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

Study on the sound transmission loss of a truncated conical shell
excited by an incident plane acoustic wave
Masoud GOLZARI1; Ali Asghar JAFARI2
1

K.N. Toosi University of Technology, Iran2

K.N. Toosi University of Technology, Iran

ABSTRACTIn this paper, a theoretical model is proposed to study on the sound transmission loss of a thin-walled
truncated circular conical shell subjected to an incident plane acoustic wave. The shell motion is governed byLove’s theory and a convergent power series solution is used to obtain exact displacements of the shell. The
current results are firstly validated against those of the experimental and analytical works reported in theliterature. Parametric study is then conducted to investigate the effects of several important design parameters
including different boundary conditions at the ends of the shell, cone angle, incidence sound wave angle, wall
thickness and material properties of the shell on the characteristics of the sound transmission loss. The
presented model can be useful and effective in the acoustic design stage of the truncated conical shells.
Keywords: Sound transmission loss, Truncated Conical Shell, Plane sound wave

1. INTRODUCTION
Plates and shells are widely applied in many types of industries such as aerospace, automotive,
marine and military. The interaction of these structures and surrounding fluids mostly inducestransmission of undesirable vibroacoustic energy, which can lead to structural fatigue and noise
pollution in the systems. For instance, the transmission of sound power into the vehicle body such as
aircraft, train and car can be a discomfort for the passengers. Therefore, the acoustic transmission
problem of plates and shells has attracted extensive research attention and numerous works on thetransmission of sound through panels and cylindrical shells have been performed for decades.
The early study for calculating the sound transmission through panel structures can be traced backto the work carried out by Beranek and Work (1). Koval (2) studied the sound transmission loss ofaircraft fuselage panels and the influences of panel curvature, fuselage pressurization and external air
flow on the acoustic behavior of the structure. Lee and Kim (3) developed a theoretical model toinvestigate on the sound transmission characteristics of a thin plate stiffened by equally spaced line
stiffeners. Xin et al. (4, 5) theoretically and experimentally obtained the transmission loss of thindouble-panel partitions with finite dimensions. They showed that the effect of boundary conditions ismore considerable at low frequencies. The transmission loss of triple-panel structures lined withporous layers was studied by Liu (6). Mana and Sonti (7) presented a theoretical model to compute theacoustic transmission through a finite perforated panel set. Qiao et al. (8) studied on the soundinsulation of a periodically rib-stiffened double panel with porous lining.
The prediction of the acoustic transmission through thin-walled cylindrical shells was firstly
performed by Smith (9). For an aircraft fuselage, Koval analytically studied the sound transmission
into a thin cylindrical shell under flight conditions (10) an orthotropic shell (11), and a laminatedcomposite cylindrical shell (12). Lee and Kim (13, 14) carried out analytical and experimental
analyses for the transmission loss of thin single- and double-walled cylindrical shells. The influencesof geometrical and mechanical properties of the shells were also explored by them. Zhou et al. (15)analytically investigated on the transmission loss of sandwich cylindrical shells with infinite length.
Magniez et al. (16) proposed an analytical model for calculating the sound transmission loss oforthotropic sandwich cylindrical shells lined with porous layer. Golzari and Jafari (17) studied on the
1
2

4684

mgolzari@mail.kntu.ac.ir
ajafari@kntu.ac.ir

sound insulation performance of triple- and multi-walled sandwich cylindrical shell with poroelastic
layers and air gaps in a diffuse sound field.
Conical shells are extensively used in many practical applications such as aircrafts, submarines,
rockets, and tanks. However, study on the sound transmission behavior of these highly complicatedstructures has received a little attention for decades. In this regard, it can be referred to the work
experimentally performed by Vipperman et al. (18) who measured the acoustic transmission through atruncated conical shell. Therefore, the main purpose of this study is to propose a theoretical model forinvestigating on the sound transmission through truncated circular conical shells and the influences ofdifferent boundary conditions at the ends of the shell, cone angle, incidence sound wave angle,
thickness and material properties of the shell on the transmission loss characteristics.

