Wave Equation

Q. A transverse wave is described by the equation, $y=A\sin 2\pi (nt-\frac{x}{{\lambda }_0})$. The maximum particle velocity is equal to $3$ times the wave velocity, if

1. ${\lambda }_0=\frac{\pi A}{3}$
2. ${\lambda }_0=\frac{2\pi A}{3}$
3. ${\lambda }_0=\pi A$
4. ${\lambda }_0=3\pi A$

Q. Two particles are executing simple harmonic motion of the same amplitude $A$ and angular frequency $\omega$ along the $x-$axis. Their mean position is separated by distance $X_o(X_o>A)$. If the maximum separation between them is $(X_o+A)$, the phase difference between their motion is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{2}$

Q. The frequency of vibration $f$ of a mass $m$ suspended from a spring of spring constant 𝐾 is given by a relation of this type $f=C{m}^x{K}^y$, where $C$ is a dimensionless quantity. The value of $x$ and $y$ are $P$ and $Q$, respectively.
Find $(P+Q)$

Q. The amplitude of wave disturbance propagating in positive $x-$direction is given by $y=\frac{1}{1+x^2}$ at $t=0$ and $y=\frac{1}{1+(x-1{)}^2}$ at $t=2\text{s}$. The wave speed is

1. $0.5\text{m/s}$
2. $1.5\text{m/s}$
3. $2\text{m/s}$
4. $1\text{m/s}$

Q.

a source of sound frequency 600 hz is placed inside water .the speed of sound in water is 1500m/s and in air is 300m/s.the frequency if sound recorded by an observer , who is standing in air is.

Q. A pipe $20cm$ long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a source of $1237.5Hz$ ?(sound velocity in air $=330m/s)$

Q.

Suppose refractive index $\mu$ is given as $\mu =A+\frac{B}{{\lambda }^2}$ where $A$ and $B$ are constants and $\lambda$ is wavelength, then dimensions of $B$ are same as that of

1. Wavelength

2. Volume

3. Pressure

4. Area

Q.

A transverse harmonic wave on a string is described by

y(x, t) = 3.0 sin (36t + 0.018x + π/4)


Where x and y are in cm and t in s. The positive direction of x is from left to right.

(a) Is this a travelling wave or a stationary wave?

If it is travelling, what are the speed and direction of its propagation?

(b) What are its amplitude and frequency?

(c) What is the initial phase at the origin?

(d) What is the least distance between two successive crests in the wave?

Q. What is amplitude?

Q.

A block of mass 0.5 kg hanging from a vertical spring executes simple harmonic motion of amplitude 0.1 m and time period 0.314 s. Find the maximum force exerted by the spring on the block.

Q.

A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest a total distance of 0.62m. The fisherman sees that the wave crests are spaced 6.0 m apart. What would be the equation of such a wave?

1. y = 2.5 sin (0.62t + 2.5x)

2. 

3. 

4. y = 6 sin (2.5t - 0.62x)

Q. Two sound waves of wavelength $1\text{m}$ and $1.01\text{m}$ in a gas produce $10$ beats in $3\text{s}$. The velocity of sound in the gas is

1. $330\text{m/s}$
2. $337\text{m/s}$
3. $360\text{m/s}$
4. $300\text{m/s}$

Q. Which of the following equations represents a transverse wave travelling along Y-axis?

1. y = A sin kycosωt

2. x = A sin (ky - ωt)

3. y = A sin (kx - ωt)

4. x = A sin (ky - λt)

Q. For a travelling wave represented by equation,
$y=0.1\sin \pi (x-330t+\frac{2}{3})$ (SI units)
The phase difference between particles at $x_1=3\text{m}$ and $x_2=3.5\text{m}$ is

1. $\frac{\pi }{2}$
2. $2\pi$
3. $\pi$
4. $\frac{3\pi }{2}$

Q. A particle of mass 4 kg moves in simple harmonic motion and its potential energy U varies with position x as shown in the figure. The time period of oscillation of the particle is $\frac{2\pi }{\alpha }$ second. Find the value of $\alpha$.___

Q.

Which of the following is correct for X-rays?

1. It is a beam of electrons.

2. It is electromagnetic waves.

3. It is a positively charged particle.

4. It is a negatively charged particle.

Q. A string of length $l$ fixed at both ends vibrates in resonance with a tuning fork of frequency $f$ at two successive values of tension $T_1$ and $T_2$ in the string. Find the specific mass (mass per unit length) of the string.

