PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Generalized method of describing acoustic duct-like system
as a multi-port
Anna SNAKOWSKA, Jerzy JURKIEWICZ
AGH University of Science and Technology Krakow, Poland

ABSTRACTThe paper presents general method of analysing acoustic mufflers or other duct-like systems assuming
propagation of a multimode wave and applying the multiport method. Apart from the main propagating tube
and expansion chambers the analysed device may contain segments of annular duct (concentric or without
the central tube) branch resonators, cavities and cavity diaphragms. Systems of such complicated geometricalstructure have been widely analysed under the assumption that the wave propagating through any element is
the plane wave. Thus the problem could have been analysed by means of the two-dimensional transfer
matrices (or others equivalents) derived from conditions of conservation of the acoustic pressure and thevolume velocity across each discontinuity. However, the assumption of the pure plane propagation acrossevery element of the acoustic system imposes strong limitations on the excitation frequency or the radius of
the duct segments as the Helmholtz number should not exceed the value of 1.84 which represents thenon-dimensional cut-on frequency of the first circumferential mode. If the excitation and the system areaxially symmetric the limited value of the Helmholtz number raises to 3.83. The analysis is carried out bymeans of the scattering matrix which can finally be transformed into transmission matrix.
Keywords: Acoustical multi-port, Scattering matrix, Multimode wave, Muffler

1. INTRODUCTION
The aim of the paper is to present a general method of analysing acoustic systems of complicatedgeometry assuming propagation of a multimode wave and applying the multiport method. Th isanalysis will be carried out within the linear theory which allows not only to split the system into a
network of sub-systems but also to apply the procedure consisting of subsequently reducing the rangeof the admittance or the scattering matrix describing the system as a whole. In the carried out analysis
each sub-system (multi-port) corresponds to a certain wave transmission path. The procedure can beapplied step by step to each of the internal sub-systems leading finally to a matrix representing only theexternal ports, most frequently the tail and the outlet pipe. Thus the paper presents general method toanalyse the acoustic mufflers’ or other duct-like systems to which the multi-port method can be
efficiently applied.
Apart from the main propagating duct and expansion chambers the analysed device may containsegments of annular duct (concentric or not with the central tube) branc h resonators, cavities andcavity diaphragms. Systems of such complicated geometrical structure have been widely analysed
under the assumption that the wave propagating through any element is the plane wave [ref] andneglecting the generation of some other duct modes at the duct discontinuity (junction). Thus the
problem might have been analysed by means of the two-dimensional matrices (impedance, admittance,scattering, transmission or others equivalents) derived from conditions of conservation of the acou sticpressure and the volume velocity across each discontinuity. It is well known that the assumption of theplane wave propagation across every element of the acoustic system imposes strong limitations on the
excitation frequency or the radius of the duct segments as the Helmholtz number should not exceed thevalue 1.84 which represents the non-dimensional cut-on frequency of the first circumferential mode. If
the excitation and the system are axially symmetric the limited value of the Helmholtz number rai ses
to 3.83. However, this approach applied to the simplest muffler composed only of one expansion
chamber led to significant errors in calculating the transmission loss (TL) for frequencies considerably5276

lower than the limited ones [1]. The source of this discrepancy is an insufficient adjustment of theboundary condition on the junction being a consequence of applying the conditions of conservation of
the acoustic volume velocity across each discontinuity. The presented method in which at a certain
point of the calculations the sub-systems are analysed separately allows fulfilling the boundary
condition with prescribed accuracy without significant increase of the computational costs.
The first attempt was to model the muffler design in the low frequency li mit (only the plane wave
propagation) as a two-port in analogy to the four-pole electric circuit [2-4]. In this case the analogy canbe applied straightforwardly. More recent was application of the electric network method to model
muffler design assuming propagation of a multimode wave [4-6] which introduced a necessity todefine what will be called the acoustic multi-port. Abom [8] and some other authors [2-7] define theacoustic multi-port depending on the number of state variables which completely describe the state ofa certain black box. Thus, the definition is not related to the number of the connected ducts and the
topology of the acoustic system. This approach has several advantages, one of which is that it provides
information on the number of degrees of freedom i.e. on the size of the matrix representing
relationship between ingoing and outgoing waves (input and output variables, in general). Anotherpossibility is to relate the definition to the number of joints (straight ducts most frequently) [ 9] whichis compatible with the topology of the acoustic system and also is independent of the carried out
frequency range analysis. This approach will be applied subsequently
The analysis is carried out by means of the scattering matrix which can be finally transformed into
the transmission matrix to calculate the transmission loss.