2. FORMULATION OF THE VIBROACOUSTIC PROBLEM
The geometry and coordinate system of a truncated circular conical shell considered in this study
are shown in Figure 1. R 1 and R 2 are the radiuses at the smaller and larger ends, respectively, α is thesemi-vertex angle, L is the height, L s is the slant height, and h is the wall thickness. The boundary
conditions at the ends of the thin isotropic shell are simply supported. In the curvilinear coordinate
system, the x and ߠ respectively show the meridional and circumferential directions, and z is normal
to the shell surface. A harmonic plane acoustic wave impinges on the external surface with theincidence angle ߚ with respect to N, and it is partially transmitted into the inner cavity, which isassumed with no reverberation.

Figure 1 – A schematic sketch of the sound transmission through the truncated conical shell
The equations of motion of the thin-walled truncated conical shell based on Love’s shell theory are
written as follows (19):
߲ଶ‫ݑ‬
(1)
‫ܮ‬ଵ ሼ‫ݑ‬ǡ ‫ݒ‬ǡ ‫ݓ‬ሽ ൌ ߩ௦ ݄ ଶ
߲‫ݐ‬
߲ଶ‫ݒ‬
(2)
‫ܮ‬ଶ ሼ‫ݑ‬ǡ ‫ݒ‬ǡ ‫ݓ‬ሽ ൌ ߩ௦ ݄ ଶ
߲‫ݐ‬
߲ଶ‫ݓ‬
(3)
‫ܮ‬ଷ ሼ‫ݑ‬ǡ ‫ݒ‬ǡ ‫ݓ‬ሽ ൅ οܲ ൌ ߩ௦ ݄ ଶ߲‫ݐ‬where u, v and w are, respectively, the displacement components at the neutral surface in x, ߠ and z
directions. ߩ௦ is the material density of the conical shell. The expression οܲ ൌ ሺܲூ ൅ ܲோ ሻ െ ்ܲ
denotes the pressure difference applying on the shell, in which P I, PR and P T are the incident, reflected
and transmitted sound pressures, respectively. The partial differential operators L1 , L 2 and L 3 are also
given in Appendix.
As shown in Figure 2, to determine the acoustic pressures at the shell and fluid media interfaces, theconical shell is divided into N parts which are narrow enough to be considered locally cylindrical, in
which the quantity of the sound pressures on the conical segment equals that on the equivalentcylindrical counterpart with the same mean radius R0,i and length li. Hence the pressure waves in thecylindrical coordinate system can be written as (15, 17):ܲ௜ூ ሺ‫ݎ‬ǡ ߠǡ ܺǡ ‫ݐ‬ሻ ൌ ෍ஶ‫݌‬ூ ߝ௡ ሺെ݆ሻ௡ ‫ܬ‬௡ ሺ݇ଵ௥ ‫ݎ‬ሻ ܿ‫ݏ݋‬ሺ݊ߠሻ ሾ݆ሺ߱‫ ݐ‬െ ݇ଵ௑ ሺܺ௜ ൅ ܺሻሿ

௡ୀ଴

(4)

ஶ

ୖ
ܲ௜ୖ ሺ‫ݎ‬ǡ ߠǡ ܺǡ ‫ݐ‬ሻ ൌ ෍ ‫݌‬௡ǡ௜‫ܪ‬௡ଶ ሺ݇ଶ௥ ‫ݎ‬ሻ௡ୀ଴ሺ݊ߠሻ ሾሺ߱‫ ݐ‬െ ݇ଶ௑ ܺሻሿ

(5)

4685

ஶ
୘
ܲ௜୘ ሺ‫ݎ‬ǡ ߠǡ ܺǡ ‫ݐ‬ሻ ൌ ෍ ‫݌‬௡ǡ௜‫ܪ‬௡ଵ ሺ݇ଷ௥ ‫ݎ‬ሻ
௡ୀ଴

(6)

