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

Theoretical model for the prediction of sound radiated from
unbaffled long enclosure with ground effectWeiping YANG1; Yatsze CHOY2
1, 2

The Hong Kong Polytechnic University, Hong Kong, P. R. China

ABSTRACTA theoretical model is presented for the prediction of sound radiated from an unbaffled long enclosure withground effect. This geometrical arrangement forms an idealized representation of traffic facilities such as
tunnels and railway stations where sound propagates along the enclosures and radiates to the outside throughthe openings at both ends. The sound fields inside and outside the enclosures should be accurately predicted
and thoroughly analyzed so as to determine an appropriate noise control strategy. In this paper, the Fouriertransform technique and the mode matching method are firstly applied to transform the intractable boundary
value problem into a scalar modified Wiener-Hopf equation. The solution of which contains infinitely many
unknowns satisfying infinite linear algebraic equations susceptible to numerical treatment. Good agreement
is found between the solutions obtained by the finite element method (FEM) and the proposed technique in awide frequency range. Then, the far-field directivity patterns of the outside acoustic fields are emphaticallyillustrated and the formation mechanisms of lobes, zeros and the after radiation are discussed in details. In theend, the advantages and the application prospects of the proposed method are summarized.
Keywords: Sound radiation, Unbaffled long enclosure, Wiener-Hopf technique

1. INTRODUCTION
In order to predict and control the noise level inside tunnels and railway stations, much attention
has been focused in recent years on the mechanisms of sound propagation and dissipation inside long
enclosures (1). Hard rough surfaces (2) and sound-absorbing materials (3) are found to be effective in
reducing this kind of noise. However, in literatures, the enclosures were usually assumed to be infinite
long and only the sound field deep inside the enclosures were considered. Reflection waves due to theabrupt size change of the openings were ignored which should be taken into consideration in practicalapplications as they contribute significantly to the total sound field especially near the opening.
As the noise is difficult to be transmitted outside through the wall, it will concentrate at both endsof the long enclosures and radiate to the outside through the portals. If there are residents nearby, theirliving conditions will be severely deteriorated. Under such circumstance, the pollution caused by theradiated sound become prominent. When the prediction of sound level radiated from long enclosures isrequired, recourses are often made to simplify the geometry in which the openings are imagined to be
surrounded by infinite baffle, thereby eliminating the diffraction effect of edge. In this condition, theRayleigh integral (4) can be applied. However, it is only available to calculate the sound field in frontof the opening while the radiation to the back side which is heavily influenced by the edge diffraction
cannot be predicted. Besides, the model assumes that the outside region is free field, while in practicaltunnels, the ground cannot be ignored which divides the total acoustic domain into a semi-infinite one
and makes the problem more complicated.
The main difference between the problem at hand and that in the literatures is that the unbaffled
opening of the long enclosure and the ground effect are taken into account simultaneously in order tomodel the real traffic facilities. However, due to the geometrical complexity, this boundary value
problem cannot be solved analytically for the lack of known conditions in the natural domain. During
the past decades, several attempts were done by researchers to find the solution of the problem fromdifferent perspectives. In 1998, the Research Committee of Road Traffic Noise in Acoustic Society of1
2

