Wave and Acoustics | Physics JEE previous year questions with Solutions

Q 1 :

A plane progressive wave is given by $y = 2$ $\cos$ $2 π (330 t - x) m .$ The frequency of the wave is [2024]

340 Hz
165 Hz
330 Hz
660 Hz

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(3)

$y = 2$ $\cos$ $2 π (330 t - x) m$

$y = A$ $\cos$ $(ω t - k x)$

Comparing $ω = 2 π \times 330$

$2 πf = 2 π \times 330 \Rightarrow f = 330$ $H z$

Q 2 :

The equation of a transverse wave travelling along a string is $y (x, t) = 4.0 \sin [20 \times 10^{- 3} x + 600 t] mm$ , where x is in the mm and t is in second. The velocity of the wave is [2025]

+ 30 m/s
– 60 m/s
– 30 m/s
+ 60 m/s

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(3)

$y = 4 \sin (20 \times 10^{- 3} x + 600 t)$

Here, $ω$ = 600 and k = $20 \times 10^{- 3}$

$∴ v = \frac{ω}{k} = \frac{600}{20 \times 10^{- 3}}$

$= 30 \times 10^{+ 3} mm / s = 30 m / s$

and direction is towards –ve x-axis

$∴$ v = – 30 m/s

Q 3 :

A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x = 0 at t = 0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave? [2025]

y = 2 cos (0.83x – 3.35t) cm
y = 2 sin (0.83x – 3.5t) cm
y = 2 cos (3.35x – 0.83t) cm
y = 2 cos (0.13x – 0.5t) cm

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(1)

$v = \frac{distance}{time}$

$v = \frac{1.2}{0.3} = 4 cm / s$

$k = \frac{2 π}{λ} = \frac{2 π}{7.5} = \frac{4 π}{15} = 0.83$

$v = \frac{ω}{k} \Rightarrow ω = v k = 4 \times \frac{4 π}{15} = 3.35$

At t = 0, x = 0, there is a crest

So, y = A cos (kx – $ω$ t)

$∴$ y = 2 cos (0.83x – 3.35t) cm

Q 4 :

Displacement of a wave is expressed as $x (t) = 5 \cos (628 t + \frac{π}{2}) m$ . The wavelength of the wave when its velocity is 300 m/s is [2025]

5 m
3 m
0.5 m
0.33 m

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(2)

$x (t) = 5 \cos [628 t + \frac{π}{2}] m$

Wave velocity v = 300 m/s

$v = \frac{ω}{k}$

$300 = \frac{628}{K} \Rightarrow K = \frac{628}{300} m^{- 1}$

$\frac{2 π}{λ} = \frac{628}{300} \Rightarrow λ = \frac{2 \times 3.14 \times 300}{628}$

$\Rightarrow λ = 3 m$

Q 5 :

The equation of a wave travelling on a string is y = sin [20 $π$ x + 10 $π$ t], where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is : [2025]

5.0 cm
20 cm
10 cm
2.5 cm

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4

(1)

$k = 20 π = \frac{2 π}{λ} \Rightarrow λ = 10 cm$

Minimum distance between 2 points having same speed is $\frac{λ}{2}$ .

Minimum distance $= \frac{λ}{2} = 5 cm$

Q 6 :

Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section R is $v_{1}$ , and that in the other string having radius of cross section R/2 is $v_{2}$ . Then $\frac{v_{2}}{v_{1}} =$ [2025]

$\sqrt{2}$
2
8
4

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(2)

$v = \sqrt{\frac{T}{μ}} = \sqrt{\frac{T}{ρ π R^{2}}}$

$\Rightarrow \frac{v_{2}}{v_{1}} = \frac{R_{1}}{R_{2}} = 2$

Q 7 :

A steel wire with mass per unit length $7.0 \times 10^{- 3} {kg m}^{- 1}$ is under tension of 70 N. The speed of transverse waves in the wire will be [2023]

$200 π m/s$
$100 m/s$
$10 m/s$
$50 m/s$

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(2)

$v = \sqrt{\frac{T}{μ}} = \sqrt{\frac{70}{70 \times 10^{- 3}}} = 100 m/s$

Q 8 :

A transverse harmonic wave on a string is given by $y (x, t) = 5 \sin (6 t + 0.003 x)$ where $x$ and $y$ are in cm and $t$ in second. The wave velocity is ______ ${ms}^{- 1}$ . [2023]

(20)

$y (x, t) = 5 \sin (6 t + 0.003 x)$

$k = {0.003 cm}^{- 1}, ω = 6 rad/s, v = \frac{ω}{k}$

$\Rightarrow \frac{6}{0.003 \times 10^{2}} = {20 ms}^{- 1}$

Q 9 :

