Wave Equation

Q.

The displacement of the particle at $x=0$ of a stretched string carrying a wave in the positive x-direction is given by $f\left(t\right)=A\sin (t/T)$. The wave speed is $v$. Write the wave equation.

Q.

The equation of a wave traveling on a string stretched along the X-axis is given by$y=A{e}^{-{\left(\frac{x}{a}+\frac{t}{T}\right)}^2}$.

(a) Write the dimensions of $A,aandT.$

(b) Find the wave speed.

(c) In which direction is the wave traveling?

(d) Where is the maximum of the pulse located at$t=T?Att=2T?$

Q. A sinusoidal wave travelling in the positive x-direction has an amplitude of $10\text{cm}$, wavelength $30\text{cm}$ and frequency $6\text{Hz}$. The vertical displacement of the medium at $t=0$ and $x=0$ is also $10\text{cm}$.

1. Wave number is $\frac{\pi }{15}{\text{cm}}^{-1}$
2. $\text{Time period of wave is}\frac{1}{6}\text{s}$
3. $\text{Value of phase constant is}\frac{\pi }{2}$
4. $\text{Speed of wave is}1.8\text{m/s}$

Q. Which of the following options represent a travelling wave? Here, $y$ is the displacement of the particle, $x$ is the position of the particle from origin and $t$ is time.

1. $y=2\cos \left(3x\right)\sin \left(10t\right)$
2. $y=2\sqrt{x-5t}$
3. $y=2\sin (5x-2t)+2\cos (5x-2t)$
4. $y=5\sin (100\pi t-5x)$

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

1. x = A sin (ky - $\omega$t)

2. y = A sin (kx - $\omega$t)

3. y = A sin kycos$\omega$t
4. x = A sin (ky - $\lambda$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. 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. 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. $1.5\text{m/s}$
2. $2\text{m/s}$
3. $1\text{m/s}$
4. $0.5\text{m/s}$

Q. Identify the option that correctly represents given wave function.
$y=2\sqrt{3}\sin \pi (x-2t+\frac{1}{6})$

1. 
2. 
3. Both of the above graphs are possible
4. None of the two graphs are possible

Q. At t = 0, a transverse wave pulse travelling in the positive x direction with a speed of 2 m/s in a wire is described by the function $y=\frac{6}{x^2}giventhatx\ne 0$. Transverse velocity of a particle at x = 2 m and t = 2 s is

1. 3 m/s
2. –3 m/s
3. 8 m/s
4. –8 m/s

Q. At time $t=0,$ equation of a wave pulse is $y(x,0)=\frac{5}{3+(x-4{)}^2}$ and at $t=2s$, equation of the same wave pulse is $y(x,2)=\frac{5}{3+(x+8{)}^2}$. Here, $y$ is in $\text{mm}$, $x$ is in $\text{m}$ and $t$ is in $\text{sec}$. The wave speed is

1. $5\text{m/s}$
2. $6\text{m/s}$
3. $6\text{mm/s}$
4. $3\text{mm/s}$

Q. A pulse moving along $x-$axis is represented by the wave function $y(x,t)=\frac{2}{(x-2t{)}^2+1}$ where $x$ and $y$ measured in centimeters and $t$ is measured in seconds. The shape of pulse $y(x,0)=\frac{2}{x^2+1}$ is shown in the figure below. Which of the following correctly represents the shape of pulse at $t=1\text{s}$.

1. 
2. 
3. 
4. 

Q. Equation for a wave on a string is given by $y=0.03\sin (7.86x-2t)$ where $y$ and $x$ are in metres and $t$ is in seconds. At $t=0$, find the value of displacement (approximately) for a particle on string at $x=0.1\text{m}$.

1. $2.1\text{cm}$
2. $4\text{cm}$
3. $3\text{cm}$
4. $7.86\text{cm}$

Q.

A simple harmonic wave is represented by the relation $y(x,t)=a_{0}sin2\pi (vt-\frac{x}{\lambda })$ If the maximum particle velocity is three times the wave velocity, the wavelength $\lambda$ of the wave is

1. $\frac{\pi a_0}{3}$

2. $\frac{2\pi a_0}{3}$

3. $\pi a_0$

4. $\frac{\pi a_0}{2}$

Q. A wave equation which gives the displacement of particle along $y-$ direction is represented as $y={10}^4\sin (60t+2x)$, where $x$ and $y$ are in $\text{metre}$ and $t$ in $\text{sec}$. Among the following choose the correct statement.

