Wave and Acoustics | Physics JEE previous year questions with Solutions

Q 1 :

Two waves of intensity ratio 1:9 cross each other at a point. The resultant intensities at that point, when

(a) Waves are in coherent is $I_{1}$

(b) Waves are coherent is $I_{2}$ and differ in phase by 60. If $\frac{I_{1}}{I_{2}} = \frac{10}{x}$ then $x =$ ___________ . [2024]

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(13) For incoherent wave $I_{1} = I_{A} + I_{B} \Rightarrow I_{1} = I_{0} + 9 I_{0}$

$I_{1} = 10 I_{0}$

For coherent wave $I_{2} = I_{A} + I_{B} + 2 \sqrt{I_{A} I_{B}} \cos 60^{0}$

$I_{2} = I_{0} + 9 I_{0} + 2 \sqrt{9 I_{0}^{2}} \cdot \frac{1}{2} = 13 I_{0}$

$\frac{I_{1}}{I_{2}} = \frac{10}{13}$

Q 2 :

In the resonance experiment, two air columns (closed at one end) of 100 cm and 120 cm long, give 15 beats per second when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is: [2025]

335 m/s
370 m/s
340 m/s
360 m/s

1

2

3

4

(4)

Fundamental frequency in close organ pipe, $f_{0} = \frac{v}{4 l}$ , $f_{1} = \frac{v}{4 l_{1}}$ and $f_{2} = \frac{v}{4 l_{2}}$

Beat $△ f = (f_{1} - f_{2}) = \frac{v}{4} (\frac{1}{l_{1}} - \frac{1}{l_{2}})$

$15 = \frac{v}{4} (\frac{1}{1} - \frac{1}{1.2})$

$\Rightarrow v = 360 m / s$

Q 3 :

Two harmonic waves moving in the same direction superimpose to form a wave x = a cos (1.5t) cos (50.5t) where t is in seconds. Find the period with which they beat (close to nearest integer) [2025]

6 s
4 s
1 s
2 s

1

2

3

4

(4)

The given equation can be written as

$x = \frac{a}{2} \cos [1.5 + 50.5] t + \frac{a}{2} \cos [50.5 - 1.5] t$

$x = \frac{a}{2} \cos [52 t] + \frac{a}{2} \cos [49 t]$

Here, $2 π f_{1} = 52 and 2 π f_{2} = 49$

$f_{1} = \frac{52}{2 π}, f_{2} = \frac{49}{2 π}$

$\Rightarrow △ f = f_{Beat} = f_{1} - f_{2} = \frac{3}{2 π} Hz$

$∴ T_{Beat} = \frac{1}{f_{Beat}} = \frac{2 π}{3} \sec = 2.09 \sec$

$\Rightarrow T_{Beat} \approx 2 \sec$

Q 4 :

The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, $y_{1} (x, t) = 4 \sin (k x - ω t)$ and $y_{2} (x, t) = 2 \sin (k x - ω t + \frac{2 π}{3})$ , are: (Take the angular frequency of initial waves same as $ω$ ) [2025]

$[6, \frac{2 π}{3}]$
$[6, \frac{π}{3}]$
$[\sqrt{3}, \frac{π}{6}]$
$[2 \sqrt{3}, \frac{π}{6}]$

1

2

3

4

(4)

$A = \sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$

Here, $ϕ = \frac{2 π}{3}$

$A = \sqrt{4^{2} + 2^{2} + 2 \times 4 \times 2 \cos \frac{2 π}{3}} = 2 \sqrt{3}$

$\tan α = \frac{2 \sin ϕ}{4 + 2 \cos ϕ} = \frac{1}{\sqrt{3}} \Rightarrow α = π / 6$

Q 5 :

Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is ______ degree. [2023]

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(120)

$2 A \cos (\frac{Δ ϕ}{2}) = A$

$\cos (\frac{Δ ϕ}{2}) = \frac{1}{2} \Rightarrow \frac{Δ ϕ}{2} = 60 °$

Q 6 :

The displacement equations of two interfering waves are given by

$y_{1} = 10 \sin (ω t + \frac{π}{3}) cm$

$y_{2} = 5 [\sin (ω t) + \sqrt{3} \cos (ω t)] cm$ respectively.

The amplitude of the resultant wave is ________ cm. [2023]

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(20)

$y_{2} = 5 (\sin ω t + \sqrt{3} \cos ω t)$

$= 10 \sin (ω t + \frac{π}{3})$

$Thus the phase difference between the waves is 0$

$So A = A_{1} + A_{2} = 20 cm$

Q 7 :

The fundamental frequency of vibration of a string stretched between two rigid supports is 50 Hz. The mass of the string is 18 g and its linear mass density is 20 g/m. The speed of the transverse waves so produced in the string is ______ ${m s}^{- 1}$ . [2023]

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(90)

[IMAGE 121]

$Fundamental frequency = 50 Hz$

$\frac{mass}{length} = 20 g/m$

$mass = 18 g$

$length of string = \frac{18}{20} m = \frac{9}{10} m$

$From diagram \frac{λ}{2} = ℓ$

$\Rightarrow λ = 2 ℓ = \frac{9}{5} m$

$Again, speed v = f λ = 50 \times \frac{9}{5} = 90 m/s$

Topic Question Set

Q 1 :

Two waves of intensity ratio 1:9 cross each other at a point. The resultant intensities at that point, when

(a) Waves are in coherent is $I_{1}$

(b) Waves are coherent is $I_{2}$ and differ in phase by 60. If $\frac{I_{1}}{I_{2}} = \frac{10}{x}$ then $x =$ ___________ . [2024]

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Q 2 :

In the resonance experiment, two air columns (closed at one end) of 100 cm and 120 cm long, give 15 beats per second when each one is sounding in the respective fundamental modes. The velocity of sound in the air column is: [2025]

Q 3 :

Two harmonic waves moving in the same direction superimpose to form a wave x = a cos (1.5t) cos (50.5t) where t is in seconds. Find the period with which they beat (close to nearest integer) [2025]

Q 4 :

The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, $y_{1} (x, t) = 4 \sin (k x - ω t)$ and $y_{2} (x, t) = 2 \sin (k x - ω t + \frac{2 π}{3})$ , are: (Take the angular frequency of initial waves same as $ω$ ) [2025]

Q 5 :

Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is ______ degree. [2023]

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Q 6 :

The displacement equations of two interfering waves are given by

$y_{1} = 10 \sin (ω t + \frac{π}{3}) cm$

$y_{2} = 5 [\sin (ω t) + \sqrt{3} \cos (ω t)] cm$ respectively.

The amplitude of the resultant wave is ________ cm. [2023]

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Q 7 :

The fundamental frequency of vibration of a string stretched between two rigid supports is 50 Hz. The mass of the string is 18 g and its linear mass density is 20 g/m. The speed of the transverse waves so produced in the string is ______ ${m s}^{- 1}$ . [2023]

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