**Questions - Answers**

** Very Short Answer Questions**

**1. What does a wave represent?**

**A:** A wave is a disturbance which moves in space, transports, energy and momentum from one point to another without transmitting matter.

**2.** Distinguish between Longitudinal and Transverse waves.

**A:**

**3.** What are the parameters used to describe a progressive harmonic waves?

**A:** The parameters of progressive wave are 1) Amplitude 2) Angular Velocity 3) Frequency 4) Time Period (T) 5) Wave length (λ) 6) Velocity (V) 7) Phase (Ø) 8) Propagation constant (k).

**4.** Obtain an expression for the wave velocity in terms of these parameters.

**5.** Using dimensional analysis obtain an expression for the speed of transverse waves in a stretched string?

**A:** Speed of transverse wave (V) T^{a} m^{b} where T is tension in the string, m is linear density. Dimensional formula for V = [LT^{−1}],

T = [MLT^{−2}], m = [ML^{−1}]

[LT^{−1}] = [MLT^{−2}]^{a} [ML^{−1}]^{b}

= [M^{a} L^{a} T^{−2a }M^{b} L^{−b}]

[LT^{−1}] = [M^{a + b} L^{a - b} T^{−2a }]

M^{0} = M^{a + b} ⇒ a + b = 0 ______ (1)

L^{1} = L^{a - b} ⇒ a^{}_{−}b = 1 ______ (2)

T^{−1} = T^{2a } ⇒ -1 = -2a

⇒ a = ______ (3)

From (1), (2) & (3)

a = , b = ^{ -}

**6.** Using dimensional analysis obtain expression for the speed of sound waves in a medium.

**A:** Velocity of Wave (V) = λ^{a} T^{b}

V = [LT^{−1}], λ = [L], T = [T]

[LT^{−1}] = [L^{a} T^{b}] ⇒ L^{1} = L^{a} ⇒ a = 1

T^{−1} = T^{b} = +b = -1

**7.** What is the principle of super position of waves?

**A:** Two or more waves overlap at a point, the resultant displacement of any particle is equal to the vector sum of displacement of all the waves.

**8.** Under what conditions will a wave be reflected?

**A:** If a wave incident on a boundary between two media, a part of incident wave returns back into the initial medium (reflection) and a part of incident wave will be refracted. During refraction they obey Snell's Law wave will be reflected if it falls on rigid surface. Because at rigid surface the particles of the medium does not vibrate.

**9.** What is the phase difference between the incident and reflected waves when the wave is reflected by a rigid boundary?

**A:** At rigid boundary the phase difference between incident and reflected waves is π. That is a crest reflects as through and vice - versa.

**10.** What is a stationary or standing wave?

**A:** When two waves of same amplitude, frequency and velocity moving in opposite directions are superposed, the phenomenon of standing waves is observed.

**11.** What do you understand by the term node and antinode?

**A:** The points at which particles never displace from their mean position (minimum displacement) are called as node. The points at which the displacements very between zero and maximum in opposite direction are called antinodes.

**12.** What is the distance between a node and an antinode?

**A:** Distance between a node & an antinode is .

**13.** What do you understand by natural frequency or normal mode of vibration?

**A:** An elastic body vibrates with a definite constant amplitude and definite single frequency known as its natural frequency and vibrations are known as natural vibrations.

**e.g.:** A tuning fork struck with a rubber hammer vibrates with its natural frequency.

**14.** What are harmonics?

**A:** The frequencies which are integral multiples of the fundamental frequency are called harmonics.

**15.** A string is stretched between two rigid supports. What frequencies of vibrations are possible in such a string?

**A:** If a string is stretched between two rigid supports, the possible frequencies are f, 2f, 3f,.... so on, where 'f' is natural frequency of vibration of string.

**16.** If the air column in a long tube, open at both ends, is set in vibration. What harmonics are possible?

**A:** The frequencies of harmonics present in an open pipe are integral multiples of fundamental or natural frequency of the air column. They are f, 2f, 3f, ............

