Numerical sound field analysis considering atmospheric conditions

Transcription

1 Numerical sound field analysis considering atmospheric conditions Satoshi Ogawa 1 and Yasuhiro Oikawa 2 1,2 Department of Intermedia Art and Science, Waseda University Okubo, Shinjuku-ku, Tokyo , Japan ABSTRACT Some systems such as an outdoor public address system and radio acoustic sounding system create loud sounds. The sounds need to be propagated to the desired location for efficient public address. However, sound is greatly affected by atmospheric conditions particularly for long-distance propagation, and it is important to comprehend atmospheric effects. Numerical methods based on geometric acoustics or linear wave acoustics are widely used today for sound field analysis. However, these methods cannot easily consider atmospheric conditions. In addition, we cannot analyze nonlinear sound fields by these methods. In the present paper, we propose a method for analyzing sound fields considering atmospheric conditions and nonlinearity. Equations for numerical sound field analysis, with which we can consider various atmospheric conditions and nonlinearity, are derived from the equations of continuity in fluid dynamics, the Navier Stokes equations and the law of the conservation of energy. We present numerical calculations of acoustic radiation characteristics and sound fields based on the equations we have obtained, and we use the finite-difference time-domain method to solve them. The effects of atmospheric conditions such as wind, viscosity and nonlinearity can be seen in the results, and thus, the sound generation and sound wave propagation can be comprehended in detail. Keywords: fluid dynamics, atmospheric condition, nonlinearity, outdoor public address 1. INTRODUCTION There have been many studies on electro acoustic transducers, and there are many loudspeakers around us. Recently, we have become highly interested in not only the quality of sound but also its directivity, because it is important to maintain environmental quality, energy saving, security and safety. There are some situations in which loud sounds are generated outdoors, for example, an outdoor public address system or a radio acoustic sounding system (RASS), which is an atmospheric temperature gradient measuring method using radar and high-amplitude sound wave [1]. For efficient public address, these outdoor systems are required to propagate sound to a specific desired location. However, outdoor sound propagation at long distances is greatly affected by atmospheric conditions such as wind, temperature gradients and viscosity. Therefore, it is important to comprehend the effects of atmospheric conditions. For long distance public address systems, flat panel speakers and parametric loudspeakers have attracted attention. These loudspeakers are able to realize a long-distance sound propagation without 1 ogststtk@akane.waseda.jp 2 yoikawa@waseda.jp 1

2 a loud sound source by generating plane waves. A parametric loudspeaker radiates a high-power amplitude-modulated (AM) or frequency-modulated (FM) ultrasonic sound and that produces an audible sound using a nonlinear phenomenon of the sound field. Therefore, it is also important to comprehend nonlinear effects in the sound field. Numerical methods based on geometry acoustics or linear wave acoustics are widely used for sound field analysis today [2]. However it is difficult to calculate sound fields numerically considering atmospheric conditions. In addition, it is also difficult to consider the nonlinearity of the air due to a large amplitude sound. Hence, we need to start from the equations of fluid dynamics to analyze sound field. In the present paper, we propose a numerical analysis method that consider atmospheric conditions and nonlinearity. First, we derive wave propagation equations from the equations of fluid dynamics: the equations of continuity, the Navier Stokes equation and the law of conservation of energy. Next, acoustic radiation characteristic and sound propagation are calculated from these equations by means of the finite-difference time-domain (FDTD) method. Numerical simulations are performed for several kinds of sound sources, and the effect of atmospheric conditions and nonlinearity can be seen in the results. Then sound generation and sound wave propagation can be comprehended in detail. 2. BASIC EQUATIONS First, wave propagation equations considering the effects of wind and viscosity are derived from the equation of fluid dynamics. Next, nonlinear wave propagation equations are derived in order to calculate large-amplitude sound propagation. 2.1 Linear Equations of Dissipative Fluid To understand sound propagation from the stand point of fluid dynamics, we use three equations of fluid dynamics as the fundamental equations [3-5], namely the equation of continuity the Navier Stokes equation and the law of the conservation of energy Dρ + ρ V = 0, (1) Dt ρ DV Dt = P + η 2 V + ζ + η 3 V, (2) ρt DS Dt = κ T + η 2 ( v i + v j 2 x j x i 3 δ v l ij ) + ζ( V) 2, (3) x l where ρ is the atmospheric density, V = (u v w) T, D is the Lagrangian derivative defined as: Dt D Dt = t + V = t + u x + v y + w z, (4) η is the share viscosity, ζ is the bulk viscosity and κ is the heat capacity ratio. The atmospheric pressure is P = P ρ, S, (5) and entropy S = S(P, T). The effects of wind are modeled by the advection term V. If a flow field is in a steady equilibrium state, ρ, T, S, P and V can be separated into the equilibrium value and the fluctuation value. Therefore we get ρ = ρ 0 + ρ, T = T 0 + T, S = S 0 + S, P = P 0 + p and V = U + v. (6) 2

