Acoustic Phenomena - Class 12 Physics

Topics to be covered on Mechanical Waves:

1. Speed of Sound wave in Any Medium
2. Speed of Sound in Solid
3. Speed of Sound in Liquid
4. Speed of Sound in Air or Gas
5. Factors Affecting the speed of Sound in Gases or Air

The branch of physics that deals with me the process of production , Transmission and reception of sound is called Acoustics.

Pressure Amplitude and Musical Sound

The branch of physics that deals with the process of the production, transmission and reception of sound is called acoustics. Some of the important fields of acoustic are

1. The design of acoustical instruments.
2. Electroacoustic i.e microphones, amplifiers, loudspeakers etc.
3. Architecture acoustics dealing with the designs and construction of buildings, music balls, recording rooms in radio and television stations.
4. Musical acoustic dealing with the design of musical instruments.

Pressure Amplitude

A longitudinal wave can be represented by a sinusoidal wave with a definite frequency, wavelength and amplitude. Such waves, travelling in x-direction can be represented as

$$ y = a\sin (\omega t – kx) \dots (i) $$

Where a = amplitude, ω = angular frequency, and k = wave number. The displacement y in a longitudinal wave is along the direction of wave travel and so, x and y are parallel to each other. The amplitude is the maximum displacements of a particle in the medium from its equilibrium position which is also called displacement amplitude.

Let \(\Delta P\) be the instantaneous pressure fluctuation in a sound wave at a point x at time t. The absolute pressure at that point is \( P + \Delta P\) where P is the atmospheric pressure. Consider a cylinder of air of cross-section area A and axis along the direction of wave travel as shown in the figure. Volume of the cylinder having length \(\Delta x\) at normal condition is \( V = A\Delta x\) when there is waves. When a wave is present, left cross-section of the cylinder displaces through a distance y₁ and the right cross-section through a distance y₂. If y₂ > y₁, the volume of the cylinder increases and this cause a decrease of the pressure. If y₂<y₁, the volume of cylinder decreases causing an increase in pressure. The pressure fluctuation depends on the difference between the displacements at neighboring points in the medium. So, the change in volume,

$$ \begin{align*} \Delta &= A(y_2 –y_1) = a\Delta y \\ \text {In the limit as} \: \Delta x \rightarrow 0, \: \\ \text {the fractional change in volume dV/V is} \\ \frac {dV}{V} &= \lim_{\Delta x \to 0} \frac {A\Delta y}{A\Delta x} = \frac {\delta y}{\delta x} \dots (ii) \\ \end{align*} $$ The pressure variation in the cylinder due to fractional change in volume is given by $$ \begin{align*} \Delta P &= -B \frac {\Delta V}{V} \dots (iii) \\\end{align*} $$

$$ \begin{align*} \text {where B is the bulk of modulus of air. }\\ \text {From equations} \: (ii)\: \text {and} \: (iii), \: \text {we have} \\ \Delta P = -B\frac {\delta y}{\delta x} = Bak\cos (\omega t – kx) \dots (iv) \\ \text {Let} \Delta P_m = Bak, \end{align*} $$  the pressure amplitude which is the maximum increase or decrease in pressure due to wave. The above equation can be written as $$ \begin{align*}\Delta P = \Delta P_m \cos (\omega t – kx) \dots (v) \\ \end{align*} $$

The compressions where the points have lowest pressure and density are points of zero displacement and the rarefaction where the points have lowest pressure and density are points of zero displacement. Substituting the value B from equation $$ \begin{align*} v = \sqrt {B/ \rho } , \text {the pressure amplitude is given by} \\ \Delta P_m = Bak = v^2 \rho ka \dots (vi) \\ \end{align*} $$

Thus pressure amplitude is directly proportional to the displacement amplitude and this amplitude is very small.

Musical Sound and Noise

Musical Sound

A desirable sound that produces a pleasing effect on the listeners is called musical sound. Such sound is produced by regular and periodic vibrations. Sounds produced by tuning a fork, flute, piano etc are music sounds.

Noise

The sound that produces an unpleasant effect on the listener is called noise. It is an unpleasant discontinuous sound produced by an irregular succession of disturbances. All sounds other than musical notes are noises.