w.p.yang@connect.polyu.hk
yatsze.choy@polyu.edu.hk

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Japan established the ASJ model (5) to predict the sound radiated from tunnel opening based on theprincipal of energy conservation. However, it can only be applied as an approximate approach forengineering applications as the correction terms for the edge diffraction and the ground effect in themodel are either from experimental data or empirical formula (6). In addition, numerical methods
based on the FEM (7, 8) were proposed to transform the semi-infinite acoustic domain outside theenclosure into a waveguide one by introducing a perfectly matched layer (PML) outside the original
geometry. Then, the radiated sound field can be easily obtained through the traditional mode matching
method. However, their computational efficiency are limited by the size of the geometry which limits
their application in dealing with problems regarding traffic noise.
Wiener-Hopf technique is fairly a standard method to solve certain type of linear partial differential
equations subjected to mixed boundary conditions on semi-infinite geometries. The exact solution to
the problem of plane wave radiation from a cylindrical duct to the free field was obtained using themethod by H. Levine (9). Then, it was extensively applied to solve problems regarding sound radiationfrom waveguide (10). Based on the existing literatures, we try to extend its application to obtain the
explicit solution of the sound fields inside and outside a semi-infinite long enclosure. In the model, theunbaffled opening as well as the ground effect will be considered simultaneously which are seldom
addressed before. In addition, the accuracy and efficiency of the proposed method will be comparedwith the FEM to highlight its availability and advantages in predicting traffic noise.

2. FORMULATION
2.1 Boundary Value Problem
Consider a two-dimensional semi-infinite long enclosure as shown in Figure 1. The thickness of thewall is zero and all the surfaces are assumed to be acoustical rigid for simplicity. The origin of thecoordinate system is at the intersection of the opening and the ground. Imaginary interfaces I and II are
depicted for analysis convenience. They divide the total sound field into three sub-regions A, B and C
which will be analyzed separately later.

Figure 1 – Schematic diagram of sound radiation from unbaffled long enclosure with ground
The total sound pressure field can be expressed by the following piecewise function
 p A  x, y 
x   ,    y   Ly ,  pT  x, y    pi  x, y   p B  x, y 
x   , 0   y   0, L y x   0,    y   0, Ly  pC  x, y (1)where Ly denotes the height of the outer wall. p A ( x, y ) , pB ( x, y ) and pC ( x, y ) stand for the scattered soundfields in sub-regions A, B and C, respectively. Assume that a plane acoustic wave is incident from theleft-hand side of the long enclosure which is given bypi  x, y   eikx(2)where k denotes the free space wave number. The time dependent factor eit with  being the angularfrequency is well known and will be suppressed throughout. Noted that in any physical medium, loss
is inevitable. Therefore, the ideal lossless medium which is often used in theoretical analysis, can be2058

regarded as the limiting case with vanishingly small loss. It is convenient to retain a small but nonzero
loss in the model for further study, namely, letk  k1  ik2 , k1  k2  0(3)So, the purely lossless medium is considered as the limiting case when k2 tends to be zero. The totalsound field satisfies the Helmholtz equation
 2 pT  x, y   k 2 pT  x, y   0(4)2
where  denotes the two-dimensional Laplace operator. The total sound field is to be determined by
the boundary conditions at the surfaces p A  x, Ly  y  0, x   , 0(5)
p B  x , L y  y  0 ,x    , 0 (6)pB  x, 0  y  0,x   , 0 (7)pC  x, 0  y  0,x   0,  (8)

the continuity relations at the imaginary interface I
p A  x, Ly   pC  x, Ly  ,x  0,  

(9)

p A  x, Ly  y  pC  x, Ly  y , x   0,  and the continuity relations at the imaginary interface II
pi  0, y   pB  0, y   pC  0, y  , y  0, Ly   pi  0, y   pB  0, y   x  pC  0, y  x , y   0, Ly 

(10)
(11)
(12)

The acoustic field of interest involves boundaries at infinity and geometrical singularity at the edge
which may give rise to several mathematically acceptable solutions of the Helmholtz equation. Only
one of them, however, is consistent with the anticipated physical phenomenon. In order to ensure theuniqueness of the solution, the Sommerfeld radiation condition at infinity
lim r pT  r,  r  ikpT  r,    0, r  x 2  y 2
(13)
r 
should be satisfied. Besides, according to the edge condition, the acoustic energy stored in any finite
neighborhood of the edge must be finite (11).
Up to now, the boundary value problem has been well described in the natural domain. Due to theexistence of semi-infinite region, however, the solution cannot be directly obtained. The Wiener-Hopf
technique in conjunction with the mode matching method will be applied to transform the intractableproblem into the spectral domain and obtain the Wiener- Hopf equation which can be solved throughanalytic continuation procedures. Essentially, this mixed method depends on expanding the soundfield inside the enclosure by normal modes and applying the Fourier transform technique elsewhere
which permits us to obtain the solution by the continuous relations at the opening.