3 In linear acoustics, ρ/ρ 0 can be approximated as 0 because ρ is sufficiently smaller than ρ 0. Using Eq. (6), Eqs. (1 5) can be written in the form: Dρ Dt + ρ 0 v = 0, (7) ρ 0 Dv Dt = p + η 2 v + ζ + η 3 v, (8) ρ 0 T 0 DS Dt = κ T, (9) D Dt = t + V = t + U + v and (10) p = ( p ρ ) Sρ + ( p S ) ρs = c 0 2 ρ + ( p S ) ρs. (11) We can assume that the process of sound propagation is adiabatic because the effect of viscosity in the air is sufficiently small in sound propagation. Therefore, the speed of sound and temperature fluctuation approaches c 0, and then: Substituting Eq. (12) into Eq. (9), we get T = 1 D 2 T c 2 0 Dt 2. (12) κ S = ρ 0 T 0 c 2 0 From Eq. (13), the formulas of thermodynamics DT Dt. (13) T = T p s p, (14) c v = T S T ρ, (15) c p = T S T p, (16) and Maxwell's relation we get c p c v = T ρ 2 p T ρ ρ T p, (17) T = 1 ρ, p S ρ 2 (18) 0 S P p S = p S ρ S ρ κ ρ 0 T 0 c 0 2 DT Dt = p κ T Dp S ρ ρ 0 T 0 c 2 0 p S Dt = κ T 0 p S ρ T p S v = κ ρ 0 2 T 0 p S ρ ρ S p v 3

4 = κ ρ 0 2 T 0 p T ρ T S ρ ρ T p T S p v = κ 1 c v 1 c p v, (19) where c v is the specific heat at constant volume and c p is the specific heat at constant pressure. Substituting Eq. (19) into Eq. (11), we get: p = c 0 2 ρ κ 1 c v 1 c p v. (20) Equations (7), (8) and (20) are the wave propagation equations for analyzing the sound field affected by wind and viscosity. However, nonlinearity cannot be considered in these equations. 2.2 Nonlinear Equations for a Dissipative Fluid Using Eq. (6), a quadratic approximation of Eq. (5) can be written as p = c 0 2 ρ p ρ 2 S ρ 2 + ( p S ) ρs. (21) When we assume that the sound wave propagates in the air, the nonlinear coefficient β is where γ = c p /c v is specific heat ratio. From Eqs. (19), (21) and (22), we have β = 1 + ρ 0 2c p ρ 2 ρ =ρ 0 = γ + 1 2, (22) p = c 0 2 ρ + βc 0 2 ρ 0 ρ 2 κ 1 c v 1 c p v. (23) where ρ/ρ 0 is not negligible, because in nonlinear acoustics, ρ is not sufficiently smaller than ρ 0. Then Eq. (1) can be written in the form Dρ Dt + (ρ 0 + ρ ) v = 0, (24) and Eq. (2) can be written in the form: (ρ 0 + ρ ) Dv Dt = p + η 2 v + ζ + η 3 ( v) = p + ζ η 2 v + ζ + η 3 v. (25) We can assume a sound field with irrotational flow because the transverse wave attenuates in a sufficiently shorter distance than a wavelength. Under this condition v = 0, then we have ρ 0 + ρ Dv Dt = p + ζ η 2 v. (26) Equations (24 26) are the wave propagation equations for analyzing the sound field affected by wind, viscosity and nonlinearity. 3. NUMERICAL SIMULATION BY FDTD METHOD 3.1 Acoustic Radiation Characteristics The acoustic radiation characteristics of a point source, flat panel speaker and parametric loudspeaker are calculated numerically. Equations (24 26) are solved by the FDTD method with second-order central difference. Mur s first-order absorbing boundary condition is adopted for an open-air boundary. The vibration velocity of loudspeakers and the atmospheric conditions are shown in Tables 1 and 2, respectively. A flat panel speaker and a parametric loudspeaker, each with a width of 0.3 m, are set parallel the x axis and they have vibration velocity in the z axial direction. 4

