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 between source and listener will result in a time delay of seconds (where is the speed of sound propagation).
The delay line length can be determined as
, where is the digital sample period (and is the sampling rate in samples per second).
Note that the quantity 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.
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, as shown in Fig. 3
Figure 3:
A damped, spherical traveling-wave simulator (frequency-dependent losses).
As mentioned earlier, the amplitude of spherical pressure waves in air is proportional to (due to spherical spreading), where 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.