Acoustic Phenomena

Acoustic Phenomenom

The study of mechanical wave of longitudinal behaviour (audible sound, infra sonic, ultra sonic sound) in solid, liquid and gases is called acoustic and corresponding wave phenomena is called acoustic phenomenon.

Pressure Amplitude

The maximum value by which pressure of a medium increases or decreases when wave propagates through the medium is called pressure amplitude. It is denoted by symbol (Po).

Let us consider an imaginary air cylinder BC of a small length 'Δ x' having cross sectional area 'A'. The volume of a cylinder is given by,
$$V = A \Delta x............(i)$$
When wave propagates through the cylinder the air particles gets displaced from their mean position. Let y1 and y2  be the displacement of particle along left and right respectively. The new volume of a cylinder is given by ,
$$V' = A (\Delta x + y_{2} - y_{1})............(ii)$$

Now the change in volume of cylinder is,
$$\Delta V = V' - V$$
$$= A (\Delta x + y_{2} - y_{1} - \Delta x)$$
$$=A (y_{2} - y_{1})$$

The change in volume per unit orginal volume of the cylinder is given by,

$$\frac{\Delta V}{V} = \frac{A (y_{2} - y_{1})}{A \Delta x}$$
$$\frac{\Delta V}{V} = \frac{y_{2} - y_{1}}{ \Delta x}$$
$$\frac{\Delta V}{V} = \frac{\Delta y}{ \Delta x}$$
For cylinder of small length,
$$\frac{\Delta V}{V} = \frac{d y}{ d x}$$

Let, B be the bulk modulus of elasticity of air then,
$$B = \frac{Stress(P)}{Strain \frac{\Delta V}{V}}$$
$$P = - B (\frac{\Delta V}{V})$$
(-ve) sign indicates that when pressure increases volume of the medium decreases.

Using eqn (iii) we get,
$$P = - B \frac{dy}{dx}$$

Since we know,
y = a Sin (
ωt - kx)
$$\frac{dy}{dx} = a \cos (\omega t - kx) . (-k)$$
$$\frac{dy}{dx} = -ak \cos (\omega t - kx)$$

Putting this value in eqn(iv) we get,
P = B ak Cos (
ωt - kx)

Since, Po = Bak is the pressure amplitude
P = Po Cos (
ω- kx)

This is the required pressure equation.

Comparing eqn(v) and (vi) we  see that when displacement increases press will decrease and vice versa.

Musical Sound

The sound that gives pleasing effect to our ear is called musical sound. It is regular, continuous, long lived sound without sudden change in amplitude. Example: sound from guitar, flute etc.

Noise

The sound that gives unpleasing effect to our ear is called noise. It is irregular, discontinuous amd short lived with sudden change in amplitude. Example: Noise from a classroom, sound produced from vehicles etc.

Characteristics of musical sound

1. Pitch
It is the characteristic of musical sound which helps us to distinguish a sharp or shrill sound from hoarse or grave sound. Hoarse sound is of low pitch and shrills  sound is of high pitch. It depends upon freuency of sound as well as the relative motion  between source of sound and listener. It is subjective quantity and can't be measured. It's value is zero for a deaf person. Example: The voice of a child or a lady is shriller than that of a man.
Similary, humming of mosquito is of high pitch and roaring of lion is of low pitch.
Pitch \propto frequency
2. Loudness
It is a characteristic of a musical sound which helps us to distunguish loud sound from the faint one both having same pitch. It depends upon intensity of the sound. Higher the intensity of the sound greater will be the loudness of sound and viceversa.
It is also a subjective quantity which can't be measured and it's value is zero for a deaf person. Beside intensity the loudness of sound depends upon amplitude of sounding body, density of the medium, surface area of the source of sound, distance between source of sound and listener, direction of motion of air etc.
Loudness \proptp frequency
3. Quality of sound
It is that characteristic of a musical sound whcih distunguish between sound of same pitch and loudness produced by different musical instrument. It is also subjective in character and depends upon number of harmonics, their order, and their relative intensities.
Due to quality of sound we can reconize our friends voice without seeing him/her.

