# Investigation of adequacy of the analytical model of sound field in rectangular room/Akustinio lauko staciakampiame kambaryje analizinio modelio adekvatumo tyrimas.

1. Introduction

In free space the propagation of sound will radiate equally in all directions and should be described by particles or by waves. In closed rooms however the walls with their acoustic properties influence the character of the sound propagation. Additional difficulties can arise when sound travels through the screen in the room. The more reflective the surfaces of a room, the more a sound wave will be reflected and therefore the longer it will take before it is fully absorbed. There is a direct relationship between the reverberation time in a room and the absorption of the surfaces. A model for sound wave propagation in a room with heterogeneous medium, for example, a screen in the room leads to more or less efficient method for solving the wave equation, for example using the Boundary Element Method (BEM) and finds an approximate solution to the wave equation by solving the system of equations resulting from discretizing the surfaces into patches. The problem with this method is that the surface mesh must be fine enough to account for phase differences. Also, it is difficult to adapt the mesh of each surface to capture the irregularities and discontinuities of the sound field. Another possibility is to use the Finite Element Method (FEM), where the wave equation is being solved by dividing the enclosure into the elements [1]. Then the wave equation is expressed by the discrete set of linear equations for these elements. On the other hand, FEM also allows modeling energy transmission between the separate surfaces. The adequacy of the acoustic field in the room with heterogeneous medium to the real acoustic field was analyzed using FEM model and shows that the suggested theoretical model created on the basis of FEM is adequate to the real processes registered in the testing laboratory [2].

Another possibility is to describe the sound field in a room by sound particles moving around along sound rays. Such a geometrical model is using the simulation of sound in large rooms, for example the Ray tracing method and the Image Source Method. Ray tracing methods [3] find propagation paths between a source and receiver by generating rays emanating from the source (or receiver) position and following them individually as they propagate through the environment. Although this method is very general and simple to implement, it is subject to aliasing artifacts as the space of rays is sampled discretely. For instance, receiver position and diffracting edges are often approximated by volumes of space (in order to admit intersections with infinitely thin rays), which can lead to false hits and paths counted multiple times. More often, important propagation paths may be missed by all samples. Moreover, ray tracing is very compute intensive, usually taking minutes to hours to compute a receiver-dependent solution.

Image source methods compute specular reflection paths by considering virtual sources generated by mirroring the location of the sound source over each surface of the environment. The key idea is that a direct path from each virtual source has the same directionality and length as a specular reflection path. Thus, specular reflection paths can be modeled up to any order by recursive generation of virtual sources. This method is simple for rectangular rooms [4]. However, for every new receiver location, each of the virtual sources must be checked to see if it is visible to the receiver, since the specular reflection path might be blocked by a polygon or intersect a mirroring plane outside the polygon [5]. As a result, this method is practical only for computing very few specular reflections from stationary sources in simple environments. The beam tracing methods [6] find propagation paths from a source by tracing beams (i.e., bundles of rays) through a 3D polyhedral environment. In general, a set of beams is constructed that completely covers the space of rays from the source. For each beam, polygons are considered for intersection in order from front to back. The advantage of this approach is that it allows finding all propagation paths up to the termination criteria. The disadvantage is that the geometric operations required for beam tracing are more complex than for the individual paths.

This paper is intended to present a new possibility to describe the propagation of sound in an enclosed heterogeneous medium and using analytical method for the calculation of relative displacement [7, 8] of air points under the action of the sound source in a known place in a room. The created technique and obtained results of the theoretical calculation were compared to the results of practical experiment. The adequacy of the sound field model and the real room 's acoustic field was analyzed.

2. The analytical model

The analytical model of room acoustic field is based on the mathematical model derived by the calculation of relative displacements of particles of air under the action of sound source in the Cartesian system of axis. The series of calculation that are carried out are shown schematically in Fig. 1. Starting at the bottom of the figure a model of a room is created for example shown in Fig. 2. The system of axes xy are fixed and axes [x.sub.i], [y.sub.i] can translate with respected xy with the speed of sound in air. So the motion of particles with respect to the frame [x.sub.i], [y.sub.i] is relative.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The structural model was set up in which walls of room are absolutely rigid and in equilibrium; the external volume (air mass) forces were neglected; the sound propagation is adiabatic and sound source parameters are known; the number and location particles depend on the known frequency and speed of sound in air. The aim is to calculate the relative displacement of air points under the action of the sound source in a known place in a room.

3. Mathematical model

The mathematical model is based on analytical calculation of relative displacements due to sound impact action, resulting from Hamilton principle [7, 8].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[F.sub.L] = [F.sub.LX] + [F.sub.LY] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[F.sub.x] = [partial derivative]F/[partial derivative]x (7)

[F.sub.y] = [partial derivative]F/[partial derivative]y (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [[rho].sub.o] is density of air; c is speed of sound in air; [bar.X], [bar.Y] is projections of external surface force (line unit is subjected to that force) on coordinate axes. In the case of sound source, taking into account that the pressure of sound source is the same in all directions ([bar.X] = [bar.Y]); F = F(x, y); q = q(t).

The function F is selected on the basis of the boundary conditions, i. e. it should fit for the room presented in Fig. 2. The duration of the sound source equal reverberation time T by Sabine's reverberation formula [9] is

T = 0.161 V/A (10)

where V is the room volume in cubic meters; A is the total absorption in square meters.

