Wave Equation and Wave Speed

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The displacement $y$ of a particle in a medium can be expressed as $y = 1 0^{- 6} sin (100 t + 20 x + \frac{π}{4}) m$ where, $t$ is in second and $x$ in metre. The speed of the wave is

2000 ms^{- 1}

(B)

5 ms^{- 1}

(C)

20 ms^{- 1}

(D)

5 π ms^{- 1}

JEE Main 2021

LEVELJEE Main

The mass per unit length of a uniform wire is $0.135 g/cm$ . A transverse wave of the form $y = - 0.21 sin (x + 30 t)$ is produced in it, where $x$ is in metre and $t$ is in second. Then, the expected value of tension in the wire is $x \times 1 0^{- 2} N$ . Value of $x$ is ……… (Round-off to the nearest integer)

JEE Main 2019

LEVELJEE Main

A progressive wave travelling along the positive x-direction is represented by $y (x, t) = A sin (k x - ω t + ϕ)$ . Its snapshot at $t = 0$ is given in the figure.

(A)

- \frac{π}{2}

(B)

π

(C)

0

(D)

\frac{π}{2}

JEE Main 2011

LEVELJEE Main

The transverse displacement $y (x, t)$ of a wave on a string is given by $y (x, t) = e^{- (a x^{2} + b t^{2} + 2 ab x t)}$ . This represents a

(A)

wave moving in

- x

direction with speed

\frac{b}{a}

(B)

standing wave of frequency

b

(C)

standing wave of frequency

1/ b

(D)

wave moving in

+ x

direction with speed

\frac{a}{b}

JEE Main 2019

LEVELJEE Main

A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle $6 0^{\circ}$ with ground level. But he finds the aeroplane right vertically above his position. If $v$ is the speed of sound, then speed of the plane is

(A)

\frac{3}{2} v

(B)

v

(C)

\frac{2 v}{3}

(D)

\frac{v}{2}

JEE Main 2020

LEVELJEE Main

A uniform thin rope of length 12 m and mass 6 kg hangs vertically from a rigid support and a block of mass 2 kg is attached to its free end. A transverse short wavetrain of wavelength 6 cm is produced at the lower end of the rope. What is the wavelength of the wavetrain (in cm) when it reaches the top of the rope?

For a transverse wave travelling along a straight line, the distance between two peaks (crests) is $5 m$ , while the distance between one crest and one trough is $1.5 m$ . The possible wavelengths (in metre) of the waves are

(A)

1, 2, 3, \dots

(B)

1, \frac{1}{3}, \frac{1}{5}, \dots

(C)

1, 3, 5, \dots

(D)

\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \dots

JEE Main 2011

LEVELJEE Main

Statement I: Two longitudinal waves given by equations— $y_{1} (x, t) = 2 a sin (ω t - k x)$ and $y_{2} (x, t) = a sin (2 ω t - 2 k x)$ will have equal intensity. Statement II: Intensity of waves of given frequency in same medium is proportional to the square of amplitude only.

(A)

Statement I is false, Statement II is true

(B)

Statement I is true, Statement II is false

(C)

Statement I is true, Statement II is true; Statement II is the correct explanation of Statement I

(D)

Statement I is true, Statement II is true; Statement II is not correct explanation of Statement I

LEVELJEE Main

A wave travelling along the x-axis is described by the equation $y (x, t) = 0.005 cos (α x - β t)$ . If the wavelength and the time period of the wave are $0.08 m$ and $2.0 s$ respectively, then $α$ and $β$ in appropriate units are

(A)

α = 25.00 π, β = π

(B)

α = \frac{0.08}{π}, β = \frac{2.0}{π}

(C)

α = \frac{0.04}{π}, β = \frac{1.0}{π}

(D)

α = 12.50 π, β = \frac{π}{2.0}

JEE Main 2021

LEVELJEE Main

The amplitude of wave disturbance propagating in the positive x-direction is given by $y = \frac{1}{1 + x ^{2}}$ at time $t = 0$ and $y = \frac{1}{1 + ( x - 2 ) ^{2}}$ at $t = 1 s$ , where $x$ and $y$ are in metre. The shape of wave does not change during the propagation. The velocity of the wave will be ...... m/s.