2. APPLICATION OF THE THEORY OF THE MULTI-PORTS TO ACOUSTICS
2.1 Electrical and acoustical multi-ports
The multiport method applied in acoustic origins from electric network analysis, in which a port
connects the poles of a black box and describes the relation between incident and outgoing wave s
represented by the state variables. Most frequently the chosen state variables are the electric current
and the voltage in electrical multi-ports and the sound pressure and the acoustic velocity in acoustical.
In the electric circuits analysis the number of poles corresponds to the number of the degrees of
freedom represented by flowing electric currents each of which needs a separate wire (joint in
topological representation) and is compatible with the number of joints appearing in an acousticalsystem as far as propagation of the plane wave only is considered (the s o-called low frequency
approximation.

Figure 1 – Acoustic 3-port with joints of different size, N=N1+N2+N3 port in acousto-electrical analogies
However substantial differences originate from the fact that in general many modes can propagate
in a duct depending on the duct size (dimensions in rectangular, radius in cylindrical ect.) and theexcitation frequency. For example, for a port with 3 connected joints represented by ducts ofdifferentiated geometry, the total number of modes can be N=N1+N 2 +N3 , depending additionally on
the frequency. Thus, within the straightforwardly applied acoustical-electrical analogies it will becalled N-port, with N depending on frequency. It seems that in the sound field analysis the number ofjoints is essential, as it determines the geometry (topological representation) of the system underconsideration, and that is why we have introduced the concept of an acoustic n-port defined by the
number of ducts coupled to the tested element.
As a result a selected acoustical sub-system connected to some others by n-joints (ducts) will becalled the acoustical n-port independently on the radiuses of different ducts and the frequency of
excitation.

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2.2 Basic assumptions
An acoustic system can be analysed as an acoustic multi-port (Fig. 2) under the following
assumptions:
1. The phenomena fulfil the assumptions of linearity.
2. Subsequent multi-ports (sub-systems) are connected by duct-like elements of a constant crosssections called joints.
3. It allows describing the acoustic pressure as a superposition of duct modes which means
decomposition into a series of eigenfunctions.
4. The wave propagating in a joint can be decomposed into a wave going out of the black box
element and going into it.
5. The joints are long enough to consider only the propagating waves at a selected cross section
i.e. neglect the near field effects appearing close to the system discontinuity.
6. The selected duct cross section fulfilling the above assumptions is called a port.
7. There is a certain freedom in choosing the position of the port as the change in its localizations
will result only in a change of phase of the state variables.
8. At a port the wave can be described by a one column vector of modal sound pressures andacoustic velocities (impedance or admittance approach) or, equivalently by the vectors of theingoing and outgoing modal sound pressures (scattering matrix approach S).
In what fallows the scattering matrix approach will be applied.

a)b)Figure 2 – a) Topological representation of an exemplary acoustical systems: A, B, C, D – sub-systems(multi-ports), A and B – three-ports, C and D –two-ports, joints with marked ports: 1 and 2 – external, 3, 4, 5
and 6 – internal; b) at a given port the acoustic pressure of each wave mode outgoing from one sub-system
equals the acoustic pressure of the corresponding in-going mode of the consecutive sub-system.