Difference between Musical and Noise

Characteristics of Musical Sound

Pitch

It is a sensation experienced by a listener. It is not a subjective term which cannot be measured quantitatively. The pitch of a sound depends upon the frequency of vibration of air. If the frequency of a sound is high, its pitch is high and if the frequency of sound is low. The high pitch sound is shrill and the low pitch is grave. The voice produced by ladies and children is high pitched.

Loudness

Loudness in a subjective sensation depends on the intensity and sensitiveness of the listener’s ear. The intensity of sound is sound energy flowing per second per unit perpendicular to the direction of travel of the sound. Loudness or intensity depends upon the following factors:

1. The amplitude of the vibration of source: Greater is the amplitude of vibration of the source, larger is the intensity and sound.
2. The motion of the medium: If the wind is blowing in the direction of propagation of sound, the loudness of the sound increases.
3. The presence of other bodies: The loudness of sound is increased due to the presence of other bodies near the source of a sound.
4. The surface area of the vibrating body: The greater the surface area of a vibrating body, the larger is loudness of sound.
5. The frequency of sound: The loudness of a vibrating body is directly proportional to the square of the frequency of the vibrating body.

Quality of Timer

It is a measure of the complexity of sound which enables us to distinguish between two sounds of the same pitch and loudness produced by two different sources. The note produced by an instrument does not contain a single frequency but consists of a fundamental and overtones with smaller intensity.

Difference between Intensity of Sound and Loudness

Threshold of Hearing

The lowest intensity of sound that can be heard by an ear is called the threshold of hearing. The magnitude of hearing sensation depends upon both the intensity and frequency of the sound. The threshold of hearing is 10^-12 Wm^-2 for a pure tone of frequency 1 kHz.

Intensity of Sound

Intensity of sound is defined as the average rate of transfer of energy per unit area cross a surface perpendicular to the direction of propagation of the sound. So, we can write intensity of sound,

$$ \begin{align*} I = \frac PA \\ \end{align*} $$ where P is the times rate of energy transfer of power of the sound wave and A, the surface area intercepting the sound. Consider a sound wave traveling in x-direction. Displacement of a particle when the wave is passing through the medium is $$ \begin{align*} y = a\sin (\omega t – kx) \dots (ii) \\ \end{align*} $$

The speed of an oscillating particle of the medium is $$ \begin{align*} \\ u = \frac {dy}{dt} = a \omega \cos (\omega t – kx ) \dots (iii) \\ \end{align*} $$

The power is equal to the product of force and velocity. So, the power per unit area is equal to the product of pressure and velocity.

$$ \begin{align*} \\ \text { So,} \\ \Delta P \times u = [Bak\cos (\omega t – kx)] \times [a\omega \cos (\omega t – kx)] \\ = Bk\omega a^2 \cos ^2 (\omega t – kx) \\ \end{align*} $$

Since the intensity is the time average rate, the average value of the function

$$ \begin{align*}\: \cos ^2 (\omega t – kx) \: \text {over a period T} = 2\pi /\omega: \text {is}\: \frac 12 \: \text {and so} \\ I = \frac 12 Bk\omega a^2 \dots (iv) \\ \text {As} \: \omega =2\pi f = 2\pi v/\lambda , \: \text {and} \: v^2 = B/\rho, \: \text {we have} \\ I = \frac 12 \sqrt {B\rho } \omega ^2 a^2 \dots (v) \\ \text {This equation can be modified as} \\ I = \frac 12 \rho v^2 k \omega a^2 = \frac 12 \rho v\omega ^2 a^2 \dots (vi) \\ \end{align*} $$

The intensity of sound can be expressed in terms of pressure amplitude as

$$ \begin{align*} \\ I = \frac 12 Bk\omega a^2 \\ \text {or,} \: I = \frac {\omega \Delta P_m^2 }{2Bk} = \frac {v\Delta p_m^2}{2B} \dots (vii) \\ \text {As} \: v^2 = B/\rho , \: \text {then} \\ I = \frac {\Delta P_m^2}{2\rho v} = \frac {\Delta P_m^2}{2\sqrt {\rho B}} \dots (viii) \\ \end{align*} $$

Comparing equations (v) and} (vii),we see that the sound wave of the same intensity but different frequency have different amplitudes a but the same pressure amplitude, \(\Delta P_m. \) 