Intensity of sound

The intensity of sound at any point into a medium can be defined as amount of energy crossing the point per unit area per unit time. It is denoted by symbol 'I'. It is given by,
$$I = \frac{E}{A. t}$$
It's unit is watt per meter square.

Expression for intensity of sound

The displacement of vibrating layer of air is given by,
y = a Sin
ωt ............(i)
where, a = amplitude,
ω= angular velocity

The rate of change of displacement of a particle is called velocity.
$$v= \frac{dy}{dt} = \frac{d}{dt}(a \sin \omega t)$$
= a
ω cos ωt

The K.E of a vibrating layer of air is given by,
$$KE = \frac{1}{2} m v^{2}$$
$$= \frac{1}{2} m (a \omega \cos \omega t)^{2}$$
$$= \frac{1}{2} m a^{2} \omega^{2} \cos^{2} \omega t$$

Then mamimum value of KE is given by,
$$(K.E)_{max} = \frac{1}{2} m a^{2} \omega^{2}$$ (For Cos2
ωt = 1)

This maximum value of K.E is equal to the total energy of wave. So, E= ½ m a2 ω2

Now intensity of a wave,
$$I = \frac{E}{A t}$$
$$= \frac{m a^{2} \omega^{2}}{2 A t}$$
$$= \frac{1}{2} V \rho a^{2} \omega^{2} \frac{1}{A t}$$
$$= \frac{1}{2} A l \rho a^{2} \omega^{2} \frac{1}{A t}$$
$$= \frac{1}{2} \rho a^{2} \omega^{2} \frac{l}{t}$$

Since, l/t = v
I = ½ v
ρ a2 ω2

This is the required expression for intensity of wave. Since, v, ρ and ω are constant,
I \propto a2

Hence, intensity of sound is directly proportional to sqaure of amplitude.

Threshold of hearing

The minimum intensity of sound that can be heard by a normal ear is called threshold of hearing. It is denoted by symbol 'Io'. For a frrequency range of 1000Hz , it's value is 10-12 watt per meter square.

Intensity level

The logarithium of intensity of sound to the intensity during threshold of hearing is called intesnity level.
$$Intensity\ level (\beta) = log (\frac{I}{I_{o}})$$

Let, L and Lo be the loudness of sound corresponding to intensity I and Io. Then,
$$L \propto \log{I}$$
$$L = k \log{I} ............(i)$$

Similarly,
$$L_{o} \propto \log{I_{o}}$$
$$L_{o} = k \log{I_{o}} ............(ii)$$

Subtracting eqn (ii) from (i),
$$L - L_{o} = k \log{I} - k \log{I_{o}}$$
$$L - L_{o} = k (\log{I} - \log{I_{o}})$$
$$L - L_{o} = k \log{(\frac{I}{I_{o}})}$$

The value of loudness during threshold of hearing is assumed to be zero. Then,
$$L = k \log{(\frac{I}{I_{o}})}$$

Since, k = 1
$$L = \log{(\frac{I}{I_{o}})} = \beta$$

The unit of intensity level is bel. and decibel(dB).

1 bel

The intensity level of sound is to be 1 bel when the intensity of sound is equal to 10 times the intensity during threshold of hearing.
i.e For I = 10 Io

1 Decibel

Decibel is the smallest unit of intensity level. 1 decibel can be defined as 1/10th of 1 bel.

Beat

The phenomenon of alternative rise or fall in the intensity of resultant wave formed by the superposition of two waves having equal amplitude but slightly different frequencies is called beat. The time interval between two succesive beat is called beat period and reciprocaal of beat period is called beat frequency.

