Analysisofairbornesoundinsulationandimpactsoundpressurelevelprovidedbyasinglepartitioncontainingaheterogeneity ARTICLEINPRESS

JOURNAL OF SOUND AND VIBRATION

Journal of Sound and Vibration 300 (2007) 800–816

Analysis of airborne sound insulation and impact sound pressure level provided by a single partition containing a heterogeneity

A. Pereira, A. Tadeu

Department of Civil Engineering, University of Coimbra, Pinhal de Marrocos, 3030-290 Coimbra, Portugal Received 19 December 2005; received in revised form 7 July 2006; accepted 27 August 2006

Available online 13 November 2006

Abstract

This paper studies the acoustic behaviour provided by a single infinite elastic partition dividing an infinite acoustic medium, containing two-dimensional (2D) heterogeneities placed parallel to the layer’s surfaces, which simulate the presence of fittings such as pipes. The problem is solved in the frequency domain using the boundary element method (BEM). Only the surfaces of the heterogeneities are discretized, since 2.5D Green’s functions for the single layered media, bounded by acoustic media, are used. Time domain responses are also computed by applying a (fast) Fourier transform to the responses obtained in the frequency domain. The heterogeneities are assumed to be either rigid, free, fluid-filled or elastic-filled inclusions. The simulated models are used to study the contribution of the heterogeneities to the final airborne sound insulation and impact sound pressure level provided by a single partition. It was found that both airborne sound insulation and impact sound pressure level may be influenced by the presence of the heterogeneity at higher frequencies.

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1. Introduction

In order to accurately predict the transmission of sound through a single separating partition a large number of variables needs to be considered. These variables include: the physical properties of the panel (mass, internal damping, elasticity modulus, Poisson’s ratio); the finiteness of the element; the mounting conditions, and the non-diffuseness of the test room. The simulation of all the acoustic phenomena involved in the sound transmission would lead to highly complex mathematical models, and so quite a number of works found in the literature adopt simplifications to deal with the problem. Among these is the well-known theoretical Mass Law, where the element is assumed to behave like a group of infinite juxtaposed masses with independent displacement and null damping forces. Sewell[1]and Sharp[2]have proposed simplified models for the frequencies below, in the vicinity of and above the coincidence effect to calculate the airborne sound insulation of single panels, improving the prediction accuracy relative to Mass Law. More recently, Tadeu et al.[3,4]developed an algorithm for predicting the airborne sound insulation provided by a single partition based on the definitions of pressure and displacement potentials, which are combined so as to verify the

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doi:10.1016/j.jsv.2006.08.039

Corresponding author. Department of Civil Engineering, Polo II of the University of Coimbra, Rua Luı´s Reis Santos, 3030-788 Coimbra, Portugal. Tel.: +351 239 797 201; fax: +351 239 797 190.

E-mail address:[email protected] (A. Tadeu).

boundary conditions at the fluid/solid interfaces. Jong-Hwa et al. [5] revisit the problem of resonant transmission related to the sound insulation of rectangular finite panels in an infinite baffle at frequencies below the critical frequency by using the general modal expansion method followed by Sewell. They investigated the validity of neglecting the resonant transmission components in the prediction of transmission loss by calculating the differences between the total transmission loss and the non-resonant transmission loss.

Alba et al.[6]revised the expression for an infinite impervious layer in a diffuse field and adjusted it in order to include the energy loss mechanisms (internal and coupling losses) produced in the layer. These authors further developed algorithms which can be used to obtain the mechanical properties of the layer with the appropriate set-up conditions, if databases are available.

Knowledge of the impact sound pressure level expected from partitions is also important at the design stage.

