 0 0 0

d , ,1 0 ,d

,

z Sz

S S

p x y zu S S

jk ρ c z

u S

(A.1, A.2)

, , 0 ,S Sp x y z p S (A.3)

and the homogeneous rigid-wall condition (at z = L) given by

X-Y

0 0 0

d , ,1 0 ,dz

z L

p x y zu S

jk ρ c z

(A.4)

where zu is the acoustic particle velocity in the chamber along the z direction, Sp and Su denote the acoustic pressure and piston velocity (assumed uniform) over the cross-section of the source port S. The acoustic pressure field inside the rectangular chamber obtained by imposing the rigid-wall condition at its X-Z faces (i.e., 0y and y H planes) and Y-Z faces (i.e., 0x and x B planes) is given by [13, 31]

, , , ,

j j1 2, ,

0,1,2... 0,1,2...

, , cos cos e e ,z n m z n mn m

k z k zn m n m

n m

n x m yp x y z A AB H

(A.5)

where 1

,n mA and 2,n mA denote the amplitudes of the forward and backward progressive waves, respectively, corresponding to the

(m, n) transverse mode. On substituting Eq. (A.5) into Eq. (A.4) and simplifying the resultant equation yields [13, 31]

, , , , , ,

j j 2j1, , ,

0,1,2... 0,1,2...0 0 0

1 cos cos e e e ,z n m z n m z n mn m

k z k z k Lz n m z n m

n m

n x m yu A kk ρ c B H

(A.6)

where 2 2

2, , 0 ,z n m

n mk kB H

(A.7)

is the axial wave number corresponding to (n, m) transverse mode. The difference between the two analytical approaches (1) and (2) can be readily appreciated from Eq. (A.6); this analytical approach does not expresses the acoustic field along the z direction as discrete axial modes, rather, the wavenumber , , z n mk varies in a continuous manner with excitation wavenumber

0.k On the other hand, in the Green’s function solution given by Eq. (6), i.e., the first analytical approach, wave propagation along the z direction is given by discrete axial modes 0, 1, 2,..., .l Equation (A.6) is now substituted into Eqs. (A.1) and (A.2) and making use of orthogonality of rigid-wall modes gives the following expression for the modal coefficient 1

,m nA

, ,

1 0 0 0, 2 2

, , 2j

0 0

cos cos d d.

1 e cos d cos d

S

z n m

S SS

n m S y Hx Bz n m k L

x y

m y n x x yH Bk ρ cA u

k n x m yx yB H

(A.8)

Equation (A.8) is back-substituted (along with the expression for 2

,m nA ) into Eq. (A.5) to yield

, , , , , ,j j 2j

0 02 2

0 , ,

0 0

cos cos d d cos cos e e e, , 1

cos d cos

z n m z n m z n m

S

k z k z k LS S

S

y Hx BS S S z n m

x y

m y n x m y n xx yH B H Bp x y z k c

ρ u S S k n x m yxB H

, ,

0,1,2.. 0,1,2... 2j

.

d 1 e z n m

n m

n m k Ly

(A.9)

Final D

raft

25

The acoustic pressure response given by Eq. (A.9) is further integrated over the cross-sectional area of the receiver port R (located on the same X-Y end face, implying 0R Sz z ) and divided by it to obtain the cross-impedance parameter RSZ which on the use of standard trigonometric identities simplifies to

, ,0 02 2

, ,

0 0

cos cos d d cos cos d dcot

cos d cos d

S R

S S R RS Sz n m

RS y Hx BS R z n m

x y

m y n x m y n xx y x yH B H Bk Ljk cZ

S S k n x m yx yB H

0,1,2.. 0,1,2...

.n m

n m

(A.10)

It is noted that RSZ parameter given by Eq. (A.10) is purely imaginary regardless of the excitation wavenumber 0k as

may be verified using the identity , , , , coth cot ,z n m z n mjk L j k L see Ref. [46].

The cross-impedance RSZ parameter given by Eq. (12a) for the configuration shown in Fig. 2 obtained by solving the inhomogeneous 3-D Helmholtz equation subject to homogeneous rigid-wall conditions is now expanded in terms of the axial plane wave and transverse modes shown hereunder.

