SOUND

1. Periodic Motion of Particles

Before we move on to study the nature and transmission of sound, we need to understand the different types of vibratory or oscillatory motions.

A motion, such as that of the earth around the sun, the movement of the hands of the clock etc., is referred to as periodic motion, since the motion of the object repeats itself at regular intervals of time.Periodic motions are simply motions which repeats itself after a fixed period of time.

A to-and-fro motion, such as the swinging of a pendulum, vertical oscillations of a mass suspended from a spring etc., is referred to as harmonic motion.

A harmonic motion in which the amplitude and time period of oscillation remains constant is particularly referred to as simple harmonic motion (SHM). In a SHM the acceleration of the body or particle executing the motion is directly proportional to its displacement from the mean position and is directed towards the mean position. The total mechanical energy(the sum of kinetic and potential energy) of the particle is conserved.

PLANCESS CONCEPTS

When an object moves back and forth repeatedly over the same path, it is said to be oscillating or vibrating. An oscillatory motion is always periodic but a periodic motion may not be oscillatory.

For example : particle moving in a circle is a periodic motion but not oscillatory.

Vipul Singh

AIR 1, NSTSE 2009

rePresentative samPle PhysiCs: sound

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1.1 Graphical representation of simple harmonic motion, its characteristics and relations

Consider a simple pendulum suspended by means of a thread from a rigid support and allowed to vibrate in a vertical plane as shown in Fig. 3.1. ‘O’ is the rest position or the mean position of the bob of the pendulum and ‘A’ and ‘B’ are its extreme position. If the direction of motion of the bob towards ‘A’ is taken as positive, then the direction towards ‘B’ is negative.

The pendulum oscillates to and fro and the time taken for one complete oscillation is known as time period (T).

The magnitudes of the displacements from mean position is maximum when the bob is at either ‘A’ or ‘B’ and this maximum displacement of the vibrating particle from its mean position is known as ‘amplitude’ (A).

A graph plotted between the displacement of the bob from its mean position and the time, is as shown in Fig. 3.2.

Fig. 3.2

As the time increases, displacement increases to the maximum of ‘A’ at t = T/4 and then the bob comes to mean position at t = T/2 and so displacement is zero. It continues to move towards negative side, and when the time is t = 3T/4 its displacement is equal to amplitude. When t = T, it comes back to mean position completing one full vibration.

The number of vibrations the pendulum bob makes in unit time is known as frequency (n) and is measured in hertz (Hz).

The time period and the frequency are related as 1n

T= 

Here, we find that the graph is in the form of a wave that we see on the surface of water.

Fig. 3.1

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1.1.1 Wave Motion:

When a pebble is thrown into still water circular ripples are formed which spread out in all directions on the surface of water from the point where the stone hit the water surface. Thus, the kinetic energy of the stone is transferred to the water and that energy is distributed to the entire water in the pond in the form of ripples or waves. To check whether water moves along with ripples produced or not, we can observe a floating object like a cork or a leaf placed on the surface of water. As the ripples move in all possible directions on the surface of water from the point where the disturbance is produced, the leaf which is floating on the surface of water vibrates up and down, but does not have lateral translational motion along the surface of water. We even observe that the leaf does not start vibrating till the first ripples reaches it from the point of disturbance. This is the characteristic of the propagation of waves.

Fig. 3.3

The energy is transmitted from one point to another without actual translational motion or transport of the particles across the medium. Thus a “wave is a disturbance produced at a point in a medium or a field and is transmitted to other parts of the medium or the field without the actual translational motion of the particles”. The transfer of energy in the form of waves is known as “wave motion”.

A pulse is a disturbance lasting for a short duration.

A wave on the other hand is a sustained disturbance lasting for a longer duration, like waves on the surface of water.Before we proceed to study wave motion in greater detail let us first review the terms and physical quantities associated with wave motion.

Fig. 3.4

Crest is the point of maximum displacement of a particle in upward direction.

Trough is the point of maximum displacement of a particle in downward direction.

Amplitude is the maximum displacement of the particle either upwards or downwards.

Wavelength (λ) is the distance between any two successive crests or troughs.

Time period (T) is the time taken by a particle to complete one oscillation or vibration.