3. THE SCATTERING MATRIX APPLIED TO ACOUSTICAL SYSTEM COMPOSEDOF MULTI-PORTS
3.1 Theoretical basis
Let us consider an acoustical system composed of some multi-ports as presented in Fig.2. Our goal
is to analyse its property by means of the scattering matrix. The final result is to derive the dependence
between the state variables of the external ports (1and 2 in Fig. 2). In the scattering matrix approach itmeans the relation between the modal acoustic pressures of the ingoing and outgoing waves. In manytechnical applications, such as muffler design it allows to predict the transmission loss and analyse its
variation due to changes in the muffler geometry.
In general, considering sources present within the acoustical system the modal sound pressures ofthe outgoing wave modes 𝒐𝒖𝒕𝑷 depends on the scattering matrix 𝑺 and the pressures of the internalsound sources 𝒔𝒓𝑷𝒐𝒖𝒕

𝑷 = 𝑺 × 𝒊𝒏𝑷 + 𝒔𝒓𝑷(1)

where 𝒐𝒖𝒕𝑷, 𝒊𝒏𝑷, 𝒔𝒓𝑷 are represented by one column matrices composed of modal pressure amplitudes
at a selected duct cross sections where the phenomena of the near field can be omitted.

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[ 𝒐𝒖𝒕𝑷𝟏 ][ 𝒊𝒏𝑷𝟏 ]𝒊𝒏𝒐𝒖𝒕

𝑷 = [ 𝒊𝒏𝑷𝟐 ](2)

𝑷 = [ 𝒐𝒖𝒕𝑷𝟐 ][ 𝒊𝒏𝑷𝟑 ][][ 𝒐𝒖𝒕𝑷𝟑 ][]The procedure leading to the final result relies, as will be presented below, on reducing step by step
the dimension of the scattering matrix representing the system as a whole by subsequently eliminating
the selected ports.
Let us assume that in the analysed system we can distinguish two sub-systems which constitute afive-port and are connected by a joint in a form of a straight duct merging port 2 with port 4. The
presented method relies on reducing the range of the multi-port to a lower number by joining port 2
with port 4.
Next step is denoting these ports as a and b and the corresponding modal sound pressure matrices
by 𝑷𝒂 and 𝑷𝒃 . Applying the relations and introducing new symbols to better distinguish the selected
joint
𝒊𝒏

𝑷𝒂 = 𝒐𝒖𝒕𝑷𝒃 = 𝑷𝑨𝒊𝒏

𝑷𝒃 = 𝒐𝒖𝒕𝑷𝒂 = 𝑷𝑩

(3)equation Eq. 1 takes the form
𝒐𝒖𝒕

𝑷𝟏
𝑺𝟏,𝟏
𝒐𝒖𝒕
𝑺𝒂,𝟏
𝑷𝑩
𝒐𝒖𝒕
𝑷𝟑 = 𝑺𝟑,𝟏
𝒐𝒖𝒕
𝑺𝒃,𝟏
𝑷𝑨
𝒐𝒖𝒕
[𝑺𝟓,𝟏
[⏟ 𝑷𝟓 ] ⏟

𝑺𝟏,𝒂
𝑺𝒂,𝒂
𝑺𝟑,𝒂
𝑺𝒃,𝒂
𝑺𝟓,𝒂

𝑺𝟏,𝟑
𝑺𝒂,𝟑
𝑺𝟑,𝟑
𝑺𝒃,𝟑
𝑺𝟓,𝟑
𝑺

𝒐𝒖𝒕𝑷

𝑺𝟏,𝒃
𝑺𝒂,𝒃
𝑺𝟑,𝒃
𝑺𝒃,𝒃
𝑺𝟓,𝒃

𝒔𝒓
𝑷𝟏
𝒊𝒏
𝑷𝟏
𝑺𝟏,𝟓
𝒔𝒓
𝒊𝒏
𝑷𝒂
𝑺𝒂,𝟓
𝑷𝑨
𝒔𝒓
𝒊𝒏
𝑺𝟑,𝟓 × 𝑷𝟑 + 𝑷𝟑
𝒊𝒏
𝑺𝒃,𝟓
𝒔𝒓
𝑷𝑩
𝑷𝒃
𝒊𝒏
𝑺𝟓,𝟓 ] [⏟𝑷𝟓 ] [ 𝒔𝒓𝑷 ]
⏟ 𝟓
𝒊𝒏𝑷