JEE Main 2021

LEVELJEE Main

A sound wave of frequency $245 Hz$ travels with the speed of $300 ms^{- 1}$ along the positive X-axis. Each point of the wave moves to and fro through a total distance of $6 cm$ . What will be the mathematical expression of this travelling wave?

(A)

y (x, t) = 0.03 [sin 5.1 x - (0.2 \times 1 0^{3}) t]

(B)

y (x, t) = 0.06 [sin 5.1 x - (1.5 \times 1 0^{3}) t]

(C)

y (x, t) = 0.06 [sin 0.8 x - (0.5 \times 1 0^{3}) t]

(D)

y (x, t) = 0.03 [sin 5.1 x - (1.5 \times 1 0^{3}) t]

JEE Main 2019

LEVELJEE Main

Equation of travelling wave on a stretched string of linear density $5 g/m$ is $y = 0.03 sin (450 t - 9 x)$ , where distance and time are measured in SI units. The tension in the string is

(A)

5 N

(B)

12.5 N

(C)

7.5 N

(D)

10 N

JEE Main 2019

LEVELJEE Main

A small speaker delivers $2 W$ of audio output. At what distance from the speaker will one detect $120 dB$ intensity sound? [Take, reference intensity of sound as $1 0^{- 12} W/m^{2}$ ]

(A)

40 cm

(B)

20 cm

(C)

10 cm

(D)

30 cm

JEE Main 2016

LEVELJEE Advanced

A uniform string of length 20 m is suspended from a rigid support. A short wave pulse is introduced at its lowest end. It starts moving up the string. The time taken to reach the support is (Take, $g = 10 ms^{- 2}$ )

(A)

2 π 2 s

(B)

2 s

(C)

22 s

(D)

2 s

JEE Main 2020

LEVELJEE Main

A transverse wave travels on a taut steel wire with a velocity of $v$ when tension in it is $2.06 \times 1 0^{4}$ N. When the tension is changed to $T$ , the velocity changed to $v /2$ . The value of $T$ is close to

(A)

10.2 \times 1 0^{2}

(B)

5.15 \times 1 0^{3}

(C)

2.50 \times 1 0^{4}

(D)

30.5 \times 1 0^{4}

JEE Main 2020

LEVELJEE Main

Speed of a transverse wave on a straight wire (mass $6.0 g$ , length $60 cm$ and area of cross-section $1.0 mm^{2}$ ) is $90 ms^{- 1}$ . If the Young's modulus of wire is $16 \times 1 0^{11} Nm^{- 2}$ , the extension of wire over its natural length is

(A)

0.01 mm

(B)

0.04 mm

(C)

0.03 mm

(D)

0.02 mm

JEE Main 2019

LEVELJEE Advanced

A heavy ball of mass $M$ is suspended from the ceiling of a car by a light string of mass $m (m << M)$ . When the car is at rest, the speed of transverse waves in the string is $60 ms^{- 1}$ . When the car has acceleration $a$ , the wave speed increases to $60.5 ms^{- 1}$ . The value of $a$ , in terms of gravitational acceleration $g$ is closest to

(A)

\frac{g}{20}

(B)

\frac{g}{5}

(C)

\frac{g}{30}

(D)

\frac{g}{10}

JEE Main 2019

LEVELJEE Main

A travelling harmonic wave is represented by the equation $y (x, t) = 1 0^{- 3} sin (50 t + 2 x)$ , where $x$ and $y$ are in metre and $t$ is in second. Which of the following is a correct statement about the wave?