ሺ݊ߠሻ ሾሺ߱‫ ݐ‬െ ݇ଷ௑ ܺሻሿǡ݅ ൌ ͳǡʹǡ Ǥ Ǥ ǡ ܰ

where p I, p R and p T are the amplitudes of the incident, reflected and transmitted sound waves,respectively. r, ߠ and capital letter X are, respectively, the radial, circumferential and axial directionsin the cylindrical coordinate system. ω is the angular frequency of the incident sound wave,  ൌ ξെͳ,n is the circumferential mode number, ε n is the Neumann factor in which ε n=1 for n=0 and ε n=2 for
n=1,2,… . ‫ܬ‬௡ , ‫ܪ‬௡ଵ and ‫ܪ‬௡ଶ are the Bessel function of the first kind, Hankel function of the first kindand Hankel function of the second kind of order n, respectively. Since the travelling waves within thesystem are driven by the incident wave, the axial wavenumbers in the fluid media must be equal, i.e.
k 1X=k 2X=k3X. Also k 1X=k 1sin(α+β) and k1r =k 1cos(α+β), in which k 1=ω/c. c is the speed of sound in thefluid media.

Figure 2 – A schematic sketch showing the narrow conical segments
The truncated conical segments in Figure 2 are connected together by using eight continuity
equations at each intersection. These boundary conditions can be formulated as follows:‫ݑ‬௜ ȁ ௟ೞǡ೔ ൌ ‫ݑ‬௜ାଵ ȁ ௟ೞǡ೔శభ ǡ‫ݒ‬௜ ȁ ௟ೞǡ೔ ൌ ‫ݒ‬௜ାଵ ȁ ௟ೞǡ೔శభ ǡ‫ݓ‬௜ ȁ ௟ೞǡ೔ ൌ ‫ݓ‬௜ାଵ ȁ ௟ೞǡ೔శభ
(7)
௫ୀ

ଶ

௫ୀି

௫ୀ

ଶ

߲‫ݓ‬௜ାଵ
߲‫ݓ‬௜
ǡ
ฬ ௟ ൌ
ฬ
ೞǡ೔
߲‫ ݔ‬௫ୀ߲‫ ݔ‬௫ୀି௟ೞǡ೔శభଶ‫ܯ‬௫௫ǡ௜ หܰ௫௫ǡ௜ ห௫ୀ௟ೞǡ೔ ൌ ܰ௫௫ǡ௜ାଵ หଶǡ௟ ൌ ‫ܯ‬௫௫ǡ௜ାଵ ห௟௫ୀ ೞǡ೔௫ୀି ೞǡ೔శభ
ଶ
ଶ

௫ୀି

ଶ

ܸ௫௭ǡ௜ หଶ೗௫ୀ ೞǡ೔

௫ୀିൌ ܸ௫௭ǡ௜ାଵ ห

మ

௫ୀ

ଶ

௟ೞǡ೔శభ ǡଶ೗௫ୀି ೞǡ೔శభమܰ௫ఏǡ௜ ห

௫ୀ

ଶ

௫ୀି

௟ೞǡ೔ ൌ ܰ௫ఏǡ௜ାଵ หଶଶ௟௫ୀି ೞǡ೔శభଶଵ డெೣഇ

where:ܸ௫௭ ൌ ܳ௫௭ ൅ ோሺ௫ሻ

డఏ

(8)
(9)

