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Sound transmission of a spherical sound wave through a finite plate
Journal of Sound and Vibration 410 (2017) 209e216
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Sound transmission of a spherical sound wave through a finite plate Bilong Liu a, b, *, Yan Jiang b, Daoqing Chang b, ** a b
School of Mechanical Engineering, Qingdao University of Technology, 777 Jialingjiang Road, 266520 Qingdao, PR China Key Laboratory of Noise and Vibration Research, University of Chinese Academy of Sciences, 100080 Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 March 2017 Received in revised form 3 August 2017 Accepted 16 August 2017
For an incident plane wave on an infinite plate, a doubling of mass or frequency adds 6 dB to the sound transmission loss (TL), but for an incident spherical wave on an infinite plate, a doubling of mass or frequency adds only 3 dB to the TL. In reality, the discrepancies of the sound transmission due to plane wave and spherical wave incidence might not be so huge, since the influences resulted from the plate size and the distance between the source and the plate cannot be ignored. In this article, the sound transmission of a spherical wave through a finite plate is theoretically analyzed through the modal expansion method. The transmission losses for typical plates are illustrated and as well are compared with that of the mass laws due to normal and spherical wave incidence, respectively. The effects of parameters such as the size of the plate, the distance between the source and the plate, and the horizontal shift of the plate are investigated. An indicator for the estimation of the TL through a finite plate due to a point source is given for the potential of practical applications. © 2017 Elsevier Ltd. All rights reserved.
Handling Editor: L. G. Tham Keywords: Sound transmission loss Vibro-acoustics Plate vibration
1. Introduction Sound transmission through a single-leaf plate has been widely reported [1e5]. One of the well-known features of the sound transmission through the plate is the so-called mass law. In the low frequency range or far below the critical frequency of the plate, the mass per unit area of the plate is the only plate parameter determining the sound transmission performance and a doubling of mass or frequency adds 6 dB to the TL. It also should be noted that the TL for a normal incidence on the plate is about 5 dB higher than that for a diffuse field incidence. The mass law is usually derived from plane wave incidence on an infinite plate. Little attention has been paid to the sound transmission due to spherical wave incidence. Motoki Yair et al. [6e9] derived a mass law under the condition that a spherical wave is incident on an infinite plate. By ignoring the distance between the source and the plate due to the infinite assumption of the plate, the spherical wave incidence mass law can be described as a 3 dB increase provided that the mass per unit area or frequency is doubled. Apart from this apparent mass law difference in compare with that of normal incidence, the sound transmission loss for spherical wave incidence on the infinite plate is about 20 dB less than that due to normal wave incidence at 1000 Hz,where the critical frequency of the infinite plate is assumed much higher.
* Corresponding author at: School of Mechanical Engineering, Qingdao University of Technology, 777 Jialingjiang Road, 266520 Qingdao, PR China. ** Corresponding author. E-mail addresses: [email protected] (B. Liu), [email protected] (D. Chang). http://dx.doi.org/10.1016/j.jsv.2017.08.020 0022-460X/© 2017 Elsevier Ltd. All rights reserved.
210
B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
Though the theory reveals the huge differences of sound transmission properties due to plane wave and spherical wave incidences, in reality the plate is always in finite size and the influences resulted from the plate size and the distance between the source and the plate cannot be ignored. It is therefore interesting to examine how these parameters affect sound transmission through a finite plate due to spherical wave incidence. In this article, the task started from the development of a numerical approach on how to predict the sound transmission of a spherical wave through a finite plate. Then it is followed by numerical analysis and discussion in Section 3. The TL for typical plates is illustrated and the effects of parameters are discussed. Finally, an indicator for the estimation of the TL through the finite plate due to spherical wave incidence is proposed and it might be useful in practical situations. 2. Theory Consider a simply supported, rectangular plate lying in the plane, z ¼ 0, occupying the region 0 x l1 ; 0 y l2 ; within an infinite flat baffle. This plate is excited by spherical wave incidence from a point source, located at ðx0 ;y0 ;z0 Þ, as shown in Fig. 1. The governing equation of the plate displacement is given in Ref. [4]:
! v4 w v4 w v4 w v2 w D þ 2 þ ¼ 2pi 2pr : þ m p vx4 vx2 vy2 vy4 vt 2
(1)
Where w is the normal displacement of the plate, pr is the acoustic pressure radiated by the plate, D ¼ Eh3 =12ð1v2 Þ is the bending stiffness, E the Young’s modulus, v the Poisson ratio, h the thickness of the plate, and mp the surface density of the plate. Using the modal expansion method, the displacement of the plate, the incident and the radiated sound pressure can be expanded as
wðx; yÞ ¼
X
Wmn fmn ðx; yÞ;
(2)
m;n
pr ¼
X
pr mn fmn ðx; yÞ;
(3)
pi mn fmn ðx; yÞ;
(4)
m;n
pi ¼
X m;n
Where fmn ðx; yÞ is the (m,n)th normalized modal shape, given by
2 mpx npy fmn ðx; yÞ ¼ pffiffiffiffiffiffiffiffi sin sin : l1 l2 l1 l2
Substituting Eqs. (2)e(4) into Eq. (1), and taking the loss factor of the plate into consideration, it reads
Fig. 1. Schematic of a spherical sound wave transmission through a rectangular plate.
