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Page 1 ï~~THE EFFECT OF AIR TURBULENCE ON SOUND AND ITS APPLICATION TO MUSICAL SIGNAL PROCESSING Matthew Bruns Gary Kendall School of Music Northwestern University Evanston, IL. 60208 USA ABSTRACT The effect of atmospheric air turbulence on acoustic signals is modeled with the goal of producing a novel DSP technique for music and sound production. The building block of the turbulence model is called a 'turbule', a localized eddy with spherical symmetry. At the current stage of development, an operational model of a turbule is implemented in Matlab, and the effect of the turbule has been evaluated for a variety of atmospheric conditions. Analysis of the results from this model will pave the way for the creation of a digital effects module built on a network of turbules, one that can exaggerate the effect of air turbulence when needed for expressive purposes. 1. INTRODUCTION We all encounter air turbulence in everyday life and we all have many associations with the experience of wind whether it be a light breeze or a dangerous gale. Turbulence affects the propagation of the sound waves through the medium of air and produces characteristic kinds of changes to the acoustic signal. Within the field of acoustic research there have been several studies that have examined the effect of atmospheric air turbulence in specific situations. For example, while one study modeled sonic boom propagation  through turbulence, another considered the effects of turbulence on steady-state sounds . Furthermore, some studies considered sound propagation through turbulent fields over listener-receiver distances greater than 100 m  while others observed the effects at close distances of less than 16 m . Despite the wide range of these studies, none of them considered how air turbulence might affect common acoustic signals such as music. As pointed out by Wilson, et al. , air turbulence functions as an acoustic filter as sounds of certain wavelengths scatter from the turbulence in the path of sound propagation. If one can determine the effects of air turbulence on each part of the aural spectrum, it is possible to create a digital signal-processing system that simulates the acoustic properties of air turbulence. Because of its chaotic nature, it is difficult to develop a model of air turbulence that consistently accounts for the effects for a wide range of frequencies and listener receiver geometries. First, there is a broad range of energies and length-scales involved in turbulence, making it necessary to determine the scale of turbulence that is most relevant to sound propagation near the ground . Second, it is challenging to obtain consistent field measurements since turbulence is dynamic and hard to isolate. Hallberg et al.  outline several of these difficulties, focusing on issues such as microphones, anemometers, and the effect of the ground. The simulation of turbulent sound fields can be achieved with a computer simulation of sound propagating through a 'turbule'. Goedecke and Auvermann  suggest the idea of a turbule, or localized eddy with spherical symmetry, as a means of representing turbulence. In other words, they treat turbulence as discrete pockets of temperature and wind velocity variations, enabling them to generate specific geometries between sound propagation and turbulence. In support of this method, McLeod et al.  use a single turbule to model the interaction of an acoustic pulse with turbulence. They are able to validate most of their predictions with outdoor measurements. There are several advantages to using the turbule model. Because of the discreteness of the turbule as well as its symmetry, it is possible to scale the turbule to different sizes, accounting for the different length-scales of atmospheric turbulence. Furthermore, one can account for the wide range of energies involved in turbulence by adjusting the meteorological parameters within the turbule. Finally, one can arrange several turbules in a series to create more complicated interactions between sound and turbulence. This study employs a two-dimensional turbule model in order to simulate atmospheric air turbulence. The goal is to produce a novel effect that may be useful in music and sound processing. At this stage, the system consists of a computer simulation of the propagation of an impulse through a turbule. The architecture of this simulation is discussed in section 2 while the specific conditions of the test simulations are summarized in section 3. After calculating the impulse response for a variety of turbules, the musically relevant effects of turbulence are discussed in section 4, while section 5 covers the implementation of musical signal-processing modules that can be used with current music software. 2. SYSTEM DESCRIPTION The computer simulation of sound propagation through a turbule is accomplished with a Matlab simulation that creates a model of a two-dimensional, virtual environment. This virtual environment approximates outdoor conditions close to the surface of the earth but far enough away from the ground to neglect ground reflections. It is assumed that atmospheric air is the sole medium of sound conduction with the following relation for the speed of sound as suggested by Kuttruff : c = 331.4+0.6*T (1)