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Frequency (n) is the number of oscillations or vibrations made by a particle in one second. 1nT

= . The S.I. unit of frequency is hertz (Hz). 1 hertz = 1 s–1

Velocity of a wave is the speed with which the wave propagates in the medium. A wave takes time equal to its time period (T) to travel a distance equal to its wavelength (λ). So,

Wave velocity Wavelength of the wavev= 

Time period of the wave T= ⇒

λ orv = n × λ

Phase: The motion of the vibrating particles and their direction is described in terms of its phase. Thus particles in the same phase would be exactly at the same distance from their mean positions and have the same instantaneous velocity at any given moment.

If the motion of two particles is such that their displacement, motion and velocity are dissimilar to each other, then they are said to have phase difference. If two particles have same magnitude of displacement from mean position and velocity but the direction of these vector quantities are opposite to each other, then they are said to be out of phase.

PLANCESS CONCEPTS

Suppose a mechanical wave is moving in a fixed direction say from left to right. Each particle copies the motion of another particle at its left with a time delay of x/v, where x is the separation between the two particles and v is the wave speed.

Vipul Singh

AIR 1, NCO

Transmission of energy:It is found that certain type of waves require a medium for propagation, e.g., water waves, sound waves etc., whereas there exist waves which do not require a medium for their propagation, e.g., light waves.

The direction of vibration of particles differ from the direction of wave motion from one type of wave to another. Similarly some waves move endlessly in a medium whereas some are confined between two points.

Based on these factors, waves can be classified into different types as follows:

(i) Classification based on the necessity of medium - Mechanical waves and Electromagnetic waves.

Mechanical waves are the waves which require a material medium for their propagation. They are also called ‘elastic waves’ as the main cause for their propagation in the medium is a property of the medium called ‘elasticity’.

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If an applied force on a body changes its shape or size or both, and when the force is taken away, if the body regains its original shape and size, the body is said to be ‘elastic’ and its property to regain its original shape and size after the applied force is removed is known as ‘elasticity’.

the important point here is that how much a body can can stretch is not elasticity but to regain its shape is elasticity.

Electromagnetic waves are the waves which do not require an elastic medium for their propagation. They can propagate through media as well as vacuum. Light waves are an example of electromagnetic waves.

Fig. 3.5

(ii) Classification based on the direction of vibration of particles with respect to the direction of wave motion - Transverse and Longitudinal waves.

When a mechanical wave propagates from one place to another in a medium, the direction of vibration of particle of the medium can be either parallel or perpendicular to the direction of wave motion.

If the direction of vibration of the particles of the medium is parallel to the direction of wave motion, such a wave is called a ‘longitudinal wave’ and if it is perpendicular to the direction of wave motion such a wave is called a ‘transverse wave’.

(a) Longitudinal wave:Consider a long spring clamped at one of the ends, placed on a horizontal surface of a table in straight position, as shown in Fig. 3.5(a). The distance between any two adjacent rings along the length of the spring is constant. If the spring is slightly pulled and then released, the spring begins to vibrate. It can be observed that any two adjacent rings in some parts of the spring come very close to each other, while in other parts they move apart as shown in Fig. 3.5 (b).

The regions where the rings are very close to each other are called ‘compressions’ and the regions where they are far apart are called ‘rarefactions’. The wave set in the spring is a

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longitudinal wave as the direction of vibration of particles (here rings) is parallel to the direction of wave motion.

So a longitudinal wave moves in a medium in the form of compressions and rarefactions. Whenever compressions and rarefactions are transmitted through a medium, a change in the volume of the medium takes place in those locations. Due to elasticity of the medium, it regains its original volume. Thus longitudinal waves can be set in a medium that opposes change in volume. Since, all the states of matter, solids, liquids and gases have the property to oppose change in volume, longitudinal waves can propagate in solids, liquids and gases.

PLANCESS CONCEPTS

When two or more waves simultaneously pass through a point, the disturbance at the point is given by the sum of the disturbances each wave would produce in absence of the other wave(s). This is known as the principle of superposition.