(5)
B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
i mp h 2 umn 1 þ jhs u2 Wmn ¼ pi mn pr mn ; 2
211
(6)
where hs is the plate material loss factor, umn is the natural frequency of the plate, and pi mn and pr mn are modal forces corresponding to incidence and radiation. The eigenfrequency and the modal forces in Eq. (6) are
u2mn
D ¼ mp
"
2 2 4 # mp 4 mp np np þ2 þ ; l1 l1 l2 l2
(7)
Zl2 Zl1 i
pi fmn ðx; yÞdxdy;
p mn ¼ 0
(8)
0
pr mn ¼ r0 c0
X
Zm0 n0 ;mn Vm0 n0 ;
(9)
m0 n0
where Zm0 n0 ;mn is a dimensionless modal impendance describing the contributions from the modal velocity component, and Vm0 n0 is the modal amplitude of plate velocity. The imaginary part of Zm0 n0 ;mn is equivalent to adding a virtual mass on the surface. The image part can be omitted here, as air is light in comparison with the plate. sm0 n0 ;mn , the real part of Zm0 n0 ;mn , is modal radiation efficiency, and by neglecting the cross parts it can be expressed as smn according to Ref. [10]. smn can be calculated by following formula
smn ¼
64k2 ab
p6 m2 n2
8 p p > Z2 Z2 > < 0
0
92 cos a cos b > > = 2 2 sin sin ih i sinqdqd4; h 2 2 > > > : ða=mpÞ 1 ðb=npÞ 1 > ;
(10)
where a ¼ kl1 sinqcos4, b ¼ kl2 sinqsin4. If m is odd, use cosða=2Þ; otherwise, use sinða=2Þ; if n is odd, use cosðb=2Þ; otherwise, use sinðb=2Þ. Substituting Eqs. (8), (9) into Eq. (6), the modal amplitude of the plate velocity can be expressed as
Vmn ¼ juWmn ¼ Ymn pimn ;
(11)
where Ymn is the modal admittance, given by
Ymn ¼
i1 2juh 2 umn 1 þ jhemn u2 ; mp
(12)
and hemn is the effective loss factor
hemn ¼ hs þ
2r0 c0 smn u : mp u2mn
(13)
The pressure of spherical wave can be written as
pi ¼ iur0 Q
expðikrÞ ; 4pr
(14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Q is the strength of sound source, and r ¼ ðx x0 Þ2 þðy y0 Þ2 þz0 2 is the distance from point source to particle on the plate. The total sound power incidence is given by:
Ji ¼
A A jQ j2 r0 c0 k2 W¼ : 4p 4p 8p
Where W is the total radiant power of the point source. If the point source is above the center of the rectangle plate, the solid angle is:
(15)
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B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
1 1 1 tan b 1 tan b tan sin cos a sin tan B C tan a tan a B C A ¼ 4B C; @ A 1 1 tan a 1 tan a þtan sin cos b sin tan tan b tan b 0
(16)
l1 l2 where d is the distance between the source and the plate, tana ¼ 2d and tanb ¼ 2d . If the source is not above the center the panel, the solid angle cannot be directly calculated according to Eq. (16), the solid angle should be derived from addition or subtraction of centered source results. The total transmitted sound power is:
Jt ¼
1 Re 2
Z A
X 1 pr v dA ¼ r0 c0 smn jYmn j2 jpi mn j2 : 2 mn
(17)
The transmission loss for a spherical wave incidence, TLspherical , can be defined by
J AjQ j2 k2 TLspherical ¼ 10log10 i ¼ : P 2 2 i Jt 16p2 mn smn jYmn j jp mn j
(18)
3. Numerical examples and discussion 3.1. Comparison with mass laws In the calculation, a simply-supported and infinite-baffled plate is taken into account. A point source is assumed above the center of the plate to generate spherical waves. The plate parameters such as Young’s modulus, thickness, density, Poisson’s ratio and loss factor used in the calculation are respectively given by E ¼ 7:1 1010 Pa, h ¼ 2mm, r ¼ 2700kg=m3 , v ¼ 0:3, hs ¼ 0:01. The air density and the sound speed are r0 ¼ 1:21kg=m3 and c0 ¼ 340m=s, respectively. The critical frequency of the plate f0 is about 5900 Hz. The mass law for normal wave incidence is given by Ref. [3]:
"
TLspherical ¼ 10log10 1 þ
mp u 2r0 c0
2 # :
(19)
The mass law for an infinite wall driven by a spherical wave is given in Ref. [9] :
TLspherical ¼ 10log10
mp u 2r0 c0
mp u 10log10 tan1 : 2r0 c0
The above mass laws are plotted to compare with that of the numerical examples in the following figures.