(4)

𝒔𝒓𝑷

At this stage it is possible to calculate 𝑷𝑨 and 𝑷𝑩 performing the following matrix operations:
𝑺𝒂,𝒂 𝑺𝒂,𝒃
𝑷
𝑷
𝑬[ 𝑩] = [] × [ 𝑨] + [ ]
(5)
𝑷𝑨
𝑷𝑩
𝑺𝒃,𝒂 𝑺𝒃,𝒃
𝑭
where:
𝑺𝒂,𝟏
𝑬
[ ]=[
𝑺𝒃,𝟏
𝑭

𝑺𝒂,𝟑
𝑺𝒃,𝟑

𝒊𝒏
𝑷𝟏
𝒔𝒓
𝑺𝒂,𝟓
𝑷
𝒊𝒏
] × [ 𝑷𝟑 ] + [ 𝒔𝒓 𝒂 ]
𝑺𝒃,𝟓
𝑷𝒃
𝒊𝒏
𝑷𝟓

(6)

and so[𝑺𝒂,𝒂 𝑺𝒂,𝒃 − 𝑰
𝑷𝑬] × [ 𝑨] = − [ ]𝑺𝒃,𝒂 − 𝑰 𝑺𝒃,𝒃
𝑷𝑩
𝑭

(7)

The above considerations lead to the algorithm composed of some steps described below.
3.2 Algorithm for reducing the range of the scattering matrix
In what fallows the algorithm for merging two ports of an acoustical system is derived. The
algorithm consists of the subsequent calculations: Construction of the U matrix such as5279

𝑺𝒂,𝒂 𝑺𝒂,𝒃 − 𝑰
𝑼=[]𝑺𝒃,𝒂 − 𝑰 𝑺𝒃,𝒃(8)



Derivation of the inverse matrix 𝑻 = −𝑼−𝟏 divided into segments corresponding to matrix 𝑺
segments
𝑻𝒂,𝒂 𝑻𝒂,𝒃
𝑻=[] = −𝑼−𝟏
𝑻𝒃,𝒂 𝑻𝒃,𝒃(9)



Modification of the sub-matrices of the 𝑺 matrix according to the formula
𝑺′ 𝒊,𝒌 =𝑺𝒊,𝒌+𝑺𝒊,𝒂 × 𝑻𝒂,𝒃 × 𝑺𝒃,𝒌+𝑺𝒊,𝒂 × 𝑻𝒂,𝒂 × 𝑺𝒂,𝒌
+𝑺𝒊,𝒃 × 𝑻𝒃,𝒃 × 𝑺𝒃,𝒌+𝑺𝒊,𝒃 × 𝑻𝒃,𝒂 × 𝑺𝒂,𝒌(10)

where both indices i and k do not take values a and b 𝑖 ≠ 𝑎, 𝑖 ≠ 𝑏 and 𝑘 ≠ 𝑎, 𝑘 ≠ 𝑏.
Modification of the source sub-matrices 𝒔𝒓𝑷 according to the formula
𝒔𝒓

