 # Reverberation

Reverberation refers to the combined efforts of multiple echoes that takes place in a concert hall or room and is heard by a listener. Furthermore, reverberation exists in almost all indoor spaces but does not exist in most outdoor venues. Moreover, if the walls of the room are hard, there would be a repetition of echoes which may cause the muddling of the sounds.

## Introduction to Reverberation

Reverberation refers to the collection of reflected sounds that take place from an enclosure’s surfaces like in an auditorium. Furthermore, the desirable property of auditoriums is to such an extent that it facilitates the overcoming of the sound intensity’s inverse square law dropoff in the enclosure. However, in case it is excessive, the sound would become muddy.

The reverberation time of a room shows the duration of the time period for which acoustic energy remains in a room. Furthermore, experts usually define it as the time it takes for the energy density or acoustics intensity to decrease by a factor of one million (60 dB). Since a whisper is about 40dB and a reasonably loud clap is about 100 dB (SPL), one can estimate a room’s reverberation time by clapping and then listening to how long one can hear the sound that remains from the clap. ### How do we Measure Reverberation?

In an existing room, one can go on-site to carry out measurement of the reverberation time by using a sound level metre and loudspeaker. Furthermore, one can also calculate the reverberation time by making use of the Sabins Formula whose attribution is to Wallace Clement Sabine. Moreover, the basis of this equation is the volume of the space as well as the total amount of absorption within a space.

Sabins refers to the total amount of absorption within a space. Most noteworthy, in order to calculate Sabins, a consideration for the surface area and absorption must take place for every material within a space.

To obtain the sabin values, we use the volume (length x width x height) of the room as well as the surfaces materials. Moreover, one can seek help from a table that contains the sabins at each octave band frequency for materials used in construction. Finally, the addition of the sound absorption to the calculation can take place.

### Formula of Reverberation

Formula for Sabins:

a = Σ S α

Where:

Σ = sabins (total room absorption that exists at a given frequency)

S = surface area of material (feet squared)

α = sound absorption coefficient that exists at a given frequency or the NRC

After the calculation of a, one can make use of the Sabine Formula for calculating the reverberation time.

Sabine Formula:

RT60 = 0.049 V/a

Where:

RT60 = Reverberation Time

V = volume of the space (feet cubed)

a = sabins (total room absorption that is present at a given frequency)

### Derivation of the Formula of Reverberation

The derivation of a simple formula for the estimation of a room’s reverberation time takes place by considering a simple model for the room. Let’s begin with a room made with perfectly reflecting walls that has a length L, width W, and height H.

Furthermore, there is a window on one wall with a total area A. Moreover, the sound is lost if it goes out the window but otherwise it would just bounce off the walls. Also, the window is a model for area A, an effective absorbing.

Now focus on the sound that travels horizontally along the length L with the speed of sound, c. The total time it takes for the sound to travel from the window wall and return is 2L/c.

Every time the sounds travels from the window wall and returns, the fraction of the sound waves which hits the window is lost. Furthermore, this means that a fraction A/WH is lost. Moreover, WH refers to the area of the window wall.

So, for each time interval T = 2L/c, there is a loss of fraction A/WH of the sound energy. In other words,  (1-A/WH) of the sound remains. Moreover, for a time 2T, there is a loss of fraction twice.

After the first, there would be (1-A/WH) left, and after the second, a loss of a fraction A/WH of that or (1-A/WH)(A/WH) more takes place.  Consequently, what remains is a total of 1-A/WH-(1-A/WH)(A/WH).

On combining these together, what would be left is (1-A/WH)2. Moreover, for a total time of nT, the fraction that would be left is (1 – A/WH)n.

Now one must find the value of nT when (1-A/WH)n = 1/1,000,000. Taking the log (base ten) of both sides, the result would be
n log(1-A/WH) = -6
In case the A/WH is smaller than one, that is A/WH << 1, then one must use the approximation
log(1-A/WH) = -A/WH .
This is a good approximation because room absorption is almost always small enough. Solving for n,
n = 6 WH/A
Hence, nT is then
nT = 12LWH/cA = 12/c times the room’s volume, V = LWH, whose division take place by the (effective) absorbing area, A. Most importantly, the result would be the same if the window was on any of the other walls.

Now, the expectation is that 1/3rd of the energy would travel in each of the three dimensions. As such, the expectation is that the loss would be about 1/3rd of the result. Consequently, the expectation is that sound energy would be three times longer.

Hence, reverberation time, TR = 36 V/cA.

This approximate formula is very close to the result that is derived experimentally by W. C. Sabine and later derived in more detail by W. S. Franklin. The measurements with more careful derivation yield approximately 55 V/cA.

## FAQs on Reverberation

Question 1: Explain what is reverberation in physics?

Answer 1: Reverberation refers to the combined efforts of multiple echoes that would happen in a concert hall or room and is heard by a listener. Furthermore, reverberation does not exist in most outdoor venues but exists in almost all indoor spaces.

Question 2: What is reverberation time in the field of Physics?

Answer 2: The reverberation time of a room indicates the duration of the time period for which there would be the presence of acoustic energy in a room. Furthermore, experts usually define it as the time it takes for the energy density or acoustic intensity to diminish by a factor of one million (60 dB).

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