2.2 Wiener-Hopf Equation and Its Solution
When applying the Wiener-Hopf technique, the Fourier integral and its inverse transformation are
required which are defined as follows


1
i x
F  , y    f  x, y  e i x dx, f  x, y  (14) F  , y  e d 2 where f ( x, y ) stands for arbitrary function in the natural domain and F ( , y) denotes the correspondingtransformed function in the spectral domain. In Eq.(14), the complex parameter     i is the Fouriertransform variable. Since pA ( x, y) satisfies the Helmholtz equation, its Fourier transform with respect
to x gives
 2 PA  , y  y 2  K 2   PA  , y   0(15)
where K    k 2   2 . The sound pressure of region A in the spectral domain can be divided into twoparts based on the definition of half range Fourier transform as
0





0

PA  , y   PA  , y   PA  , y    p A  x, y  e i x dx   pA  x, y  e i x dx

(16)

The plus and minus signs in the superscript of the transformed function indicate that they are analytic2059

functions in upper and lower half complex plane, respectively. By taking into account the following
asymptotic behavior of outgoing wave at infinity

p A  x, y   O e ik x , x  (17)

one can show that P ( , y ) and P ( , y ) are regular functions in the upper half complex plane definedby   k2 and the lower half complex plane defined by   k2 , respectively. The general solution ofEq.(15) satisfying the radiation condition in the spectral domain reads

A


A

PA  , y   A   eiK   y  Ly



(18)

where A( ) is an unknown coefficient to be determined. Obviously, the square root function K ( ) is
double-valued in the complex plane. Therefore, it is necessary to specify its branches so as to uniquely
define it. As shown in Figure 2, for the function K ( ) , there are two branch points k and two branchcuts along    k to    k  i and    k to   k  i , so that K (0)  k .

Figure 2 – Branch cuts of the square root function in the complex plane
Using the derivative of Eq.(18) with respect to y and the transformed form of Eq.(5) we havePA  , Ly   iK   A  

(19)

where the dot specifies the derivative with respect to y. Similarly, we convert the Helmholtz equation
for region C into the spectral domain by using the half-range Fourier transform technique 2 PC  , y  y 2  K 2   PC  , y   f  y   i g  y (20)
where the new functions in Eq.(20) are defined asf  y   pC  0, y  x , g  y   pC  0, y (21)
The general solution of Eq.(20) can be attained by using the method of constant variationPC  , y   B   cos  K   y   C   sin  K   y  1y

 f  t   i g  t   sin  K   y  t  dt (22)K    
0where B   and C   are unknown coefficients. Using the transformed form of Eq.(8), the coefficientscan be determined. Then, the sound field of region C in the spectral domain reads
L
 

cos  K   y PC  , y   
P

,
L

 f  t   i g  t   cos  K    Ly  t   dt 
 A
y

K   sin  K   Ly  
0

(23)
y



1

y

 f  t   i g  t   sin  K   y  t   dt
K    
0Noted that the left-hand side of Eq.(23) is regular in the upper half complex plane. However, theregularity of the right-hand side of the equation is violated by the presence of simple poles occurringat the zeros of denominator, namely,   m satisfying
K  m  sin  K  m  Ly   0, Im  m   Im  k , m  0,1, 2,...