𝑷′𝒊 =

𝒔𝒓

𝑷𝒊

+𝑺𝒊,𝒂 × 𝑻𝒂,𝒃 × 𝒔𝒓𝑷𝒃
+𝑺𝒊,𝒂 × 𝑻𝒂,𝒂 × 𝒔𝒓𝑷𝒂
+𝑺𝒊,𝒃 × 𝑻𝒃,𝒃 × 𝒔𝒓𝑷𝒃
+𝑺𝒊,𝒃 × 𝑻𝒃,𝒂 × 𝒔𝒓𝑷𝒂(11)

where 𝑖 ≠ 𝑎, 𝑖 ≠ 𝑏.
Finally the formula for the multi-port of a range reduced by two (from a five-port to a three-port)
described by the scattering matrix is obtained
𝑺′ 𝟏,𝟏
𝑷𝟏[ 𝒐𝒖𝒕𝑷𝟑 ] = [𝑺′𝟑,𝟏
𝒐𝒖𝒕⏟ 𝑷𝟓⏟𝑺′𝟓,𝟏𝒐𝒖𝒕

𝒐𝒖𝒕𝑷

𝑺′ 𝟏,𝟑
𝑺′𝟑,𝟑
𝑺′𝟓,𝟑
𝑺′

𝒔𝒓
𝒊𝒏
𝑷′𝟏
𝑺′𝟏,𝟓
𝑷𝟏
𝑺′𝟑,𝟓 ] × [ 𝒊𝒏𝑷𝟑 ] + [ 𝒔𝒓𝑷′𝟑 ]
𝒊𝒏
𝒔𝒓
𝑺′𝟓,𝟓
⏟ 𝑷𝟓
⏟ 𝑷′𝟓
𝒊𝒏𝑷

(12)

𝒔𝒓𝑷′

The above presented algorithm can be applied to any two ports represented by matrices 𝑺𝑨 and 𝑺𝑩 ,not necessarily connected at the first stage. In the first step the scattering matrix comprising both ports
has to be constructed
𝑺 𝟎
𝑺𝑨𝑩 = [ 𝑨
]
(13)
𝟎 𝑺𝑩
The next step consists of merging these two ports applying the above algorithm.

4. THE SCATTERING MATRIX OF SOME MUFFLERS’ ELEMENTS
The elements encountered most frequently in the mufflers’ design are straight ducts of given lengthand different cross-section, hard or finite impedance lids, duct outlets – baffled or unflanged etc. In astraight duct the sound pressure of a mode n can be divided at a given cross section into parts
representing waves going upstream and downstream.
4.1 Duct terminations as an acoustical one-port
If the duct is terminated by a lid of a given acoustic impedance the termination can be analysed as
an acoustical one-port with the scattering matrix
𝑦𝑛 − 𝑌𝑆𝑛𝑘 = (
)(14)
𝑦𝑛 + 𝑌 𝑛𝑘

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where 𝑦𝑛 is the modal admittance 𝑦𝑛 = 𝑘𝑧,𝑛 𝑘 −1 , and 𝑌 is the non-dimensional acoustic admittance ofthe lid surface, and 𝑛𝑘 is the Kronecker delta function, which means that the matrix 𝑺 is diagonal.
For an infinite duct or a duct terminated with non-reflecting surface or anecoic termination the
scattering matrix 𝑺 = 𝟎 as there are no out-going waves.
For an open end of a duct the scattering matrix is composed of the reflection and transformation
coefficients, which have been derived analytically for a circular flanged [10] or unflanged [11, 12] duct.
4.2 Straight duct of a given length as an acoustical two-port
In a straight duct the sound pressure of a mode n can be divided at a given cross section into parts
representing waves going upstream and downstream
𝑃𝑛 (𝑧) = 𝑃𝑛+ (𝑧) + 𝑃𝑛− (𝑧)(15)

𝑃𝑛+ (𝑧) = 𝑃𝑛+ 𝑒 −𝑖𝑘𝑧,𝑛 𝑧

(16)

where

and 𝑃𝑛+ is the mode amplitude, and 𝑘𝑧,𝑛 is the axial wave number. Thus the scattering matrix for a duct oflength 𝑑 is0
diag[ exp(−𝑖𝑘𝑧,𝑛 𝑑)]𝑺=[]diag[ exp(−𝑖𝑘𝑧,𝑛 𝑑)]
0(17)