Page 2 ï~~where c is the speed of sound in m/sec and T is the air temperature in centigrade. Simply adding additional speed to c accounts for the influence of wind. Inside Matlab, the virtual environment is divided into a 2401 x 2401 grid of equally spaced points. Alford et al.  suggest that the spacing must allow at least ten spatial samples per upper half-power wavelength in order to avoid "grid dispersion," the phenomenon in which the coarseness of a grid results in some frequencies propagating faster than others. In this study, the smallest separation in the simulation is equivalent to 9.235e-4 meters in a real environment, providing enough spatial resolution to allow at least forty samples per wavelength of a tone at 10 kHz under any temperature and wind conditions imposed. Computer power and memory limit the upper cutoff to 10 kHz. 2.1. 2D Finite Difference Wave Equation Although Goedecke and Auvermann  use ray tracing techniques in their implementation of the turbule model, the two-dimensional, finite difference approximation of the acoustic wave equation was judged more appropriate for this simulation. This approach offers advantages such as the ability to account for diffraction, the ability to account for all angles of incidence, and the ability to sample the wave front at any point in the virtual space. In the finite difference approximation, the acoustic pressure at a particular point in space is determined by the pressure values at surrounding spatial samples and from the current and previous time samples. The following equation implements the time stepping (leapfrog) method outlined by Landau et al. : pt+l =Cx At (t P -2*Pt +Py =X, - 2, Px,y +P -~ signals. Therefore, the pressure is set to zero at the boundaries, and the size of the virtual environment is enlarged to allow the impulse to completely propagate through the turbule before any reflections interfere. At t = 0, there is no sound pressure, no ambient wind, and an ambient temperature of 200 centigrade. The sound source is introduced into Px,y at t = 1. To ensure stability of the equation, the following constraint suggested by Lines et al.  is imposed: c*At 1 Ax - (3) where c is the speed of sound in m/sec, At is the sampling period in seconds, and Ax is the finite difference in meters. 2.2. Constructing the Turbule The turbule has circular symmetry and contains wind velocities and a temperature that differ from ambient atmospheric conditions. For simplicity, the turbule remains fixed in space, and there is no ambient wind. Inside the turbule, the temperature is homogeneous. As an initial approximation of a circular eddy of wind, the wind geometry depicted in Figure 1 is applied to the turbule. In this particular arrangement, both the xcompnent and the y-component of the wind velocity are applied to each spatial sample within the circular boundary of the turbule. The amplitudes of the components are uniform throughout the turbule. Other possible geometries are left for future experiments. x-axis cy-x2 9 Pxy +Px -, 2,y+l (2) +2Px,y -Px,1 where P is the acoustic pressure in N/m2, x and y are the spatial indices, t is the time index, At is the finite time difference equal to the sampling period (1/705600 sec), Ax and Ay are the finite spatial differences equal to our spatial resolution (9.235e-4 meters), and c is the speed of sound in rm/sec. To account for the directional components of the wind, the speed of is treated as a two-dimensional vector rather than as a scalar. In order to simulate free-field conditions, the boundary conditions must ensure that the sound pressure completely dissipates at infinity. Since the virtual environment is finite in extent, total absorption is required at the boundaries to simulate such conditions. In practice, it is not possible to achieve perfect absorption in a finite simulation space. Higdon  suggests boundary conditions that approximate perfect absorption at the boundaries; however, his method proves inadequate for the precision required for musical Figure 1. Sample structure of the wind velocity components inside the two-dimensional turbule. The vectors are arranged to approximate a circular eddy. 2.3. Choosing the Sound Source Both the temporal and spectral effects of turbulence are important to this work; therefore, the sound sources must have transients that account for all frequencies in the audible range. In this regard, an acoustic impulse provides the ideal source because it occurs over a short interval of time and includes all frequencies. But, because of the finite sampling rate, the impulse must be band-limited to avoid energy above 10 kHz, the cutoff of the simulation. Matlab is used to create a sinc function of amplitude 1 with a Gaussian envelope to ensure a more finite duration. See Figure 2.
