Different Forms of Equations

Q. The wave equation of a pulse is given by $y=\frac{3}{2+(x-4t{)}^2}$ where y is in metre and t in second. The wave velocity of the pulse is

1. $+4m{s}^{-1}$
2. $-4m{s}^{-1}$
3. $-2m{s}^{-1}$
4. $+2m{s}^{-1}$

Q. A progressive wave travelling along the positive $x$ direction is represented by $y(x,t)=A\sin (kx-\omega t+\varphi )$. Its snapshot at $t=0$ is given in the figure.


For this wave, the phase constant $\varphi$ is :

1. $-\frac{\pi }{2}$
2. $\pi$
3. $0$
4. $\frac{\pi }{2}$

Q. The equation of a transverse wave on a stretched string is given by $y=0.05\sin 2\pi (\frac{t}{0.002}-\frac{x}{0.1})$, where $x$ and $y$ are expressed in metre and $t$ in second. The speed of the wave is

1. $100{\text{ms}}^{-1}$
2. $50{\text{ms}}^{-1}$
3. $200{\text{ms}}^{-1}$
4. $400{\text{ms}}^{-1}$

Q. Which of the following expressions represents a simple harmonic progressive wave?

1. $\text{A sin}\omega \text{t cos kx}$
2. $\text{A sin}(\omega t-kx)$
3. $\text{Acos kx}$
4. $\text{A sin}\omega t$

Q. Two sources $S_1$ and $S_2$ separated by $2$ meters vibrate according to the equations $y_1=0.03\sin \pi t$ and $y_2=0.02\sin \pi t$ where $y_1$ and $y_2$ are in meters. They send out waves of velocity $1.5\frac{\text{m}}{\text{s}}$. The magnitude of the resultant motion of particle , collinear with $S_1$and $S_2$ and at the middle of $S_1{S}_2$ will be :

1. $0.05\text{m}$
2. $0.01\text{m}$
3. $0.11\text{m}$
4. $0.06\text{m}$

Q. A table is revolving on its axis at $5$ revolution per sec. A sound source of freq. $1000Hz$. is fixed on the table at $70cm$ from the axis. The minimum frequency heard by a listener standing at a distance very far from the table will be (speed of sound $352\text{m/s}$).

1. $\text{941 Hz}$
2. $\text{352 Hz}$
3. $\text{1066 Hz}$
4. $\text{1000 Hz}$

Q.

The magnetic field in a travelling electromagnetic wave has a peak value of 20 nT. The peak value of electric field strength is

1. 9 V/m

2. 12 V/m

3. 3 V/m

4. 6 V/m

Q. An electromagnetic wave is represented by the electric field $\overrightarrow{E}=E_0\hat{n}\sin [\omega t+(6y-8z\left)\right]$ Taking unit vectors in $xy$ and $z$ directions to be $\hat{i},\hat{j},\hat{k}$, the direction of propogation $\hat{s}$ is given by:

1. $\hat{s}=\frac{-4\hat{k}+3\hat{j}}{5}$
2. $\hat{s}=\frac{2\hat{j}-3\hat{k}}{5}$
3. $\hat{s}=\left(\frac{-3\hat{j}+4\hat{k}}{5}\right)$
4. $\hat{s}=\frac{3\hat{i}-4\hat{j}}{5}$

Q. Four harmonic waves of equal frequencies and equal intensities $I_0$ have phase angles $0$, $\frac{\pi }{3}$, $\frac{2\pi }{3}$, and $\pi$. When they are superposed, the intensity of the resulting wave is $n{I}_0$. The value of $n$ is

Q. A transverse wave is propagating along $+x$ direction. At $t=2s$, the particle at $x=4$ $m$ is at $y=2mm$. With the passage of time its $y$ coordinate increases and reaches to a maximum of $4mm$. The wave equation is (using $\omega$ and $k$ with their usual meanings)

1. $y=4\sin \left(\omega (t+2)+k(x-2)+\frac{\pi }{6}\right]$
2. $y=4\sin \left(\omega (t+2)+kx+\frac{\pi }{6}\right]$
3. $y=4\sin \left(\omega (t-2)+k(x-4)+\frac{5\pi }{6}\right]$
4. $y=4\sin \left(\omega (t-2)-k(x-4)+\frac{\pi }{6}\right]$

Q. A wave is represented by $y=0.1\sin (100\pi t-kx),$ where $y,t,x$ are in SI units. If wave velocity is $100\text{m/s},$ its wave number is

1. $1{\text{m}}^{-1}$
2. $2{\text{m}}^{-1}$
3. $\pi {\text{m}}^{-1}$
4. $2\pi {\text{m}}^{-1}$

Q. A plane electromagnetic wave having a frequency $\nu =23.9\text{GHz}$ propagates along the positive z-direction in free space. The peak value of the Electric Field is $60\text{V/m}$. Which among the following is the acceptable magnetic field component in the electromagnetic wave ?

1. $\overrightarrow{B}=2\times {10}^7\sin (0.5\times {10}^{3}z+1.5\times {10}^{11}t)\hat{i}$
2. $\overrightarrow{B}=2\times {10}^{-7}\sin (0.5\times {10}^{3}z-1.5\times {10}^{11}t)\hat{i}$
3. $\overrightarrow{B}=2\times {10}^{-7}\sin (0.5\times {10}^{3}x+1.5\times {10}^{11}t)\hat{j}$
4. $\overrightarrow{B}=60\sin (0.5\times {10}^{3}x+1.5\times {10}^{11}t)\hat{k}$

Q.