These poles can be eliminated by imposing that their residues are zero. This gives

2060

(24)

m
PA  m , Ly   Qm  f m  i m g m  1

(25)

where the coefficients f m and g m are defined asy fm 
1  f t g    cos  K  m  t  dt
 m  Qm 0  g  t  

L

(26)

The coefficient Qm is the norm of these series expansion which is defined as
Ly

Ly

Qm   cos 2  K  m  t  dt 20

(1   0 m )(27)

where  denotes the delta function. Consider now the transformed pressure continuity at the imaginaryinterface I and using Eq. (16) and Eq.(23) we have
L
PA  , Ly 
1


P

,
L

(28)
A 
y
  f  t   i g  t   cos  K   t  dt
K 2    L  
K   sin  K   L y  0ywhere the kernel function is expressed as
L   

sin  K   Ly   iK   Lye
K  

(29)

Owing to Eq.(26) the defined functions can be expanded into cosine series as follows
 f  t    f m      cos  K  m  t  g  t   m0  gm 

(30)

Substituting Eq.(30) into Eq.(28) and evaluating the resultant integral, one obtains the scalar modified
Wiener-Hopf equation of the second kind which is valid in the strip k2    k2 asm
PA  , Ly 
 1

(31)

P

,
L


f

i

g
 m
A 
y
m
K 2    L  
 2   m2
m 0
The solution of Eq.(31) can be obtained through the classical Wiener-Hopf procedure (12) as
f m  i m g mmPA  , Ly     1
 k   m  L  m  k    L  
(32)
2 m    m m0Here the split function is regular and free of zeros in the upper half plane which is given byL   

sin  kLy k    iK    K   Ly  exp ln 


k

 




 2  i  i Ly    i Ly   exp  1  C  ln   1 
 exp  kL  2     m 
 m 1   m 
 y


(33)

where 𝐶 is the Euler’s constant given by 𝐶 = 0.57721 …
2.3 Determination of the Sound Fields
As can be seen, Eq.(32) contains infinite number of unknown coefficients. To determine them, thewell-known mode matching method will be applied. The scattered sound field in region B can be
expressed in terms of normal modes aspB  x, y    an cos  K  n  y  ei n x

(34)

n0

where the eigenvalues are evaluated by the boundary conditions as2

 n 
n
K  n  , n  k 2  , n  0,1, 2,...
 L 
Ly
 y 

(35)

Based on the continuity relations at the opening, we have the following identities f cos  K   y   ik  i a  cos  K   y 

m 0

m

m

n 0



n

n

n



 g cos  K   y   1   a cos  K   y 

m 0

(36)

m

m

n0

n

n

(37)

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Multiply both sides of the equations by cos  K  s  y  and integrate from zero to Ly in terms of y, using
the orthogonality of the modal functions, we havef 0  ik 1  a0  , s  0, f s  i s as , s  0(38)
g0  1  a0 , s  0, g s  as , s  0(39)
On the other hand, combine Eq.(25) and Eq.(32) together when a new index 𝑠 is used, we havef m  i m g msmQs  f s  i s g s  1    1
 k   m  L  m  k   s  L  s (40)
2 m  s   m m0Substitute Eq.(38) and Eq.(39) into Eq.(40) we have the following identity
as   1s 1

kL  k  L  s 
Qs s

(41)

which are the modal coefficients of region B. Then, by Eq.(34) the sound field inside the enclosure canbe determined. The radiated sound field in region A can be obtained by taking the inverse Fouriertransform of PA  , y  asp A  x, y  

1 PA  , Ly   iK   y  Ly  i xee d2  iK  (42)

where  is a straight line parallel to the real axis lying in the strip k2    k2 . In order to perform theasymptotic evaluation of the equation via the saddle point technique, let us change the variables andexpress the radiated sound field in cylindrical polar coordinate system as shown in Figure 1.
  k cos w, x  r cos  , y  r sin (43)Then, the radiated field can be determined (12) as
p A  r ,  

ei 4
2

2kL   k cos   L  k  e

ik sin  L ye ikrkr(44)

Up to now, the sound fields inside and outside the semi-infinite long enclosure are obtained. Next, theFEM will be applied to validate the proposed model.