2 2 1,2,...0 2

0

2

0 0

Axial plane wave modes

1 2

cos d d cos d dS R

S R

l

S S R RS S

RSS R

S SBHL k l k

L

n x n xx y x yB B

n lB

jk cZS S

=

2 1,2,3,... = 0,1,2,... 2

0 0

Transverse modes along the direction and the cross-modes

ε

cos d d cosS

n l

n ln l

S SS

x

k BHLL

m y m yx yH H

=

2 2 1,2,3,... = 0,1,2,... 2

0 0

Transverse modes along the direction and the cross-modes

d d

ε

cos

R

R Rm l S

m lml

y

x y

m l k BHLH L

= =

2 2 2 1,2,3,... = 1,2,3,... = 0,1,2,... 2

0

Transverse modes a

d d cos d d

ε

S R

S S R Rn m l S S

n m lnml

m y m yx y x yH H

n m l k BHLB H L

long the and directions and the cross-modes

.

x y

(A.11)

On substituting the following trigonometric identity [46] in Eq. (A.11)

, ,22 2

1,2,3,...

cot 1 2 , ,z n mP

k LP

(A.12a, b)

followed by algebraic manipulation of the resultant expression using Eq. (A.7) yields the following

Final D

raft

26

0

0

, ,0

1,2,3,... , ,0 0

0 0

Transverse modes along th

Axial plane wave modes

cot

cos d d cos d dcot

εS R

S R

S S R Rn S Sz n

n z n n

RSS R

k LS SBH k

n x n xx y x yB Bk L

k BH

jk cZS S

,0,

1,2,3,... ,0, 0

e direction and the cross-modes

Transverse modes

cos d d cos d dcot

εS R

S S R Rm S Sz m

m z m m

x

m y m yx y x yH Hk L

k BH

= , ,

1, = 1,2,3,... , ,

along the direction and the cross-modes

cos d d cos d dcot

εS R

S S R Rm S Sz n m

n m z n m nm

y

m y m yx y x yH Hk L

k BH

2,3,...

Transverse modes along the and directions and the cross-modes

,

n

x y

(A.14)

where

ε 1 for 0 ,

0.5 for 0, 0 , 0, 0 ,

0.25 for 0, 0 .

nm m n

m n m n

m n

(A.15a-c)

A comparison of Eq. (A.14) with Eq. (A.10) will reveal that the former is actually an expanded form of the latter,

thereby demonstrating that the two analytical methods for obtaining the acoustic pressure response (or the Z matrix parameters) of a flow-reversal rectangular expansion chamber are mathematically equivalent [47]. In fact, the mathematical equivalence of these two analytical methods can also be shown for the straight-flow and the cross-flow rectangular chamber configurations. Incidentally, the [Z] matrix (for an end-inlet and end-outlet system) based on the 1-D axial plane wave theory [13, 28] can also be derived from Eqs. (A.10) or (A.11) by ignoring the modal summation expressions corresponding to the transverse modes and retaining only the first modal summation series.

REFERENCES

1. Munjal, M.L.: A simple numerical method for three-dimensional analysis of simple expansion chamber mufflers of rectangular as well as circular cross-section with a stationary medium. J. Sound Vib. 116, 71-88 (1987)

2. Ih, J. -G.: The reactive attenuation of rectangular plenum chambers. J. Sound Vib. 157, 93-122 (1992) 3. Venkatesham, B., Tiwari, M., Munjal, M.L.: Transmission loss analysis of rectangular expansion chamber with arbitrary

location of inlet/outlet by means of Green’s function. J. Sound Vib. 323, 1032-1044 (2009) 4. Chu, C.I., Hua, H.T., Liao, I.C.: Effect of three-dimensional modes on acoustic performance of reversal flow mufflers

with rectangular cross-section. Comput. Struct. 79, 883-890 (2001) 5. Wu, C.J., Wang, X.J., Tang, H.B.: Transmission loss prediction on SIDO and DISO expansion-chamber mufflers with

rectangular section using the collocation approach. Int. J. Mech. Sci. 49, 872-877 (2007) 6. Zhou, W., Kim, J.: Formulation of four poles of three-dimensional acoustic systems from pressure response functions with

special attention to source modelling. J. Sound Vib. 219, 89-103 (1999) 7. Kadam, P., Kim, J.: Experimental formulation of four poles of three-dimensional cavities and its application. J. Sound

