Expression for Standing Waves

Q.

What are Standing or stationary waves?

Q. A uniform rope of mass $0.1\text{kg}$ and length $2.45\text{m}$ hangs from a ceiling. The time taken by a transverse wave to travel through the full length of the rope is
[Take $g=9.8{\text{m/s}}^2$]

1. $2.0\text{s}$
2. $1.2\text{s}$
3. $1.0\text{s}$
4. $2.2\text{s}$

Q. Vibration in a string of length $60\text{cm}$ fixed at both ends is represented by the equation, $y=4\sin \left(\frac{\pi x}{15}\right)\cos \left(96\pi t\right)$, where $x$ and $y$ are in $\text{cm}$ and $t$ in $\text{s}$. The number of loops formed in vibrating string will be

1. $4$
2. $6$
3. $3$
4. $5$

Q. A point mass is subjected to two simultaneous sinusoidal displacements along the $x$ - direction: $x_1\left(t\right)=A\sin \omega t$ and $x_2\left(t\right)=A\sin (\omega t+\frac{2\pi }{3})$. Adding a third sinusoidal displacement $x_3\left(t\right)=B\sin (\omega t+\varphi )$ brings the mass to a complete rest. The values of $B$ and $\varphi$ are respectively:

1. $\sqrt{2}A$ and $\frac{3\pi }{4}$
2. $A$ and $\frac{4\pi }{3}$
3. $\sqrt{3}A$ and $\frac{5\pi }{6}$
4. $A$ and $\frac{\pi }{3}$

Q. The equation of a standing wave in a stretched string is given by $y=5\sin \left(\frac{\pi x}{3}\right)\cos \left(40\pi t\right)$, where $x$ and $y$ are in $\text{cm}$ and $t$ is in $\text{sec}$. The separation between two consecutive nodes is (in $\text{cm}$)

1. $6$
2. $4$
3. $1.5$
4. $3$

Q. What is the difference between the equation of waves y=Asin(wt-kx) & y=Asin(kx-wt).

Q. A source of frequency 10 kHz when vibrated over the mouth of a closed organ pipe is in resonance at 300 K. The beats produced per second when temperature rises by 1 K is

Q. The equation of a stationary wave is $y=0.8\text{cos}\left(\frac{\pi x}{20}\right)\text{sin}200\pi t$, where $x$ is in $\text{cm}$ and $t$ is in seconds. The separation between a successive node and antinode will be

1. $20\text{cm}$
2. $10\text{cm}$
3. $30\text{cm}$
4. $40\text{cm}$

Q.

In sound waves, a displacement node is a pressure antinode snd vice versa.Explain why?

Q.

A transverse harmonic wave on a string is described by $y(x,t)=3.0sin(36t+0.018x+\frac{\pi }{4})$
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. If in a stationary wave the amplitude corresponding to antinode is $4\text{cm}$, then the amplitude corresponding to a particle of medium located exactly midway between a node and an antinode is

1. $2\text{cm}$
2. $\sqrt{2}\text{cm}$
3. $1.5\text{cm}$
4. $2\sqrt{2}\text{cm}$

Q. A string of length $1.5\text{m}$ with both its ends clamped is vibrating in fundamental mode. Amplitude at the centre of string is $4\text{mm}$. What is the minimum distance between two points having amplitude $2\text{mm}$?

1. $1\text{m}$
2. $0.75\text{m}$
3. $1.25\text{m}$
4. $0.50\text{m}$

Q. Equation of a stationary wave is given by $y=5\cos \left(\frac{\pi x}{25}\right)\sin \left(100\pi t\right)$. Here, $x$ is in $\text{cm}$ and $t$ in $\text{s}$. Node will not occur at distance

1. $25\text{cm}$
2. $62.5\text{cm}$
3. $12.5\text{cm}$
4. $37.5\text{cm}$

Q. A string with a mass density of $4\times {10}^{-3}\text{kg/m}$ is under a tension of $360\text{N}$ and is fixed at both ends. One of its resonance frequencies is $375\text{Hz}$. The next higher resonance frequency is $450\text{Hz}$. If the mass of the string is represented by $x\times {10}^{-3}\text{kg}$, find $x$

