Intro to Standing Waves

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

Q. When a string is divided into three segments of length $l_1,l_2$ and $l_3$, the fundamental frequencies of these three segments are ${\nu }_1,{\nu }_2$ and ${\nu }_3$ respectively. The original fundamental frequency $\left(\nu \right)$ of the string is

1. $\frac{1}{\nu }=\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}+\frac{1}{{\nu }_3}$

2. $\frac{1}{\sqrt{\nu }}=\frac{1}{\sqrt{{\nu }_1}}+\frac{1}{\sqrt{{\nu }_2}}+\frac{1}{\sqrt{{\nu }_3}}$

3. $\sqrt{\nu }=\sqrt{{\nu }_1}+\sqrt{{\nu }_2}+\sqrt{{\nu }_3}$

4. $\nu ={\nu }_1+{\nu }_2+{\nu }_3$

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. A progressive wave of frequency $500\text{Hz}$ is travelling with a velocity $360{\text{ms}}^{-1}$. How far are two points, ${60}^{\circ }$ out of phase with each other ?

1. $0.06\text{m}$
2. $0.12\text{m}$
3. $0.18\text{m}$
4. $0.24\text{m}$

Q. Doppler's effect is not applicable for

1. Sound waves
2. Ultrasonic waves
3. Infrasonic waves
4. matter waves

Q.

What is the phase difference between two points on a wavefront?

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

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

Q. When a stationary wave is formed, its frequency is

1. Same as that of the individual waves
2. Thrice as that of the individual waves
3. Twice as that of the individual waves
4. Half as that of the individual waves

Q. Two travelling waves $y_1=\text{A sin [k(x - ct)]}$ and $y_2=\text{A sin}\left[k\right(x+ct\left)\right]$ are superimposed on string. The distance between adjacent nodes is

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

Q.

A standing wave is formed when _________.

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. Four independent waves are represented by the following equations:
$X_1=a_1\sin \omega t...\left(1\right)$
$X_2=a_1\sin 2\omega t...\left(2\right)$
$X_3=a_1\sin {\omega }_{1}t...\left(3\right)$
$X_4=a_1\sin (\omega t+\delta )...\left(4\right)$
Interference is possible between waves represented by the equations,

1. $3\text{and}4$
2. $1\text{and}2$
3. $2\text{and}3$
4. $1\text{and}4$

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. Which one of the following configurations has the highest fin effectiveness?

1. 
   Thin, closely spaced fins
2. 
   Thin, widely spaced fins
3. 
   Thick, widely spaced fins
4. 
   Thick, closely spaced fins

Q. In a stationary wave, when particles of a string passes through their mean position then kinetic energy of these particles at antinodes is

1. Minimum
2. Maximum
3. Zero
4. Infinite

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.

A travelling harmonic wave on a string is described by

(a) What are the displacement and velocity of oscillation of a point at x = 1 cm, and t = 1 s? Is this velocity equal to the velocity of wave propagation?

(b) Locate the points of the string which have the same transverse displacements and velocity as the x = 1 cm point at t = 2 s, 5 s and 11 s.

Q. Two travelling waves produces a standing wave represented by equation, $y=1.0\text{mm}\cos \left[\right(1.57{\text{cm}}^{-1}\left)x\right]\sin \left[\right(78.5{\text{s}}^{-1}\left)t\right]$. The node closest to the origin in the region $x>0\text{cm}$ will be at $x=$____$\text{cm}$.

Q.

For the wave described in Exercise 15.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?

Q. In a sonometer wire, the tension is maintained by suspended a $50.7\text{kg}$ mass from the free end of the wire. The suspended mass has a volume of $0.0075{\text{m}}^3$. The fundamental frequency of vibration of the wire is $260\text{Hz}$. If the suspended mass is completely submerged in water, the fundamental frequency will become $\text{Hz}$.

Q.

Which of the given below options is not a characteristic of the waves.

1. Amplitude

2. Frequency

3. Wavelength

4. None

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. A wire is attached to a pan of mass $100\text{g}$ that contains a $1\text{kg}$ mass as shown in figure. When the wire is plucked, the wire vibrates at a fundamental frequency of $110\text{Hz}$. An additional mass $M$ is then added to the pan and a fundamental frequency of $130\text{Hz}$ is detected. What is the mass $M$?

1. $0.22\text{kg}$
2. $0.44\text{kg}$
3. $0.66\text{kg}$
4. $0.55\text{kg}$

Q.
In given YDSE experiment a plane wavefront of $\lambda =600nm$ is incident on slits at angle $\theta ={30}^{\circ }$ as shown in figure. Slit separation is $d=0.5mm$, then

1. We get maximum intensity at O
2. Central maxima is $\frac{D}{\sqrt{3}}$ distance above O on screen
3. Central maxima is $\frac{D}{2}$ distance above O on screen
4. Intensity at O is not zero

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. To obtain an interference pattern, the necessary condition is that the two sources should be,

1. coherent
2. narrow
3. close to each other
4. of same amplitude

Q.

Does the natural frequency change with the mass?

Q. The equation $y=A{\cos }^2(2\pi nt-2\pi \frac{x}{\lambda })$ represents a wave with

1. Amplitude $A/2$, frequency $2n$ and wavelength $\lambda /2$
2. Amplitude $A/2$, frequency $2n$ and wavelength $\lambda$
3. Amplitude $A$, frequency $2n$ and wavelength $2\lambda$
4. Amplitude $A$, frequency $n$ and wavelength $\lambda$

Q. Statement-1: All points on a wavefront vibrate in same phase with same frequency
Statement-2: Two sources are said to be coherent if they produce waves of same frequency with a constant phase difference.

1. Statement-1 is false, Statement -2 is true
2. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
3. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1
4. Statement-1 is true, Statement-2 is false