**17.** If the air column in a tube, closed at one end, is set in vibration. What harmonics are possible?

**A:** The frequencies of harmonics present in a closed pipe are 3f, 5f, 7f ............ (2n − 1)f, where 'f' is natural frequency.

**18.** What are beats?

**A:** When two sound waves of nearly equal frequency are superposed, they will create a waxing (maximum) and waning (minimum) of sound. This effect is called beats. Beat frequency = n_{1} ~ n_{2}

**19.** Write down an expression for beat frequency and explain the terms there in.

**A:** Beat frequency = n_{1} ~ n_{2}

n_{1}, n_{2} are frequencies of two sound waves.

**20.** What is ''Doppler effect"? Give an example.

**A:** The apparent change in frequency of source of sound due to relative motion between the source and observer is known as ''Doppler effect".

**e.g.:** The whistle of approaching train appears to have high pitch. Similarly as the train moves away from us the pitch of whistle sound decreases.

**21.** Write down an expression for the observed frequency when both source and observer are moving relative to each other in the same direction.

**A:** When both source and observer are moving relative to each other in the same direction, the apparent frequency is

**SHORT ANSWER QUESTIONS**

**1.** What are transverse waves? Explain with an example.

**A:** If the particles of the medium vibrate perpendicular to the direction of propagation of the waves, the waves are called transverse waves. These waves can propagate through solids and liquids.

**e.g.:** Vibrations in strings, ripples on water surface and electromagnetic waves. During transverse vibrations, the particles may move upwards (or) downwards from their mean positions.

The uppermost point of the wave, where the particle is having maximum positive displacement is called crest and the lowest point, where the particle is having maximum negative displacement is called ''trough". Crests and troughs are appear alternatively. The velocity of the particle is maximum at mean position and zero at extreme position. The distance between two adjacent crests (or) troughs represents the wavelength (λ) of the wave.

**2.** What are longitudinal waves? Give illustrative examples of such waves?

**A:** In longitudinal waves particles of the medium vibrate parallel to the direction of propagation of the wave. These waves can propagate through solids, liquids and gases.

**e.g.:** Waves on springs, sound waves. When a longitudinal wave travel through a medium, the medium divided into alternate compressions (C) and rarefractions (R). The region where the particles of the medium are crowded is called a compression, where the pressure and density are maximum. The other region where the particles of the medium are widely separated is called rarefraction, there the pressure and density are minimum. In air sound waves travel as longitudinal waves.

**3.** Write an expression for a progressive harmonic wave and explain the various parameters used in the expression.

**A:** A plane progressive wave can be represented by

y = A Sin (ωt ± Ø)

where y = displacement at any given time

A = amplitude of the wave

ω = angular velocity

Ø = phase constant

+ve indicates wave is propagating in negative X − direction. −ve indicates wave is

propagating in positive X - direction. Other forms are

y = A Sin (2πnt ± Kx) (... Ø = Kx)

Where K is called propagation constant.

**4.** Explain the modes of vibration of a stretched string with examples.

**A:** Consider a wire of length 'l' is stretched between points A and B. Let 'T' be the tension in the string. If the wire vibrates as a single loop, then frequency of vibrations is known as fundamental frequency and is denoted by (υ).

υ1 is called fundamental frequency (or) 1st harmomic.

If the string vibrates in two loops, then frequency of the string is known as first overtone or second harmonic, is given by

In stretched string the ratio of harmonics are 1 : 2 : 3 ............

** e.g.:** Sonometer.

**5.** Explain the modes of vibration of an air column in an open pipe.

**A:** Consider open pipe of length 'l'. In fundamental mode of vibration two antinodes and one node is formed. Then

l = => λ_{1} = 2l

In the next mode of vibration three antinodes and two nodes are formed. Then length of the pipe

Similarly in the next mode four antinodes and three nodes are formed. Then length of pipe

Hence in case of open pipe, harmonics are in the ratio of 1 : 2 : 3 : ....

