J. Acoust. Soc. Jpn. (E) 1, 1 (1980) Method of estimating the reverberant sound absorption coefficient of the absorbing structure composed of different plane porous materials Kyoji Fujiwara and Yasuo Makita Department of Acoustic Design, Kyushu Institute of Design, 226 Shiobaru, Minami-ku, Fukuoka, 815 Japan (Received 11 May 1979) A method of estimating the reverberant sound absorption coefficient of an absorbing structure composed of several different plane porous materials without measurements in a reverberation room is studied. It is found that the reverberant sound absorption coefficient of this structure may be calculated in terms of the acoustic parameters of the constituent materials, their shapes, the mounting conditions and the dimensions of the room. The validity of the method is verified by experiments. PACS number: 43. 55. Dt, 43. 55. Ev 1. INTRODUCTION An absorbing structure composed of several different porous materials is often utilized as a means of adjustment of the reverberation time and reduction of the sound pressure level in a room. The reverberant sound absorption coefficient of the absorbing structure varies to a large extent with its mounting condition. Therefore, the reverberant sound absorption coefficients of test pieces of absorbing structures with specific mounting conditions have to be measured in a reverberation room. The lack of data of reverberant sound absorption coefficients results from this troublesome procedure of the measurement and diminishes the freedom of choice of absorbing structure in the acoustic design of a room. The present authors have studied the method of calculation of the reverberant sound absorption coefficient of an absorbing structure composed of single plane porous material.1-3) The aim of this paper is to make it possible to calculate the reverberant sound absorption coefficient of an absorbing structure composed of several different plane porous materials in terms of the acoustic properties of the constituent materials, their shapes, the mounting conditions and the dimensions of the room without measuring it in a reverberation room. 2. ANALYSIS BY THE GEOMETRICAL THEORY 2.1 Absorbing Structure Composed of n Different Plane Porous Materials Placed Adjacent Side by Side The geometry of an absorbing structure composed of n different plane porous materials placed adjacent side by side is shown in Fig. 1. Each constituent material, for example, the material i is mounted at an average distance hi from the wall of the reverberation room. When the back space of the materials is not partitioned (hereafter, let this case be Case A), the space in the room is assumed to be virtually separated into (n+1) parts; space 0, i and the projected area of this material on the wall and has volume Vi. When the back space of the materials is partitoined (hereafter, let this case be Case B), space i is the space enclosed by the material
J. Acoust. Soc. Jpn. (E) 1, 1 (1980) where E0 and Ei are the sound energy densities in space 0 and i respectively; W is the sound power of a source placed in space 0; a0j and aij are constants determined by the physical parameters of each space and its boundaries, but when space i is not adjacent to space j, aij is zero. Being solved Eq. (1), i.e., the simultaneous differential equations with constant coefficients, the decaying process of energy density in space 0 is given as follows. (2) where Fig. 1 Geometry of an absorbing structure constructed in a reverberation room. the partitions and the wall and has volume Vi. Space 0 is the remaining space of the room in both cases mentioned above and has volume V0. The case where Case A and Case B are mixed is also able to be treated, though very complex. According as the material is mounted at a distance from the wall, the sound field in the back space of the material takes three different states; (a) when the material is mounted far from the wall, the diffusivity and the sound energy density in the back space are almost equal to those in space 0, (b) when the material is a little nearer to the wall, the sound field in the back space is almost diffuse as well as in space 0, but the sound energy density in the back space is different from that in space 0, (c) when the material is quite near to the wall, the sound field in the back space is coherent and the material is in it. This section is concerned with the states (a) and (b), the third one will be discussed in the next chapter. In the two states (a) and (b), the sound fields in hij are given by the following simultaneous equations being solved, provided that h0j is equal to unity, are the roots of following equation, diffuse. Hence the relation between sound energy E00 is the energy density of space 0 at steady state, densities in each space and their time rates of increase may be expressed by the following simul- and Hi0 is the ratio of the steady state energy density of space i, E0i to E00, viz., taneous differential equations, Hi0=E0i/E00 (1)
K. FUJIWARA and Y. MAKITA: REVERBERANT SOUND ABSORPTION COEFFICIENT For example, let the case where two different (i,j=0,1,2) are given by In the case where the source is in space 0, the following inequality holds for normal mounting condition of materials, and the Eq. (2) is approximated to The reverberation time in space 0 is obtained as (5) (8) Inserting the value of T into the following equation yields the reverberant sound absorption coefficient (6) where rmi is the sound energy fraction reflected by fraction absorbed by the material i for random i to the wall, Swij is the area of the boundary surface between the spaces i and j except for the area of the material, Smi is the one side area of the where V is the summation of the volume Vi (i= material i. When spaces 1 and 2 are separated from space 0 by the partition which the sound energy sound velocity. T is the reverberation time when does not penetrate as shown in Fig. 2, i.e., Case B, the room is empty, and sound absorption coefficient of the surface of the bare room, S is the inner surface area of the bare room and V is the total volume of the room. (9) Fig. 2 Geometry of an absorbing structure composed of two different materials when the backing condition is Case B. efficient of the partition between space 0 and spaces 1 and 2. When space 1 is insulated from space 2 by changed to 39