Vipul Singh

AIR 1, NCO

(a) Transverse wave:When we take a long string along the horizontal position and vibrate it at one end in a direction perpendicular to the length of the string, a wave form is set in the string. In the Fig. 3.6 the string is shown by a dotted line. It is also called as mean or rest position. Here the wave moves in the horizontal direction whereas the particles of the string vibrate in the perpendicular direction (vertical).

Fig. 3.6

The displacement of the vibrating particle is measured from the mean position. The particles at positions ‘A’ and ‘B’ have maximum displacement in the upward direction and these points are known as ‘crest’. Similarly the particles at positions ‘C’ and ‘D’ have maximum downward displacement and these points are known as ‘troughs’. As the direction of particle vibration is perpendicular to the direction of wave motion, the wave set in the string is a transverse wave.

Thus when a transverse wave is set in a medium, a series of crests and troughs propagate through the medium. These crests and troughs change the shape of the medium and due to elasticity, the medium

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regains its original shape. Hence, transverse waves can be set in a medium which opposes change in shape. For this reason, transverse waves can propagate only in solids and at the surface of the liquids but not through liquids and gases. Consider the cross section of water surface when waves are propagating through its surface as shown in the Fig. 3.7. The dotted line indicates the rest position of the water surface. As the wave propagates from the left to the right, the water particles vibrate up and down forming crests and troughs. The displacement of the particles at ‘A’ and ‘C’ from the mean position is equal and their direction of motion is the same. Thus their status of vibration with respect to the direction of motion and the displacement from the mean position, which is known as ‘phase’ is equal.When the particles at ‘A’ and ‘B’ are considered, their magnitudes of displacement from their mean position are equal but their direction of motion is opposite. So they are said to be ‘out of phase’.

Fig. 3.7

The minimum distance between the particles of the medium which are in the same phase is called ‘wavelength’ of the wave, and is denoted by the Greek letter ‘λ’ (lambda). So the distance between ‘A’ and ‘C’ or that between ‘C’ and ‘E’ is the wavelength (λ). By the time the particle at ‘A’ completes one vibration i.e., after one time period (T), the wave advances by one wavelength (λ). So the velocity of

propagation of the wave is given by vTλ

= . As 1 nT

= (the frequency of the wave),

so v = nλ

The velocity of the vibrating particles is not constant throughout their vibration. It is minimum at the extreme positions and maximum at the mean position. But the velocity of the wave propagating through the medium is constant.

The wave considered in Fig.3.8is a transverse wave, and it produces crests and troughs. Similarly when a longitudinal wave such as a sound wave propagates through a medium like gas, it causes compressions and rarefactions while propagating through the medium, causing change in density and pressure throughout the medium.

The graph of pressure (p) or density (d), of a gas, taken along the Y-axis versus the distance of the element of vibrating gas from the source of sound taken along the X-axis is as shown in Fig.3.8.

At positions ‘A’ and ‘E’ which correspond to compressions, the density and pressure of a gas are maximum and are more than the normal values. Similarly at positions ‘C’ Fig. 3.8

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and ‘G’ which correspond to rarefactions, the density and pressure of a gas are minimum and are less than the normal values. The positions, ‘B’, ‘D’, ‘F’ and ‘H’ show normal pressure and density of the gas.

PLANCESS CONCEPTS

The velocity of the particle executing S.H.M. decreases, as it moves from the mean to extreme position. The acceleration of a particle executing S.H.M. increases as it moves from the mean position to extreme position.

Vipul Singh

AIR 1, NCO

Comparative study of transvers:

Transverse waves Longitudinal waves

1. The direction of vibration of particles is perpendicular to the direction of propagation of a wave.

1. The direction of vibration of particles is parallel to the direction of propagation of a wave.

2. The wave propagates in the form of crests and troughs.

2. The wave propagates in the form of compressions and rarefactions.

3. These waves can travel through solids and on surface of liquids only, as the propagation of these waves causes change in the shape of the medium.

3. These waves can pass through solids, liquids and gases also, as the propagation of these waves causes change in the volume of the medium.

4. As there is no variation of volume, there is no variation in the density of the medium while the wave propagates through it.

4. When the wave propagates through a medium, there is a change in the volume and this causes a variation in the density.

5. There is no difference in pressure created in the medium while the wave propagates.

5. Propagation of longitudinal waves causes pressure difference in the medium.

(iii) Classification based on the limitations of motion- Progressive and stationary waves.

