**Physics 4830 Course Notes Thurs
9/20/01**

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**Last time:**
Source Width, Reverberation, dBs

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__Reverberation Time__ is the time it takes the sound to decay away 60
dBs.

**Outline**

- Flutter echoes and other problems
- Typical reverberation times
- Sabine's formula

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**Next Time: **Harmonics

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**Things to Avoid:**

__Flutter echo:__ A common problem occurs when there is distinct path of
reflection off two hard walls. One hears fast, successive echoes, or
"flutter echoes."

__Two distinct decay times:__ This occurs when there are two distinct
chambers, or two distinct paths of reflection, which decay on two fairly
different time scales.

__Reverberation Time__ is the time it takes the sound to decay away 60
dBs.

__Typical reverberation times__:

Practice room: V = 27 m^{3 }, T = 0.6 s (
3m x 3m x 3m)

Rehearsal room: V = 600 m^{3} , T_{ }=
1.4 s ( 6m x 10m x 10 m)

Large concert hall: V = 20,000 m^{3 }, T_{
}= 2.2 s (20m x 32m x 32m)

Reverberation times should be consistent with visual cues, which tell us what the size of the room is. A desirable reverberation time also depends on the type of music. Chamber music (a string quartet, for example) benefits from a shorter reverberation time than orchestral music. See Figure 23.7

It is also important that the lower frequencies have a longer reverberation time. This is what gives warmth to the sound. See Figure 23.8

__Sabine's formula__

T = 0.16 V / A

where T is the reverberation time, V is the volume of the room and A is the surface area of absorbing material.

This formula is fairly general and we will talk shortly about how to determine A in this equation. But, for now let's assume that the walls in our room are completely absorbing. That is, any sound that hits the wall is absorbed and there is no reflected sound. We can then exactly derive Sabine's formula. This is the most math I will use in the course and I will keep this derivation short.

Let's say we play a steady tone for a long time in our room the sound energy will build up and we will call the energy density: e. We will assume the energy density is constant through out the room

e = sound energy density Joules/m^{3}

The power (Joules/sec) being dissipated in the walls will be the area of the walls times the intensity (Watts/m2) of the sound hitting the walls.

P = A I

The power lost to the walls must equal the time rate of change (calculus!) of the energy.

d/dt (e V) = - P = - AI

V is constant and the intensity equals the energy density times the speed of sound divided by 4. We divide by 4 because sound is going out in all directions, left, right, forward, backward. We do not divide by 6. Why not? Because 4 gives a better answer (closer to what is actually measured). The physical reason is that sound is not really radiating uniformly in all directions so the number must be something less than 6. Sound tends to reflect in a planar surface, which is consistent with using 4 instead of 6.

de/dt V = - A e/4 v

de/dt = - Av/(4V) e

e = e_{0} exp ( - Av/(6V) t )

ln (e/e_{0}) = - A v / (6 V)
T

T = -ln (10^{-6}) x 4 / 344 V / A

= 13.8 x 4 / 344 V/A = 0.161 V/A

Can you tell me what assumptions went into this derivation?

__Absorption Coefficients__

In reality, surfaces do not purely absorb or purely reflect.

The absorption coefficient is the amount of power absorbed divided by the amount of incident power.

The way we estimate the absorbing area, A, in Sabine's formula is by weighting absorbing areas by their absorption coefficient.

A = C_{1} x A_{1} + C_{2} x A_{2} + C_{3}
x A_{3} + …

V = the total volume of the room in cubic meters

A_{i} = surface area

C_{i} = absorption coefficients

A is the total "fully-absorbing" surface area of the room. Each surface is weighted by its absorption coefficient. You add up all surfaces. Each surface contributes its surface area times its absorption coefficient.

These calculations can become quite involved. You have to take into account
windows, doors, hallways, the audience, empty seating, etc. Every surface has a
different absorption coefficient.

C is the fraction of energy lost each time the sound wave reflects off the surface. The value of A ranges from 0-1. 0.01 would be highly reflective, 0.3 is fairly absorbing.

Sabine's formula is approximate. However, it works reasonably well if the average absorption coefficient is less than 0.15. That is,

A / A_{total} < 0.15

It also gives a way to compare various options when choosing size, shape and materials. There are computer programs which do "ray tracing." That is, the computer simulates the motion of the wave fronts and how they reflect off the surfaces and how they are absorbed (CATT Acoustic is the program I am familiar with). These programs also have "diffusion." That is, a way to mimic how sound diffuses by scattering off of complex objects. Ray tracing is also used to simulate how light reflects off objects for three-dimensional visualization and animation. Another common practice is to make a scale model of the concert hall and test the model with higher frequency sound sources so that the wavelengths are in proportion to the size of the model. This was not done for Philharmonic Hall!