**6.** What do you understand by resonance? How would you use resonance to determine the velocity of sound in air?

**A:** If the natural frequency of a vibrating body is equal to the frequency of periodic force, then the body vibrates with increasing amplitude. This phenomenon is called resonance. Consider a closed tube in which length of air column can be changed.

The first resonating length of air column is l1 + e = ..................... (1)

Where 'e' is called end correction.

The second resonating length of air column is

**7.** What are standing waves? Explain how standing waves may be formed in a stretched string?

**A:** Two progressive waves of same amplitude, same frequency travelling in opposite direction superimpose over each other produce stationary or standing wave.

When a stretched string is fixed at both ends is plucked perpendicular to the length, then resulting transverse wave travels along the length of string and get reflected at rigid end. The reflected wave is having same amplitude and frequency as incident wave. The waves are represented by

y_{1} = A sin (ωt - kx) y_{2} = -A sin (ωt + kx)

According to principle of superposition

y = y_{1} + y_{2} = A sin (ωt - kx) - A sin (ωt + kx)

By plucking the stretched wire at various positions standing waves are produced as shown in figure.

**8.** Describe a procedure for measuring the velocity of sound in stretched string.

**A:** Let us consider a string of mass 'M', length 'l' and linear density 'µ' is fixed between two rigid supports with some tension 'T'. When it is plucked at the middle, the transverse wave is produced. Speed of this wave depends on the restoring force in the medium (provided by tension 'T'), and on the intertial property (provided by linear density). The velocity of transverse wave

But, practically velocity of wave is measured by adjusting the length of string until the stationary wave is produced in it. Then V = n λ or υ λ

**9.** Explain using suitable diagrams, the formation of standing waves in a closed pipe. How may this be used to determine the frequency of a source of sound?

**A:** A closed pipe is a cylindrical tube with air as medium inside it. One end is closed and other end is open.

Let a longitudinal wave be sent through a closed pipe. It gets reflected at closed end. Both incident and reflected waves superposed along the length of the pipe and stationary waves formed. Node is formed at closed end, where the displacement is zero. At open end the air particles are free to vibrate, so antinode is formed.

The frequency of sound υ can be calculated by

The ratio of frequencies in closed pipe are υ_{1} : υ_{3} : υ_{5} ........... = 1 : 3 : 5 : .............

**10.** What are 'beats'? When do they occur? Explain their use, if any.

**A:** When two sound notes of nearly equal frequency travelling in the same direction superposed to produce regular waxing and waning in the intensity of combined wave. This effect is known as "beats".

Beat frequency = υ_{1} − υ_{2}

**Formation of beats**

Let us take two tuning forks of nearly same frequencies 256 Hz and 254 Hz.

When two sound waves are in same phase, then maximum intensity of sound is produced. After second, first tuning fork completes 64 vibrations while second fork comopletes 63 vibrations. At this instant waves are not in same phase, produce minimum intensity of sound. After sec (or) sec first fork completes 128 vibrations, second fork completes 127 vibrations. The waves again meet in phase, produce waxing (or) maximum intensity of sound. After seconds, the waves again meet in opposite phase. So, minimum sound is heard. After or 1 second the number of vibrations are 256, 254 respectively, intensity of sound becomes maximum.

Here the intensity will become two times maximum and two times minimum.

That is two beats will be heard in 1 second.

Beat frequency is 2 beats per second equal to the difference of frequencies of two tuning forks.

**Uses**

1) Beats can be used to tune musical instruments.

2) Dangerous gases in mines can be detected by using beats.

**11.** What is "Doppler effect"? Give illustrative examples.

**A:** The apparent change in frequency of sound due to relative motion between source and observer is called Doppler effect.

**Examples:**

1) When a train is approaching, the pitch of horn of train to be increased and that of train is going away the pitch of sound decreases.