Intensity Level (Weber-Frechner Law)

The intensity of sound which the human ear can hear from 10^-12 W/m² to 1 W/m². Experiment shows that to produce an apparent doubling in loudness, the intensity of sound must be increased by a factor of 10. Intensity level \(\beta \) is defined as

$$ \beta = \log _{10} \frac {I}{I_0} $$

Where I is the measured intensity and I₀ is a reference intensity chosen as 10^-12 W/m², threshold of hearing at 1000 Hz. The sound intensity levels are expressed in bels in SI-units, a unit named to honour Alexander Graham Bell. However, it is a larger unit and a smaller unit, called the decibel is used.

$$ \beta = (10\: dB) \log _{10} \frac {I}{I_0} \: [\because 1 bel =10\: dB ]$$

Beats

When two sound waves of slightly different frequencies but similar amplitudes are produced simultaneously, the loudness increases and decreases periodically. This phenomenon is called the beat. The periodic variation in the intensity of sound at a point, due to the superposition of two sound waves of slightly different frequencies travelling in the same direction, is called beats. The time interval in which one beat occurs is called beat period while the number of beat per second is called beat frequency.

Mathematical Derivation for Beat Frequency

Suppose two waves of frequencies f₁ and f₂ and each of amplitude ‘a’ are travelling in a medium in the same direction. The equations of the waves are

$$ \begin{align*} y_1 = a\sin (\omega _1 t – k_1x) \dots (i) \\ y_2 = a\sin (\omega _2 t – k_2x) \dots (ii) \\ \text {where,} \: \omega _1 = 2\pi f_1 \: \text {and} \: \omega _2 = 2\pi f_2.\end{align*} $$

When two waves superpose

$$ \begin{align*} \text{at x} = o, \: \text {then } \\ y_1 = a\sin \omega _1 t \dots (iii) \\ y_1 = a\sin \omega _2 t \dots (vi) \\ \end{align*} $$

And from the superposition principle the resultant displacement at that point is given by

$$ \begin{align*}\\y = y_1 + y_2 \\ = a\sin \omega _1 t + a\sin \omega _2 t \\ =2a\sin \left (\frac {\omega _1 + \omega _2}{2}\right ) t \cos \left (\frac {\omega _1 - \omega _2}{2}\right )t \\ =2a\cos 2\pi \left (\frac {f_1 – f_2}{2} \right ) t \sin 2\pi \left (\frac {f_1 + f_2}{2} \right ) t \\ \text {or,} \: y = A\sin 2\pi \left (\frac {f_1 + f_2}{2} \right ) t \\ \text {where A} \: = \: 2a\cos 2\pi \left (\frac {f_1 – f_2}{2} \right ) t. \: \\\end{align*} $$ The above equation shows that resultant wave has an amplitude A that depends on time t and varies with a frequency of f $$ \begin{align*}= \frac {f_1 – f_2}{2}. \end{align*} $$

Condition for Maxima

The resultant amplitude A will be maximum when

$$ \begin{align*} \: \cos 2\pi \left ( \frac {f_1 – f_2}{2}\right ).t \: \text {is maximum. That is,} \\ \cos 2\pi \left ( \frac {f_1 – f_2}{2}\right ).t = \pm 1 \\ \text {or,} \: \cos 2\pi \left ( \frac {f_1 – f_2}{2}\right ).t = \cos n\pi \\ \text {where n} = 0, 1, 2, 3, 4, \dots \\ \text {So,} \: 2\pi\left ( \frac {f_1 – f_2}{2}\right ).t = n\pi \\ \text {or,} \: t = \frac {n}{f_1 – f_2} \\ \text {or,} \: t = 0, \frac {1}{f_1 – f_2}, \frac {2}{f_1 – f_2}, \frac {3}{f_1 – f_2}, \dots \\ \end{align*} $$

The time interval between two consecutive maxima is the period and it is given by

$$ \begin{align*} \\ T = \frac {1}{f_1 – f_2} – 0\: \text {or,} \:\frac {2}{f_1 – f_2} - \frac {3}{f_1 – f_2} \: \text {or,} \dots = \frac {1}{f_1 – f_2} \\ \therefore T = \frac {1}{f_1 - f_2} \\ \text {Hence frequency of maxima} \: = \frac 1T = f_1 – f_2 \\ \end{align*} $$