The equation of wave is given by $Y = 10^{- 2} \sin 2 π (160 t - 0.5 x + \frac{π}{4})$

where $x$ and $Y$ are in $m$ and $t$ in $s$ . The speed of the wave is ________ ${km h}^{- 1}$ . [2023]

(1152)

$V = \frac{ω}{k} = \frac{160}{0.5} m/s = \frac{160}{0.5} \times \frac{18}{5} km/h = 1152 km/h$

Q 10 :

Two strings (A, B) having linear densities $μ_{A} = 2 \times 10^{- 4} kg/m$ and $μ_{B} = 4 \times 10^{- 4} kg/m$ and lengths $L_{A} = 2.5 m and L_{B} = 1.5 m$ respectively are joined. Free ends of A and B are tied to two rigid supports C and D, respectively creating a tension of 500 N in the wire. Two identical pulses, sent from C and D ends, take time $t_{1} and t_{2}$ respectively to reach the joint. The ratio $t_{1} / t_{2}$ is [2026]

1.18
1.90
1.08
1.67

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(1)

$Given L_{A} = 2.5 m, L_{B} = 1.5 m, T = 500 N$

$v_{A} = \sqrt{\frac{T}{μ_{A}}} = \sqrt{\frac{500}{2 \times 10^{- 4}}} = 5 \sqrt{10} \times 10^{2} m/s$

$v_{B} = \sqrt{\frac{T}{μ_{B}}} = \sqrt{\frac{500}{4 \times 10^{- 4}}} = 5 \sqrt{5} \times 10^{2} m/s$

$t_{1} = \frac{L_{A}}{v_{A}} = \frac{2.5}{5 \sqrt{10}} \times 10^{- 2} s$

$t_{2} = \frac{L_{B}}{v_{B}} = \frac{1.5}{5 \sqrt{5}} \times 10^{- 2} s$

$∴$ $\frac{t_{1}}{t_{2}} = \frac{2.5}{5 \sqrt{10}} \times \frac{5 \sqrt{5}}{1.5} = \frac{5}{3} \times \frac{1}{\sqrt{2}} = \frac{1.66}{1.41} = 1.18$

Q 11 :

The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young’s modulus of the bar material are increased by 0.5% and 1%, respectively then the speed of the wave is changed approximately to ________ m/s. [2026]

401
398
399
402

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(1)

$V_{sound} = \sqrt{\frac{Y}{ρ}}$

$\frac{Δ V}{V} \times 100 = \frac{1}{2} (\frac{Δ Y}{Y} \times 100) - \frac{1}{2} (\frac{Δ ρ}{ρ} \times 100)$

$= \frac{1}{2} \times 1 - \frac{1}{2} \times 0.5$

$\frac{Δ V}{V} \times 100 = \frac{1}{4}$

$Δ V = \frac{1}{4} \times \frac{V}{100}$

$Δ V = 1 m/s$

$V_{final} = 400 + 1 = 401 m/s$

Q 12 :

A travelling wave is described by the equation $y (x, t) = 0.05 \sin (8 x - 4 t) m$ . The velocity of the wave is: - [all the quantities are in SI unit]

$4 {ms}^{- 1}$
$2 {ms}^{- 1}$
$0.5 {ms}^{- 1}$
$8 {ms}^{- 1}$

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(3)

$V = \frac{ω}{k}$

Q 13 :

A heavy but uniform rope of length L is suspended from a ceiling. A particle is dropped from the ceiling at the instant when the bottom end is given a transverse wave pulse. Where will the particle meet the pulse?

at a distance $\frac{2 L}{3}$ from the bottom
at a distance $\frac{L}{3}$ from the bottom
at a distance $\frac{3 L}{4}$ from the bottom
at a distance $\frac{L}{2}$ from the bottom

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(2)

For dropped body $L - x = \frac{1}{2} g t^{2}$ , for pulse $t = 2 \sqrt{\frac{x}{g}}$ , $x$ is the distance from bottom free end of row.