Page 3 ï~~3.2. Impulse Responses The effects of air turbulence on the acoustic impulse are measured by sampling the data both before and after the propagation through the turbule. The pre-turbule sample point lies just outside the turbule at the 00 angle of incidence, while the post-turbule sampling points lie just outside the turbule at 180Â~, 1500, and 2100, respectively (see Figure 3). The samples from all four points are converted to the frequency domain. In order to isolate the effect of the turbule, the pre-turbule spectrum is divided by one of the post-turbule spectrums, resulting in a frequency domain transfer function in the form of the following equation: Figure 2. The acoustic impulse used as the sound source in the turbulence simulation. It is constructed from a sinc function with a Gaussian envelope. 2.4. Propagation Geometry In this simulation, the center of the turbule coincides with the center of the virtual environment. The left most point on the radius is defined to be 00 (opposite of unit circle). The impulse propagates from the left such that it enters the turbule at 00. See Figure 3. 3. RUNNING THE SIMULATION Within the flexible virtual environment, it is possible to test the effects of turbulence in a variety of contexts. This section covers the specific parameters used and how the effects of turbulence on sound are measured. For the following processes, a MacBook Pro laptop running OS X version 10.4.11 is used. turbule set # radius (fin x-comp y-comp temp diff. diff) wind (m/s) wind (m/s) (Co) 1 100 10 10 5 2 100 13 13 7 3 100 20 20 10 4 200 10 10 5 5 200 13 13 7 6 200 20 20 10 7 400 10 10 5 8 400 13 13 7 9 400 20 20 10 10 800 10 10 5 11 800 13 13 7 12 800 20 20 10 Table 1. Parameter sets for turbules simulated: radius of turbule in finite differences, components of wind velocity in m/sec, and temperature difference between turbule and ambient conditions in degrees centigrade. 3.1. Turbule Variations Different length-scales of turbulence are represented by a variety of turbule radii. For a given turbule radius, several wind velocities and temperature variations are tested. Table 1 shows several sets of the wind and temperature conditions for the four turbule radii tested. H[k] = X[k] Y[k] (4) where H is the frequency domain impulse response of the turbule, X is the pre-turbule spectrum, Y is one of the post-turbule spectrums, and k is the index of the discrete frequencies. After finding H[k], one can find the time domain impulse response of the turbule if necessary. Figure 3. Sampling geometry inside virtual environemt. Point 1 at 00 samples pre-turbule propagation while points 2, 3, and 4 at 180Â~, 1500, and 2100, respectively, sample post-turbule propagation. 4. RESULTS 4.1. Magnitude Response Calculating the magnitude response of each turbule is one of the primary methods of determining the effects of turbulence on sound. Figure 4 shows some sample magnitude responses from parameter sets 4-6 (sampled at 2100). As one would expect, some frequency components of the sound source are attenuated while are others are reinforced. Increasing the wind velocity and temperature in the turbule increased the deviation in magnitudes of certain frequencies, resulting in a range of magnitudes greater than 12 dB in this example. 4.2. Phase Delay In addition to magnitude response of the turbule, the phase delay reveals important information concerning the effects of turbulence on sound. Figure 5 shows the phase delay of the four turbule sizes from parameter sets 1, 4, 7 & 10 (sampled at 1500). From the figure it is evident that increasing the radius of the turbule increases the phase delay to a total of 1 msec for frequencies below 1000 Hz. This suggests that the