Condition for Minima

The resultant amplitude A will be minimum when

$$ \begin{align*} \: \cos 2\pi \left (\frac {f_1 – f_2}{2}\right )t \: \text {is miniimum. } \\ \text {That is,} \\ \cos 2\pi \frac {f_1 – f_2}{2}.t = 0 \\ \text {or,} \: \cos 2\pi \left (\frac {f_1 – f_2}{2}\right ). t = \cos (2n + 1)\pi/2 \\ \text {where n} = 0, 1, 2, 3, 4, \dots \\ \text {So,} \: 2\pi\left ( \frac {f_1 – f_2}{2}\right ).t = (2n + 1)\pi/2 \\ \text {or,} \: t = \frac {2n + 1}{2(f_1 – f_2)} \\ \text {or,} \: t = \frac {1}{2(f_1 – f_2)}, \frac {3}{2(f_1 – f_2)}, \frac {5}{2(f_1 – f_2)}, \dots \\ \end{align*} $$

So, the time interval between two consecutive minima i.e. beat period is given as,

$$ \begin{align*} \\ T = \frac {3}{2(f_1 – f_2)} - \frac {1}{2(f_1 – f_2)} \: \text {or,} \: \frac {5}{2(f_1 – f_2)} - \frac {3}{2(f_1 – f_2)} \\ = \frac {1}{f_1 – f_2} \\ \text {Hence, frequency of minima} = \frac 1T = f_1 – f_2, \text {same as the frequency of maxima. } \end{align*} $$.So, the beat frequency is equal to the difference of frequencies of two sound waves.

Doppler Effect

The apparent change in frequency of sound wave due to the relative motion of source of sound of sound and observer is called Doppler’s effect. For example: You hear the high pitch of the siren of approaching ambulance and you notice dropping of pitch sudenly as ambulance passes you whic is dpppler effect. This phenomenon was first derived by Australian Scientist Doppler. So, it is Doppler’s effect.

Let ‘v’ be the velocity of sound ‘λ’ be the wavelength of sound wave and ‘f’ be the frequency.

$$\text {Then,} \: f = \frac {v}{\lambda } $$

Cases

1. When source of sound moves towards the Observer in rest

   When source of sound moves towards observer in rest, then wavelength of sound decreases. The apparent change in wavelength is given by
   $$ \begin{align*} \lambda ‘ &= \frac {v – u_s}{f} \\ \text {If} \: ‘f’' \: \text { be the apparent change in frequency, Then} \\ f’ &= \frac {v}{\lambda ‘} = \frac {v}{(v –v_s)/f} \\ f’ &= \frac {v}{v – u_s} \times f \dots (i) \\ v &= \text {velocity of sound} \\ u_s &= \text {velocity of source} \\ \lambda ‘ &= \text {changed wavelength} \\ f &= \text {frequency of sound wave} \\ \text {Since,} \\ v>v –u_s \: i.e \: f’>f \\ \end{align*} $$ So, frequency increases when source wave is towards the observer in rest.

2. When source of sound moves away from the Observer in rest

   When source of sound moves away from the observer in rest, the wavelength of sound wave. Therefore, apparent change in wavelength is given by
   $$ \begin{align*} \lambda ‘ &= \frac {v +u_s}{f} \\ v &= \text {velocity of sources} \\ u_s &= \text {velocity of sound} \\ \lambda ‘ &= \text {changed wavelength} \\ f &= \text {frequency of sound wave} \\\text {If f’ be the apparent change in frequency.} \\ \text {Then,} \: fi &= \frac {v}{\lambda } = \frac {v}{(v +u_s)/f} \\ f’ &= \frac {v}{v +u_S} \times f \dots (ii) \\ \text {Since,} \\ v<v + u_s \: i.e \: f’<f \\ \end{align*} $$ So, frequency decreases then source moves away from the observer in rest.

3. When observer moves towards the source in rest

   When observer towards the source in stationary then relative velocity of sound wave to the observer is v +u_o.
   $$ \begin{align*} f’ &= \frac {\text {relative velocity of sound}}{\text {wavelength}} \\ &= \frac {v + u_0}{v/f} = \left ( \frac {v + u_0}{v} \right ) f \\ \therefore f’ &= \left ( \frac {v + u_0}{v} \right ) f \dots (iii) \\ v +u_0 > v \: i.e \: f’ >f \\ \end{align*} $$ So, frequency increases when observer moves towards the source in rest.