Vib. 307, 578-590 (2007) 8. Li, X., Hansen, C.H.: Comparison of models for predicting the transmission loss of plenum chambers. Appl. Acoust. 66,

810-828 (2005) 9. Cummings, A.: The attenuation of lined plenum chambers in ducts: I. Theoretical models. J. Sound Vib. 61, 347-373

(1978)

Final D

raft

27

10. Cummings, A.: The attenuation of lined plenum chambers in ducts: II. Measurements and comparison with theory. J. Sound Vib. 63, 19-32 (1979)

11. Cummings, A.: Low frequency acoustic transmission through the walls of rectangular ducts. J. Sound Vib. 61, 327-345 (1978)

12. Venkatesham, B., Tiwari, M., Munjal, M.L.: Prediction of breakout noise from a rectangular duct with compliant walls. Int. J. Acoust. Vib. 16, 180-190 (2011)

13. Munjal, M.L.: Acoustics of Ducts and Mufflers. Wiley, Chichester (2014) 14. Chiu, M.C., Chang, Y.C., Cheng, H.C., Tai, W.T.: Shape optimization of mufflers composed of multiple rectangular fin-

shaped chambers using differential evolution method. Arch. Acoust. 40, 311-319 (2015) 15. Munjal, M.L., Sreenath, A.V., Narasimhan, M.V.: Velocity ratio in the analysis of linear dynamical systems. J. Sound Vib.

26, 173-191 (1973) 16. Pan, J., Elliott, S.J., Baek, K.H.: Analysis of low frequency acoustic response in a damped rectangular enclosure. J. Sound

Vib. 223, 543-566 (1999) 17. Ali, A., Rajkumar, C., Yunus, S.M.: Advances in acoustic eigenvalue analysis using boundary element method. Comput.

Struct. 56, 837-847 (1995). 18. Eriksson, L.J.: Higher-order mode effects in circular ducts and expansion chambers. J. Acoust. Soc. Am. 68, 545-550

(1980) 19. Eriksson, L.J.: Effect of inlet/outlet locations on higher order modes in silencers. J. Acoust. Soc. Am. 72, 1208-1211

(1982) 20. Yi, S.I., Lee, B.H.: Three-dimensional acoustic analysis of a circular expansion chamber with side inlet and side outlet.

J. Acoust. Soc. Am. 79, 1299-1306 (1986) 21. Yi, S.I., Lee, B.H.: Three-dimensional acoustic analysis of a circular expansion chamber with side inlet and end outlet. J.

Acoust. Soc. Am. 81, 1279-1287 (1987) 22. Mimani, A., Munjal, M.L.: 3-D acoustic analysis of elliptical chamber mufflers having an end inlet and a side outlet: An

impedance matrix approach. Wave Motion 49, 271-295 (2012) 23. Mimani, A., Munjal, M.L.: Acoustical behavior of single inlet and multiple outlet elliptical cylindrical chamber muffler.

Noise Control Eng. J. 60, 605-626 (2012) 24. Yu, X., Tong, Y., Pan, J., Cheng, L., Sub-chamber optimization for silencer design. J. Sound Vib. 351, 57-67 (2015) 25. Chiu, M.C., Chang, Y.C.: Shape optimization of multi-chamber cross-flow mufflers by SA optimization. J. Sound Vib.

