Acoustics

Waves in three dimensions

Point source radiates spherical wave into air. Energy is radiating uniformly in all directions. The sound intensity, I, at a distance r is equal to the power dicvided by the surface area of a sphere of radius r

The pressure equation for a diverging spherical wave

p = (A/r)cos(wt - kr)

where A is constant.

Sound intensities and intensity levels at different distance from the source

If r2 = 2r1 then I2 = (1/4)I1.

In terms of intensity levels

b2 - b1 = 10log(I2/I1) = 10log(r1/r2)2

b2 - b1 = 20log(r1/r2)

So if r2 = 2r1

then b2 - b1 = 20log(1/2) = - 6dB i.e. intensity level decreases by 6dB.

 

Interference of sound

The superposition principle applies if two or more waves pass through the same medium simultaneously. Then the resultant wave’s displacement or sound pressure at any point x will be the addition of the displacement or pressure of the component waves.

p(x,t) = p1(x,t) + p2(x,t)

Where the interaction of the waves leads to cancellation or reinforcement it is called destructive or constructive interference respectively.

Sound pressure due to i.e. two waves travelling in the same medium at the same time

p1 =A1cos(wt+a1) and p2 = A2cos(wt+a2)

Algebraically

rresultant = P cos (wt+a)

 

where d is the phase difference between P1 and P2

 

If A1 =A2, P2 = 2A2(1 +cosd) and P =2Acos(d/2)

Since I ~P2

I ~ 4A2cos2(d/2)

When I = 0, as observed in destructive interference, the waves are out-of-phase.

Constructive interference is used to describe the in-phase condition, when I =4A2.

Reflection and Transmission of Waves

For pressure to be continued on reflection and transmission

Ai = Ar +At

Reflection coefficient

R = Ar/Ai

Transmission coefficient

T = At/Ai

Electrical analogy for a propagating wave

  1. pressure due to wave is taken to correspond to voltage,
  2. velocity to current, and
  3. the acoustic characteristic impedance associated with the medium in which a wave is propagating to the electric circuit impedance.

Since the characteristic impedance of medium is the ratio of the pressure and the velocity of wave

Z = cr (measured in Rayl)

Using Z the reflection coefficient

And transmission coefficient

Since the intensity goes as Pm2/2rc the sound power reflection coefficient

ar= Ir/It =R2

And the sound power transmission coefficient

at = It/Ii = 4ZtZi/(Zt+ Zi)2

Standing Waves

When a wave is incident on a boundary, the complete acoustic field will result from the superposition of the incident and reflected waves.

For the simplest case of total reflection, the amplitudes of the incident and reflected pressures

Pmi = Pmr = Bo

Resultant = Bosin(wt-kx) +Bosin(wt+kx) = 2Bosin(wt) cos(kx)

Example:

A standing wave is produced by reflections from two plane parallel boundaries separated by a distance L, with one wall at x = 0, and the other x = L.

The pressure at the walls must be maximum (nodes at minimum displacements)

cos(kx) =1

cos(kL) =1

Hence kL = np; n = 1,2,3,… or k = np/L

Thus, only certain values of k (or wavelengths since k = 2p/l or frequency since f=c/l) are allowed as a result of boundary conditions.

Partial Reflection

Standing wave formed by incident and partially reflected wave with r as a fraction of the incident wave amplitude in the reflected wave

pi = Pi sin(wt — kx)

pr = r Pi sin(wt-kx)

pi + pr =Pi(1 + r) sin(wt)cos(kx) — Pi(1 — r) cos(wt) sin(kx)

If those two result in standing wave

maximum at Pi(1 + r)

and

minimum at Pi(1 — r)

The ration of max/min is called the Standing Wave Ratio (S.W.R.)

The reflection coefficient, ar, which is the function of energy reflected

 

Modulation of Waves

Amplitude modulation is used in commercial broadcasting.

Amplitude modulation occurs in the phenomenon of beats. If two sinusoids having frequencies close together are combined, then the resultant is an amplitude-modulated wave.

Let the two sinusoids be

x1 = Asinwt

and

x2 = A sin[(w+Dw)t]

where the amplitudes are made equal for simplicity and Dw<<w.

The resultant therefore is

Equation represents a wave of frequency w modulated by a frequency Dw/2.

Frequency modulation is used in electronics (sound in TV, instrumentation tape recorders, etc.).

 

Three-dimensional sound waves in a room (enclosure)

If the regular reflection takes place at the walls of the room a complicated three-dimensional standing wave system will be set up.

The frequencies of the various resonant modes are calculated using suitable boundary conditions at the walls.

Since acoustic pressure and wave displacement are always out of phase with each other the acoustic pressure will be a maximum at rigid walls.

3-d wave equation in terms of acoustic pressure

Compared with plane wave

p = P cos (wt —kx)

The wave number (vector) k is now 3-d with magnitude

The numbers nx, ny and nz are the mode numbers. They are called the axial, tangential and obligue resonant mode numbers.

Example

The frequencies, f, of the resonant modes of a rectangular room of dimensions lx, ly, lz are given by

Where c = 340ms-1 (speed of sound in air) and nx, ny and nz are the mode numbers.

Calculate the frequencies of the four lowest tangential and axial (two of each) modes of a room 7m x 5m x 4m.

 

Diffuse Sound Field

A diffuse sound field is defined as one having a constant value of the sound energy density (energy per unit volume).

Resonant modes are therefore excluded since they cause large fluctuations of the energy from point to point in the sound field.

The energy density may be steady or may grow or decay with time.

Steady state condition

Power of sound source = Power absorbed by the enclosure walls

Power = Ia A

Where Ia is the intensity of the absorbed sound, and A is the total area of the walls.