The beat frequency is equal to the difference in frequencies of two super imposing waves. It is denoted by symbol fb.
For f1 > f2 , fb = f1 - f2

Expression for beat frequency

Let us consider two waves having equal amplitude 'a' but slightly different frequencies f1 and f2 propagating into a medium. The displacement of a particle due to these waves is given by,

y1 = a Sin ω1 t
or, y1 = a Sin 2
π f1 t ............(i)

Also,
y2 = a Sin
ω2 t
or, y2 = a Sin 2
π f2 t ............(ii)

Using principle of superposition of wave we get,

y = y1 + y2
$$= a (\sin 2 \pi f_{1} t + \sin 2 \pi f_{1} t)$$
$$= a 2 \sin(\frac{2 \pi f_{1} t + 2 \pi f_{2} t}{2}) \cos(\frac{2 \pi f_{1} t - 2 \pi f_{2} t}{2})$$
$$= a 2 \sin 2 \pi (\frac{ f_{1} + f_{2}}{2})t . \cos 2 \pi (\frac{ f_{1} - f_{2}}{2})t$$
$$= A \sin 2 \pi f t ............(iii)$$

where, $$f = (\frac{f_{1} + f_{2}}{2})$$ be the frequency of resultant wave
$$A = 2 a \cos 2 \pi (\frac{f_{1} - f_{2}}{2}) t$$ be the amplitude of resultant wave.

Condition of maxima(Maximum Intensity)

For maximum intensity, amplitude of resultant wave should be maximum. For this,
$$\cos 2 \pi \frac{(f_{1}-f_{2})}{2} t = \pm 1$$
$$\cos \pi (f_{1}-f_{2}) t_{n} = \cos n \pi$$ where n= 0,1,2...
$$(f_{1}-f_{2}) t_{n} = n$$
$$t_{n} = \frac{n}{f_{1}-f_{2}}$$

The time interval between two succesive maxima is called beat period.
Beat period(T) = t2 - t1
$$= \frac{2}{f_{1}-f_{2}} - \frac{1}{f_{1}-f_{2}}$$
$$= \frac{1}{f_{1}-f_{2}}$$

Beat Frequency
Frequency of maxima = 1\ t = f1 - f2 ............(4)

Condition of Minima

For minimum intensity, amplitude of resultant wave should be minimum. For this,
$$\cos 2 \pi \frac{(f_{1}-f_{2})}{2} t = 0$$
$$\cos \pi (f_{1}-f_{2}) t_{n} = \cos (2n+1) \frac{\pi}{2}$$ where n= 0,1,2...
$$(f_{1}-f_{2}) t_{n} = \frac{(2n+1)}{2}$$
$$t_{n} = \frac{(2n+1)}{2(f_{1}-f_{2})}$$

The time interval between two succesive mainima is called beat period.
Beat period(T) = t2 - t1
$$= \frac{5}{2(f_{1}-f_{2})} - \frac{3}{(f_{1}-f_{2})}$$
$$= \frac{1}{f_{1}-f_{2}}$$

Beat Frequency
Frequency of maxima = 1\ t = f1 - f2 ............(5)

From eqn (4) and (5) we see that frequency of maxima and minima which is equal to difference in frequency of two super imposing waves.
Beat frequency(fb) = f1 - f2

Doppler's Effect

The phenomena of apparent change in frequency of sound due to the relative motion between source of sound and the observer is called Doppler's Effect. Example: The sound of engine of train is more shriller when it is approaching towards a stationary observer than as it passes away.

1. Observer in motion and source at rest
• Observer moving towards source

Let us consider a source of sound S and an observer O. The source of sound is at rest while observer is moving towards source with velocity uo . Let v and λ be the velocity and wave length of sound emitted by the source. Then the frequency of sound is given by,
$$f = \frac{v}{\lambda}............(i)$$
Since, source is at rest . wave length of sound remains constant. Hence, apparent change in frequency is due to the change in velocity of sound with respect to the motion of observer.
The relative velocity of sound w.r.t. observer is,
v' = v - (-uo)
or, v' = v + uo ............(ii)
The apparent frequency ,
$$f^{'} = \frac{v^{'}}{\lambda^{'}}= \frac{v + u_{o}}{\lambda}$$(Since, wave length is constant)
$$f^{'} = \frac{v + u_{o}}{\frac{v}{f}}$$
$$f^{'} = (\frac{v + u_{o}}{v}) .f$$
Since, v + uo > v ; f' > f
Hence, apparent frequency increases when an observer moves towards a stationary source of sound.
• Observer moving away from the source