Some waves start at the point of origin of the waves and progress endlessly into other parts of the medium. Such waves are known as ‘Progressive waves’.

such as transverse waves and electromagnetic waves.

Consider a progressive transverse water wave moving from left (point P) to right and striking a hard surface at ‘Q’ as shown in Fig. 3.9(a). It then gets reflected at ‘Q’, and travels towards ‘P’. Thus the two waves, one going from ‘P’ to ‘Q’ and the other going from ‘Q’ to ‘P’ overlap resulting in the

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formation of ‘nodes’ and ‘antinodes’. Points, where the displacement of a vibrating particle of the medium is zero or minimum are called ‘nodes’ (shown as N in Fig. 3.9 (b)) and points, where the displacement of the vibrating particles is maximum are called ‘antinodes’ (shown as A in Fig. 3.9 (b)).

On the whole, the wave appears to be standing or stationary, contained between two positions ‘P’ and ‘Q’ and so called as ‘standing’ or ‘stationary waves’.

Fig. 3.9

Thus a ‘progressive wave is a wave which is generated at a point in a medium and travels to all parts of the medium infinitely carrying the energy’ and a ‘stationary wave is a wave which is formed by a combination of two similar progressive waves traveling in opposite directions’.

Comparative study of progressive and stationary waves:

Progressive waves Stationary waves

1. These waves start at a point and move indefinitely and infinitely to all parts of the medium or space.

1. These waves appear to be standing at a place and are confined between two points in a medium or space.

2. These waves transmit energy from one place to another.

2. These waves store energy in them.

3. The energy possessed by these waves is kinetic in nature.

3. The energy associated with these waves is potential in nature.

4. These waves contain crests and troughs or compressions and rarefactions.

4. These waves contain nodes and antinodes.

5. All the particles in the wave have equal amplitudes.

5. Different particles in the wave have different amplitudes.

6. There is a continuous phase difference between the particles in the waves.

6. The phase difference between the particles in a given loop in the wave is zero.

(contd.)

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SOLVED EXAMPLES

Example 1.A source of wave produces 40 crests and 40 troughs in 0.4 second. Find the frequency of the wave.

Sol. According to given data:

(i) Number of crests and troughs produced by the wave = 40

(ii) Time taken by wave to propagate up to 40 wave forms t = 0.4 s

And we want to find the frequency n =?

The solution is as follow:

Number of waves produced in one second

= 400.4 s

= 100 s–1;

∴Frequency of the wave = 100 Hz (∵ frequency is the measure of repetition of wave forms i.e. single crest & trough in a wave)

Example 2. A person has a hearing range from 20 Hz to 20 kHz. What are the typical wavelengths of sound waves in air corresponding to these two frequencies? Take the speed of sound in air as 1344 ms− .

Sol. According to given data:

(i) Hearing range = 20Hz to20 kHz (= 20000 Hz)

(ii) Speed of sound in the air = 344 m s–1

For a wave, Wavelength = Wave velocityFrequency

or λ =vn

. So, for frequency n = 20Hz = 20 /s

1

1

344 ms

20 s

−

−λ = =17.2m

and forfrequencyn= 20000 Hz = 20000 s–1, 1

1

344 ms

20000 s

−

−λ = = 0.0172m = 1.72 cm

(contd.)

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EXERCISE — 1 (For School Examinations)

Fill in the blanks

Direction: Complete the following statements with an appropriate word/term to be filled in the blank space(s)

Q.1 Sound waves having the frequency in range are audible to human being.

Q.2 The wavelength of a sound from a tuning fork of frequency 330 Hz is nearly cm

Q.3 Velocity of sound in vacuum is .

Q.4) The total energy E of sound is related to their frequency as .

Q.5 Longitudinal waves cannot pass through .

Q.6 The frequency of the particles oscillating in a medium is the frequency of waves in the medium.

Q.7 A transverse wave is made up of .

Q.8 A longitudinal wave is made up of .

Q.9) Wave motion involves the transport of .

Q.10 Velocity = frequency × .

True / False

Direction: Read the following statements and write your answer as true or false.

Q.11 Longitudinal waves are produced in all the three states.

