19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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1 9 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 007 SCREEN USE POSSIBILITIES AS PROTECTION FROM TRANSPORT NOISE AND PROSPECTS OF THEIR IMPROVEMENT Danielius Gužas ; Robertas Klimas ; Vaclovas Tricys,, Šiauliai university, Vilniaus st. 4, LT-549, Šiauliai, Lithuania danielius.guzas@fondai.com matlsiauliai@takas.lt vaclovas.tricys@cr.su.lt Keywords: screens, sound insulation and absorption ABSTRACT For transport noise reduction, various screens are widely applied. In the urban zones, where possible, the uninhabitable house or various buildings intended for recreation and purposes other than work, like garages, workshops and other business and trade edifices, are often used. In the suburban areas, the residential buildings and recreation zones are protected from the noise propagating from highways by the special screens designed for that purpose and placed on the roadsides. Different acoustical, technical and aesthetical requirements are applied in respect of the screens. Screens currently used for this purpose not always comply with the above-mentioned requirements. Our work is aimed at improving the said qualities of screens. For that purpose theoretical sound insulation calculation methods for improvement of the efficiency of screen elements are being used to improve sound insulation, absorption and other parameters. Our proposal is to install special elements into screen constructions that will improve the acoustical properties of the screens. The stress is laid on the damping of low-frequency sound reaching the screen barriers. INTRODUCTION Screens for noise reduction have been in application since of old. Primarily, they were used for noise reduction in premises. Acoustical screens as to their form may be plane (linear in plan [, or of more complicated configuration: with the application of additional elements for increasing the efficiency of noise reduction [, 4. In the first turn, acoustical screens perform the function of protection from the source of direct sound. In addition, if the surface of the screen is coated with the sound-absorbing material or special elements, such additional elements have an effect on the field of the sound reflected from the screen. Attenuation of direct sound by the screen of dimensions, exceeding lengths of a sound wave, is based on the fact that the area of a sound shadow is being formed behind the screen, into which only a part of sound waves, enveloping the edges of the screen in accordance with the laws of diffraction, enters. The application of acoustical plane screens without sound-absorbing constructions is less effective. To increase the efficiency of acoustical screens, they are usually coated with the soundabsorbing materials on the side, facing the sound source. Thus, the efficiency of the screens depends greatly on sound-absorbing materials or constructions, which may be mounted in the screen body. In this report we shall analyze the possibilities of special sound-absorbing and sound-insulating elements in the construction of screens while increasing the efficiency at low frequencies of the noise source. THEORY In the walls of screens it is possible to install our proposed special elements (see Fig., which reflect very insignificantly the sound waves falling onto the screen, whereas sound when propagating through that element becomes attenuated, especially at low frequencies. For that

2 purpose, we shall analyze here sound reduction when it propagates through a duct, composed of separate widening and narrowing ducts of variable cross-sections [5. Re -ik(x-a e ik(x-a S S(x Duct I Duct II Duct III 0 a b D e ik(x-b S x Figure. Disposition scheme of cross-section joint relatively to coordinate axes SOUND INSULATION OF VARIABLE CROSS-SECTION DUCT Let us assume that a duct of variable cross-section is located between points x=a and x=b. Its length is d=b a. The law of cross-section area variation is described by the function S(x. The medium of all ducts is characterized by the density p and sound propagation velocity c. Let us analyse harmonic oscillations with cyclic frequency ω=πf. The temporal member exp(-iωf is going to be omitted. Settled single-dimensional medium oscillations inside the duct of variable cross-section are described by Webster equation d p ( x S ( x dp ( x + + k p ( x = 0 (Eq. dx S( x d( x where p (x sound pressure, k=?lc wave number, S (x = ds(x ldc derivative S(x. Equation ( is a linear differential equation with variable coefficient f (x = S (xls(x by the first derivative of sound pressure p (x. General solution (Eq. may be written in linear combination of two functions f (xand f (x, satisfying (Eq. : ( x ( kx + C ( kx p = ϕ (Eq. where C and C constant defined by boundary conditions. Solution (Eq. forms a fundamental system only when their determinant ( kx ϕ ( kx ( kx ϕ ( kx ϕ w[ ϕ ( x, ϕ ( x = (Eq. ϕ is not equal to null. Thus, if f and f satisfy the equation (Eq. and their determinant w 0, so sound pressure inside the duct of variable cross-section is written as (Eq.. Vibration velocity v(x in accordance with Euler equation equals ( x i p i i v ( x = = Cϕ( kx Cϕ ( kx, (Eq. 4 ipω x where f (kxf (kx derivative by argument kx. Let us assume that from the first duct to the duct of variable cross-section falls a plane wave of individual amplitude, the sound pressure of which may be written as ( x = cxp[ ik( x a p na. 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID

3 Let us introduce a reflected wave as ( x Rcxp[ ik( x a p otp =. General sound field of the first duct is equal to the sum of falling and reflected wave ik ( ka a ik( x a x e + p = Rc. (Eq. 5 Oscillation velocity of particles by Euler equation equals ikx kx v ( x = e Rc (Eq. 6 pa Sound field in the third duct is described only by the running to the infinity wave p v ik = ( x b, x D e ik ( x b x = D e (Eq. 7 Unknown amplitudes R, C, C and D are found by boundary conditions by x=a and x=b expressing sound pressure and oscillation velocity equality on the right and on the left from these cross-sections. The only amplitude of wave D having passed to the third duct is of interest to us. Because of this by x=a we exclude from the boundary condition reflection factor R: + R = Cϕ ( ka + Cϕ ( ka, R = C ϕ( ka C ϕ ( ka. By adding these equations we get [ ( ka ϕ ( ka + C ϕ ( ka ϕ ( ka [ =. (Eq. 8 By x=b p (h=p (b and v (b=v (b where from ( kb + Cϕ ( kb D = 0, ( kb + ( kb D = 0 (Eq. 9 Equations (Eq. 8 and (Eq. 9 from heterogeneous system of linear equations. Its solution for D may be found by the Cramer s formula D = D. The determinant of the system is [ ϕ ( ka ϕ ( ka [ ϕ ( ka ϕ ( ka 0 ϕ ( kb ϕ ( kb ϕ ( kb ϕ ( kb By its expansion we get + [ ϕ( ka ϕ( kb + ϕ ( ka ϕ( kb ϕ ( ka ϕ ( kb ϕ ( ka ϕ ( kb ( ka ϕ ( kb + ϕ ( ka ϕ ( kb ϕ ( ka ϕ ( kb ϕ ( ka ϕ ( kb [ ϕ + (Eq. 0 The determinant is ( kb ϕ ( kb ( kb ϕ ( kb ( ϕ ϕ ϕ ϕ w ϕ ( kb, ϕ ( kb ϕ D = = = [. (Eq. ϕ e. i., is equal to redoubled determinant of. 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID

4 Not the transfer coefficient by pressure D is of interest in the problem but sound transfer coefficient by energy τ, defining the sound energy passing from the first duct to the third one. In moving plane wave which pressure amplitude equals A, energy flow equals q=a /. In incident wave single amplitude s q =/,and full energy Q =q S /. In previous wave amplitude equals D, q = D / and Q S D = /. Sound transfer coefficient by energy Q S D S D τ = = =. (Eq. Q S S where D and are defined by the formulas (Eq. and (Eq. 0. MUTUAL RATIO FOR A VARIABLE CROSS-SECTION DUCT Common ratio obtained allow us to define the transfer coefficient π by arbitrary law of crosssection area variation S(x. Solutions (Eq. ϕ (x andϕ (x by preset S(x are defined. By substituting them to (Eq. 0 and (Eq. meanings D and as well as π are found. The value of these expressions is not limited by it. They help us to link the solution of problems on sound transfer through a duct of variable cross-section of direct and reciprocal direction. Let us analyse the problem where a plane wave of single amplitude Pna falls on the duct of variable cross-section from the side of the third duct. As it propagates to the direction of negative axis x, so P na = exp[ ik( x b, and P otp = exp[ ik( x b. Sound pressure of the third duct equals ik( x b ik( x b p = e + Re, (Eq. and oscillation velocity ik( x b ik( x b v = e + Re. (Eq. 4 Pressure and velocity of variable cross-section duct p and v may be written as in (Eq. and (Eq. 4. The passed wave in the first duct is p v ik = ( r a x D e ik ( x a x = D e From boundary conditions by x=b we get + R = + R = C,. (Eq. 5 ( kb + Cϕ ( kb ϕ ( kb ( kb Let us eliminate R, subtracting from the first equation the second one: [ ( kb + iϕ ( kb + C ϕ ( kb + iϕ ( kb [ =. (Eq. 6 Two other equations we get from the condition by x=a: p (a=p (a and v (a=v (a. Then. ( ka + Cϕ ( ka D = 0, ( ka + ( ka + id = 0. (Eq. 7 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 4