Fig. 2. TL of a spherical wave incidence through a 1m 1m plate.
(20)
B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
213
The predicted TL of a 1m 1m squared plate in the frequency range from 100 Hz to 10,000 Hz is illustrated in Fig. 2. Similar to that of a finite plate due to the diffuse field incidence, a dip at the critical frequency of the plate due to coincidence effect and an increase at very low frequencies due to boundary effects can be clearly observed. Apart from that, it is found that the TL of the plate under spherical wave incidence depends heavily on the distance between the source and the plate. When the distance between the source and the plate increases, more incident wave energy approaches to the normal incidence, therefore the TL is closer to the mass law of the normal incidence. If the effects of the boundary conditions of the plate on the very low frequency are ignored, the TL of the plate is always lying in between two curves given by the mass law due to the spherical wave incidence and the normal wave incidence, respectively. If the size of plate increases and in the high frequency range, the modal numbers used in the calculation will increase rapidly, the calculation time will become very long and then the modal summation approach is not able to handle, therefore frequency range from 100 Hz to 2500 Hz is taken into consideration in the following figures for the rest of plates. This frequency range used in calculation is rational since in most noise transmission problems, the low- and mid- frequency range is more important than high- frequency range. For much bigger plates, lets saying 3m 3m and 5m 5m as it illustrated in Fig. 3, again it is observed that below the critical frequency of the plate, the TL lies between two curves given by the mass law due to the spherical wave incidence and the normal wave incidence, respectively. Meanwhile, the TL slightly decreases in comparison with that of a smaller plate, since a larger plate increases oblique incident waves when the distance of the source is fixed. Similarly to that of a finite plate due to the diffuse-field incidence, boundary effects are weakened as the sizes of panel are enlarged, and in the calculated frequency range from 100 Hz to 2500 Hz, no increase at low frequencies is observed. To evaluate the quantitative influences resulted from the plate size and the distance between the source and the plate, the root-mean-square error (RMSE) is introduced as
Fig. 3. TL of a spherical wave incidence through the plates with the size of (a)3m 3m and (b)5m 5m.
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B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn i¼1 TLnormal;i TLnumerical; i RMSE ¼ ; n
(21)
where TLnormal;i is the transmission loss of the i th 1/3 octave center frequency for normal wave incidence mass law and TLnumerical;i is that for spherical wave incidence through the finite plate, respectively. The calculated RMSE versus the parameter l=d is shown in Fig. 4. Apparently the RMSE increases in proportion to l=d, or alternatively with the increase of the plate size and the reduction of the distance between the plate and the source. Ignoring boundary effects, the RMSE is less than 1.3 when l=d is smaller than 1, and that is to say, the discrepancies between the normal wave incident mass law and the predicted TL due to spherical wave incidence are less than 1.3 dB in average. For a relatively small plate installed in the frame/wall, the TL will be influenced in the low frequency range due to plate size and boundary installing conditions. The details are usually complicated, and we call it “boundary effects” in general. Combining results in Fig. 2, Fig. 3 and Fig. 4, it may be concluded that if l=d < 1, the TL of spherical wave incidence through finite square plates can be estimated by the mass law of the normal wave incidence in practical situation without much deviation. When the distance between the plate and the source is extremely close, an example of the predicted TL is given in Fig. 5. The distance is set as 0:01m in calculation, relatively a trivial gap in comparison with the size of plates. One may expect that this extreme case is similar to that of an infinite plate due to spherical incidence, since the solid angle of the incidence waves approaches to 90 degree as it does for an infinite plate. According to Fig. 5, however, thought the TL drops along with the addition size of the plate as expected, the TL is always above the spherical wave incidence mass law significantly. This result indicates the sound transmission behavior of finite plates due to spherical incidence is substantially different from that of an infinite plate. The mass law for spherical wave incidence through an infinite plate is just an idealized model, and it may lead to errors if it is utilized in the evaluation of a finite plate in practical situation. 3.2. Influence of point source shifting horizontally In practice, for example in an aircraft, the influence of the side-panel positions on the sound transmission could be an interesting problem. The problem might be simplified by investigating a plate in different horizontal positions described in Fig. 6. The plate has the size of 3m 3m and the vertical distance between the plate and the point source is assumed as 3m. While the horizontal positions of the plate relative to the point source are assumed as 0 m, 4m, 8m and 16m, respectively. The results in Fig. 7 indicate that the TL decreases significantly along with the increase of horizontal distance between the point source and the center of the plate. The decrease of the TL doesn’t mean that the transmitted energy would be increased by the horizontal shifting. When the horizontal distance between the source and the plate increases, more oblique of the waves are incident on the plate, and therefore the sound power incident on the plate is smaller. As demonstrated in Fig. 8(a), the incident sound power decreases significantly when the distance between the plate and the source increases and the solid angle deceases. According to Fig. 8(b), the decrease of incident power plays a decisive role on the variations of the TL. 4. Conclusion A model for the sound transmission of a spherical wave through a finite plate is established in this paper. The result indicates the sound transmission behavior of finite plates due to spherical wave incidence is substantially different from that of an infinite plate. Unlike an infinite plate, the TL of a spherical wave through a finite plate depends heavily on the size of the
Fig. 4. RMSE between normal incidence mass law and numerical results.