2) Doppler shift is also expressed in terms of wavelenghts. If Radiation is shifted towards red colour called redshift then its wavelength increases. It indicates the stars are moving away and universe in expanding.

**Long Answer Questions **

**1.** Explain the formation of stationary waves in stretched strings and hence deduce the laws of transverse waves in stretched strings.

**A:** Two progressive waves of same amplitude, same frequency travelling in opposite direction superimpose over each other produce stationary or standing wave. When a stretched string is fixed at both ends is plucked perpendicular to the length, then resulting transverse wave travels along the length of string and get reflected at rigid end. The reflected wave is having same amplitude and frequency as incident wave.

The waves are represented by

y_{1} = A sin (ωt - kx) y_{2} = -A sin (ωt + kx)

According to principle of Superposition

y = y_{1} + y_{2} =A sin (ωt - kx) - A sin (ωt + kx)

By plucking the stretched wire at various positions standing waves are produced s shown in figure.

**Equation of fundamental frequency:**

In the first mode of vibration the wire vibrates with one loop.

**Laws of vibrations in stretched string:**

I Law: The fundamental frequency of vibrating string is inversely proportional to the length of the string, when tension and linear density are constant.

II Law: The fundamental frequency of vibrating string is directly proportional to square root of tension in the string.

when length, linear density are constant.

III Law: The fundamental frequency of vibration is inversely proportional to square root of linear density, when length, tension are constant.

**2. **Explain the formation of stationary waves in air column closed in open pipe. Derive the equation for the frequencies of harmonics produced.

**A:** Consider open pipe of length 'l'. In fundamental mode of vibration two anti nodes and one node is formed. Then

In the next mode of vibration three antinodes and two nodes are formed. Then length of pipe

Similarly in the next mode four antinodes and three nodes are formed. Then length of pipe

Hence in case of open pipe, harmonics are in the ratio of 1 : 2 : 3 : ....

**3.** How are stationary waves formed in closed pipes? Explain various vibrations and obtain relations for their frequencies.

**A:** A closed pipe is a hallow cylindrical pipe is closed at one end and open at other end. consider a pipe of length l. Let V be the velocity of sound in air. When longitudinal waves are produced in closed pipe, it gets reflected at closed end and a node is formed at closed end, antinode is formed at open end. In the first mode of vibration one node and one antinode is formed and its frequency is called fundamental frequency.

The fundamental frequency or first harmonic is

In the next mode of vibration two nodes and two antinodes are formed. Then,

υ_{5} is called 5th harmonic (or) second overtone. The ratio of harmonics in closed pipe are 1 : 3 : 5 : ........

**4. **What are beats? obtain an expression for beat frequency. Where and how are beats made use of?

**A:**

**Beats:** When two sound notes of nearly equal frequencies travelling in the same direction superposed to produce regular waxing and waning in the intensity of combined wave. This effect is known as beats.

Beat frequency = n_{1} ~ n_{2} or υ_{1} ~ υ_{2}

**Expression for beat frequency:**

Let us consider two waves y_{1} and y_{2} of nearly equal frequency n_{1} and n_{2}, each of amplitude 'a' superpose each other then resultant wave is

y = y_{1} + y_{2} = a sin ω_{1} t + a sin ω_{2} t

= a [sin 2Πn_{1} t + sin 2Πn_{2} t]

**Importance of Beats**

1) Beats can be used in tuning of musical instruments.

2) Beats are used to produce special effects in cinematography.

3) Beats are used in detecting dangerous gases in mines.

4) Beats are used in hetrodyne receivers range.

**5.** What is Doppler effect? Obtain an expression for the apparent frequency of sound heard when the source is in motion with respect to an observer at rest.

**A: Def:** The apparent change in frequency of sound due to relative motion between the observer and the source of sound is called "Doppler effect".

Let 'S' is a source of sound moving, away with a velocity 'Vs' from stationary observe. Let 'υ_{0}' be the frequency of sound produced by the source and 'T_{0}' be the time period. Let the observer has a device to count the number of crest compressions of wave produced.