Page 4 ï~~radius of the turbule correlates with the effected part of the spectrum of sound. Parameter Set 4 ("low") Parameter Set 5 ("med") 12. 0 -4 1. Parameter Set 6 ("high") 0 12 -4 0 2k 4k 6k 8k 10k Frequency (Hz) Figure 4. Magnitude responses for parameter sets 4-6 (sampled at 2100). The turbulence created a range of magnitudes > 12 dB. Increasing the wind velocity and temperature increased the range. Turbule Radius = 100 Turbule Radius = 200 0 1 Turbule Radius = 400 Turbule Radius = 800 A 1 0 100 1000 10k 0 100 1000 10k Frequency (Hz) Figure 5. Phase delay of four turbule radii from parameter sets 1,4,7 & 10 (sampled at 1500). The phase delay increases for frequencies < 1 kHz as the radius of the turbule increases for a total delay of 1 msec. 5. CONCLUSIONS The turbule model has been successfully implemented in order to demonstrate the effects of atmospheric turbulence on acoustic signals. The impulse responses of simulated turbules have been calculated and examined. In the next stage of the project, a library of turbule responses will be created, each representing a unique size and unique atmospheric condition. This data will be used to re-implement turbules for audio signal processing by simulating the turbule's properties through convolution of the impulse response with the audio signal. For a digital effects module to be practical, it must allow the user to control and scale the magnitude and quality of the turbulence in real-time. This will be accomplished by building a dynamic network of turbules, a network that includes turbules of different sizes, wind velocities and temperatures. The user of such a module will be able to simulate air turbulence through a variety of atmospheric conditions. One possible way to allow user scalability of turbulence in real-time is to convert the audio signal and the library of impulse responses to the frequency domain and multiply the spectrums before converting back to the time domain. In the frequency domain, it is possible to cross-fade between different impulse responses without many of the artifacts that would result from a cross-fade between impulse responses in the time domain. The proposed turbulence model is not restricted to simulating only real-world conditions. The magnitude of air turbulence can be exaggerated to whatever extent is useful in order to convey the impression of air turbulence. A digital effect does not necessarily depend on realism to be practical. We anticipate that the physical modeling of turbulence via the turbule can be extended to simulate exaggerated conditions. The most fascinating aspect of turbulence, however, is its chaotic behavior over time. Future work would focus on creating a time-variant DSP module that captures the dynamic effects of turbulence for a given set of turbule parameters in addition to allowing the user to scale the turbule size, wind, and temperature. 6. REFERENCES  Alford, R.M., K. R. Kelly, and D. M. Boore, "Accuracy of Finite-Difference Modeling of the Acoustic Wave Equation". Geophysics, 1974. 39(6): p. 834-842.  Blanc-Benon, P., B. Lipkens, L. Dallois, M. Hamilton, D. Blackstock, "Propagation of finite amplitude sound through turbulence: Modeling with geometrical acoustics and the parabolic approximation". JASA, 2002. 111(1): p. 487-498.  Goedecke, G.H., R C. Wood, H. J. Auvermann, V. E. Ostashev, D. I. Havelock, and C. Ting, "Spectral Broadening of Sound Scattered by Advecting Atmospheric Turbulence". JASA, 2001. 109(5): p. 1923-1934.  Goedecke, G.H. and H.J. Auvermann., "Acoustic Scattering by Atmospheric Turbules". JASA, 1997. 102(2): p. 759-771.  Hallberg, B., C. Larsson, and S. Israelsson, "Some Aspects on Sound Propogation Outdoors". Acustica, 1988. 66: p. 109-112.  Higdon, R.L., "Absorbing Boundary Conditions for Difference Approximations to the MultiDimensional Wave Equation". Mathematics of Computation, 1986. 47(176): p. 437-459.  Kuttruff, H., Room Acoustics. 2nd ed. 1979, London: Applied Science Publishers LTD. 309.  Landau, R.H., M. J. Piez, and C. C. Bordeianu, Computational Physics: Problem Solving with Computers. 2nd, Revised and Enlarged ed. 2007, Weinheim: Wiley-VCH. 593.  Lines, L., R. Slawinski, and R. Bording, "A Recipe for Stability Analysis of Finite-Difference Wave Equation Computations". Geophysics, 1999. 64(3): p. 967-969.  McLeod, I.D., C. G. Don, and G. G. Swenson, "Acoustic Pulse Propogation in an Outdoor Turbulent Atmosphere". JASA, 2004. 116(5): p. 2855-2863.  Wilson, D., J. Brasseur, and K. Gilbert, "Acoustic Scattering and the Spectrum of Atmospheric Turbulence". JASA, 1999. 105(1): p. 30-34.