5 Transfer coefficient D may be defined by Cramer s formula D = D /. The determinant of the system (Eq. 6, (Eq. 7 is By its expansion we get + i [ ϕ ( kb + iϕ ( kb [ ϕ ( kb + iϕ ( kb 0 ϕ ( ka ϕ ( ka ϕ ( ka ϕ ( ka [ ϕ ( ka ϕ ( kb + ϕ ( ka ϕ ( kb ϕ( ka ϕ ( kb ϕ ( ka ϕ ( kb ( ka ϕ ( kb + ϕ ( ka ϕ ( kb ϕ ( ka ϕ ( kb ϕ ( ka ϕ ( kb [ ϕ + (Eq. 8 The determinant is ( ka ϕ ( ka ( ka ϕ ( ka ϕ D = = W [ ϕ ( ka, ϕ ( ka. (Eq.9 ϕ Sound transfer coefficient by energy is S τ = (Eq. 0 S It follows from the comparison of equations (Eq. 0 and (Eq. 8 that determinants of equations systems (Eq. 8 ( Eq. 9 and (Eq. 6 ( Eq. 7 are equal to each other. Because of it the ratio of sound transfer coefficients τ and τ equals τ S W [ ϕ ( kb, ϕ ( kb = τ S. (Eq. W [ ϕ ( ka, ϕ ( ka As it follows from formula Liouville [ ratio of Vronsky s determinant b S ( x [ dx W ϕ kb, ϕ kb S( x a α = W [ ϕ ( ka, ϕ ( ka Let us calculate the integral appearing in the above formula. = b S x I dx. As S ( x ds( x / dx a S( x =, so S x dx = ( ds / dx dx = ds. From here = S b S b I ds / S = InS = InS b / S( a = InS / S. Ratio of Vronsky s determinant S( a S( a As in (Eq. is a square a, so e. i., 9 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 5 = e ins / S S α = e =. S τ S S = S τ S =, τ = τ. (Eq. Thus, it makes no difference from which side sound falls on the duct of variable cross-section as the amount of energy which passes through it remains the same. This statement is related to a group reciprocity theorems. NUMERICAL AND EXPERIMENTAL INVESTIGATION Computations of sound insulation of exponential diffuser of different cross-section area ratio S /S have been carried out. In Fig. calculation results for S /S =,4,6,8,6 are presented. At abscissa axis unlimited frequency kd=ω/ω 0 is put aside, while at ordinate axis sound insulation

6 R in db. From the diagrams we see that the higher ratio S /S the greater R is. At frequencies higher the critical ω 0 sound insulation R quickly falls to null. The elements shown in the Figure were tested after their mounting into special blocks. An analysis of theoretical and experimental results showed that on applying cone or exponential elements it is possible to reduce the noise at a frequency of 4-50 Hz by 8- db, in some cases even more. R, db =S /S a=s/s Figure. Scheme of sound insulation quantity dependence upon diffuser areas 4 Figure. The element of the screen: cone; flange; screw; 4 inserts CONCLUSIONS. In the screens applied, special elements, which insignificantly reflect sound waves and insulate well low-frequency noise, may be installed.. Theoretical and experimental investigations have proved that sound on propagating through ducts of variable cross-sections is reflected, and the energy of low-frequency sound reduces.. An analysis of theoretical and experimental results showed that on applying come or exponential elements its is possible to reduce the noise at a frequency of 4-50 Hz by 8 db, in some cases even more. References: [ Maekawa Z., Lord P. Environmental and Architectural Acoustics. 004, 77 p. (in Japan and UK. [ Maekawa Z., et al.: Some Problems on Noise Reduction by Barriers Proc. Sympo. Noise Prevention (Miskolc [ Danielius Gužas, Robertas Klimas, Vaclovas Tricys. Influence of Rigidity of Acoustic Shield Walls into Sound Insulation. Mechatronic Systems and Materials, 006. Trans Tech Publications Inc. Publishers in Science and Engineering Solid State Phenomena. Volume until, p Science Citation Index. ISBN: [4 Danielius Gužas, Robertas Klimas, Vaclovas Tricys: Construction of Acoustic Shields and Their Properties for Noise and Vibration Reduction. nd International Conference Mechatronic Systems and Materials (MSM 006 August September 006 Krakow, Poland, p. 0. [5 L. Borisov, D. Gužas. Sound Insulation in Machine-Building (Moscow, Mashinostroenie, 990 (in Russian, p th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 6