4. When observer moves away from the source in rest

   When observer moves away from the source in rest then relative velocity of sound wave to the observer is v + u_0.
   $$ \begin{align*} f’ &= \frac {\text {relative velocity of sound}}{\text {wavelength}} \\ &= \frac {v - u_0}{v/f} = \left ( \frac {v - u_0}{v} \right ) f \\ \therefore f’ &= \left ( \frac {v- u_0}{v} \right ) f \dots (iii) \\ \text {Since, } \\ v -u_0 < v \: i.e \: f’ <f \\ \end{align*} $$ So, frequency decreases when observer moves away from the source in rest.

5. When source and observer moves towards each other

   When the source and observer are approaching towards each other with the velocity u_s and u_o respectively, then
   $$ \begin{align*} \text {velocity of the waves relative to the observer,} v_r = v + u_0, \\ \text {and apparent wavelength,} \lambda ‘ = \frac {v – u_s}{f} \\ \text {Putting these values in equation} \: (v) \: \text {we get} \\ f’ &= \frac {v’}{\lambda ‘} = \frac {v + u_0}{(v – u_s)/f} = \frac {v + u_0}{v –v_s}f \\ \therefore f’ &= \frac {v + u_0}{v –v_s}f \dots (v)\\ \text {Since, } \\ v -u_0 < v + u_s \: i.e \: f’ >f \\ \end{align*} $$ So, frequency increases when source and observer towards each other

6. When source and observer moves away from each other

   When the source and observer moves away from each other with the velocity u_s and u_o respectively, then
   $$ \begin{align*} \text {velocity of the waves relative to the observer,} v_r = v - u_0, \\ \text {and apparent wavelength,} \lambda ‘ = \frac {v + u_s}{f} \\ \text {Putting these values in equation} \: (v) \: \text {we get} \\ f’ &= \frac {v’}{\lambda ‘} = \frac {v - u_0}{(v + u_s)/f} = \frac {v - u_0}{v +v_s}f \\ \therefore f’ &= \frac {v - u_0}{v + v_s}f \dots (vi)\\ \text {Since, } \\ v -u_0 < v + u_s \: i.e \: f’ < f \\ \end{align*} $$ So, frequency decreases when source and observer moves away from each other .

7. When source leads the observer

   When the source and observer moves in same direction and the source is leads the observer, then
   $$ \begin{align*} \text {velocity of the waves relative to the observer,} v_r = v + u_0, \\ \text {and apparent wavelength,} \lambda ‘ = \frac {v + u_s}{f} \\ \text {Putting these values in equation} \: (v) \: \text {we get} \\ f’ &= \frac {v’}{\lambda ‘} = \frac {v + u_0}{(v + u_s)/f} = \frac {v + u_0}{v +v_s}f \\ \therefore f’ &= \frac {v + u_0}{v + v_s}f \dots (vii)\\ \text {So, frequency will change depending on}\: u_o \: \text {and} \: u_s. \\ \end{align*} $$

Doppler Effect, Waves and Noise Pollution

Effect of Medium

If the medium travels to the direction of sound wave then frequency of sound increases and if medium travels in opposite direction of sound wave then frequency of sound decreases. Apparent change in frequency, if direction of medium and sound wave is same, given by

$$ \begin{align*} F’ &=\left [ \frac {(v + v_m) – u_0}{(v + v_m) –u_s} \right ] \times f \\ \end{align*} $$

Apparent change in frequency, if direction of medium and sound wave is opposite to each other. Then,

$$ \begin{align*} F’ &=\left [ \frac {(v - v_m) – u_0}{(v - v_m) –u_s} \right ] \times f \\ \text {where,} \: v_m &= \text { velocity of medium} \\ v &= \text {velocity of sound} \\ u_o &= \text {velocity of observer} \\ u_s &= \text {velocity of source} \\ f’ &= \text {change in frequency} \\ \end{align*} $$

Limitation of Doppler Effect

• The Doppler’s principle can only be applied in the cases where the relative velocity between the source and observer is less than the velocity of sound.
• This principle is not applicable if the source moves towards the observer with supersonic velocity.