where i=1,2,..,N-1, ls,i is the slant length of the ith cone and ܴሺ‫ݔ‬ሻ ൌ ܴ଴ ൅ ‫ߙ݊݅ݏݔ‬.
At the interfaces between the truncated conical segments and the fluid media, the particle velocitiesof the fluid media in the z direction normal to the shell surface must equal to the normal velocity of theshell. Hence, the following relations are obtained:
߲‫ݓ‬௜
୍
ୖ
୍
ୖ
(10a)
൫ܸ௥ǡ௜
൅ ܸ௥ǡ௜൯ ߙ ൅ ൫ܸ௑ǡ௜൅ ܸ௑ǡ௜
൯Ƚ ൌ
߲‫ݐ‬
߲‫ݓ‬௜
୘
୘
(10b)
ܸ௥ǡ௜
ߙ ൅ ܸ௑ǡ௜Ƚ ൌ
߲‫ݐ‬where Vr and V X are the sound particle velocities in the radial and longitudinal directions of thecylindrical coordinate system, respectively.
The boundary conditions at the ends of the truncated conical shell , i.e. x=0,Ls , for the simply
supported (S), clamped (C) and free (F) boundary conditions are defined in equations (11)-(13),respectively, as follows:(11)‫ ݒ‬ൌ Ͳǡ‫ ݓ‬ൌ Ͳǡܰ௫௫ ൌ Ͳǡ‫ܯ‬௫௫ ൌ Ͳ߲‫ݓ‬(12)
‫ ݑ‬ൌ Ͳǡ‫ ݒ‬ൌ Ͳǡ‫ ݓ‬ൌ Ͳǡ
ൌͲ
߲‫ݐ‬
(13)
ܰ௫௫ ൌ Ͳǡܰ௫ఏ ൌ Ͳǡ‫ܯ‬௫௫ ൌ Ͳǡܸ௫௭ ൌ ͲGeneral solution in the form of power series for the displacement components of the each truncatedconical part can be expressed as:ஶஶ‫ݑ‬ሺ‫ݔ‬ǡ ߠǡ ‫ݐ‬ሻ ൌ ෍ ܷሺ‫ݔ‬ሻ ሺ݊ߠሻ ሺ߱‫ݐ‬ሻwhere:

௡ୀ଴

4686

(14)

௠ୀ଴
ஶ

௡ୀ଴
ஶ

‫ݒ‬ሺ‫ݔ‬ǡ ߠǡ ‫ݐ‬ሻ ൌ ෍ ܸሺ‫ݔ‬ሻ ሺ݊ߠሻ ሺ߱‫ݐ‬ሻܷሺ‫ݔ‬ሻ ൌ ෍ ܽ௠ ‫ ݔ‬௠where:

ܸሺ‫ݔ‬ሻ ൌ ෍ ܾ௠ ‫ ݔ‬௠
௠ୀ଴

(15)

ஶ

‫ݓ‬ሺ‫ݔ‬ǡ ߠǡ ‫ݐ‬ሻ ൌ ෍ ܹሺ‫ݔ‬ሻ ሺ݊ߠሻ ሺ߱‫ݐ‬ሻ௡ୀ଴ஶ

where:

ܹሺ‫ݔ‬ሻ ൌ ෍ ܿ௠ ‫ ݔ‬௠

(16)

௠ୀ଴

in which a m, b m and c m are the constants.
By using equations 4-6, 10 and 14-16, and substituting them into the governing equations given inequations 1-3, it is found that the displacements of the conical parts can be obtained in terms of eight
unknown constants a 0, a 1, b 0, b 1, c0 , c 1 , c 2 and c 3 as follows:‫ݑ‬ଵ ሺ‫ݔ‬ሻ ‫ ଼ݑ ڮ‬ሺ‫ݔ‬ሻ‫ݑ‬଴ ሺ‫ݔ‬ሻܷሺ‫ݔ‬ሻ
(17)቎ ܸሺ‫ݔ‬ሻ ቏ ൌ ቎ ‫ݒ‬ଵ ሺ‫ݔ‬ሻ ‫ ଼ݒ ڮ‬ሺ‫ݔ‬ሻ ቏ ‫ ܠ‬൅ ቎ ‫ݒ‬଴ ሺ‫ݔ‬ሻ ቏ܹሺ‫ݔ‬ሻ
‫ݓ‬ଵ ሺ‫ݔ‬ሻ ‫ ଼ݓ ڮ‬ሺ‫ݔ‬ሻ‫ݓ‬଴ ሺ‫ݔ‬ሻwhere up,i (x), v p,i(x) and w p,i (x) (p=0,1,2,…,8; i=1,2,…,N) are the base functions and x=[a 0 a 1 b 0 b 1 c0c 1 c2 c 3] T .
Substituting equation 17 into the boundary conditions at the intersections between the conical strips
provided in equations 7-9 and the boundary conditions at both ends of the truncated conical shell, here
simply supported boundary condition in equation 11, leads to the following matrix equation for eachcircumferential mode number n:(18)‫ ܠۯ‬ൌ ‫܊‬
By solving the matrix equation 18, the displacement components of the truncated conical shell are
determined, which are used to obtain the reflected and transmitted sound pressures from equations 10.
Finally, the transmission loss is defined in below (14, 15):୘σஶ௡ୀ଴ ܹ௡
(19) ൌ െͳͲ
୍ܹwhere the incident sound power WI and the transmitted sound power Wn T are:
ͳ(20)
ܹ ୍ ൌ  ൜න ୍ܲ ሺܸ ୍ ሻ‫ܵ݀ כ‬୧୬ ൠ
ʹ
‫כ‬
ͳ
߲‫ݓ‬
(21)
ܹ௡୘ ൌ  ቊන ܲ୘ ൬ ൰ ݀ܵ୭୳ ቋʹ߲‫ݐ‬
where Re ሼήሽ and ሺ‫כ‬ሻ represent the real part and complex conjugate of the argument, respectively. S inis the plane projection of the conical shell surface and ݀ܵ௢௨ ൌ ሺܴ଴ ൅ ‫ߙݔ‬ሻ݀‫ߠ݀ݔ‬. ܸ ୍ ൌ ୍ܲ Τߩ୍ ୍ܿ is theparticle velocity of the incident plane sound wave, in which ߩ୍ is the density of external fluid. In arandomly incidence field, the average transmission loss is also defined by (14):ఉౢ౟ౣ σஶ୘௡ୀ଴ ܹ௡
(22)ୟ୴ ൌ െͳͲ ቈʹ නቆቇ ߚ ߚ݀ߚ቉
୍ܹ଴where β lim is the limiting angle of incident sound wave, in which above this angle it is assumed that no
acoustic wave can cross the conical shell.