B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
215
Fig. 5. TL of spherical wave incidence through plates when d ¼ 0:01m.
Fig. 6. Schematic of the horizontal positions of the plate relative to the source.
Fig. 7. TL of a spherical wave incidence through a 3m 3m plate when the horizontal positions of the plate relative to the point source are set as 0m, 4m, 8m and 16m.
plate and the position of the plate relative to the source. If the effects resulted from the boundary conditions of the plate on very low frequencies are ignored, the TL is always lying between two curves given by the mass law due to the spherical wave incidence and the normal wave incidence, respectively. When the distance between the source and the plate increases, the TL is closer to the normal incidence mass law. For the plates investigated, one may conclude that if the length of the plate side is less than the distance between the source and the plate (l=d < 1), the TL of a spherical wave through a finite plate can be estimated by the normal incidence mass law. This conclusion is useful in practical situations when the TL is needed to be estimated due to spherical wave incidence. The horizontal shift of the point source relative to the plate reduces the sound reduction index obviously due to the increase of the oblique wave incidences. However, the shift increases the distance between the source and the plate and therefore decreases the sound power incident on the plate, and the transmitted sound power would be reduced as well.
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B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
Fig. 8. (a)Incident sound power level, (b)transmitted sound power level; when the horizontal positions of the plate relative to the point source are set as 0m, 4m, 8m and 16m.
Acknowledgments The authors gratefully acknowledge the financial support from NSFC Grant 11374326 and State Key Development Program for Basic Research of China (973 Program) Grant 2012CB720204. References [1] A. Nilsson, B. Liu, Vibro-Acoustics, Volume 2, Science Press, Springer-Verlag, Beijing, Berlin Heidelberg, 2016 (Vibro-Acoustics, Volume 2, by ISBN 9783-662-47933-9). [2] A. De Bruijn, Influence of diffusivity on the transmission loss of a single leaf wall, J. Acoust. Soc. Am. 47 (3A) (1970) 667e675. [3] I.L. Ver, L.L. Beranek, Noise and Vibration Control Engineering: Principles and Applications[M], Wiley, 2006. [4] B. Liu, L. Feng, A. Nilsson, Sound transmission through curved aircraft plates with stringer and ring frame attachments, J. Sound Vib. 300 (3) (2007) 949e973. [5] D. Takahashi, Effects of plate boundedness on sound transmission problems, J. Acoust. Soc. Am. 98 (5) (1995) 2598e2606. [6] M. Villot, C. Guigou-Carter, Airborne sound insulation: case of a small airborne sound source close to a wall, in: Proceedings of the 18th International Congress on Acoustics (ICA 2004), Kyoto (Japan), 2004. [7] K. Sakagami, S. Nakanishi, M. Daido, et al., Reflection of a spherical sound wave by an infinite elastic plate driven to vibration by a point force, Appl. Acoust. 55 (4) (1998) 253e273. [8] D. Takahashi, Y. Furue, K. Matsuura, Vibration and sound transmission of an elastic plate by a spherical sound wave, J. Acoust. Soc. Jpn. 35 (1979) 314e321 (in Japanese). [9] M. Yairi, T. Koga, K. Takebayashi, et al., Transmission of a spherical sound wave through a single-leaf wall: mass law for spherical wave incidence, Appl. Acoust. 75 (2014) 67e71. [10] C.E. Wallace, The acoustic radiation damping of the modes of a rectangular plate, J. Acoust. Soc. Am. 81 (6) (1987) 1787e1794.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Sound transmission of a spherical sound wave through a finite plate Bilong Liu a, b, *, Yan Jiang b, Daoqing Chang b, ** a b
School of Mechanical Engineering, Qingdao University of Technology, 777 Jialingjiang Road, 266520 Qingdao, PR China Key Laboratory of Noise and Vibration Research, University of Chinese Academy of Sciences, 100080 Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 March 2017 Received in revised form 3 August 2017 Accepted 16 August 2017
For an incident plane wave on an infinite plate, a doubling of mass or frequency adds 6 dB to the sound transmission loss (TL), but for an incident spherical wave on an infinite plate, a doubling of mass or frequency adds only 3 dB to the TL. In reality, the discrepancies of the sound transmission due to plane wave and spherical wave incidence might not be so huge, since the influences resulted from the plate size and the distance between the source and the plate cannot be ignored. In this article, the sound transmission of a spherical wave through a finite plate is theoretically analyzed through the modal expansion method. The transmission losses for typical plates are illustrated and as well are compared with that of the mass laws due to normal and spherical wave incidence, respectively. The effects of parameters such as the size of the plate, the distance between the source and the plate, and the horizontal shift of the plate are investigated. An indicator for the estimation of the TL through a finite plate due to a point source is given for the potential of practical applications. © 2017 Elsevier Ltd. All rights reserved.