Q 14 :

A thick uniform rope of Length L is hanging from a rigid support. A transverse wave of wavelength $λ_{0}$ is set up in the middle of the rope. The wavelength of the wave as it reaches the top most point is

$2 λ_{0}$
$\sqrt{2} λ_{0}$
$\frac{λ_{0}}{\sqrt{2}}$
$λ_{0}$

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(2)

$Here frequency f is constant. Speed of wave$

$v = \sqrt{\frac{T}{μ}}$

$\Rightarrow \frac{v_{1}}{v_{2}} = \frac{λ_{1}}{λ_{2}} = \sqrt{\frac{T_{1}}{T_{2}}} = \sqrt{\frac{M g / 2}{M g}}$

$\Rightarrow λ_{2} = \sqrt{2} λ_{1} = \sqrt{2} λ_{0}$

Q 15 :

The equation of a plane progressive wave is given by $y = 5 \cos π (200 t - \frac{x}{150})$ where $x$ and $y$ are in cm and $t$ is in second. The velocity of the wave is _____ m/s. [2026]

120
150
200
300

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(4)

$y = 5 \cos π (200 t - \frac{x}{150})$

$v = \frac{coeff of t}{coeff of x} = \frac{200}{\frac{1}{150} \times 100} = 300 m/s$

Q 16 :

A transverse wave on a string is described by $y = 3 \sin (36 t + 0.018 x + \frac{π}{4})$ . Where $x, y$ are in cm and $t$ in seconds. The least distance between the two successive crests in the wave is ______ cm. (Nearest integer) ( $π = 3.14$ ) [2026]

(349)

$y = 3 \sin (36 t + 0.018 x + \frac{π}{4})$

$Comparing with the standard equation$

$y = A \sin (ω t \pm k x + ϕ)$

$λ = \frac{2 π}{k} = \frac{2 \times 3.14}{0.018} = \frac{6.28}{0.018}$

$λ = 348.9 cm$

$Nearest integer=349 cm$

Topic Question Set

Q 1 :

A plane progressive wave is given by $y = 2$ $\cos$ $2 π (330 t - x) m .$ The frequency of the wave is [2024]

Q 2 :

The equation of a transverse wave travelling along a string is $y (x, t) = 4.0 \sin [20 \times 10^{- 3} x + 600 t] mm$ , where x is in the mm and t is in second. The velocity of the wave is [2025]

Q 3 :

A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x = 0 at t = 0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave? [2025]

Q 4 :

Displacement of a wave is expressed as $x (t) = 5 \cos (628 t + \frac{π}{2}) m$ . The wavelength of the wave when its velocity is 300 m/s is [2025]

Q 5 :

The equation of a wave travelling on a string is y = sin [20 $π$ x + 10 $π$ t], where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is : [2025]

Q 7 :

A steel wire with mass per unit length $7.0 \times 10^{- 3} {kg m}^{- 1}$ is under tension of 70 N. The speed of transverse waves in the wire will be [2023]

Q 8 :

A transverse harmonic wave on a string is given by $y (x, t) = 5 \sin (6 t + 0.003 x)$ where $x$ and $y$ are in cm and $t$ in second. The wave velocity is ______ ${ms}^{- 1}$ . [2023]

Q 9 :

The equation of wave is given by $Y = 10^{- 2} \sin 2 π (160 t - 0.5 x + \frac{π}{4})$

where $x$ and $Y$ are in $m$ and $t$ in $s$ . The speed of the wave is ________ ${km h}^{- 1}$ . [2023]

Q 11 :

The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young’s modulus of the bar material are increased by 0.5% and 1%, respectively then the speed of the wave is changed approximately to ________ m/s. [2026]

Q 12 :

A travelling wave is described by the equation $y (x, t) = 0.05 \sin (8 x - 4 t) m$ . The velocity of the wave is: - [all the quantities are in SI unit]

Q 13 :

A heavy but uniform rope of length L is suspended from a ceiling. A particle is dropped from the ceiling at the instant when the bottom end is given a transverse wave pulse. Where will the particle meet the pulse?

Q 14 :

A thick uniform rope of Length L is hanging from a rigid support. A transverse wave of wavelength $λ_{0}$ is set up in the middle of the rope. The wavelength of the wave as it reaches the top most point is

Q 15 :

The equation of a plane progressive wave is given by $y = 5 \cos π (200 t - \frac{x}{150})$ where $x$ and $y$ are in cm and $t$ is in second. The velocity of the wave is _____ m/s. [2026]

Q 16 :

A transverse wave on a string is described by $y = 3 \sin (36 t + 0.018 x + \frac{π}{4})$ . Where $x, y$ are in cm and $t$ in seconds. The least distance between the two successive crests in the wave is ______ cm. (Nearest integer) ( $π = 3.14$ ) [2026]

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Topic Question Set

Q 1 : A plane progressive wave is given by y=2 cos 2π(330t-x)m. The frequency of the wave is [2024]

Q 2 : The equation of a transverse wave travelling along a string is y(x,t)=4.0 sin [20×10–3x+600 t] mm, where x is in the mm and t is in second. The velocity of the wave is [2025]

Q 3 : A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x = 0 at t = 0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave? [2025]

Q 4 : Displacement of a wave is expressed as x(t)=5 cos (628t+π2) m. The wavelength of the wave when its velocity is 300 m/s is [2025]

