Simulating Sound Wave Propagation

Sound waves travel at a speed of approximately 345 meters per second. As a result, there is a time delay for sound to travel from an emitting source to a listener some distance away. This is more obvious when the distance between the sound source and listener is large, for example when observing fireworks from a few kilometers away.

The time delay that results from this finite speed of propagation can be implemented in a discrete-time simulation with a delay line.

A distance $d$

between source and listener will result in a time delay of $d / c$

seconds (where $c$

is the speed of sound propagation).

The delay line length can be determined as $M = d / (c T_s)$

, where

is the digital sample period (and $f_s$

is the sampling rate in samples per second).

Note that the quantity $c T_s$

represents the distance traveled by sound in a single sample period, which is about 7 millimeters at a sample rate of 48000 Hz.

In this way, we can simulate the propagation of traveling-waves of sound over a specified distance.

To simulate damped traveling-waves, we should include terms that represent the loss experienced over the distance traveled per unit delay, as represented in Fig. 1.

**Figure 1:** A damped traveling-wave simulator.
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For efficiency, distributed damping constants can be “commuted” (assuming linearity) and implemented at a few (or just one) discrete points in the system, as shown in Fig. 2.

**Figure 2:** An efficient damped traveling-wave simulator (frequency-independent losses).
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In reality, these losses will be frequency dependent (typically more losses at higher frequencies) and thus more accurately represented with appropriately designed lowpass digital filters, as shown in Fig. 3

**Figure 3:** A damped, spherical traveling-wave simulator (frequency-dependent losses).
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As mentioned earlier, the amplitude of spherical pressure waves in air is proportional to $1/d$

(due to spherical spreading), where $d$

is the distance of the wavefront from its source. Thus, when simulating the pressure of spherical wavefronts, an additional “spreading” factor should be added as shown in Fig. 3.