Although the final responses provided by the loads that act in an acoustic or in an elastic medium differ, the dynamic behaviour of partitions may be similar. Ver [7] determined a relation between airborne sound insulation and impact sound pressure level provided by partitions. Gerretsen[8]initially proposed a model to predict airborne sound insulation including flanking transmission. His model requires knowing the direct sound reduction index provided by a partition and takes into account the boundary conditions by means of a structural reverberation time and the vibration level differences across junctions, which are calculated from in situ measurements. Since a major part of the model concerns vibration transmission and radiation, he later extended it to predict impact sound pressure level [9].

Most of the works found in the literature refer to the prediction of homogeneous partitions. However, the acoustic behaviour of real partitions may be influenced by the presence of heterogeneities, which are often present inside. These may be other building elements such as heat, sewerage, drainage or water pressure pipes or pipes that host electrical cables and wires. These are not taken into account during design, and when the performance of walls is tested the results may not match the predictions. Therefore it seems important to see if the presence of those elements can influence the sound insulation of partitions. This work aims to contribute to the analysis of this problem by developing and applying a boundary element method (BEM) model to assess the acoustic behaviour of single partitions infinite in two directions (infinite plates) dividing an infinite acoustic medium. The influence of heterogeneities in the elastic partition on the acoustic behaviour is studied by determining airborne sound insulation and impact sound pressure levels. Results for a single layer solution without discontinuities are used as a reference. The algorithm uses a Boundary Element formulation where only the heterogeneity needs to be discretized, since Green’s functions for single layered media are used. These heterogeneities may be rigid, free, fluid-filled or elastic-filled inclusions. The Green’s functions’ solutions are the analytical solutions developed by Tadeu et al. [3,4]to predict airborne sound insulation provided by a single panel. The computations are performed in the frequency domain, assuming that harmonic line loads act either in the fluid or in the elastic medium. Time domain responses are then obtained by applying an Inverse Fourier Transform. Wave propagation features occurring in a single layered medium with a heterogeneity are investigated by analysing the responses provided by different sets of receivers placed in the acoustic medium.

The influence of the direction of the load on impact sound pressure level is studied. Different positions and sizes of the heterogeneity are modelled and the resulting sound level is analysed.

The next section outlines the problem formulation, indicating the Green’s functions for a single layer medium, and the BEM. Then the procedure used to calculate time domain responses is summarized. The simulations are described next, and finally the results are discussed.

2. Problem formulation

Consider an elastic layer of thickness h, infinite along its plane (x and zdirections), dividing an infinite acoustic medium, as inFig. 1. The acoustic medium has a mass densityrf, a Lame´ constantlfand permits a dilatational wave velocity a_f ¼ ffiffiffiffiffiffiffiffiffiffiffiffi

l_f=r_f

q . The material properties of the elastic medium are the density rs, Poisson’s ratio nsand a shear modulus ms. In this medium the propagation occurs following compressional waves with a velocitya_s¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2m_sð1n_sÞ=r_sð12n_sÞ

p and shear waves with a velocityb_s¼ ffiffiffiffiffiffiffiffiffiffiffi m_s=r_s

p . The internal material losses are considered by using a complex shear modulus and a complex Lame´ constant. The shear

A. Pereira, A. Tadeu / Journal of Sound and Vibration 300 (2007) 800–816 801

modulus is computed asm¼m_rð1þiZÞ, wheremrcorresponds to the classic modulus andZis the loss factor.

The Lame´ constant is written in the same form as the shear modulus.

In this paper the influence of a heterogeneity in the elastic partition on sound pressure level is studied by inserting a cylindrical circular inclusion with radius R, infinite along the zdirection, into the elastic layer.