Handling Editor: L. G. Tham Keywords: Sound transmission loss Vibro-acoustics Plate vibration
1. Introduction Sound transmission through a single-leaf plate has been widely reported [1e5]. One of the well-known features of the sound transmission through the plate is the so-called mass law. In the low frequency range or far below the critical frequency of the plate, the mass per unit area of the plate is the only plate parameter determining the sound transmission performance and a doubling of mass or frequency adds 6 dB to the TL. It also should be noted that the TL for a normal incidence on the plate is about 5 dB higher than that for a diffuse field incidence. The mass law is usually derived from plane wave incidence on an infinite plate. Little attention has been paid to the sound transmission due to spherical wave incidence. Motoki Yair et al. [6e9] derived a mass law under the condition that a spherical wave is incident on an infinite plate. By ignoring the distance between the source and the plate due to the infinite assumption of the plate, the spherical wave incidence mass law can be described as a 3 dB increase provided that the mass per unit area or frequency is doubled. Apart from this apparent mass law difference in compare with that of normal incidence, the sound transmission loss for spherical wave incidence on the infinite plate is about 20 dB less than that due to normal wave incidence at 1000 Hz,where the critical frequency of the infinite plate is assumed much higher.
* Corresponding author at: School of Mechanical Engineering, Qingdao University of Technology, 777 Jialingjiang Road, 266520 Qingdao, PR China. ** Corresponding author. E-mail addresses: [email protected] (B. Liu), [email protected] (D. Chang). http://dx.doi.org/10.1016/j.jsv.2017.08.020 0022-460X/© 2017 Elsevier Ltd. All rights reserved.
210
B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
Though the theory reveals the huge differences of sound transmission properties due to plane wave and spherical wave incidences, in reality the plate is always in finite size and the influences resulted from the plate size and the distance between the source and the plate cannot be ignored. It is therefore interesting to examine how these parameters affect sound transmission through a finite plate due to spherical wave incidence. In this article, the task started from the development of a numerical approach on how to predict the sound transmission of a spherical wave through a finite plate. Then it is followed by numerical analysis and discussion in Section 3. The TL for typical plates is illustrated and the effects of parameters are discussed. Finally, an indicator for the estimation of the TL through the finite plate due to spherical wave incidence is proposed and it might be useful in practical situations. 2. Theory Consider a simply supported, rectangular plate lying in the plane, z ¼ 0, occupying the region 0 x l1 ; 0 y l2 ; within an infinite flat baffle. This plate is excited by spherical wave incidence from a point source, located at ðx0 ;y0 ;z0 Þ, as shown in Fig. 1. The governing equation of the plate displacement is given in Ref. [4]:
! v4 w v4 w v4 w v2 w D þ 2 þ ¼ 2pi 2pr : þ m p vx4 vx2 vy2 vy4 vt 2
(1)
Where w is the normal displacement of the plate, pr is the acoustic pressure radiated by the plate, D ¼ Eh3 =12ð1v2 Þ is the bending stiffness, E the Young’s modulus, v the Poisson ratio, h the thickness of the plate, and mp the surface density of the plate. Using the modal expansion method, the displacement of the plate, the incident and the radiated sound pressure can be expanded as
wðx; yÞ ¼
X
Wmn fmn ðx; yÞ;
(2)
m;n
pr ¼
X
pr mn fmn ðx; yÞ;
(3)
pi mn fmn ðx; yÞ;
(4)
m;n
pi ¼
X m;n
Where fmn ðx; yÞ is the (m,n)th normalized modal shape, given by
2 mpx npy fmn ðx; yÞ ¼ pffiffiffiffiffiffiffiffi sin sin : l1 l2 l1 l2
Substituting Eqs. (2)e(4) into Eq. (1), and taking the loss factor of the plate into consideration, it reads
Fig. 1. Schematic of a spherical sound wave transmission through a rectangular plate.