1. $\frac{4T_1{T}_2}{f^2{l}^2{(\sqrt{T_1}+\sqrt{T_2})}^2}$
2. $\frac{2T_1{T}_2}{f^2{l}^2{(\sqrt{T_1}+\sqrt{T_2})}^2}$
3. $\frac{T_1{T}_2}{4f^2{l}^2{(\sqrt{T_1}-\sqrt{T_2})}^2}$
4. $\frac{4T_1{T}_2}{f^2{l}^2{(\sqrt{T_1}-\sqrt{T_2})}^2}$

Q. A wave travelling along the positive $x-$direction having maximum displacement along $y-$direction as $1\text{m}$, wavelength $2\pi \text{m}$ and frequency of $1/\pi \text{Hz}$ is represented by:

1. $y=\sin (2\pi x-2\pi t)$
2. $y=\sin (10\pi x-20\pi t)$
3. $y=\sin (x-2t)$
4. $y=\sin (2\pi x+2\pi t)$

Q. If the speed of the wave shown in the figure is 330 m/s in the given medium, then the equation of the wave propagating in the positive x - direction will be -(all quantities are in MKS units)

1. $y=0.05\text{sin}2\pi (4000t-12.5x)$
2. $y=0.05\text{sin}2\pi (4000t-122.5x)$
3. $y=0.05\text{sin}2\pi (3300t-10x)$
4. $y=0.05\text{sin}2\pi (3300x-10t)$

Q.

Are water waves transverse or longitudinal?

Q.

The equation $y=A{\sin }^2(kx-\omega t)$ represents a wave with

1. amplitude $A$ frequency $\omega /2\mathrm{\pi }$

2. amplitude $A/2$ frequency $\omega /\mathrm{\pi }$

3. amplitude $2A$ frequency $\omega /4\mathrm{\pi }$

4. It does not represent a wave motion

Q.

A transverse wave is traveling along a string in the positive x-axis. The figure shows the photograph of the wave at an instant.
The points moving upward are _____.

1. the points moving upward _____

2. the points moving downwards_____

3. the points which have maximum velocity_____

4. the points which have zero velocity_____

Q.

A particle of mass m is attached to three springs A, B and C of equal force constants k as shown in figure (12-E6). If the particle is pushed slightly against the spring C and released, find the time period of oscillation.

Q. A wave travelling along the $x-$axis is described by the equation $y(x,t)=0.008\cos (\alpha x-\beta t)$. If the wavelength and time period of the wave are $0.16\text{m}$ and $6.0$ respectively, then $\alpha$ and $\beta$ in appropriate units are

1. $\alpha =25.00\pi ,\beta =\pi$
2. $\alpha =12.50\pi ,\beta =\frac{\pi }{2}$
3. $\alpha =12.50\pi ,\beta =\frac{\pi }{3}$
4. $\alpha =\frac{0.08}{\pi },\beta =\frac{2}{\pi }$

Q. A sinusoidal wave travelling in the positive direction on a stretched string has an amplitude $20\text{cm}$, wavelength $1.0\text{m}$ and wave velocity $5\text{m/s}$. At $x=0$ and $t=0$ , it is given that $y=0$ and $\frac{\partial y}{\partial t}<0$. Find the wave function $y(x,t)$

1. $y(x.t)=0.02\sin \left[\right(2\pi x)+(10\pi t\left)\right]\text{m}$
2. $y(x.t)=0.02\cos \left[\right(10\pi x)+(2\pi t\left)\right]\text{m}$
3. $y(x.t)=0.02\sin \left[\right(2\pi x)-(10\pi t\left)\right]\text{m}$
4. $y(x.t)=0.02\sin \left[\right(\pi x)+(5\pi t\left)\right]\text{m}$

Q. A whistle of frequency 500 Hz tied to the end of a string of length 1.2 m revolves at 400 rev/min. A listener standing some distance away in the plane of rotation of whistle hears frequencies in the range (speed of sound = 340 m/s)

1. 436 to 586
2. 436 to 674
3. 426 to 574
4. 426 to 584

Q. The resultant amplitude in interference with two coherent sources depends upon-

1. Intensity
2. Only phase difference
3. On both the above
4. None of the above

Q. A transverse sinusoidal wave is generated at one end of a long horizontal string by a bar that moves the end up and down through a distance by 2.0 cm. The motion of bar is continuous and is repeated regularly 125 times per second. If the distance between adjacent wave crests is observed to the 15.6 cm and the wave is moving along positive x-direction, and at t = 0 the element of the string at x = 0 is at mean position y = 0 and is moving downward, the equation of the wave is best described by

1. y = (1 cm) cos [(40.3 rad/m) x – (786 rad/s)t]
2. y = (2 cm) cos [(40.3 rad/m) x – (786 rad/s)t]
3. y = (1 cm) sin [(40.3 rad/m) x – (786 rad/s)t]
4. y = (2 cm) sin [(40.3 rad/m) x – (786 rad/s)t]

Q. The displacement, $y$ of a particle on a straight line is given by $y=f(x,t)$, as a function of time. Which of the following functions does not represent wave motion?

1. $y=A{\sin }^2(kx-\omega t)$
2. $y=A\sin (kx-\omega t)$
3. $y=A\sin (kx+\omega t+\frac{\pi }{10})$
4. $y=A\sin (k^2{x}^2-{\omega }^2{t}^2)$