Q 5 : The equation of a wave travelling on a string is y = sin [20 πx + 10 πt], where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is : [2025]

Q 7 : A steel wire with mass per unit length 7.0×10-3 kg m-1 is under tension of 70 N. The speed of transverse waves in the wire will be [2023]

Q 8 : A transverse harmonic wave on a string is given by y(x,t)=5sin(6t+0.003x) where x and y are in cm and t in second. The wave velocity is ______ ms-1. [2023]

Q 9 : The equation of wave is given by Y=10-2sin2π(160t-0.5x+π4) where x and Y are in m and t in s. The speed of the wave is ________ km h-1. [2023]

Q 11 : The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young’s modulus of the bar material are increased by 0.5% and 1%, respectively then the speed of the wave is changed approximately to ________ m/s. [2026]

Q 12 : A travelling wave is described by the equation y(x,t)=0.05sin(8x-4t) m. The velocity of the wave is: - [all the quantities are in SI unit]

Q 13 : A heavy but uniform rope of length L is suspended from a ceiling. A particle is dropped from the ceiling at the instant when the bottom end is given a transverse wave pulse. Where will the particle meet the pulse?

Q 14 : A thick uniform rope of Length L is hanging from a rigid support. A transverse wave of wavelength λ0 is set up in the middle of the rope. The wavelength of the wave as it reaches the top most point is

Q 15 : The equation of a plane progressive wave is given by y=5cosπ(200t-x150) where x and y are in cm and t is in second. The velocity of the wave is _____ m/s. [2026]

Q 16 : A transverse wave on a string is described by y=3sin(36t+0.018x+π4). Where x, y are in cm and t in seconds. The least distance between the two successive crests in the wave is ______ cm. (Nearest integer) (π=3.14) [2026]

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Q 1 :

A plane progressive wave is given by $y = 2$ $\cos$ $2 π (330 t - x) m .$ The frequency of the wave is [2024]

Q 2 :

The equation of a transverse wave travelling along a string is $y (x, t) = 4.0 \sin [20 \times 10^{- 3} x + 600 t] mm$ , where x is in the mm and t is in second. The velocity of the wave is [2025]

Q 3 :

A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at x = 0 at t = 0 sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave? [2025]

Q 4 :

Displacement of a wave is expressed as $x (t) = 5 \cos (628 t + \frac{π}{2}) m$ . The wavelength of the wave when its velocity is 300 m/s is [2025]

Q 5 :

The equation of a wave travelling on a string is y = sin [20 $π$ x + 10 $π$ t], where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is : [2025]

Q 7 :

A steel wire with mass per unit length $7.0 \times 10^{- 3} {kg m}^{- 1}$ is under tension of 70 N. The speed of transverse waves in the wire will be [2023]

Q 8 :

A transverse harmonic wave on a string is given by $y (x, t) = 5 \sin (6 t + 0.003 x)$ where $x$ and $y$ are in cm and $t$ in second. The wave velocity is ______ ${ms}^{- 1}$ . [2023]

Q 9 :

The equation of wave is given by $Y = 10^{- 2} \sin 2 π (160 t - 0.5 x + \frac{π}{4})$

where $x$ and $Y$ are in $m$ and $t$ in $s$ . The speed of the wave is ________ ${km h}^{- 1}$ . [2023]

Q 11 :

The speed of a longitudinal wave in a metallic bar is 400 m/s. If the density and Young’s modulus of the bar material are increased by 0.5% and 1%, respectively then the speed of the wave is changed approximately to ________ m/s. [2026]

Q 12 :

A travelling wave is described by the equation $y (x, t) = 0.05 \sin (8 x - 4 t) m$ . The velocity of the wave is: - [all the quantities are in SI unit]

Q 13 :

A heavy but uniform rope of length L is suspended from a ceiling. A particle is dropped from the ceiling at the instant when the bottom end is given a transverse wave pulse. Where will the particle meet the pulse?

Q 14 :

A thick uniform rope of Length L is hanging from a rigid support. A transverse wave of wavelength $λ_{0}$ is set up in the middle of the rope. The wavelength of the wave as it reaches the top most point is

Q 15 :

The equation of a plane progressive wave is given by $y = 5 \cos π (200 t - \frac{x}{150})$ where $x$ and $y$ are in cm and $t$ is in second. The velocity of the wave is _____ m/s. [2026]

Q 16 :

A transverse wave on a string is described by $y = 3 \sin (36 t + 0.018 x + \frac{π}{4})$ . Where $x, y$ are in cm and $t$ in seconds. The least distance between the two successive crests in the wave is ______ cm. (Nearest integer) ( $π = 3.14$ ) [2026]