3. RESULTS AND DISCUSSIONIn Table 1, the basic property parameters of the truncated conical steel shell are presented. The
boundary conditions at the ends of the shell are simply supported. The amplitude of the incident
acoustic wave is 1 Pa, and the fluid media in the outside and inside of the shell is ambient air at 20°C
with the density of 1.21 kg m -3 and the sound speed of 343.2 m s -1.
3.1 Convergence checking
Because the acoustic waves, displacement components and transmission loss are expressed asseries form, the use of inadequate number of circumferential modes n leads to unreliable outcomes.
Therefore, the convergence checking should be carried out during the analysis. Figure 3 shows thevariation of transmission loss with the circumferential mode number n at the frequencies of 200 Hz, 2
kHz and 20 kHz. The mode convergence curve changes with n until it reaches a constant value. It isalso observed that the number of modes needed for converging increases as the frequency is enhanced .
3.2 Model validation
Since the acoustic transmission through conical shells has seldom been studied as mentioned in
Section 1, Figures 4 and 5 respectively show the comparisons carried out between the current results
and those of Zhou et al. (15), and Lee and Kim (14) for circular cylindrical shells. It should be notedthat the sound transmission problem of the cylindrical shells can be achieved by substituting α=0 into
the governing equations implemented for the truncated conical shells in Section 2. From these figures,

4687

Table 1 – Property parameters of the truncated conical steel shell
SymbolDescription
Value
ߩୱ

Material density

7850 kg m-3‫ܧ‬Young’s modulus of elasticity

2 × 10 11 Pa

ߤPoisson’s ratio

0.3

ܴଵ

Small radius

0.1 m

ܴଶ

Large radius

0.144 mߙ

Semi-vertex angle

5°݄Thickness

1 mm

‫ܮ‬

Height0.5 m

Figure 3 – Convergence checking for the sound transmission loss of the truncated conical shell
it is observed that the results of the present model are in a good agreement with those of Zhou et al.
(15) and Lee and Kim (14) at middle and high frequencies. However, discrepancies particularly in the
low frequency range can be attributed to two main reasons: (a) because the analytical models proposedby Zhou and Lee are presented for the shell with infinite length, they are not cap able to take into
account the effects of the finite-size assumptions such as the implication of boundary constraints at theends. Considering the boundary condition increases the shell stiffness which leads to an improvementin the transmission loss outcomes at low frequencies. However, by increasing of the length size, theanalytical predictions approach each other at these frequencies. (b) In the Lee’s experimental setting
the creation of an incident plane acoustic wave condition by a single extern al sound source and theacoustic room without reverberation is not clear. Also, the inner cavity in the current model is assumed
to be anechoic while in the experimental model is reverberant.

