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Q. No.
A standing wave is formed by the superposition of two waves traveling in opposite directions. The transverse displacement is given by \(y(x,t)=0.5\sin\left(\frac{5\pi}{4}x\right) \cos(200\pi t).\) What is the speed of the traveling wave moving in the positive \(x\) direction?
\((x\) and \(t\) are in meters and seconds, respectively.)
1. \(180~\text{m/s}\)
2. \(160~\text{m/s}\)
3. \(120~\text{m/s}\)
4. \(90~\text{m/s}\)
Subtopic:  Standing Waves |
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Level 1: 80%+
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A pipe of length \(11~\text{cm}\) is closed at one end. The first harmonic frequency of the pipe in the air at \(0^\circ \text{C}\) is:
(the velocity of sound at \(0^\circ \text{C}=330~\text{m/s}\))
1. \(1200~\text{Hz}\)
2. \(1000~\text{Hz}\)
3. \(800~\text{Hz}\)
4. \(750~\text{Hz}\)
Subtopic:  Standing Waves |
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A string, under tension, and lying along the \(x\)-axis is set into transverse vibrations. The displacement at a point \(x\) is given by the function \(y(x,t)\) where \(t\) represents the time:    \(y(x,t)=\left ( 3~\text{mm} \right )\mathrm{sin}\left ( \frac{\pi x}{20~\text{cm}} \right ) \)\(\mathrm{cos}\left\{2\pi\left ( 100~\text{s}^{-1} \right )t \right\}\)
The frequency of vibration of the string is:
1. \(100~\text{Hz}\) 2. \(200~\text{Hz}\)
3. \(100~\pi~\text{Hz}\) 4. \(200~\pi~\text{Hz}\)
Subtopic:  Standing Waves |
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A string is fixed at both ends and set to vibrate in five loops. If the wavelength is \(8\) cm then the length of the string is:
1. \(10 \) cm
2. \(15\) cm
3. \(20\) cm
4. \(25\) cm

Subtopic:  Standing Waves |
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If the speed of sound in air is \(v,\) then the minimum possible length of the closed-end organ pipe which resonates to frequency \(f\) will be:

1. \(\dfrac{v}{2f}\) 2. \(\dfrac{v}{4f}\)
3. \(\dfrac{v}{3f}\) 4. \(\dfrac{v}{f}\)
Subtopic:  Standing Waves |
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The equation of stationary wave along a stretched string is given by y = 5sinπx3cos40πt, where x and y are in cm and t in second. The separation between two adjacent nodes is:

1.  1.5 cm

2.  3 cm

3.  6 cm

4.  4 cm

Subtopic:  Standing Waves |
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Two waves executing simple harmonic motion travelling in the same direction with the same amplitude and frequency are superimposed. The resultant amplitude is equal to the \(\sqrt 3 \) times of amplitude of individual motions. The phase difference between the two motions is:
1. \(30^{\circ}\) 2. \(45^{\circ}\)
3. \(60^{\circ}\) 4. \(90^{\circ}\)
Subtopic:  Standing Waves |
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The fifth overtone of a closed pipe is observed to be unison with third overtone of an open pipe. The ratio of the lengths of the pipes is:

1. \(9:7\) 2. \(11:8\)
3. \(12:9\) 4. \(13:10\)
Subtopic:  Standing Waves |
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When a string is divided into three segments of lengths \(l_1\), \(l_2\) and \(l_3\), the fundamental frequencies of these three segments are \(\nu_1\), \(\nu_2\) and \(\nu_3\) respectively. The original fundamental frequency (\(\nu\)) of the string is:

1. \(\sqrt{\nu} = \sqrt{\nu_1}+\sqrt{\nu_2}+\sqrt{\nu_3}\)
2. \(\nu = \nu_1+\nu_2+\nu_3\)
3. \(\dfrac{1}{\nu} =\dfrac{1}{\nu_1} +\dfrac{1}{\nu_2}+\dfrac{1}{\nu_3}\)
4. \(\dfrac{1}{\sqrt{\nu}} =\dfrac{1}{\sqrt{\nu_1}} +\dfrac{1}{\sqrt{\nu_2}}+\dfrac{1}{\sqrt{\nu_3}}\)
Subtopic:  Standing Waves |
 85%
Level 1: 80%+
AIPMT - 2012
Hints
A string, under tension, and lying along the \(x\)-axis is set into transverse vibrations. The displacement at a point \(x\) is given by the function \(y(x,t)\) where \(t\) represents the time:    \(y(x,t)=\left ( 3~\text{mm} \right )\mathrm{sin}\left ( \frac{\pi x}{20~\text{cm}} \right ) \)\(\mathrm{cos}\left\{2\pi\left ( 100~\text{s}^{-1} \right )t \right\}\)
The maximum amplitude of vibration at any point on the string is:
1. \(3~\text{mm}\)
2. \(20~\text{cm}\)
3. \(300~\text{mm}\)
4. \(15~\text{mm}\)
Subtopic:  Standing Waves |
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Level 1: 80%+
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