In addition to determine the influence of the screen in the room for sound radiation we can use the absorption coefficient [alpha]. Sound waves more or less are absorbed by a screen. The absorption coefficients express the absorption factor of materials at given frequencies [10]. We can suggest using the multiplier [zeta] and following calculation technique, for example in x direction where the screen is located at x = [x.sub.w]

[zeta] = 1 - [alpha] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

In this case, the integral differential equation (1) solved approximately by means of the iteration method, for example in x direction got the fifth approximation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Then, having applied equation (13), we can calculate approximate relative displacements of air particles

[u.sub.i] = [F.sub.i][q.sub.ui]; [v.sub.i] = [F.sub.i][q.sub.vi] (14)

Finally, taking into account relationships of acoustic quantities associated with a plane progressive acoustic sound wave [11], loudness at any point of the room is calculated.

4. Numerical examples and experimental results

In order to determine the adequacy of the created analytical model, the experimental test was done and the obtained numerical results of the theoretical modeling can be comparing to the experimental ones. All geometrical and sound source parameters of the theoretical experiment were selected through the imitation of the real experiment where the sound pressure measurements were done in the different points around the screen using device Investigator 2260 and applying to analyze the modular precise vibration and noise analyzer PULSE 3560 [12]. In order to reduce the acoustic noise the screen was used. Fig. 3 shows the general view of this screen and sound sources.

[FIGURE 3 OMITTED]

The problem simulated numerically and the principal scheme of the measurement experiment of sound pressure is sketched in Fig. 4.

[FIGURE 4 OMITTED]

For this case taking into account the model of a room for displacement analysis (Fig. 2), the geometrical values a = 3.4 m and h = 2.4 m. Let's suppose that density of air [[rho].sub.o] = 1.224 kg/[m.sup.3], speed of sound in air c = 343 m/s, frequency of sound wave v = 1000 Hz and function F is selected on the basis of the boundary conditions (Fig. 2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

All parameters for the calculation of Eqs. (15), (2) and (3) obtained in accordance with earlier created technique [10] are shown in Table.

The numerical examples of analytical model were done with different absorption coefficient [alpha]. Taking into account principal scheme of the sound pressure measurement (Fig. 4), the obtained results of the theoretical and experimental tests in different points of measurement are presented below in Figs. 5 and 6.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

5. Conclusions

The proposed and developed analytical method allows the analysis of sound field in rectangular room. The analytical method enables:

1. to calculate approximately the displacements of air particles appeared under sound impact at a specific place of the room;

2. to create precondition for operation of acoustic field in enclosure taking into account screen in specific place of the room.

6. Acknowledgement

This work was supported by Lithuanian State Scientific and Study fund, project T -87/09.

Received October 02, 2009 Accepted November 24, 2009

References

[1.] Chung, T.J. Computational Fluid Dynamics.--Cambridge: Cambridge University Press, 2002.-1042p.

[2.] Mikalauskas, R., Volkovas, V. Investigation of adequacy of the acoustical field model -Mechanika. -Kaunas: Technologija, 2009, Nr.2(76), p.46-49.

[3.] Kulowski, A. Algorithmic representation of the ray tracing technique. -Applied Acoustics, 1985, 18, p.449 469.

[4.] Allen, J.B., Berkley, D.A. Image method for efficiently simulating small-room acoustics. -J. Acoust. Soc. Am., 1979, 65, p.943-950.

[5.] Borish, J. Extension of the Image Model to Arbitrary Polyhedra. -J. Acoust. Soc. Am., 75, 6, June, 1984,

p.1827-1836.

[6.] Dadoun, N., Kirkpatrick, D.G., Walsh, J.P. The Geometry of Beam Tracing. -Proceedings of the Symposium on Computational Geometry, June, 1985, p.55-71.

[7.] Investigation of heterogeneous acoustical fields generated by several sources in the enclosed space, creation of the means implementing the model of noise field control and evaluation of their efficiency. Report of the Research and Study Found of Lithuania. Project G26/2007.-112p.

[8.] Dorosevas, V. The new analytical method of acoustic field estimation in the room.-Journal of Vibroengineering. -Vilnius: Vibromechanika, 2008, v.10, no.3, p.302 306.

[9.] Sabine, W.C. Collected Papers on Acoustics. 1993, Trade Cloth ISBN 0-932146-60-0 Peninsula Publishing, Los Altos, U. S. LCCN: 93-085708.

[10.] Harris, C.M. Handbook of Acoustical Measurements and Noise Control. Volume 1.-McGraw Hill, 1991. -1136p.

[11.] Kinsler, L.E., Frey A.R., Coppens A.B., Sanders, J. V. Fundamentals of Acoustics. -New York: John Wiley & Sons, Inc, 4th edition, 2000.-549p.

[12.] Volkovas, V, Slavickas, E.S., Gulbinas, R.J. The investigation of effectiveness of acoustical isolation of noise sources. -Mechanika 2009: Proceedings of 14th International Conference, April 2-3, 2009, Kaunas, 2009, p.445-448.

V. Dorosevas *, V. Volkovas **

* Kaunas University of Technology, Faculty Civil Engineering and Architecture, Student^ 48-411, 51367 Kaunas, Lithuania, E-mail: viktoras.dorosevas@ktu.lt

** Kaunas University of Technology, Technological Systems Diagnostics Institute, Kqstucio 27, 44312 Kaunas, Lithuania, E-mail: vitalijus.volkovas@ktu.lt
```Table

Parameters

Parameters   Value

[k.sub.1]    -1.35661
[k.sub.2]     0.678137
[k.sub.3]     2.86242
[m.sub.o]     5.52053
[k.sub.o]     0.0000373902
```
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