Doppler’s Principle in Light

When a star recedes from the earth, its spectral lines are displaced towards the red end which indicates an increase in wavelength and is called a red shift. When a star approaches the earth, the spectral lines are displaced towards the violet and there is a decrease in wavelength, the velocity known as blue shifts. If λ is the wavelength of light emitted by the tar in the stationary position, λ’ be the wavelength when it is in motion, u_s is the velocity of approach of the star and c is the velocity of light, then

$$ \begin{align*} \lambda ‘ &= \left ( \frac {c – u_s}{c} \right ) \lambda = \left ( 1 - \frac {u _s}{c} \right ) \lambda = \lambda - \frac {u_s}{c} \lambda \\ \text {or,} \: \frac {u_s}{c} \lambda &= \lambda - \lambda ‘ \\ \text {or,} \: u_s &= \frac {c(\lambda - \lambda ‘)}{\lambda } \\ \end{align*} $$

When λ and the shift (λ – λ’) are known, the velocity of star can be calculated. The measurement of red shift of light from galaxies suggests that the universe is expanding.

Application of Doppler Effect

1. Velocity of approaching or receding stars can be determined by using Doppler’s principle using the formula,
   $$ u_s = \frac {c(\lambda - \lambda ‘ )} {\lambda } $$
2. Red Shift: it has been observed that some distant nebulae are moving away with a velocity greater than 20 × 10³ km/s and the spectral lines appear to shift towards the red end of the spectrum by 200 A^o. This gives the idea that universe is expanding.
3. In radar: microwaves emitted from a radar transmitter are sent out towards the target and after reflection from the target, these waves are picked up by the receiving the station of radar. If the aeroplane is approaching the observing station, there will be an increase in frequency or decrease in frequency or decrease in the wavelength of microwaves. But is the aeroplane is moving away from the observing station, there will be a decrease in frequency or increase in wavelength.
4. In SONAR (Sound Navigation and Ranging): Doppler’s principle is used in SONAR for the detection of submarine or group of fishes under water in the sea. The SONAR makes use of ultrasounds of frequencies more than 20 kHz and the reflected signals can be used for calculating the speed approaching or receding of submarines under water.

Range of Hearing

Normal human ears can hear only those waves whose frequency lies between 20 Hz and 20,000 Hz. The sound waves having a frequency between 20 Hz and 20, 000 Hz are known as sound waves which are an audible range of frequency. The waves having a frequency less than 20 Hz and greater than 20, 000 Hz cannot be heard by human ear.

Infrasonic and Ultrasonic Waves

Infrasonic Wave

The waves of frequency less than 20 Hz are known as infrasonic waves. These waves are not audible to a human ear. For example vibration of earth’s surface and a sound of animals like elephants, rhinoceros and whales etc.

Ultrasonic Wave

The waves of frequency greater than 20, 000 Hz are known as ultrasonic waves or ultrasound. It is not audible for the human ear. Example: sound produced by jet planes, bomb blasting etc. In the case of bat, wings of bat produce ultrasound which is transmitted towards the prey. It receives quickly and which position is the prey, so, the bat can easily catch the prey.

Uses of Ultrasound

1. For welding plastics
2. To diagnosis of human disease of human disease (Eg: cancer in stomach)
3. To check the development of unborn baby
4. To kill bacteria in the liquid
5. To find faults and cracks in metals
6. To determine the depth of a sea

Noise Pollution

The sound of loudness greater than 45 Db is harmful to human ear which produce noise pollution. According to WHO, the loudness of city must be balanced up to 45 Db.

Effects of Noise Pollution

1. It causes permanent deafness.
2. It causes a loss in heavy power.
3. It causes a headache, blood pressure etc.
4. It resists doing work properly.
5. It causes disease like neurosis, insomnia, and hypertension, behavioral and emotional stress.
6. It interferes with a speech of people.

Control of Noise Pollution

1. Generating awareness about the cause of pollution.
2. Designing and fabricating silencing devices in aircraft engines or any other machines.
3. Room walls should be covered with sound absorbers.
4. Workers exposed to noise should be given bear plugs.
5. It can be controlled by banning the air horn.
6. It can be controlled by running loudspeaker, radio and another music system at low volumes.

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