(5)
B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
i mp h 2 umn 1 þ jhs u2 Wmn ¼ pi mn pr mn ; 2
211
(6)
where hs is the plate material loss factor, umn is the natural frequency of the plate, and pi mn and pr mn are modal forces corresponding to incidence and radiation. The eigenfrequency and the modal forces in Eq. (6) are
u2mn
D ¼ mp
"
2 2 4 # mp 4 mp np np þ2 þ ; l1 l1 l2 l2
(7)
Zl2 Zl1 i
pi fmn ðx; yÞdxdy;
p mn ¼ 0
(8)
0
pr mn ¼ r0 c0
X
Zm0 n0 ;mn Vm0 n0 ;
(9)
m0 n0
where Zm0 n0 ;mn is a dimensionless modal impendance describing the contributions from the modal velocity component, and Vm0 n0 is the modal amplitude of plate velocity. The imaginary part of Zm0 n0 ;mn is equivalent to adding a virtual mass on the surface. The image part can be omitted here, as air is light in comparison with the plate. sm0 n0 ;mn , the real part of Zm0 n0 ;mn , is modal radiation efficiency, and by neglecting the cross parts it can be expressed as smn according to Ref. [10]. smn can be calculated by following formula
smn ¼
64k2 ab
p6 m2 n2
8 p p > Z2 Z2 > < 0
0
92 cos a cos b > > = 2 2 sin sin ih i sinqdqd4; h 2 2 > > > : ða=mpÞ 1 ðb=npÞ 1 > ;
(10)
where a ¼ kl1 sinqcos4, b ¼ kl2 sinqsin4. If m is odd, use cosða=2Þ; otherwise, use sinða=2Þ; if n is odd, use cosðb=2Þ; otherwise, use sinðb=2Þ. Substituting Eqs. (8), (9) into Eq. (6), the modal amplitude of the plate velocity can be expressed as
Vmn ¼ juWmn ¼ Ymn pimn ;
(11)
where Ymn is the modal admittance, given by
Ymn ¼
i1 2juh 2 umn 1 þ jhemn u2 ; mp
(12)
and hemn is the effective loss factor
hemn ¼ hs þ
2r0 c0 smn u : mp u2mn
(13)
The pressure of spherical wave can be written as
pi ¼ iur0 Q
expðikrÞ ; 4pr
(14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Q is the strength of sound source, and r ¼ ðx x0 Þ2 þðy y0 Þ2 þz0 2 is the distance from point source to particle on the plate. The total sound power incidence is given by:
Ji ¼
A A jQ j2 r0 c0 k2 W¼ : 4p 4p 8p
Where W is the total radiant power of the point source. If the point source is above the center of the rectangle plate, the solid angle is:
(15)
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B. Liu et al. / Journal of Sound and Vibration 410 (2017) 209e216
1 1 1 tan b 1 tan b tan sin cos a sin tan B C tan a tan a B C A ¼ 4B C; @ A 1 1 tan a 1 tan a þtan sin cos b sin tan tan b tan b 0
(16)
l1 l2 where d is the distance between the source and the plate, tana ¼ 2d and tanb ¼ 2d . If the source is not above the center the panel, the solid angle cannot be directly calculated according to Eq. (16), the solid angle should be derived from addition or subtraction of centered source results. The total transmitted sound power is:
Jt ¼
1 Re 2
Z A
X 1 pr v dA ¼ r0 c0 smn jYmn j2 jpi mn j2 : 2 mn
(17)
The transmission loss for a spherical wave incidence, TLspherical , can be defined by
J AjQ j2 k2 TLspherical ¼ 10log10 i ¼ : P 2 2 i Jt 16p2 mn smn jYmn j jp mn j
(18)
3. Numerical examples and discussion 3.1. Comparison with mass laws In the calculation, a simply-supported and infinite-baffled plate is taken into account. A point source is assumed above the center of the plate to generate spherical waves. The plate parameters such as Young’s modulus, thickness, density, Poisson’s ratio and loss factor used in the calculation are respectively given by E ¼ 7:1 1010 Pa, h ¼ 2mm, r ¼ 2700kg=m3 , v ¼ 0:3, hs ¼ 0:01. The air density and the sound speed are r0 ¼ 1:21kg=m3 and c0 ¼ 340m=s, respectively. The critical frequency of the plate f0 is about 5900 Hz. The mass law for normal wave incidence is given by Ref. [3]:
"
TLspherical ¼ 10log10 1 þ
mp u 2r0 c0
2 # :
(19)
The mass law for an infinite wall driven by a spherical wave is given in Ref. [9] :
TLspherical ¼ 10log10
mp u 2r0 c0
mp u 10log10 tan1 : 2r0 c0
The above mass laws are plotted to compare with that of the numerical examples in the following figures.