Figure 4 – Comparison of the present results with

Figure 5 – Comparison of the present results withthe analytical results of Zhou et al. (15), β=45°

the experimental measurements and analyticalresults of Lee and Kim (14)4688

3.3 Parametric study
In Figure 6, the transmission loss outcomes of the truncated conical shell with free -free (F-F),
simply-simply (S-S) and clamped-clamped (C-C) boundary conditions are presented. It can be seenthat the effect of boundary constraint through changing the shell stiffness is more considerable at thefrequencies below 1000 Hz. The highest and lowest transmission loss amplitude are generally
predicted by the fully clamped and free boundary conditions, respectively. However by disappearing
the influence of the boundary conditions, there is no significant behavior for the sound transmission
losses at higher frequencies and the tendency of the curves is similar.

Figure 6 – Effect of the boundary conditions on the transmission loss of the truncated conical shell, β=45°
Figure 7 shows the transmission loss of the truncated conical shell at different semi -vertex angles
α=0°, 5°, 10°. It should be noted that α=0° is corresponded with the cylindrical shell with the mean
radius 0.122 m, which equals to the mean radius of the conical shell with the parameters given in Table
1. It is concluded that the transmission loss at frequencies below 400 Hz is reduced as the cone angle
increases, specifically for the cylindrical shell compared with the conical shell. This can be caused by
reducing the shell stiffness when the cone angle is enhanced. The mass law states that the soundtransmission loss increases with the mass or frequency and decreases by the acoustic impedance of thefluid media (17). Therefore, increasing of the cone angle leads to slightly superior transmission loss at
higher frequencies as a result of the mass increase.
Figure 8 represents the acoustic transmission of the truncated conical shell at different incidence
angles. The transmission of sound into the shell is significantly increased in the frequency range 250
Hz to 2800 Hz as the incidence angle decreases. However, an improvement in the transmission loss isobserved at other frequencies.

Figure 7 – Effect of the cone angle on the

Figure 8 – Effect of the incidence acoustic wave

transmission loss of the truncated conical shell,

angle on the transmission loss of the truncatedconical shell
R1=0.1 m, L=0.5 m, β=45°
Figure 9 illustrates the effect of the shell thickness on the transmission loss of the tr uncated conicalshell. By increasing of the thickness, the transmission of sound into the inner cavity is reduced over awide frequency range. This is because both the shell stiffness and shell mass enhance. However, it
should be noted that an enhancement in the shell thickness is not always an appropriate perspective toreach superior sound insulation performance because of the design restriction in the weight,
construction procedure and costs.

4689

A comparison between the transmission loss outcomes for three types of materials includingaluminum (with density 2700 kg m -3 , Young modulus 69 GPa and Poisson’s ratio 0.33), steel and brass(with density 8553 kg m -3, Young modulus 104 GPa and Poisson’s ratio 0.37) is carried out in Figure
10. It is shown that the best transmission loss at low and middle frequencies is achieve by the conicalsteel shell, whereas the conical shell made of brass has the highest amplitude of the transmission loss
at high frequencies. Because steel and brass have the most stiffness and ma ss, respectively.

Figure 9 – Effect of the shell thickness on the the

Figure 10 – Effect of the shell material properties ontransmission loss of the truncated conical shell,

the transmission loss of the truncated conical shell.

