Using Delay to 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 with a delay line such as that shown in Fig. 11.

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 which represent the loss experienced over the distance traveled per unit delay, as represented in Fig. 14.

**Figure 14:** 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. 15.

**Figure 15:** 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.

**Figure 16:** A damped traveling-wave simulator.
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The amplitude of spherical pressure waves in air is proportional to $1/r$

, where

is the distance of the wavefront from its source. This scaling coefficient is attributable to the spreading of a given energy over an expanding spherical surface (the surface area of a sphere is equal to $4 \pi r^2$ and pressure amplitudes are proportional to the square root of energy).

On the other hand, guided planar wavefronts (such as in pipes) do not experience this attenuation because their wavefronts do not expand.