312, 526-550 (2008) 26. Selamet, A., Ji, Z.L.: Acoustic attenuation performance of circular expansion chambers with single-inlet and double-

outlet. J. Sound Vib. 229, 3-19 (2000) 27. Banerjee, S., and Jacobi, A.M.: Transmission loss analysis of single-inlet/double-outlet (SIDO) and double-inlet/single-

outlet (DISO) circular chamber mufflers by using Green’s function method. Appl. Acoust. 74, 1499-1510 (2013) 28. Mimani, A., Munjal, M.L.: Acoustical analysis of a general network of multi-port elements – An impedance matrix

approach. Int. J. Acoust. Vib. 17, 23-46 (2012) 29. Yang, L., Ji, Z.L.: Acoustic attenuation analysis of network systems by using impedance matrix method. Appl. Acoust.

101, 115-121 (2016) 30. Easwaran, V., Gupta, V.H., Munjal, M.L.: Relationship between the impedance matrix and the transfer matrix with

specific reference to symmetrical, reciprocal and conservative systems. J. Sound Vib. 161, 515-525 (1993) 31. Mimani, A.: 1-D and 3-D analysis of multi-port muffler configurations with emphasis on elliptical cylindrical chamber,

Ph.D. Thesis, Indian Institute of Science, Bangalore (2012). 32. Denia, F.D., Albelda, J., Fuenmayor, F.J., Torregrosa, A.J.: Acoustic behaviour of elliptical chamber mufflers. J. Sound

Vib. 241, 401-421 (2001) 33. Kim, J., Soedel, W.: General formulation of four pole parameters for three-dimensional cavities utilizing modal expansion

with special attention to the annular cylinder. J. Sound Vib. 129, 237-254 (1989) 34. Selamet, A., Ji, Z.L.: Acoustic attenuation performance of circular expansion chambers with offset inlet/outlet: I.

Analytical approach. J. Sound Vib. 213, 601-617 (1998) 35. Selamet, A., Ji, Z.L.: Acoustic attenuation performance of circular flow-reversing chambers. J. Acoust. Soc. Am. 104,

2867-2877 (1998) 36. Atkinson, K.E.: An introduction to numerical analysis. John Wiley & Sons, Singapore (2004). 37. Abom, M.: Measurement of the scattering-matrix of acoustical two-ports. Mech. Syst. Signal Pr. 5, 89-104 (1991) 38. Mimani, A., Munjal, M. L.: Acoustic end-correction in a flow-reversal end chamber muffler: A semi-analytical approach.

Accepted for publication in the J. Comput. Acoust. on the 1st October, 2015. 39. Mimani, A., Munjal, M.L.: On the role of higher-order evanescent modes in end-offset inlet and end-centered outlet

elliptical flow-reversal chamber mufflers. Int. J. Acoust. Vib. 17, 139-154 (2012). 40. Chaitanya, P., Munjal, M.L.: Effect of wall thickness on the end corrections of the extended inlet and outlet of a double

tuned expansion chamber. Appl. Acoust. 72, 65-70 (2011) 41. Lai, P. C. -C., Soedel, W.: Two-dimensional analysis of thin, shell or plate like muffler elements. J. Sound Vib. 194, 137-

171 (1996)

Final D

raft

28

42. Li, L., Guang-Xu, C., Lau, S.K., Yun, L., Zhao-Lin, G., Transmission loss investigation of two-dimensional thin mufflers by the modal expansion method using a new line source model. Acta Acust. united Ac. 97, 54-61 (2011)

43. Mimani, A., Munjal, M.L.: Transverse plane-wave analysis of short elliptical end-chamber and expansion-chamber mufflers. Int. J. Acoust. Vib. 15, 24-38 (2010)

44. Jones, P.W., Kessissoglou, N.: A numerical and experimental study of the transmission loss of mufflers used in respiratory medical devices. Acoust. Aust. 38, 13-19 (2010)

45. Munjal, M.L.: Plane wave analysis of side inlet/outlet chamber mufflers with mean flow. Appl. Acoust. 52, 165-175 (1997)

46. Ablowitz, M.J., Fokas, A.S.: Complex Variables: Introduction and Applications, Cambridge University Press, New York (2003)

47. Myint-U, T., Debnath, L.: Linear Partial Differential Equations for Scientists and Engineers. Birkhauser, Boston (2007)