Expression for Standing Waves

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. 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 $\text{centimeter}$ and $t$ in $\text{seconds}$. The number of loops formed in vibrating string will be

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

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

1. same as
2. twice
3. half

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

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.

The vibrations of a string fixed at both ends are described by the

$y=\left(5.00mm\right)sin\left[\right(1.57c{m}^{-1}\left)x\right]sin\left[\right(314s^{-1}\left)t\right]$

If the length of the string is 10.0 cm, locate the nodes and the antinodes. How many loops are formed in the vibration?

1. nodes - 0, 2cm, 4 cm, 6 cm, 8 cm, 10 cm

   Anti-nodes - 1 cm, 3 cm, 5 cm, 7 cm, 9 cm

2. nodes - 0, 1 cm, 3 cm, 9 cm

   Anti-nodes - 2 cm, 4 cm, 6 cm, 8 cm, 10 cm

3. nodes - 0, 4 cm, 8 cm, 10 cm

   Anti-nodes - 1 cm, 5 cm, 9 cm

4. None of these

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

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 $y=\text{A sin}\left(\frac{20\pi x}{3}\right)\text{cos (1000}\pi \text{t)}$ 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 $y=\text{A sin}\left(\frac{20\pi x}{3}\right)\text{cos (1000}\pi \text{t)}$ 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. Spacing between two successive nodes in a standing wave on a string is $x$. If tension in the string is doubled keeping frequency constant, then find the new spacing between successive nodes.

1. $\frac{x}{\sqrt{2}}$
2. $\sqrt{2}x$
3. $\frac{x}{2}$
4. $2x$

Q. Spacing between two successive nodes in a standing wave on a string is $x$. If tension in the string is doubled keeping frequency constant, then find the new spacing between successive nodes.

1. $\frac{x}{\sqrt{2}}$
2. $\sqrt{2}x$
3. $\frac{x}{2}$
4. $2x$

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 $\text{centimeter}$ and $t$ in $\text{seconds}$. Node will not occur at distance

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