The magnetic field in a traveling electromagnetic wave has a peak value of $20\mathrm{nT}.$ What is the peak value of electric field strength?

Q.

A radio wave has a maximum magnetic field induction of ${10}^{-4}T$ on arrival at a receiving antenna. The maximum electric field intensity of such a wave is

1. Zero

2. $3\times {10}^4\raisebox{1ex}{$V$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$

3. $5.8\times {10}^{-9}\raisebox{1ex}{$V$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$

4. $3.3\times {10}^{-13}\raisebox{1ex}{$V$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$

Q. A transverse wave is propagating on a stretched string. The rate of transfer of energy in a wave is

1. directly proportional to the tension in the string
2. inversely proportional to the tension in the string
3. directly proportional to the square root of tension in the string
4. None of these

Q. A transverse wave is propagating through a medium. The potential energy associated with vibrating particle is maximum 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 the above

Q. Average total energy density for a transverse wave is $16\text{J/m}$. The average kinetic energy density for the corresponding wave will be

1. $4\text{J/m}$
2. $2\text{J/m}$
3. $8\text{J/m}$
4. $16\text{J/m}$

Q. An electromagnetic wave contains non-zero energy density associated with it. It has both electric and magnetic fields associated with it. Then

1. In free space with RMS value of electric field as $E$, the average energy density associated with electric field is $\frac{{ϵ}_0{E}^2}{2}$.
   
2. In free space with RMS value of magnetic field as $B$, the average energy density associated with magnetic field is $\frac{B^2}{2{\mu }_0}$.
3. Contribution of electric field to average energy density is double to that by magnetic field.
4. Both options $\left(\text{A}\right)$ and $\left(\text{B}\right)$ are correct.

Q. A wave pulse is described by $y(x,t)=\left[4e^{-(2x-10t{)}^2}\right]\text{m}$. The speed of this wave pulse is -

1. $15\text{m/s}$
2. $5\text{m/s}$
3. $20\text{m/s}$
4. $10\text{m/s}$

Q. If potential energy density is twice the average rate of transmission of potential energy. The speed of transverse wave is

1. $2\text{m/s}$
2. $5\text{m/s}$
3. $1\text{m/s}$
4. $0.5\text{m/s}$

Q. Two transverse waves in a given medium are represented by the curve as shown in the figure. Find the ratio of their intensities. All the measurements are in $\text{S.I}$ units.

1. $\frac{10}{11}$
2. $\frac{24}{5}$
3. $\frac{25}{36}$
4. $\frac{25}{18}$

Q. Two waves propagating in a given medium are represented by the curve shown below.

Find the ratio of average potential energy per unit length of wave $1$ is to wave $2$. All measurements are in $\text{S.I}$ unit.

1. $1$
2. $1/2$
3. $1/4$
4. $1/8$

Q. A plane progressive transverse wave at an instant is shown below.

Choose an correct option.

1. $\text{K.E}$ associated with particle $A$ is maximum
2. $\text{P.E}$ associated with particle $C$ is maximum
3. $\text{P.E}$ associated with particle $B$ is maximum
4. $\text{K.E}$ associated with particle $D$ is minimum

Q. For a transverse wave
Statement $1$ : Total average energy density is the sum of average $\text{K.E.}$ density and average $\text{P.E.}$ density

Statement $2$ : Average $\text{K.E.}$ density is equal to the average $\text{P.E.}$ density

1. Both the statements are correct.
2. Both the statements are incorrect.
3. Statement $1$ is correct and Statement $2$ is incorrect.
4. Statement $2$ is correct and Statement $1$ is incorrect.

Q. A wave is propagating on a long stretched string. If the cross-sectional area of string is doubled then average power associated with wave becomes

1. twice of its initial value
2. half of its initial value
3. four times of its initial value
4. none of these

Q.

Is amplitude always positive

Q. A wave represented by the given equation $y=\alpha cos(kx-\omega t)$ is superposed with another wave to form a stationary wave such that the point
x = 0 is a node. The equation for the other wave is

1. y = α sin(kx - ωt)
2. y = -α cos(kx - ωt)
3. y = -α cos(kx - ωt)
4. y = -α sin(kx - ωt)

Q. A transverse wave having amplitude $A$ and angular frequency $\omega$ is travelling on a string of linear mass density $\mu$. If amplitude is doubled and angular frequency is halved, average kinetic energy per unit length will be

1. Doubled
2. Remain same
3. Halved
4. None of these.

Q. For a transverse wave propagating in a medium.
Statement $1$: At mean position, $\text{P.E.}$ is maximum.
Statement $2$: At mean position, $\text{K.E}$ is maximum.

1. Statement $1$ is correct, statement $2$ is incorrect
2. Both statements are correct
3. Both statements are incorrect
4. Statement $1$ is incorrect, statement $2$ is correct

Q. For a transverse wave propagating in a medium.
Statement $1$: At mean position, $\text{P.E.}$ is maximum.
Statement $2$: At mean position, $\text{K.E}$ is maximum.

1. Statement $1$ is correct, statement $2$ is incorrect
2. Statement $1$ is incorrect, statement $2$ is correct
3. Both statements are correct
4. Both statements are incorrect