5 Table 1 Vibration velocity of speakers. Speaker Vibration velocity [m/s] Point source sin(2π 3000t) Flat panel speaker sin(2π 3000t) Parametric loudspeaker sin 2π 3000t sin(2π 40000t) Table 2 Atmospheric conditions. Parameter Value Atmospheric pressure P 0 [Pa] Air temperature T 0 [K] 300 Ratio of specific heat γ 1.40 Gas constant R [J/kg K] 287 Speed of sound c 0 [m/s] γrt 0 Air density ρ 0 [kg/m 3 ] P 0 /RT 0 Heat capacity ratio κ Share velocity η Bulk velocity ζ 0 Specific heat at constant volume c v Specific heat at constant pressure c p (a) Point source (b) Flat-panel speaker Figure 1 Acoustic radiation characteristics of (a) a point source and (b) a flat panel speaker. Figure 1 shows the acoustic radiation characteristics in the case of point source and flat-panel speaker located in the center of the displayed area. A spherical wave is generated by the point source. The flat-panel speaker has high directivity with a main-lobe in the z axial direction and some side-lobes. Figure 2 shows the sound pressure and sound pressure level (SPL) in the case of a parametric loudspeaker. The parametric loudspeaker is driven by an AM signal with a carrier frequency of 40kHz. The radiation of a parametric loudspeaker has sharp directivity like a spot-light, and there are no side-lobes because of the parametric array, which is a virtual sound source array created by nonlinearity of the air [6 8]. Air viscosity causes attenuation of the ultrasonic waves as they propagate. 5

6 (a) Sound pressure (b) SPL Figure 2 Acoustic radiation characteristics of a parametric loudspeaker. (a) Waveform (b) Power spectrum Figure 3 Waveform and power spectrum at 50 cm distance from parametric loudspeaker. Figure 3 shows the waveform and power spectrum of sound generated by a parametric loudspeaker in the 50-cm front, which is represented by the cross mark in Fig. 2 (a). We can find peaks at 3, 6, 37, 40 and 43kHz in Fig. 3 (b). The 40-kHz component is the carrier for the AM signal of parametric loudspeaker, and the 37- and 43-kHz components are side bands. The 3- and 6-kHz components are audible sound demodulated by nonlinearity of the air. The effect of viscosity and nonlinearity can be seen in the results. 3.2 Outdoor Public Address We calculate the sound field of an outdoor public address system with a point source or a flatpanel speaker. Equations (24 26) are solved by FDTD with second-order central difference. We assume the conditions of absolute reflection by the ground surface and dry air. Mur s first-order absorbing boundary condition is adopted for an open-air boundary. A wind profile blowing from left to right is given as the function at height from ground surface, i.e.: U = u 0 ( z z 0 ) α [m/s], (27) where u 0 = 15 m/s, z 0 = 21.2 m and α = These conditions represent a strong wind condition [9]. The vibration velocity of the loudspeakers and the atmospheric conditions are shown in Tables 3 and 4, respectively. Figure 4 shows the wind profile in this simulation. Sound fields generated by a point source and a flat-panel speaker are shown in Figs. 5 and 6, respectively. Speakers are located at a height of 0.17m from the ground. The flat panel speaker, which has the width of 1.2 m, is set parallel to the x axis and has vibration velocity only in the z axial direction, which represents the radiation of sound in the sky. 6