True False

Q.12 Bells are made of metal and not of wood because the sound is not conducted by metals but is radiated.

True False

Q.13 The rate of transfer of energy in a wave depends directly on the square of the wave amplitude and square of the wave frequency.

True False

Q.14 Velocity of sound in air at the given temperature decreases with increase in pressure.

True False

(contd.)

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EXERCISE 2 — (For Competitive Examinations)

Multiple choice questions

Direction: This section contains multiple choice questions. Each questions has 4 choice (a), (b), (c) and (d), out of which only one is correct.

Q.1. A thunder clap is heard 5.5 second after the lightening flash. The distance of the flash is

(Given: velocity of sound in air = 330 m/s)

(a) 1780 m (b) 1815 m

(c) 300 m (d) 3560 m

Q.2. A wave frequency 1000 Hz travels between X and Y, a distance of 600 m in 2 sec. How many wavelengths are there in distance XY.

(a) 3.3 (b) 300

(c) 180 (d) 2000

Q.3. Ultrasonic waves have frequency

(a) below 20 Hz (b) between 20 and 20,000 Hz

(c) only above 20,000 Hz (d) Only above 20,000 MHz

Q.4. A person has the audible range from 20 Hz to 20KHz. Find the wavelength range corresponding to these frequencies. Take velocity of sound as 340 m/s.

(a) 315m to 15 10 m−× (b) 311 m to 11 10 m−×

(c) 317 m to 17 10 m−× (d) 815 m to 15 10 m−×

Q.5. If you are at open-air concert and someone’s head gest between you and the orchestra, you can still hear the orchestra because

(a) sound waves pass easily through a head

(b) a head is not very large compared with the wavelength of the sound

(c) the sound is reflected from the head

(d) the wavelength of the sound is much smaller than the head.

Q.6. In a long spring which of the following type of waves can generated

(a) Longitudinal only (b) Transverse only

(c) Both longitudinal and transverse (d) Electromagnetic only

(contd.)

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Solutions

EXERCISE 1

Fill in the blanks

1. 20 to 20 KHz 2. 1

1

330 msv1m

n 330 s

−

−λ = = = 3. 0 ms-1

4. E an25. vacuum 6. the same as 7. crests and troughs

8. compressions and rarefactions 9. energy 10. wavelength

True / False

11. True. In all the three states of matter mechanical energy can be transferred & hence sound can travel in all the three states of matter.

12. False, metal are good conductor of sounds & sound travels faster in metal.

13. True

14. False, with increase in pressure the velocity of sound increases because the energy of sound wave travels faster in compressed air

15. False, 16. False 17. True 18. False

Match the columns

19. A → r; B → s; C → p; D → q.

20. A → s; B → r; C → q; D → p.

Very Short Answer Questions

21. Condition (a) is not sufficient as the direction of acceleration is opposite to displacement and that needs to be mentioned.

22. (i) At both the extreme positions K.E. is zero.

(ii) At mean position, the P.E. is zero.

(iii) Potential energy of simple pendulum P.E. = ( )2 2 21m a x

4ω −

2 21T.E. m a

2

= ω

(contd.)

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Solutions

EXERCISE 2

Multiple Choice questions

1. b Distance = velocity time = 330=1815 m

2. D as the frequency is measure of formation of a wavelength in 1 second.

∴ in time t = 2 s the wavelengths formed are 21000= 2000

3. c

4. C, as wavelength vn

λ = so the range of wavelength found to be from 16.5 m to 0.165 m.

5. b 6. c 7. a 8. a 9. b 10. c 11.b 12. (b)

13. (c) As fixed end is a node, therefore, distance between two consecutive nodes = 2λ

=10 cm

λ = 20 cm = 0.2 m. As v= nλ

v=100 x 0.2 = 20 m/s

14. (a) Velocity of sound = RTM

γ

When water vapor are present in air average molecular weight of air-vapor mixtureincreases and hence velocity increases.

15. (d) Frequency of vibration in tight string

p T

n n T2l m

= ⇒ ∝

( )n T 14% 2%

n 2T 2∆ ∆

⇒ = = × =

2

Number of beats n n100

⇒ = ∆ = ×

2100 2

100= × =

(contd.)