The values of the axial wave numbers 𝑘𝑧,𝑛 are obtained solving the wave equation with adequate
boundary condition and are known for ducts of cylindrical symmetry, plain or annular, with all threekinds of the boundary condition, and for ducts of rectangular and equilateral triangle cross-section. For
more complicated shapes the scattering matrix has to be calculated by means of some numericalmethods such as FEM, BEM or similar.
4.3

Junction of pipes with different cross-sections as an acoustical multi-port

Figure 3 – An example of an acoustical multi-port
Junction of pipes with different cross-sections constitutes an acoustical multi-port. Junction of two
pipes of different radiuses constitute an acoustical two-port, while Fig. 3 represents an acousticalthree-port consisting of a tail pipe A and an annular duct C connected to a larger pipe B.
The boundary condition of the sound pressure and the normal component of the acoustic velocity
continuity have to be fulfilled across the junction, which means𝑝𝐿 (𝒙) = 𝑝𝑅 (𝒙),𝑣𝐿 (𝒙) = 𝑣𝑅 (𝒙),(18)

where 𝒙 represents a point on a junction cross-section, indices L and R describe the left and the rightside of it and the relation is fulfilled at each point of the junction.
For the acoustical three-port in Fig. 3 it readsvA (𝒙) 𝒙 ∈ 𝐴(𝒙)vL= {vC (𝒙) 𝒙 ∈ 𝐶
(19)
0
𝒙∈𝐷
assuming that surface D is an acoustically hard surface.
The above-mentioned boundary conditions can be meet applying to the acoustic pressure andvelocity the mode matching method (MMM) [13]. At first, both quantities should be distributed intomodes in the orthogonal function database (eigenfunctions is the best choice, but also other complete
set of orthogonal functions will lead to the result). Next step is to apply linear transformation to5281

calculate the unknown distribution coefficients. This in turn allows to calculate the amplitudes ofmodes propagating to and from the multi-port and ultimately to determine the scattering matrix [9, 14].
The scattering matrix of multi-ports of irregular geometry can be calculated by means of numericalmethods or experimentally.

5. APPLICATION TO MUFFLERS ANALYSISThe above described algorithm will be applied to a muffler presented in Fig. 4 and discussed previously
in the literature [1]Figure 4 – Muffler with extended inlet and outlet pipes
The muffler is composed of elements such as: plain (A1 and A2 and B) and annular (C1 and C2)
circular ducts, ducts hard terminations (D1 and D2) and junctions (Q1 and Q2) at which the mode
matching method (MMM) has to be applied [9, 14] to fulfil the boundary conditions. The inlet pipe is
extended into the expansion chamber by one second of its length l=0.4 m and the outlet pipe by one
fourth, the radiuses are 0.04 m and 0.16 m, respectfuly (cf. [1], page 324).

Figure 5 – Muffler elements disconnected

a)

b)

c)

d)
e)
f)

Figure 6 – Calculations of the scattering matrix step by step: a) topological scheme of the muffler presented
in Fig. 5; b) the system decomposed into separate multi-ports of known scattering matrix; c) and d) and e)
consecutive steps of reducing the number of ports by applying the procedure presented in Section 3; f) final
result – the system reduced to an acoustical two-port of determined scattering matrix

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Figure 7 – Transmission loss of a muffler as in Fig. 4 calculated according to the presented algorithm(continuous line) and presented by Munjal (cf. [1], page 324, dotted line) assuming the plane wave incident
to the left-hand tail pipe;

6. Conclusions
The presented method is a general way of describing acoustic duct-like systems as multi-ports. It
can be applied to devices composed of many sub-systems not necessarily constituting a cascade. The
procedure can be implemented step by step applying the scattering matrices already derived and themode matching method at junctions. For acoustical system of more complicated geometry for which
the reflection coefficients have not been calculated the problem can be solved by means of thenumerical methods. The scattering matrix can be also derived experimentally.

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