Fig. 2. TL of a spherical wave incidence through a 1m 1m plate.
(20)
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The predicted TL of a 1m 1m squared plate in the frequency range from 100 Hz to 10,000 Hz is illustrated in Fig. 2. Similar to that of a finite plate due to the diffuse field incidence, a dip at the critical frequency of the plate due to coincidence effect and an increase at very low frequencies due to boundary effects can be clearly observed. Apart from that, it is found that the TL of the plate under spherical wave incidence depends heavily on the distance between the source and the plate. When the distance between the source and the plate increases, more incident wave energy approaches to the normal incidence, therefore the TL is closer to the mass law of the normal incidence. If the effects of the boundary conditions of the plate on the very low frequency are ignored, the TL of the plate is always lying in between two curves given by the mass law due to the spherical wave incidence and the normal wave incidence, respectively. If the size of plate increases and in the high frequency range, the modal numbers used in the calculation will increase rapidly, the calculation time will become very long and then the modal summation approach is not able to handle, therefore frequency range from 100 Hz to 2500 Hz is taken into consideration in the following figures for the rest of plates. This frequency range used in calculation is rational since in most noise transmission problems, the low- and mid- frequency range is more important than high- frequency range. For much bigger plates, lets saying 3m 3m and 5m 5m as it illustrated in Fig. 3, again it is observed that below the critical frequency of the plate, the TL lies between two curves given by the mass law due to the spherical wave incidence and the normal wave incidence, respectively. Meanwhile, the TL slightly decreases in comparison with that of a smaller plate, since a larger plate increases oblique incident waves when the distance of the source is fixed. Similarly to that of a finite plate due to the diffuse-field incidence, boundary effects are weakened as the sizes of panel are enlarged, and in the calculated frequency range from 100 Hz to 2500 Hz, no increase at low frequencies is observed. To evaluate the quantitative influences resulted from the plate size and the distance between the source and the plate, the root-mean-square error (RMSE) is introduced as
Fig. 3. TL of a spherical wave incidence through the plates with the size of (a)3m 3m and (b)5m 5m.
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn i¼1 TLnormal;i TLnumerical; i RMSE ¼ ; n
(21)
where TLnormal;i is the transmission loss of the i th 1/3 octave center frequency for normal wave incidence mass law and TLnumerical;i is that for spherical wave incidence through the finite plate, respectively. The calculated RMSE versus the parameter l=d is shown in Fig. 4. Apparently the RMSE increases in proportion to l=d, or alternatively with the increase of the plate size and the reduction of the distance between the plate and the source. Ignoring boundary effects, the RMSE is less than 1.3 when l=d is smaller than 1, and that is to say, the discrepancies between the normal wave incident mass law and the predicted TL due to spherical wave incidence are less than 1.3 dB in average. For a relatively small plate installed in the frame/wall, the TL will be influenced in the low frequency range due to plate size and boundary installing conditions. The details are usually complicated, and we call it “boundary effects” in general. Combining results in Fig. 2, Fig. 3 and Fig. 4, it may be concluded that if l=d < 1, the TL of spherical wave incidence through finite square plates can be estimated by the mass law of the normal wave incidence in practical situation without much deviation. When the distance between the plate and the source is extremely close, an example of the predicted TL is given in Fig. 5. The distance is set as 0:01m in calculation, relatively a trivial gap in comparison with the size of plates. One may expect that this extreme case is similar to that of an infinite plate due to spherical incidence, since the solid angle of the incidence waves approaches to 90 degree as it does for an infinite plate. According to Fig. 5, however, thought the TL drops along with the addition size of the plate as expected, the TL is always above the spherical wave incidence mass law significantly. This result indicates the sound transmission behavior of finite plates due to spherical incidence is substantially different from that of an infinite plate. The mass law for spherical wave incidence through an infinite plate is just an idealized model, and it may lead to errors if it is utilized in the evaluation of a finite plate in practical situation. 3.2. Influence of point source shifting horizontally In practice, for example in an aircraft, the influence of the side-panel positions on the sound transmission could be an interesting problem. The problem might be simplified by investigating a plate in different horizontal positions described in Fig. 6. The plate has the size of 3m 3m and the vertical distance between the plate and the point source is assumed as 3m. While the horizontal positions of the plate relative to the point source are assumed as 0 m, 4m, 8m and 16m, respectively. The results in Fig. 7 indicate that the TL decreases significantly along with the increase of horizontal distance between the point source and the center of the plate. The decrease of the TL doesn’t mean that the transmitted energy would be increased by the horizontal shifting. When the horizontal distance between the source and the plate increases, more oblique of the waves are incident on the plate, and therefore the sound power incident on the plate is smaller. As demonstrated in Fig. 8(a), the incident sound power decreases significantly when the distance between the plate and the source increases and the solid angle deceases. According to Fig. 8(b), the decrease of incident power plays a decisive role on the variations of the TL. 4. Conclusion A model for the sound transmission of a spherical wave through a finite plate is established in this paper. The result indicates the sound transmission behavior of finite plates due to spherical wave incidence is substantially different from that of an infinite plate. Unlike an infinite plate, the TL of a spherical wave through a finite plate depends heavily on the size of the
Fig. 4. RMSE between normal incidence mass law and numerical results.