1. It represents a wave propagating along positive $x-$ axis with a velocity of $30\text{m/s}$
2. It represents a wave propagating along negative $x-$ axis with a velocity of $120\text{m/s}$
3. It represents a wave propagating along negative $x-$ axis with a velocity of $30\text{m/s}$
4. It represents a wave propagating along negative $x-$ axis with a velocity of ${10}^4\text{m/s}$

Q. A sinusoidal wave travelling in the positive direction on stretched string has amplitude 2.0 cm, wavelength 1.0 m and wave velocity 5.0 m/s. At x = 0 and t = 0 it is given that y = 0 and $\frac{\delta y}{\delta t}<0$. Find the wave function y(x, t).

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

Q. A point source is producing sinusoidal transverse waves. If the displacement from the equilibrium position of a point $4\text{cm}$ from source is half the amplitude of wave at $t=\frac{T}{6}$ ($T$ being time period). The wavelength of travelling wave is

1. $0.48\text{m}$
2. $0.96\text{m}$
3. $0.24\text{m}$
4. $0.12\text{m}$

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. The equation of a plane progressive wave is given by
$y=0.025\sin (100t+0.25x)$
Which of the following represents the velocity of the particle of medium through which the sinusoidal wave is propagating?

1. $2.5\cos (100t+0.25x)$
2. $50\cos (100t+0.25x)$
3. $5\sin (100t+0.25x)$
4. $20\sin (50t+0.25x)$

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

Q. Identify the option that correctly represents given wave function.
$y=2\sqrt{3}\sin \pi (x-2t+\frac{1}{6})$

1. 
2. 
3. Both of the above graphs are possible
4. None of the two graphs are possible

Q. A wave pulse is traveling on a string at 2 m/s. Displacement y of the particle at x = 0 at any time t is given by $\frac{2}{(t^2+1)}$. Find the expression of the function y = (x, t), i.e., displacement of the particle at position x and time t.

1. $y=\frac{2}{{(t-\frac{x}{2})}^2+1}$
2. $y=\frac{2}{{(t-\frac{x}{4})}^2+1}$
3. $y=\frac{2}{{(t-2x)}^2+1}$
4. $y=\frac{2}{{(t-4x)}^2+1}$

Q. Find the phase velocity of the wave whose $y-x$ graph is shown at two instants.

1. $10\text{m/s}$
2. $15\text{m/s}$
3. $25\text{m/s}$
4. $20\text{m/s}$

Q.

A plane sound wave is travelling in a medium. In reference to a frame A, its equation is $y=acos(\omega t-kx)$. Which reference to frame B, moving with a constant velocity v in the direction of propagation of the wave, equation of the wave will be

1. $y=acos\left[\right(\omega +kv)t–kx]$

2. $y=-acos\left[\right(\omega -kv)t-kx]$
3. $y=acos\left[\right(\omega -kv)t-kx]$
4. $y=acos\left[\right(\omega t+kv)t+kx]$

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

Q. For the wave shown in the figure, write the equation of the wave if its position is shown at $t=0$ speed of wave is 200 m/s.

1. $y(x,t)=0.12\sin \pi [x-2000t]\text{m}$
2. $y(x,t)=0.12\sin [31.4x-6200t]\text{m}$
3. $y(x,t)=0.12\sin [3.14x-6283t]\text{m}$
4. $y(x,t)=0.12\sin [31.4x-6283t]\text{m}$

Q. For the wave shown in figure, write the equation of this wave if its position is shown at t = 0. Speed of wave is v = 30 m/s.

1. $y=\left(0.06m\right)cos\left[78.5m^{-1}\right)x+\left(2356.2s^{-1}\right)t]m$
2. $y=\left(0.06m\right)sin\left[\right(78.5m^{-1})x-(2356.2s^{-1}\left)t\right]m$
3. $y=\left(0.06m\right)sin\left[\right(78.5m^{-1})x+(2356.2s^{-1}\left)t\right]m$
4. $y=\left(0.06m\right)cos\left[\right(78.5m^{-1})x-(2856.2s^{-1}\left)t\right]m$

Q.

What is the equation of a sinusoidal wave moving in the positive x direction with wave length 6 m and time period 0.5s. The amplitude of the wave is 5 m.

1. $y=6sin(\frac{2\pi }{5}x-4\pi t)$

2. $y=5sin(\frac{\pi }{3}x-4\pi t)$

3. $y=5sin(x-t)$

4. None of these

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. A transverse wave is passing through a medium. The maximum speed of the vibrating particle occurs when the displacement of the particle from the mean position is:

1. zero
2. half of the amplitude
3. equal to the amplitude
4. none of these