Let us consider a source of sound S and an observer O. The source of sound is at rest while observer is moving away source with velocity uo . Let v and λ be the velocity and wave length of sound emitted by the source. Then the frequency of sound is given by,
$$f = \frac{v}{\lambda}............(i)$$
Since, source is at rest, wave length of sound remains constant. Hence, apparent change in frequency is due to the change in velocity of sound with respect to the motion of observer.
The relative velocity of sound w.r.t. observer is,
v' = v - uo
or, v' = v - uo ............(ii)
The apparent frequency ,
$$f^{'} = \frac{v^{'}}{\lambda^{'}}= \frac{v - u_{o}}{\lambda}$$(Since, wave length is constant)
$$f^{'} = \frac{v - u_{o}}{\frac{v}{f}}$$
$$f^{'} = (\frac{v - u_{o}}{v}) .f$$
Since, v - uo < v ; f' < f
Hence, apparent frequency decreases when an observer moves away a stationary source of sound.
2. Source in motion and observer at rest
• Source moving towards observer

Let us consider a source of sound S and an observer O. The observer is at rest while source is moving towards observer with velocity us . Let v and λ be the velocity and wave length of sound emitted by the source. Then the frequency of sound is given by,
$$f = \frac{v}{\lambda}............(i)$$
Since, source is moving towards the observer when emitted per second by the source are compressed in between space (v - us). Hence, wave length of sound always decreases. The change in wave length of sound is,
$$\lambda^{'} = \frac{v - u_{s}}{f}............(ii)$$
The apparent frequency of the sound is ,
$$f^{'} = \frac{v^{'}}{\lambda^{'}}$$
$$f^{'} = \frac{v}{\frac{v - u_{s}}{f}}$$
$$f^{'} = (\frac{v}{v - u_{s}}) .f$$
Since, v > v -us ; f' > f
Hence, apparent frequency increases when a source of sound move towards stationary observer.
• Source moving away from observer

Let us consider a source of sound S and an observer O. The observer is at rest while source is moving away from observer with velocity us . Let v and λ be the velocity and wave length of sound emitted by the source. Then the frequency of sound is given by,
$$f = \frac{v}{\lambda}............(i)$$
Since, source is moving away the observer when emitted per second by the source are compressed in between space (v + us). Hence, wave length of sound always decreases. The change in wave length of sound is,
$$\lambda^{'} = \frac{v + u_{s}}{f}............(ii)$$
The apparent frequency of the sound is ,
$$f^{'} = \frac{v^{'}}{\lambda^{'}}$$
$$f^{'} = \frac{v}{\frac{v + u_{s}}{f}}$$
$$f^{'} = (\frac{v}{v + u_{s}}) .f$$
Since, v < v + us ; f' < f
Hence, apparent frequency decreases when a source of sound move towards stationary observer.
3. Source and Observer both in motion
• Source and observer moving towards each other

Let us consider source of sound 'S' is moving towards observer 'O' moving with velocity 'uo' with velocity 'us' towards each other . Let v and λ be the velocity and and wavelength of sound emitted by the source. Then frequency of sound is given by,
$$f = \frac{v}{\lambda}$$
Since source and observer are moving towards each other, the apparent change in frequency is due to change in velocity of sound w.r.t observer as well as change in wave length .
Relative velocity of sound w.r.t observer = v- (-uo) = v + uo
Change in wave length of sound is given by,
$$\lambda^{'} = \frac{v - u_{s}}{f}$$
The apparent frequency of sound ,
$$f^{'} = \frac{v^{'}}{\lambda^{'}}$$
$$f^{'} = (\frac{v + u_{o}}{v - u_{s}}) .f$$
Since, v + uo > v - us ; f' >f
Hence apparent frequency will increase when source of sound and observer moves towards each other.

Special Thanks(Credits) : Hari Saran Regmi Sir