3. DISCUSSIONSAn enclosure of 2.5m long and 0.5m high is established in COMSOL Multiphysics, a commercial
software based on the finite element method (FEM), to validate the proposed technique. In theory, theradiated sound field extends to infinity. However, for the computational efficiency, a relatively small
calculation domain is considered. The outside acoustic domain is bounded by introducing PML which
is an artificial absorbing layer allowing sound wave to propagate out without reflection. The results ofabsolute sound pressure in region B at 800Hz obtained by the Wiener-Hopf technique and the FEM arecompared in Figure 3.

Figure 3 – Comparison of the sound fields in region B (absolute sound pressure/ Pa) obtained by
Wiener-Hopf technique (W-H) and the FEM at 800Hz
From (a) and (b) we can observe good agreement between the results obtained by the Wiener-Hopf

2062

technique and the FEM. From (c), we can find that, inside the enclosure, apart from the incident planewave, the scattered wave will contribute to the total sound field due to the impedance mismatch at the
opening. For small size enclosure with plane wave incidence in this case, the scatted field is relatively
small and simply distributed. But it cannot be ignored as it still contributes nearly 1/3 of the soundenergy. For larger size enclosure and in high frequency case, the contribution will increase.
Besides, comparison of sound pressure level (dB) between the Wiener-Hopf technique and the FEM
in the frequency range 200Hz to 2000Hz at a particular point (-0.1,0.35) near the opening edge is also
shown in Figure 4. In this case, 100 modes are taken in the calculation. Good agreement is found which
validates the correctness and accuracy of the proposed model.

Figure 4 – Comparison of sound pressure level (dB) between Wiener-Hopf technique and FEM in thefrequency range 200Hz to 2000Hz at a particular point (-0.1,0.35)
Directivity patterns of the radiated sound field in region A at different frequencies are plotted inpolar coordinate (r=5m). As shown in Figure 5, the results show good agreement despite some sharp
deeps which may come from mathematical issues. Clear lobes and zeros can be observed. The numberof lobes increases with the increasing of frequency. In front of the opening, the sound field is formedby the superposition of radiated, diffracted and reflected sound. They propagate to the receiver withdifferent phase and distance, which results in the directivity pattern. At the back, the sound field is theresult of diffraction at the opening edge. It becomes stable and stands at relatively low level.

Figure 5 – Directivity patterns of the radiated sound fields (Sound pressure level/dB) obtained by the
Wiener-Hopf technique and the FEM at different frequencies (r=5m)
We also calculate the sound fields of lager enclosures at higher frequency range, the results fromthe Wiener-Hopf technique and the FEM coincide well with each other (The results are not listed forthe limitation of pages). As the increase of geometry size and frequency, the calculation time remains
nearly the same for the proposed method as it is an analytical solution. Despite the fact that, the mode
number in the calculation will increase for larger size enclosure and high frequency so as to capture
high order mode. The calculation efficiency is still high using the Wiener-Hopf technique because it is2063

easy for computer to do mode superposition. However, the meshes for the FEM increase exponentiallywhich results in the low efficiency. So, in this condition, the Winer-Hopf technique performs muchbetter than the FEM.

4. CONCLUSIONSIn this paper, a rigorous and explicit solution is obtained for the prediction of sound fields insideand outside a semi-infinite long enclosure. In the proposed method, the unbaffled opening of theenclosure and the ground effect are taken into consideration simultaneously in order to model thepractical traffic facilities. The boundary value problem is formulated as a modified Wiener-Hopf
equation involving three sets of unknowns satisfying three infinite systems of linear algebraic
equations which can be solved numerically. Results obtained from the FEM and the Wiener-Hopf
technique are compared and discussed which indicates that the proposed method is capable ofpredicting the sound fields inside and outside a tunnel simultaneously and show better performance in
big size condition.

ACKNOWLEDGEMENTS
The authors would like to acknowledge the funding support from The Hong Kong Polytechnic
University (PolyU152029/17E) and (PolyU152666/16E). The first author would also like to thank thestudentship of The Hong Kong Polytechnic University.

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