β=45°

β=45°

4. CONCLUSIONSSound transmission loss of a thin-walled truncated circular conical shell, which is excited by anoblique incident plane sound wave, was theoretically investigated. Love’s theory was used to
implement the governing equations of the shell motion and a convergent power series solution was
employed to obtain the exact displacements of the conical shell. In order to compute the acousticpressure loadings at the structure-fluid interfaces, the shell is divided into segments which are narrow
enough to be taken into account locally cylindrical. At first, the model predictions were validated
against both their experimental and analytical counterparts provided in previous studies and in thisregard, a good agreement was achieved. Then, the effects of several key parameters were investigatedon the transmission loss, which can be summarized as follows:
The effect of the boundary constraint is found to be more significant at low frequencies. In thisregard, the comparison between the results of simply supported-simply supported, clamped-clamped
and free-free boundary conditions shows the highest and lowest transmission loss for the fully clamped
boundary condition and free boundary condition, respectively. However, a similar trend for thetransmission loss behaviors is observed at higher frequencies.
The transmission loss is reduced in the low frequency range as the cone angle increases. However,an enhancement in the amplitude of the transmission loss is observed in the middle and high frequency
bands with a rise in the vertex angle of conical shell.
Increasing of the shell thickness results in better noise insulation performance in almost the entire
frequency spectrum.
It is found that the transmitted sound power increases at middle frequencies when the incidence
angle decreases, which is opposite to low and high frequencies.
Also, the comparison of the transmission loss results of the conical shell with different types ofmaterials including brass, steel and aluminum represents superior sound insulation ability for steel at
low and middle frequencies, whereas brass has better performance in the high frequency range.
The proposed model can be considered as an effective tool in the acoustic design of the truncatedconical shells.

APPENDIX
The partial differential operators given in equations 1 to 3 are:߲ ଶ ‫ܥ ݑ‬ሺͳ െ ߤሻ ߲ ଶ ‫ܥ ݑ߲ ߙܥ ݑ‬ଶ ߙ‫ܥ‬ሺͳ ൅ ߤሻ ߲ ଶ ‫ݒ‬‫ܮ‬ଵ ሼ‫ݑ‬ǡ ‫ݒ‬ǡ ‫ݓ‬ሽ ൌ ‫ ܥ‬ଶ ൅
൅
െ
‫ݑ‬
൅
ଶ
ଶ
ܴሺ௫ሻ ߲‫ݔ‬ʹܴሺ௫ሻ ߲‫ߠ߲ݔ‬
߲‫ݔ‬
߲ߠ ଶ
ʹܴሺ௫ሻ
ܴሺ௫ሻ
‫ܥ‬ሺߤ െ ͵ሻߙ ߲‫ߙ ߙܥ ݓ߲ ߙ ߤܥ ݒ‬
൅
െ
‫ݓ‬
ଶ
ଶ
ܴሺ௫ሻ ߲‫ݔ‬
߲ߠ
ܴሺ௫ሻ
ʹܴሺ௫ሻ

4690

(A1)