When the system is excited by a point load oscillating with a frequencyoand acting in the acoustic medium at (x0,y0,z0), the incident pressure field at a point (x, y,z) can be obtained by the following expression:

s^full_3Dðo;x;y;zÞ ¼Aeⁱ

ao_f a_ft

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xx₀

ð Þ^2þð^yy0Þ^2þð^zz0Þ²

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xx₀

ð Þ²þyy₀2

þðzz₀Þ²

q , (1)

in whichAis the wave amplitude and i¼ ffiffiffiffiffiffiffi p1

. As the geometry of the model is constant along thezdirection and the source is three-dimensional (3D), the incident field can also be obtained by calculating a series of 2D problems and applying a Fourier transformation in thezdirection. With this procedure, the responses are obtained in the frequency–wavenumber domain for varying effective wavenumbers, ka_f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o=af

2

k²_z q

, with Im ka_f

p0 and kzbeing the axial wavenumber. In this kz domain, the system is excited by spatially sinusoidal harmonic line loads acting at (x₀,y₀) whose pressure field at a point (x,y) is given by

s^fullðo;x;y;k_zÞ ¼iA

2 H^{ð Þ}₀² k_af

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xx₀

ð Þ²þyy₀2

q

e^ikzz. (2)

The former 3D pressure field is then calculated as a discrete summation of 2D problems obtained using expression (2), by applying an inverse Fourier transformation, and assuming the existence of an infinite number of sources placed along the z direction at equal intervals, L. This equation converges and can be approximated by a finite sum of terms. The distance L needs to be large enough to avoid spatial contamination.

The same procedure can be applied to point loads acting within the solid medium. Each 2D incident field produced by a spatially sinusoidal harmonic line load acting at a point (xs,ys) of the elastic medium can be expressed by the displacementsGi,jfull

(where the index i¼x, y, zdefines the direction in which the load is acting, while the second index, j¼x, y, z, indicates the direction of the displacement) at a point (x, y) according to the following expressions[10]:

G^full_ij ðo;x;y;kzÞ ¼A k²_sH0b1

rB1þg²_iB2

; with i¼j¼x;y, (3)

(xs;ys;zs) αs

y (x0;y0;z0)

ρβ_ss

αf

ρ_f

R Fluid Elastic

Fluid

x z

αf

ρ_f

Fig. 1. Geometry of the problem.

A. Pereira, A. Tadeu / Journal of Sound and Vibration 300 (2007) 800–816 802

G^full_zz ðo;x;y;k_zÞ ¼A k ²_sH_0bk²_zB₀ , G^full_xyðo;x;y;kzÞ ¼G^full_yxðo;x;y;kzÞ ¼g_xg_yAB2,

G^full_iz ðo;x;y;kzÞ ¼G^full_zj ðo;x;y;kzÞ ¼ikzg_iAB1 with i¼j¼x;y,

where A¼1=4ir_so²; g_x¼ ðxx_sÞ=r; g_y¼ ðyy_sÞ=r; B_n¼kⁿ_bH_nbkⁿ_aH_na; H_na¼H^ð2Þ_n k_asr

and H_nb¼ H^ð2Þ_n kb_sr

are Hankel functions of the second kind and order n; ks¼ ffiffiffiffiffiffiffiffiffiffiffi o=b_s

p ; r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xxs

ð Þ²þyy_s2

q

; k_as¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o=a_s

2

k²_z q

with Imk_as

p0;k_bs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o=b_s

2

k²_z q

with Imk_bs

p0 and k_z¼ ð2p=LÞn.

In the same way, the scattered field produced by point loads with a heterogeneity inserted into the single layer can be evaluated by solving a sequence of 2D problems, with varying values ofkz.

2.1. Green’s functions for a single layered medium

This section briefly describes the procedure used to obtain the 2.5D Green’s functions for a single homogeneous elastic layer bounded by two fluid media, when excited by harmonic line loads with differentkz

values. These solutions have already been derived by Tadeu et al.[3]and can be expressed as the sum of the source terms equal to those in full space (which can be calculated according to above-defined expressions, (2) and (3)) and surface terms generated by the fluid/solid interfaces (interfaces a and b, as in Fig. 2). The calculation of the surface terms requires knowledge of the solid layer displacement potentials and the pressure potentials generated by the solid/fluid surfaces. These potentials are written as a superposition of plane waves by means of a discrete wavenumber representation (after applying a Fourier transform in thexdirection). The integrals of the expressions are transformed into a summation by considering an infinite number of virtual plane sources distributed along thexdirection at equal intervals,Lx. In the fluid medium, pressure potentials

Fig. 2. Definition of potentials, stresses and displacements at the interfaces.