7 Table 3 Vibration velocity of speakers. Speaker Vibration velocity [m/s] Point source 0.012sin(2π 3000t) Flat panel speaker sin(2π 3000t) Table 4 Atmospheric conditions. Parameter Value Atmospheric pressure P 0 [Pa] z Air temperature T 0 [K] z Ratio of specific heat γ 1.40 Gas constant R [J/kg K] 287 Speed of sound c 0 [m/s] γrt 0 Air density ρ 0 [kg/m 3 ] P 0 /RT 0 Heat capacity ratio κ Share velocity η Bulk velocity ζ 0 Specific heat at constant volume c v Specific heat at constant pressure c p Figure 4 Wind profile. (a) Sound pressure (b) SPL Figure 5 Outdoor public address with a point source. 7

8 (a) Sound pressure (b) SPL Figure 6 Outdoor public address with a flat-panel speaker. The sound fields are affected by the wind blowing from the left, so the direction of sound propagation is toward the right. In addition, the SPL on the windward side is smaller than that on the leeward side. We can also see the difference in the sound field near the ground between the point source and the flat-panel speaker. The flat-panel speaker produces a smaller SPL near the ground than does the point source. 4. CONCLUSIONS Wave propagation equations considering various atmospheric conditions and nonlinearity were derived from the equations of fluid dynamics, and numerical simulations were performed using these equations. The effects of atmospheric conditions and nonlinearity could be shown in the results, and so sound generation and sound wave propagation were comprehended in detail. In the future, we will focus on sound propagation considering more complex atmospheric conditions such as wind vortices and larger-scale sound propagation. We will also study outdoor public address systems that are insulated from the influence of atmospheric conditions. REFERENCES [1] K. Takahashi, Y. Matsuda, N. Matsuura, S. Kato, S. Fukao, T. Tsuda, and T. Sato, Analysis of acoustic wave fronts in the atmosphere to profile temperature and wind with a radio acoustic sounding system, J. Acoust. Soc. Am., 84(3), (1988). [2] M. Vorländer, Auralization: Fundamentals of Acoustics, Modelling, Simulation, Algorithms and Acoustic Virtual Reality (Springer Berlin Heidelberg, Berlin, 2008). [3] Bent O. Enflo and Claes M. Hedberg, Physical theory of nonlinear acoustics, Chap.2 in Theory of Nonlinear Acoustics in Fluids (Springer Netherlands, Berlin, 2004). [4] K. Naugolnykh and L. Ostrovsky, Nonlinear Wave Processes in Acoustics (Cambridge University Press, New York, 1998). [5] R.T. Beyer, Nonlinear Acoustics (Naval Sea Systems Command, Washington, DC, 1974). [6] P.J.Westervelt, Parametric End-fire Array, J. Acoust. Soc. Am., 32(7), (1960). [7] P.J.Westervelt, Parametric Acoustic Array, J. Acoust. Soc. Am., 35(4), (1963). [8] M. Yoneyama, J. Fujimoto, Y. Kawamo, and S. Sasabe, The audio spotlight: An application of nonlinear interaction of Sound waves to a new type of loudspeaker design, J.Acoust. Soc. Am., 73(5), (1983). [9] Touma, J.S., 1977, Dependence of the wind profile power law on stability for various locations, J. Air Pollution Control Association, 27(9), (1977). 8