(A)

The wave is propagating along the negative X-axis with speed

25 ms^{- 1}

(B)

The wave is propagating along the positive X-axis with speed

25 ms^{- 1}

(C)

The wave is propagating along the positive X-axis with speed

100 ms^{- 1}

(D)

The wave is propagating along the negative X-axis with speed

100 ms^{- 1}

LEVELJEE Main

The displacement $y$ of a wave travelling in the $x$ -direction is given by $y = 1 0^{- 4} sin (600 t - 2 x + \frac{π}{3})$ metre where, $x$ is expressed in metres and $t$ in seconds. The speed of the wave-motion, in $ms^{- 1}$ is

A sound absorber attenuates the sound level by 20 db. The intensity decreases by a factor of

When temperature increases, the frequency of a tuning fork

increases or decreases depending on the material

JEE Main 2021

LEVELBoard

Which of the following equations represents a travelling wave?

(A)

y = A sin (15 x - 2 t)

(B)

y = A e^{- x^{2}} (v t + θ)

(C)

y = A e^{x} cos (ω t - θ)

(D)

y = A sin x cos ω t

JEE Main 2019

LEVELJEE Main

The pressure wave $p = 0.01 sin [1000 t - 3 x] Nm^{- 2}$ , corresponds to the sound produced by a vibrating blade on a day when atmospheric temperature is $0^{\circ} C$ . On some other day when temperature is $T$ , the speed of sound produced by the same blade and at the same frequency is found to be $336 ms^{- 1}$ . Approximate value of $T$ is

(A)

1 5^{\circ} C

(B)

1 1^{\circ} C

(C)

1 2^{\circ} C

(D)

4^{\circ} C

LEVELJEE Main

The equation of a wave on a string of linear mass density $0.04 kg m^{- 1}$ is given by $y = 0.02 (m) sin [2 π (\frac{t}{0.04 (s)} - \frac{x}{0.50 (m)})]$ . The tension in the string is

(A)

4.0 N

(B)

12.5 N

(C)

0.5 N

(D)

6.25 N

JEE Advanced 1990

LEVELJEE Main

The amplitude of a wave disturbance travelling in the positive $x$ -direction is given by $y = \frac{1}{1 + x ^{2}}$ at time $t = 0$ and by $y = \frac{1}{1 + ( x - 1 ) ^{2}}$ at $t = 2 s$ , where $x$ and $y$ are in metre. The shape of the wave disturbance does not change during the propagation. The velocity of the wave is ...... m/s.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The displacement y of a particle in a medium can be expressed as y=10−6sin(100t+20x+4π​)m where, t is in second and x in metre. The speed of the wave is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A progressive wave travelling along the positive x-direction is represented by y(x,t)=Asin(kx−ωt+ϕ). Its snapshot at t=0 is given in the figure.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The transverse displacement y(x,t) of a wave on a string is given by y(x,t)=e−(ax2+bt2+2ab​xt). This represents a

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A small speaker delivers 2 W of audio output. At what distance from the speaker will one detect 120 dB intensity sound? [Take, reference intensity of sound as 10−12 W/m2]

A sound absorber attenuates the sound level by 20 db. The intensity decreases by a factor of

When temperature increases, the frequency of a tuning fork

Which of the following equations represents a travelling wave?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of a wave on a string of linear mass density 0.04 kg m−1 is given by y=0.02 (m)sin[2π(0.04 (s)t​−0.50 (m)x​)]. The tension in the string is

The displacement $y$ of a particle in a medium can be expressed as $y = 1 0^{- 6} sin (100 t + 20 x + \frac{π}{4}) m$ where, $t$ is in second and $x$ in metre. The speed of the wave is

A progressive wave travelling along the positive x-direction is represented by $y (x, t) = A sin (k x - ω t + ϕ)$ . Its snapshot at $t = 0$ is given in the figure.

The transverse displacement $y (x, t)$ of a wave on a string is given by $y (x, t) = e^{- (a x^{2} + b t^{2} + 2 ab x t)}$ . This represents a

A small speaker delivers $2 W$ of audio output. At what distance from the speaker will one detect $120 dB$ intensity sound? [Take, reference intensity of sound as $1 0^{- 12} W/m^{2}$ ]

The equation of a wave on a string of linear mass density $0.04 kg m^{- 1}$ is given by $y = 0.02 (m) sin [2 π (\frac{t}{0.04 (s)} - \frac{x}{0.50 (m)})]$ . The tension in the string is