Table 1

Potentials generated at the interfaces when the load acts in the elastic medium along thexdirection

Interfacea Interfaceb

Elastic medium

f^x;a¼E_a^n¼þNP

n¼N kn nnE^a_bA^x_n

E_d f^x;b¼E_a^n¼þNP

n¼N kn nnE^b_bE^x_n

E_d

c^x;a_x ¼0 c^x;b_x ¼0

c^x;a_y ¼Eakz^n¼þNP

n¼N E^a_c g_nB^x_n

Ed c^x;b_y ¼Eakz^n¼þNP

n¼N E^b_c g_nF^x_n

Ed

c^x;a_z ¼ Ea^n¼þNP

n¼N

E^a_cC^x_n

Ed c^x;b_z ¼Ea^n¼þNP

n¼N

E^b_cG^x_n

Ed

Fluid medium

f^x;a_f ¼ _Lⁱ

x

P

n¼þN n¼N

a²_f o²l_f

_Ea f n^f_nD^x_n

Ed f^x;b_f ¼ ⁱ

Lx

P

n¼þN n¼N

a²_f o²l_f

E^b_f n^f_nH^x_n

Ed

A. Pereira, A. Tadeu / Journal of Sound and Vibration 300 (2007) 800–816 803

(f^i;a_f ; f^i;b_f ) are defined at the interfaces, whereas in the elastic medium, the wave field is expressed by means of pressure potentials (f^i;a; f^i;b) and shear potentials (c^i;a_x; c^i;a_y ; c^i;a_z ;c^i;b_x ; c^i;b_y ; c^i;b_z ) with i¼x;y;z;f corre- sponding to the loads applied (as listed inTables 1–4).

In the expressions listed in Tables 1–4, the coefficients correspond to: E^a_f ¼e^infnjyj; E^b_f ¼e^infnjyhj; n^f_n¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k²_p

f k²_zk²_n

q with Im n^f_n p0; E^a_b¼e^injyjn ; E^b_b¼e^injnyhj; E^a_c¼e^{igj jny}; E^b_c¼e^igjnyhj; g_n¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k²_sk²_zk²_n q

Table 2

Potentials generated at the interfaces when the load acts in the elastic medium along theydirection

Interfacea Interfaceb

Elastic medium

f^y;a¼Ea^n¼þNP

n¼N

E^a_bA^y_n

Ed f^y;b¼ Ea^n¼þNP

n¼N

E^b_bE^y_n

Ed

c^y;a_x ¼Eakz^n¼þNP

n¼N E^a_c

g_n C^y_n

Ed c^y;b_x ¼Eakz^n¼þNP

n¼N E^b_c

g_n G^y_n

Ed

c^y;a_y ¼0 c^y;b_y ¼0

c^y;a_z ¼Ea^n¼þNP

n¼N kn g_nE^a_cB^y_n

Ed c^y;b_z ¼Ea^n¼þNP

n¼N kn g_nE^b_cF^y_n

Ed

Fluid medium

f^y;a_f ¼ _Lⁱ

x

P

n¼þN n¼N

a²_f o²l_f

E^a_f n^f_nD^y_n

Ed f^y;b_f ¼ _Lⁱ_x^n¼þNP

n¼N a²_f o²l_f

_Eb f n^f_nH^y_n

Ed

Table 3

Potentials generated at the interfaces when the load acts in the elastic medium along thezdirection