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Fig. 5. TL of spherical wave incidence through plates when d ¼ 0:01m.
Fig. 6. Schematic of the horizontal positions of the plate relative to the source.
Fig. 7. TL of a spherical wave incidence through a 3m 3m plate when the horizontal positions of the plate relative to the point source are set as 0m, 4m, 8m and 16m.
plate and the position of the plate relative to the source. If the effects resulted from the boundary conditions of the plate on very low frequencies are ignored, the TL is always lying between two curves given by the mass law due to the spherical wave incidence and the normal wave incidence, respectively. When the distance between the source and the plate increases, the TL is closer to the normal incidence mass law. For the plates investigated, one may conclude that if the length of the plate side is less than the distance between the source and the plate (l=d < 1), the TL of a spherical wave through a finite plate can be estimated by the normal incidence mass law. This conclusion is useful in practical situations when the TL is needed to be estimated due to spherical wave incidence. The horizontal shift of the point source relative to the plate reduces the sound reduction index obviously due to the increase of the oblique wave incidences. However, the shift increases the distance between the source and the plate and therefore decreases the sound power incident on the plate, and the transmitted sound power would be reduced as well.
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Fig. 8. (a)Incident sound power level, (b)transmitted sound power level; when the horizontal positions of the plate relative to the point source are set as 0m, 4m, 8m and 16m.
Acknowledgments The authors gratefully acknowledge the financial support from NSFC Grant 11374326 and State Key Development Program for Basic Research of China (973 Program) Grant 2012CB720204. References [1] A. Nilsson, B. Liu, Vibro-Acoustics, Volume 2, Science Press, Springer-Verlag, Beijing, Berlin Heidelberg, 2016 (Vibro-Acoustics, Volume 2, by ISBN 9783-662-47933-9). [2] A. De Bruijn, Influence of diffusivity on the transmission loss of a single leaf wall, J. Acoust. Soc. Am. 47 (3A) (1970) 667e675. [3] I.L. Ver, L.L. Beranek, Noise and Vibration Control Engineering: Principles and Applications[M], Wiley, 2006. [4] B. Liu, L. Feng, A. Nilsson, Sound transmission through curved aircraft plates with stringer and ring frame attachments, J. Sound Vib. 300 (3) (2007) 949e973. [5] D. Takahashi, Effects of plate boundedness on sound transmission problems, J. Acoust. Soc. Am. 98 (5) (1995) 2598e2606. [6] M. Villot, C. Guigou-Carter, Airborne sound insulation: case of a small airborne sound source close to a wall, in: Proceedings of the 18th International Congress on Acoustics (ICA 2004), Kyoto (Japan), 2004. [7] K. Sakagami, S. Nakanishi, M. Daido, et al., Reflection of a spherical sound wave by an infinite elastic plate driven to vibration by a point force, Appl. Acoust. 55 (4) (1998) 253e273. [8] D. Takahashi, Y. Furue, K. Matsuura, Vibration and sound transmission of an elastic plate by a spherical sound wave, J. Acoust. Soc. Jpn. 35 (1979) 314e321 (in Japanese). [9] M. Yairi, T. Koga, K. Takebayashi, et al., Transmission of a spherical sound wave through a single-leaf wall: mass law for spherical wave incidence, Appl. Acoust. 75 (2014) 67e71. [10] C.E. Wallace, The acoustic radiation damping of the modes of a rectangular plate, J. Acoust. Soc. Am. 81 (6) (1987) 1787e1794.