Q. The displacement equation for a particle in standing wave is represented as $y(x,t)=0.4\sin \left(0.5x\right)\cos \left(30t\right)$, where $x$ and $y$ are in $\text{cm}$.
Magnitude of velocity of particle at $x=2.4\text{cm}$ and $t=0.8\text{s}$ is -
$\left[\sin \right(1.2)=0.93,\text{sin}(24)=-0.9]$

1. $0.4\text{cm/s}$
2. $5.2\text{cm/s}$
3. $10.1\text{cm/s}$
4. $12\text{cm/s}$

Q. The standing wave in a medium is expressed as $y=0.2\sin \left(0.8x\right)\cos \left(3000t\right)\text{m}$. The seperation between any two consecutive points of minimum or maximum displacement is

1. $\frac{\pi }{2}\text{m}$
2. $\frac{\pi }{4}\text{m}$
3. $\frac{\pi }{6}\text{m}$
4. $\frac{5\pi }{8}\text{m}$

Q. Which of the following statements is/are true regarding a stationary wave?

1. All particles in a particular segment between two nodes vibrate in the same phase.
2. Particles in the neighbouring segments vibrate in opposite phases.
3. All the particles of the medium attain their maximum velocities simultaneously when they pass through their mean positions.
4. All the particles of the medium, except those at nodes, execute simple harmonic motion about their mean positions.

Q. A standing wave $y=\text{A sin}\left(\frac{20\pi x}{3}\right)\text{cos (1000}\pi \text{t)}$ is set up in a taut string, where $x$ and $y$ are in metre and $t$ is in seconds. The distance between first two successive points oscillating with the amplitude $\frac{A}{2}$, from the fixed end will be equal to

1. $10\text{cm}$
2. $15\text{cm}$
3. $2.5\text{cm}$
4. $4\text{cm}$

Q. A standing wave is represented by $y=a\sin \left(0.05x\right)\cos \left(100t\right)$, where $t$ is in $\text{s}$ and $x$ is in $\text{m}$, then the velocity of the constituent wave is

1. ${10}^5\text{m/s}$
2. ${10}^3\text{m/s}$
3. ${10}^2\text{m/s}$
4. $2\times {10}^3\text{m/s}$

Q. A wave disturbance in a medium is described by $y(x,t)=0.02\cos \left(10\pi x\right)\cos (50\pi t+\frac{\pi }{2})$, where $x$ and $y$ are in $\text{m}$ and $t$ in $\text{s}$. Identify the incorrect option.

1. Node occurs at $x=0.15\text{m}$
2. Antinode occurs at $x=0.25\text{m}$
3. Wavelength of the constituent waves is $0.2\text{m}$
4. Speed of the constituent waves is $5\text{m/s}$

Q. A string of length $l$ along $x-$ axis is fixed at both ends and is vibrating in second harmonic. If amplitude of incident wave is $2.5\text{mm}$, the equation of standing wave is (T is tension and $\mu$ is linear density)

1. $\left(2.5\text{mm}\right)\text{sin}\left(\frac{2\pi }{l}x\right)\text{cos}\left(2\pi \sqrt{\left(\frac{T}{\mu l^2}\right)}t\right)$
2. $\left(5\text{mm}\right)\text{sin}\left(\frac{\pi }{l}x\right)\text{cos}2\pi t$
3. $\left(5\text{mm}\right)\text{sin}\left(\frac{2\pi }{l}x\right)\text{cos}\left(2\pi \sqrt{\left(\frac{T}{\mu l^2}\right)}t\right)$
4. $\left(7.5\text{mm}\right)\text{cos}\left(\frac{2\pi }{l}x\right)\text{cos}\left(2\pi \sqrt{\left(\frac{T}{\mu l^2}\right)t}\right)$