Interfacea Interfaceb

Elastic medium

f^z;a¼Eakz^n¼þNP

n¼N E^a_b nnA^z_n

Ed f^z;b¼Eakz^n¼þNP

n¼N E^b_b

nnE^z_n

Ed

c^z;a_x ¼Ea^n¼þNP

n¼N

E^a_cB^z_n

Ed c^z;b_x ¼ Ea^n¼þNP

n¼N

E^b_cF^z_n

Ed

c^z;a_y ¼Ea^n¼þNP

n¼N kn

g_n E^a_cC^z_n

Ed c^z;b_y ¼Ea^n¼þNP

n¼N kn

g_n E^b_cG^z_n

Ed

c^z;a_z ¼0 c^z;b_z ¼0

Fluid medium

f^z;a_f ¼ _Lⁱ_x^n¼þNP

n¼N a²_f o²l_f

_Ea f n^fn

D^z_n

Ed f^z;b_f ¼ _Lⁱ

x

P

n¼þN n¼N

a²_f o²l_f

_Eb f n^f_nH^z_n

E_d

Table 4

Potentials generated at the interfaces when the load acts in the fluid medium

Interfacea Interfaceb

Elastic medium

f^f;a¼Ea^n¼þNP

n¼N

E^a_bA^f_n

Ed f^f;b¼ Ea^n¼þNP

n¼N

E^b_bE^f_n

Ed

c^f;a_x ¼Eakz^n¼þNP

n¼N E^a_c

g_n C^f_n

Ed c^f;b_x ¼Eakz^n¼þNP

n¼N E^b_c

g_n G^f_n

Ed

c^f;a_y ¼0 c^f;b_y ¼0

c^f;a_z ¼Ea^n¼þNP

n¼N kn g_nE^a_cB^f_n

Ed c^f;b_z ¼Ea^n¼þNP

n¼N kn g_nE^b_cF^f_n

Ed

Fluid medium

f^f;a_f ¼ _Lⁱ

x

P

n¼þN n¼N

a²_f o²l_f

_Ea f n^f_nD^f_n

Ed f^f;b_f ¼ _Lⁱ_x^n¼þNP

n¼N a²_f o²l_f

_Eb f n^f_nH^f_n

Ed

A. Pereira, A. Tadeu / Journal of Sound and Vibration 300 (2007) 800–816 804

with Im g_n p0;nn¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k²_pk²_zk²_n q

with Imð Þnn p0;Ea¼1=2r_so²Lx;Ed ¼e^iknðxx0Þ;kn¼ ð2p=LxÞn;kp_f ¼ o=af andkp ¼o=as. The coefficientsAⁱ_n; Bⁱ_n; Cⁱ_n; Dⁱ_n; Eⁱ_n; Fⁱ_n; Gⁱ_n; andHⁱ_n with i¼x,y, z, fare unknowns which are determined by deriving the above-defined potentials in order to calculate stresses and displacements, and then establishing the appropriate boundary conditions: continuity of normal displace- ments (u^i;a_y ¼u^i;a_y;f; u^i;b_y ¼u^i;b_y;f) and stresses (s^i;a_yy ¼s^i;a; s^i;b_yy¼s^i;b) and null tangential stresses (s^i;a_yx¼0; s^i;a_yz ¼0; s^i;b_yx¼0; s^i;b_yz ¼0) at the interfaces (seeFig. 2).

Once the unknown coefficients have been calculated, the displacements and stresses associated with the surface terms can be determined by applying partial derivatives to the potentials defined inTables 1–4. The Green’s functions for the solid/fluid formation are then obtained from the sum of the source terms and the surface terms. When this has been done, expressions for displacementsG^surf_i;j , in the elastic medium are given, as follows:

Load acting in the elastic medium in thex direction:

G^surf_xx ¼G^full_xx þEa

X

n¼þN

n¼N

A^x_nik²_n nn

E^a_bþ ig_nC^x_nik²_z g_n B^x_n

E^a_c

EdþEa

X

n¼þN

n¼N

E^x_nik²_n nn

E^b_bþ ig_nG^x_nik²_z g_nF^x_n

E^b_c

Ed,

G^surf_xy ¼G^full_xy þEa

X

n¼þN

n¼N

iknA^x_nE^a_bþiknC^x_nE^a_c

EdþEa

X

n¼þN

n¼N

iknE^x_nE^b_biknG^x_nE^b_c

Ed,

G^surf_xz ¼G^full_xz þEa

X

n¼þN

n¼N

ikzkn

nn

A^x_nE^a_bþikzkn

g_n B^x_nE^a_c

EdþEa

X

n¼þN

n¼N

ikzkn

nn

E^x_nE^b_bþikzkn

g_n F^x_nE^b_c

Ed. ð4Þ

Load acting in the elastic medium in they direction:

G^surf_yx ¼G^full_yx þEa

X

n¼þN

n¼N

iA^y_nknE^a_bþiB^y_nknE^a_c

EdþEa

X

n¼þN

n¼N

iE^y_nknE^b_bþiF^y_nknE^b_c

Ed,

G^surf_yy ¼G^full_yy þEa

X

n¼þN

n¼N

innA^y_nE^a_bþ ik²_n

g_n B^y_nþik²_z g_n C^y_n

E^a_c

EdþEa

X

n¼þN

n¼N

innE^y_nE^b_bþ ik²_n

g_n F^y_nþik²_z g_n G^y_n

E^b_c

Ed,

G^surf_yz ¼G^full_yz þEa

X

n¼þN n¼N

iA^y_nkzE^a_bþiC^y_nkzE^a_c

EdþEa

X

n¼þN n¼N

iE^y_nkzE^b_bþiG^y_nkzE^b_c

Ed. ð5Þ

Load acting in the elastic medium in thezdirection:

G^surf_zx ¼G^full_zx þEa

X

n¼þN

n¼N

ik_zk_n nn

A^z_nE^a_bþik_zk_n g_n C^z_nE^a_c

EdþEa

X

n¼þN

n¼N

ik_zk_n nn

E^z_nE^b_bþik_zk_n g_n G^z_nE^b_c

Ed, G^surf_zy ¼G^full_zy þEa

X

n¼þN

n¼N

ikzA^z_nE^a_bþiB^z_nkzE^a_c

EdþEa

X

n¼þN

n¼N

ikzE^z_nE^b_bþiF^z_nkzE^b_c

Ed, G^surf_zz ¼G^full_zz þE_a^n¼þNX

n¼N

ik²_z nn

A^z_nE^a_bþ ik²_n

g_n C^z_nig_nB^z_n

E^a_c

E_dþE_a^n¼þNX

n¼N

ik²_z nn

E^z_nE^b_bþ ik²_n

g_n G_nig_nF^z_n

E^b_c

E_d. ð6Þ

Load acting in the fluid medium:

G^surf_fx ¼Ea

X

n¼þN

n¼N

iA^f_nknE^a_bþiB^f_nknE^a_c

EdþEa

X

n¼þN

n¼N

iE^f_nknE^b_bþiF^f_nknE^b_c

Ed, G^surf_fy ¼Ea

X

n¼þN

n¼N

innA^f_nE^a_bþ ik²_n

g_n B^f_nþik²_z g_n C^f_n

E^a_c

EdþEa

X

n¼þN

n¼N

innE^f_nE^b_bþ ik²_n

g_n F^y_nþik²_z g_n G^f_n

E^b_c

Ed, G^surf_fz ¼Ea

X

n¼þN

n¼N

iA^f_nkzE^a_bþiC^f_nkzE^a_c

EdþEa

X

n¼þN

n¼N

iE^f_nkzE^b_bþiG^f_nkzE^b_c

Ed. ð7Þ

A. Pereira, A. Tadeu / Journal of Sound and Vibration 300 (2007) 800–816 805