It is a complicated problem to determine the exact effect that reflections have on a direct signal. The resulting superposition (or cancellation) depends on both the relative amplitude and phase of each wave. The amplitude of the reflected signal (and phase rotation) can vary according to the distance travelled by the wave, the wavelength of the wave, the angle of incidence with each reflective surface, and the absorbent nature of the room among other things.

For demonstrative purposes, we will consider a general equation for the time it takes a reflected wave to arrive at a listening position relative to the direct sound. The equation is given by:

*t = (d _{2} – d_{1}) / c*

where:

*d*_{1}_{ }= direct path distance (m)

*d*_{2 }= reflected path distance (m)

*c* = speed of sound (344 m/s)

At a given frequency, the relative phase of the direct signal and the reflected signal will determine how much amplitude is added or subtracted from the direct sound.

A formula for the relative phase shift is shown to be:

*P = 360 lp / λ*

where:

*lp = *the difference in path length between the reflected and direct signal (m)

*λ = *the wavelength (m)

As an example, consider a sound wave at 688 Hz, at which the wavelength is *0.5* m. If the direct signal arrives at the listening position after traversing just *1 *m, and the reflected signal reaches the listener after 2m, then the resulting path difference is just 1m. The phase shift formula would yield:

*P = 360 (1 / 0.5) = 720°*

By dividing the result by *360 °* (one full sound wave revolution), one can find the phase difference between the direct and reflected sound at the listening position, which is

*0*in the above result. A

*°**0*phase difference means the reflected wave will add to the pressure of the direct wave (superposition).

*°*Contrast this to heavy cancellation of the direct signal, which occurs when the reflected wave is completely out of phase with the direct signal. For instance, with the same sound wave at *688* Hz, and a path length difference of *0.75* m, the phase shift is:

*P = 360 (0.75 / 0.5) = 540°*

This phase shift of the reflected signal relative to the direct signal will cause the reflection then to arrive completely out phase (*180 °*) with the direct signal. This will drop the pressure of the direct signal. The amount of cancellation that takes place depends on the amplitude and phase of the reflected signal.

The amount of cancellation or superposition varies according to several factors as mentioned earlier. The absorption coefficient of the walls can change both the amplitude and phase of the reflected signal, as can the angle of incidence. To further complicate things, absorption coefficients are frequency dependent, which means that certain frequencies will reflect with more or less amplitude than others. An increase in distance travelled will also decrease the level of the reflected wave. All of these factors are the reason why sound rooms are such delicate places. Proper setup and awareness of monitoring conditions are crucial in sound room design.

### References

[1] Newell, P., *Recording Studio Design*, 3^{rd} ed., Focal Press, Burlington, MA, 2013.