At time t = 0, the source produces a crest. Let the distance between source and

observer is 'L' and velocity is 'V'.

Time taken by the crest to reach obser t_{1} = ............. (1)

The second crest is produced after a time interval 'T_{0}'

The distance travelled by the source during time interval T_{0} is 'V_{s }T_{0}'

The total distance between source & observer = L + V_{s} T_{0}

Time taken to detect 2^{nd }crest t_{2} = .............(2)

If source produces (n + 1)th crest at time nT_{0}, then time taken T_{0} detect the

i.e., The apparent frequency decreases.

If source is moving towards observer with

velocity 'V_{s}'

i.e. the apparent frequency increases.

**6.** What is doppler shift ? obtain an expression for the apparent frequency of sound heard when the observer is in motion with respect to source at rest.

**A: Doppler Shift:** The difference between apparent frequency heard by observer and actual frequency produced by the source is called 'Doppler Shift'. Let 'S' is source at rest produces a sound of constant frequency 'υ_{0}', To be the timeperiod of wave, V_{0} be the velocity of observer. At time t = 0, source produces a crest. The distance between source and observer is 'L' and velocity of sound is 'V'.

The time taken by the observer to detect the crest is t_{1} = (1)

2nd crest is produced after a time period T_{0}.

During this time observer moves a distance V_{0}T_{0}.

Time taken by observer to detect crest is t_{2}

i.e. apparent frequency υ < υ_{0}

when obeserver is moving from source take 'V_{0}' as -ve, then

i.e. apparent frequency increases.

**Problems**

**1.** A stretched wire of length 0.6 m is observed to vibrate with a frequency of 30 Hz in the fundamental mode. If the string has linear mass 0.05 kg/m. Then find,

a) Velocity of propagation of transverse waves in string

b) The tension in the string.

**Sol.** l = 0.6 m, υ = 30 Hz, µ = 0.05 kg/m

**2.** Two progressive transverse waves given by y_{1} = 0.07 sin Π (12x − 500 t) and y_{2} = 0.07 sin Π (12x + 500 t) travelling along a stretched string and form nodes and antinodes. What is the displacement at the nodes, antinodes, wavelength of standing wave?

**Sol.** y_{1} = 0.07 sin Π (12x − 500 t)

y_{2} = 0.07 sin Π (12x + 500 t)

Dilsplacement at node = 0

Displacement at antinode = 2A = 2 × 0.07 = 0.14 m

λ = 0.16 m

**3.** A vertical tube is made to stand in water so that the water level can be adjusted. Sound waves of frequency 320 Hz are sent into the top the tube. If standing waves are produced at successive water levels of 20 cm and 73 cm, what is the speed of sound waves in the air in the tube?

**Sol.** υ = 320 Hz, l_{1 }= 20 cm, l_{2} = 73 cm

V = 2n (l_{2} - l_{1})

= 2 × 320 (73 - 20) × 10^{-2}

= 640(53) × 10^{-2}

V = 339 m/s

**4.** A train sound its whistle as it approaches and a frequency of 184 Hz as it crosses a level crossing. An observer at the crossing measures a frequency of 219 Hz as the train approaches and a frequency of 184 Hz as it leaves. If the speed of sound is take to be 340 m/s. Find the speed of the train and the frequency of its whistle.

**Sol.** V = 340 m/s, υ'= 219 Hz, υ'' = 184 Hz

**5. **** **A closed organ pipe of 70 cm long is sounded. If the velocity of sound is 331 m/s, what is the fundamental frequency of vibration of the air column?

**Sol: **Given L = 70 cm = 0.7 m

V = 331 m/s

Fundamental frequency,

V 331

**6**. An open organ pipe of 85 cm long is sounded. If the velocity of sound is 340 m/s, what is the frequency of 3rd harmonic?

**Sol:** Given L = 85 cm = 0.85 m

V = 340 m/s

Frequency of 3rd harmonic

V 340