Intro to Standing Waves

Q. Consider the three waves, $z_1,z_2$ and $z_3$ as $z_1=\text{A sin}(kx-\omega t),z_2=\text{A sin}(kx+\omega t)$ and $z_3=\text{A sin}(ky-\omega t)$. Which of the following represents a standing wave?

1. $z_1+z_2$
2. $z_2+z_3$
3. $z_3+z_1$
4. $z_2+2z_3$

Q.

When the interference occurs with non-coherent sources, _______ of the superimposing waves add like_______.

Q. In stationary waves, energy is uniformly distributed.

1. False
2. True

Q. The points where displacement is zero are called nodes and the points where the displacement is maximum are called antinodes.

1. True
2. False

Q.

Where is energy stored in electromagnetic waves?

Q. Stationary waves of frequency $300\text{Hz}$ are formed in a string, in which the velocity of wave is $200\text{m/s}$. The distance between a node and the consecutive antinode is

1. $\frac{1}{6}\text{m}$
2. $2\text{m}$
3. $\frac{3}{2}\text{m}$
4. $4\text{m}$

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

1. $a\cos (\omega t-kx)$
2. $-a\cos (\omega t+kx)$
3. $a\sin (\omega t-kx)$
4. $-a\sin (\omega t-kx)$

Q. Stationary waves are produced in a $10\text{m}$ long stretched string. If the string vibrates in 5 segments and wave velocity is $20\text{m/sec}$, then the frequency is (in $\text{Hz}$)

Q. Standing wave is produced in $10\text{m}$ long stretched string. If the string vibrates in $5$ segments and wave velocity is $20\text{m/s}$, then frequency of oscillation is

1. $5\text{Hz}$
2. $2\text{Hz}$
3. $8\text{Hz}$
4. $7\text{Hz}$

Q. Stationary waves of frequency $300\text{Hz}$ are formed in a string, in which the velocity of wave is $200\text{m/s}$. The distance between a node and the consecutive antinode is

1. $\frac{1}{6}\text{m}$
2. $2\text{m}$
3. $\frac{3}{2}\text{m}$
4. $4\text{m}$

Q. Stationary waves are formed due to superposition of

1. two identical waves travelling in same directions.
2. two identical waves travelling in opposite directions.
3. two unlike waves travelling in same directions.
4. two unlike waves travelling in opposite directions.

Q. Stationary waves are produced in a $10\text{m}$ long stretched string. If the string vibrates in 5 segments and wave velocity is $20\text{m/sec}$, then the frequency is (in $\text{Hz}$)

Q. In a standing wave, the energy is confined between two nodes.

1. False
2. True

Q. A thin wire of length $99\text{cm}$ is fixed at both ends. The wire has some tension and is divided into three segments of lengths $l_1,l_2$, and $l_3$ as shown in the figure. If these segments are made to vibrate, with their fundamental frequencies respectively in the ratio $1:2:3$, then the lengths $l_1,l_2,l_3$ respectively are (in $\text{cm}$):

1. $27,54,18$
2. $18,27,54$
3. $54,27,18$
4. $27,9,18$

Q. The ends of a stretched wire of length $L$ are fixed at $x=0$ and $x=L$. If in an experiment, the displacement of the wire is $y_1=A\sin \left(\frac{\pi x}{L}\right)\sin \omega t$ and energy is $E_1$ and in another experiment, its displacement is $y_2=A\sin \left(\frac{2\pi x}{L}\right)\sin 2\omega t$ and energy is $E_2$, then
[Assume, there is no energy loss at the ends of the stretched wire]

1. $E_2=E_1$
2. $E_2=2E_1$
3. $E_2=4E_1$
4. $E_2=16E_1$

Q. A steel wire of length $1\text{m}$ and mass $0.1\text{kg}$, having a uniform cross-sectional area of ${10}^{-6}{\text{m}}^2$ is rigidly fixed at both ends. The temperature of the wire is lowered by ${20}^{\circ }\text{C}$. If the wire is vibrating in harmonic motion, pick from the following possible frequencies of motion (in $\text{Hz}$)
Given Young's modulus of steel $Y=2\times {10}^{11}{\text{N/m}}^2,\alpha =1.21\times {10}^{-5}/{}^{\circ }\text{C}$

1. 11
2. 22
3. 10
4. 20

Q. The ends of a stretched wire of length $L$ are fixed at $x=0$ and $x=L$. If in an experiment, the displacement of the wire is $y_1=A\sin \left(\frac{\pi x}{L}\right)\sin \omega t$ and energy is $E_1$ and in another experiment, its displacement is $y_2=A\sin \left(\frac{2\pi x}{L}\right)\sin 2\omega t$ and energy is $E_2$, then
[Assume, there is no energy loss at the ends of the stretched wire]

1. $E_2=E_1$
2. $E_2=2E_1$
3. $E_2=4E_1$
4. $E_2=16E_1$

Q. A thin wire of length $99\text{cm}$ is fixed at both ends. The wire has some tension and is divided into three segments of lengths $l_1,l_2$, and $l_3$ as shown in the figure. If these segments are made to vibrate, with their fundamental frequencies respectively in the ratio $1:2:3$, then the lengths $l_1,l_2,l_3$ respectively are (in $\text{cm}$):

1. $27,54,18$
2. $18,27,54$
3. $54,27,18$
4. $27,9,18$