Using Delay to Simulate 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.

• 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 $T_s = 1 / f_s$ 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. 1.

  Figure 1: A damped traveling-wave simulator.

  

• 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).

  

• 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 3: A damped traveling-wave simulator.