Q. Two identical transverse sinusoidal waves travel in opposite directions along a string. The speed of transverse waves in the string is $0.5\text{cm/s}$. Each has an amplitude of $3.0\text{cm}$ and wavelength of $6.0\text{cm}$. The equation for the resultant wave is

1. $y=6\sin \left(\frac{\pi t}{6}\right)\cos \left(\frac{\pi x}{3}\right)$
2. $y=6\sin \left(\frac{\pi x}{3}\right)\cos \left(\frac{\pi t}{6}\right)$
3. Both $\left(a\right)$ and $\left(b\right)$ may be correct
4. Both $\left(a\right)$ and $\left(b\right)$ are wrong

Q. A standing wave is maintained in a homogenous string of cross-sectional area $s$ and density $\rho$. It is formed by the superposition of two waves travelling in opposite directions given by the equations $y_1=a\sin (\omega t-kx)$ and $y_2=2a\sin (\omega t+kx)$. The total mechanical energy confined between the sections corresponding to adjacent nodes is

1. $\frac{\rho {\omega }^2{a}^2\pi s}{k}$
2. $\frac{9}{2}\frac{\rho {\omega }^2{a}^2\pi s}{k}$
3. $\frac{7}{2}\frac{\rho {\omega }^2{a}^2\pi s}{k}$
4. $\frac{3}{2}\frac{\rho {\omega }^2{a}^2\pi s}{k}$

Q.

Why liquids and gases cannot propagate transverse waves?

Q. A string of length $l$ along $x-$ axis is fixed at both ends and is vibrating in second harmonic. If amplitude of incident wave is $2.5\text{mm}$, the equation of standing wave is (T is tension and $\mu$ is linear density)

1. $\left(5\text{mm}\right)\text{sin}\left(\frac{\pi }{l}x\right)\text{cos}2\pi t$
2. $\left(5\text{mm}\right)\text{sin}\left(\frac{2\pi }{l}x\right)\text{cos}\left(2\pi \sqrt{\left(\frac{T}{\mu l^2}\right)}t\right)$
3. $\left(7.5\text{mm}\right)\text{cos}\left(\frac{2\pi }{l}x\right)\text{cos}\left(2\pi \sqrt{\left(\frac{T}{\mu l^2}\right)t}\right)$
4. $\left(2.5\text{mm}\right)\text{sin}\left(\frac{2\pi }{l}x\right)\text{cos}\left(2\pi \sqrt{\left(\frac{T}{\mu l^2}\right)}t\right)$

Q. A plane simple harmonic progressive wave given by the equation, $y=A\sin (\omega t-kx)$ of wavelength $120\text{cm}$ is incident normally on a plane surface which is a perfect reflector (acts as fixed end). If a stationary wave is formed, then the ratio of amplitudes of vibrations at points $10\text{cm}$ and $30\text{cm}$ from the reflector is

(Here $x$ is measured from reflector)

1. $1:2$
2. $1:3$
3. $2:1$
4. $3:1$

Q. For a string clamped at both ends, which of the following wave equation is valid for a stationary wave set up in it whose origin is at one of the ends of the string?

1. $y=A\sin kx\sin \omega t$
2. $y=A\cos kx\sin \omega t$
3. $y=A\cos kx\cos \omega t$
4. $y=2A\cos kx\cos \omega t$

Q. A stationary wave set up on a string have the equation $y=\left(2\text{mm}\right)\left[\text{cos}\right(6.28{\text{m}}^{-1}\left)\text{x cos}\right(\omega t\left)\right]$. This stationary wave is created by two identical waves, of amplitude $A$ each moving in opposite directions along the string. Then,

1. $A=4\text{mm}$
2. $A=2\text{mm}$
3. The smallest length of the string is $25\text{cm}$
4. The smallest length of the string is $50\text{cm}$

Q.

what is pie in terms of wave?

Q. When a stationary wave is formed, then its frequency is that of the individual waves.

1. same as
2. twice
3. half