‫ܥ‬ሺͳ െ ߤሻ ‫ܦ‬ሺͳ െ ߤሻ ଶ ߙ ߲ ଶ ‫ݒ‬‫ܥ‬ሺͳ ൅ ߤሻ ߲ ଶ ‫ܥ ݑ‬ሺ͵ െ ߤሻߙ ߲‫ݑ‬
൅
൅
ቆ
൅
ቇ ଶ
ଶ
ଶ
ʹ
߲ߠ
߲‫ݔ‬
ʹܴሺ௫ሻ ߲‫ߠ߲ݔ‬
ʹܴሺ௫ሻ
ʹܴሺ௫ሻ
ଶ
ଶ
ଶ
ଶ
‫ܥ‬
‫ݒ ߲ ߙ  ܦ‬‫ܥ‬ሺͳ െ ߤሻߙ ‫ܦ‬ሺͳ െ ߤሻ  ߙߙ ߲ ‫ܥ ݒ‬ሺͳ െ ߤሻଶ ߙ
ቆ ଶ ൅
ቇ
൅
ቆ
െ
ቇ ଶെ
‫ݒ‬
ଷ
ସ
ଶ
ʹܴሺ௫ሻ
߲ߠ ଶ
߲ߠ
ܴሺ௫ሻ
ܴሺ௫ሻ
ʹܴሺ௫ሻ
ʹܴሺ௫ሻ
‫ ߲ ߙ ܦ‬ଷ ‫ ߲ ߙ ܦ ݓ‬ଷ ‫ݓ‬‫ ߲ ߙ ߙܦ‬ଶ ‫ݓ߲ ߙ ܥ ݓ‬
െ ସ
െ
െ
൅ ଶ
(A2)
ଷ
ଶ
߲‫ ݔ‬ଶ ߲ߠ߲‫ߠ߲ݔ‬
ܴሺ௫ሻ ߲ߠ
ܴሺ௫ሻ ߲ߠ ଷܴሺ௫ሻܴሺ௫ሻ‫ܮ‬ଶ ሼ‫ݑ‬ǡ ‫ݒ‬ǡ ‫ݓ‬ሽ ൌ߲ ସ ‫ ߲ ߙܦʹ ݓ‬ଷ ‫ܦ ݓ‬ଶ ߙ ߲ ଶ ‫ܦ ݓ‬ଷ ߙ ߲‫ݓ‬‫ߙߙ ܥ ݑ߲ ߙ ߤܥ‬
െ
‫ݑ‬
െ
‫ܦ‬
െ
൅
െ
ଷ
ଶ
ଶ
ܴሺ௫ሻ ߲‫ ݔ‬ଷ߲‫ ݔ‬ସ߲‫ ݔ‬ଶ
߲‫ݔ‬ܴሺ௫ሻ ߲‫ݔ‬
ܴሺ௫ሻ
ܴሺ௫ሻ
ܴሺ௫ሻ
ସ
ଶ
ଶ
ଶ
ସ
ଶ
‫ ݓ ߲ ܦ‬Ͷ‫ݓ ߲ ߙ ܦ‬Ͷ‫ ߙ ܦ‬Ƚ ‫ ܥ‬Ƚ ߲‫ݓ ߲ ܦʹ ݒ‬‫ߙ  ܥ‬
െ ସ
െ
൅ቆ
െ ଶ ቇ
െ ଶ
െ
‫ݓ‬
ସ
ସ
ଶ
ସ
ଶ
ଶ
ଶ
ܴሺ௫ሻ
ܴሺ௫ሻ ߲ߠ
ܴሺ௫ሻ ߲ߠ
ܴሺ௫ሻܴሺ௫ሻ ߲ߠ ܴሺ௫ሻ ߲‫ߠ߲ ݔ‬ʹ‫ ߲ ߙܦ‬ଷ ‫ݓ‬‫ ߲ ߙ ܦ‬ଷ ‫ ߲ ߙ ߙܦ͵ ݒ‬ଶ ‫ݒ‬‫ ߲ ߙ ܦ‬ଷ ‫ݒ‬
൅
െ
൅
(A3)
ସ
ଷ
ଷ
ଶ
߲‫ ߠ߲ݔ‬ଶ߲ߠ ଷ߲‫ߠ߲ݔ‬߲‫ ݔ‬ଶ ߲ߠ
ܴሺ௫ሻ
ܴሺ௫ሻ
ܴሺ௫ሻ
ܴሺ௫ሻ
where ‫ ܥ‬ൌ ‫݄ܧ‬Τሺͳ െ ߤଶ ሻ and ‫ ܦ‬ൌ ‫݄ܧ‬ଷΤͳʹሺͳ െ ߤଶ ሻ are the membrane stiffness and the bending
stiffness of the truncated conical shell, respectively, in which E is the Young’s modulus and μ is the
Poisson’s ratio of the shell.
‫ܮ‬ଷ ሼ‫ݑ‬ǡ ‫ݒ‬ǡ ‫ݓ‬ሽ ൌ െ

REFERENCES
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13. Lee JH, Kim J. Analysis and measurement of sound transmission through a double-walled cylindricalshell. J Sound Vib 2002;251:631-649.
14. Lee JH, Kim J. Study on sound transmission characteristics of a cylindrical shell using analytical andexperimental models. Appl Acoust 2003;64(6):611-632.
15. Zhou J, Bhaskar A, Zhang X. The effect of external mean flow on sound transmission throughdouble-walled cylindrical shells lined with poroelastic material. J Sound Vib 2014;333:1972-1990.
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layers. J Acoust Soc Am 2018;143:3529-3544.
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