If there are present a number of surfaces of areas A1, A2, A3,… for which the absorption coefficients are a1, a2, a3, …, then the total absorption, a, is given by

 

The absorption coefficient is an important quantity in relation to room acoustics. Since different materials are involved for the proper control of sound in a room, values of their absorption coefficients are required at different frequencies.

Decay of Diffuse Sound Wave

Steady state diffuse sound field in a room and the source is suddenly switched off.

Large number of rays or wave packets propagating in all possible directions will experience reflection or absorption by the walls.

According to the statistical theory, on average a wave packet will experience n reflections by the walls per unit of time

N = Ac/4V

A-total surface area of the walls

c-velocity of sound

V-volume of the room

The energy density will gradually decay until all sound energy gets absorbed by the walls.

If Io is the intensity of sound field at t = 0, after a time t during which there will be n reflections, the intensity will be

where t is the time constant for decaying process.

The time taken for the energy density to decrease to one-millionth of its initial value, as defined by Sabine, is known as the reverberation time, Tr.

Sabine’s formula shows that Tr is directly proportional to the volume of the room, V, and inversely proportional to the total absorption

Reverberation may result in amplification and prolongation of the sound.

Near the source, within about 3m, the direct sound predominates and the room has little effect. Beyond about 10 m the reverberant sound predominates.

The reverberation time for a given enclosure changes with frequency.

Rooms used for the speech should have fairly short reverberation time, so that successive syllables do not overlap. In contrast, rooms used for music should have longer reverberation times.

For concert halls the reverberation time should increase at low frequencies to provide "warmth" of tone, and some earlier reflections to give "presence" or "attack".

Example

A large seminar room is 4 x 6 x 10 m and has a reverberation time of 1.5s.

  1. Determine the total absorption of the room.
  2. If 40 students are in the room and each is equivalent to 0.5 Sabine absorption, determine the new reverberation time of the room.
  3. If a student presenting a seminar speaks with an acoustic power output of 10uW, determine the sound pressure level in the room with and without the students.

Reflection, absorption and transmission of sound

When an acoustic wave is incident on a barrier, some of the incident energy is reflected, some absorbed and some transmitted.

Ii is the intensity of the incident wave, Ir the intensity of the reflected wave, It the intensity of the transmitted wave and Ia the absorbed intensity.

By conservation of energy

Ir + Ia + It = Ii

Hence

r + a + t = 1

Where

is the sound energy absorption coefficient, ...

i.e. sound energy reflection coefficient + sound energy absorption coefficient + sound energy transmission coefficient = 1

The absorption coefficient is a very important for room acoustic.

To control the sound in a room, values of absorption coefficients are required at different frequencies for all materials involved.

There are two standard methods for measuring absorption coefficients

  1. the impedance tube method, and
  2. the reverberation chamber method.
  1. In this method a standing wave is set up in a rigid tube of length at least 1m.

    2. A pure tone is generated by the loudspeaker and a partial standing wave is set up in the tube.

    The values of the pressure at the nodes and antinodes in this partial standing wave depend on the amount of energy absorbed by the sample being tested.

    The pressures at the nodes and antinodes are measured using probe microphone and the standing wave ratio, S.W.R., in the tube determined.

    The sound energy reflection coefficient, r,

     

     

    Because the end cap is solid no energy is transmitted into it, and so the energy transmission coefficient, t, will be zero.

    Hence in this case atube = 1 — r

     

  2. A reverberation chamber is specially designed, large "reverberant room" with highly reflective walls.

Ideally it should be such that a diffuse field can be set up.

It should have a reverberation time of at least 15 seconds (with bare walls).

First the reverberation time of the empty chamber is measured at a number of

different frequencies.

Next large panels of the material under test are placed on the floor, walls etc.

The reverberation times are then remeasured, and finally the coefficient of the test

material for each frequency is deduced.

With the room empty, the reverberation time,

where a = acA

With Xm2 of the surface covered with the material of absorption coefficient ae the total absorption of the chamber is increased to a + X(ae — ac).

The new reverberation time

 

 

Other quantities related to the acoustics of room
    1. Attenuation coefficient

      2. If the sound wave travels a distance x in air, its intensity decreases

      where aatt is called the attenuation coefficient that depends on the frequency of the sound, humidity and temperature of the air.

      (High frequency sound is absorbed more quickly than the low frequency sound.)

    2. The noise-reduction rating

 

 

 

where at is transmission coefficient.

Double brick wall has a noise-rating of 50dB.

The intensity level of the sound transmitted through this wall is 50dB less than the intensity level of the sound incident on the wall.

The frequencies at which the room responds, depend very much on the size of the room and absorption property of all surfaces.

If there were no absorption, standing waves could be set up between the walls, floor and ceiling and the sound would get louder without limit.

Determination of the attenuation coefficient — Pulse Echo Method

An acoustic wave loses energy as it travels through a specimen, and so the amplitude of the wave decays through the specimen.

In general the amplitude, A(x), of the wave after travelled a distance, x, is given by

where Ao is the initial amplitude

a is the attenuation coefficient, measured in neper per metre

In the pulse echo method the potential difference across the output of the transducer is proportional to the amplitude of the acoustic wave incident on it.

The pulse of sound travels across a specimen and gets reflected producing echoes.

The amplitudes of successive echoes decay exponentially with time, and so with distance travelled.

Using graph of ln (peak height) against (peak number) the attenuation coefficient will be determined

Example

In a single-ended pulse echo experiment using a specimen of length 6.5cm, five echoes were observed having amplitudes of 50.0, 21.0, 8.7, 3.6 and 1.5 units, with a mean time of 24.4ms between pulses.

Determine

  1. the mean velocity of sound for acoustic waves in the specimen, and
  2. the attenuation coefficient for acoustic waves in the specimen.