tively by efficient of the partition between spaces 1 and 2. (10) J. Acoust. Soc. Jpn. (E) 1, 1 (1980) Firstly, the absorbing structure of two different materials, i.e, material i and material j as shown in Fig. 4, is considered for simplicity. When the sound is incident on the front side of this structure, the effective energy fractions reflected and absorbed 2.2 Multi-layer Absorbing Structure This section is concerned with the case where n different materials are used as a multi-layer absorbing structure. It is assumed that the thickness of this structure is very thin compared with the length of perimeter of this structure and this structure behaves as a thin material. The multi-layer absorbing structure of n different materials under consideration is shown in Fig. 3. n from upper side of this structure. The notations reflected and absorbed by the material i respectively in the case where the sound is incident on the front side of this material (upper side in Fig. 3), and rmib reflected and absorbed by the material i respectively in the case where the sound is incident on the back side of this material. (11) (12) When the sound is incident on the back side of this structure, the effective energy fractions reflected Fig. 3 Geometry of a multi-layer absorbing structure. (13) (14) When the multi-layer absorbing structure of n Fig. 4 Process of the reflection and transmission in the multi-layer absorbing structure composed of two different materials. (14) are calculated for each pair of adjacent differ- obtained by the above calculations as parameters of composite materials of respective pair materials, we continue the same procedure recurrently until the effective energy fractions reflected and absorbed of the multi-layer absorbing structure are 40
K. FUJIWARA and Y. MAKITA: REVERBERANT SOUND ABSORPTION COEFFICIENT obtained. If n is odd, one of the materials of outside layer is kept out of consideration and the above procedure is applied to (n-1) different materials. From the obtained effective energy fractions reflected and absorbed of (n-1) different materials and those of the remaining material, the effective energy fractions reflected and absorbed of n multi-layer absorbing structure are obtained finally. 3. CORRECTION BY THE WAVE THEORY 3.1 Correction for the Lack of Uniformity of the Energy Distribution in the Field near to the Wall It is shown by Waterhouse4) that the sound energy is distributed into interference patterns near the reflecting boundaries in a reverberation room. The third state mentioned in the previous chapter is the state when the absorbing material is mounted in the sound field of the interference patterns. In this state, the results obtained by the geometrical theory in the previous chapter should be corrected. (A) The case of pure tone incidence: The potential energy distribution when the absorbing material i is in the sound field near to the wall is given by Fig. 5 Correction term Kn as a function of the distance h. The center frequency of band noise is 500Hz. ćmi+Ďdi=0.9. The solid curve in Fig. 5 shows the numerical example of Eq. (17) when (ćmi+Ďdi)=0.9. (B) The case of band noise incidence: The reverberant sound absorption coefficient is usually measured by the use of band noise. Therefore, Eqs. (15) and (16) must be averaged over the band width of noise. The potential and kinetic energy distributions for band noises are given as follows. and the kinetic energy distribution is given by1) (15) (18) (16) The fraction of sound energy propagated in a porous absorbing material is transformed into heat energy in proportion to the square of particle velocity, i.e., the kinetic energy of the sound. Therefore, the absorbed energy fraction Ďdi' in the field near to the wall is given by Ďdi being multiplied by the ratio of the kinetic energy to the total energy at the position of material; (17) (19) where (k2-k1) is the band width of noise. And then the correction term Kn for the band noise is given by (20) Figure 5 shows the numerical examples of Eq. (20). The dashed curve shows the case when the band width is one-third octave band, and chain curve shows the case when the band width is one octave band. 41
J. Acoust. Soc. Jpn. (E) 1, 1 (1980) (C) Influence of the phase delay of sound in the material: The correction term K given in the previous subsection (A) was obtained by the phase delay of reflected sound wave in the material being neglected.1) Here, the influence of this phase delay on the correction term is considered. When the sound passes through a sheet-type porous material, let the phase angle lagging behind the sound which would not pass through the material be ƒõ (degree). Although this angle depends slightly upon the incident angle of sound, it is assumed to be constant for simplicity. Then, the potential and kinetic energy distributions are given by Fig. 7 Geometry of the material not parallel to the wall. chain curve correspond to the cases where the phase delay angles ƒõ are 0, 10 and 30 respectively. (D) Application to the case where the material is set up not parallel to the wall: When the material i is set up on the wall with an angle ƒæi as shown (21) in Fig. 7, the correction term K must be averaged over the area of the material because the kinetic energy distribution over the area of the material is not uniform, and the energy absorbed by a small part of the material is effected by the kinetic energy density at the point where the small part exists. Then the correction term KƒÆi for this case is given by And the correction term KƒÕ is given as follows, (22) where dƒð is the area of small part of the material as a function of h. (24) (23) 3.2 Correction for the Edge Effect Figure 6 shows the numerical examples of the correction term KƒÕ. The solid curve, dashed curve and Fig. 6 Correction term KƒÕ as a function of the distance h. The frequency of incident sound wave is 500Hz. ƒámi+ƒédi=0.9. (A) Energy discontinuity in the field near to the edge of material: The edge effect is a phenomenon caused by the diffraction of sound around the edge of the material. This diffraction may be regarded as the phenomenon which arises from the nature of sound field to reduce the discontinuity of energy density distribution near to the edge. Therefore the amount of this diffraction may be assumed to be proportional to the degree of the discontinuity in the energy density distribution. When a plane sound wave is incident on a boundary line between two different materials as shown in Fig. 8, according to the geometrical theory the reflected sound wave from the material i is in the left side region of the dashed line (side Ai), and the reflected sound wave from the material j is in the right side region of the dashed line (side Aj). There exists the discontinuity of sound energy density or the discontinuity of sound intensity between these two sound fields in side Ai and side Aj. Let the 42