Test: Standing Waves(17 Oct) - JEE MCQ

Standing waves are produced by two similar waves superposing while traveling in the opposite direction

A standing wave can be produced by:

• On a string clamped at both ends.
• On a string clamped at one end and free at the other.
• When an incident wave gets reflected from a wall.
• The phenomenon is the result of interference; that is, when waves are superimposed, their energies are either added together or canceled out. In the case of waves moving in the same direction, interference produces a traveling wave.
• But a standing wave cannot be produced when two identical waves with a phase difference of π are moving in the same direction

There are two types of organ pipes:

Where n = positive integer, v = velocity of sound, and l = length of the organ pipe.
CALCULATION:
Given that: 
Fundamental frequency (f) = 140 Hz = v/2l 
So v/l = 140 × 2 = 280 Hz
Next frequency (f₁) = 2 v/2l = v/l = 280 Hz
and f₂ = 3v/2l = 3 × 280/2 = 420 Hz
Thus given organ pipe is a open organ pipe. So option 2 is correct.

• Standing Wave: In standing wave or stationary wave, i.e. a combination of two waves moves in the opposite direction with the same amplitude and frequency get superimposed and form nodes and antinodes.
• An antinode is a point on the wave where the amplitude is maximum and hence it is the wave crest.
• Node is a point where the amplitude of oscillation is zero, i.e. displacement is minimum from the mean.

• This is usually depicted for stationary waves, that is waves which have a steady pattern and have no direction of motion.

The pressure wave variation of the standing wave is opposite to normal mode.

• Closed Organ pipe (One end close): The open end becomes antinode and closed-end become a node.
  • 

EXPLANATION

• The pressure wave variation of the standing wave is opposite to normal mode.
• In a standing wave profile. At antinode, the pressure is variation is minimum (zero). i.e. A node for displacement is always an antinode for pressure. At the open-end, the wave is the node and the closed-end wave is the antinode.
• For the Third over-tone,

⇒ Option 1 is correct.

• Beat: The combination of two waves of slightly different frequencies, perceived as a periodic variation whose rate is a difference of two frequencies.
• Beat frequency, n = |f₁ - f₂|, where f₁ and f₂ are two frequencies
• The frequency of the n^th harmonic in an open pipe is given as,  where, n is the n^th harmonic, v is a velocity and L is the length of the pipe.
• The frequency of the nth harmonic in a closed pipe is given as,  where  n is a nth harmonic, v is a velocity and L is the length of the pipe.
• For frequency at fundamental nodes, we put n = 1 in both of the above formula, 
  

Explanation:
The fundamental frequency of the open pipe is given by, 
The beat frequency, 
If the length of the open pipe will increase, the frequency will decrease, then the number of beats will also decrease.

• Standing Wave: In standing wave or stationary wave, i.e. a combination of two waves moves in the opposite direction with the same amplitude and frequency get superimposed and form nodes and antinodes.
  • An antinode is a point on the wave where the amplitude is maximum and hence it is the wave crest.
  • Node is a point where the amplitude of oscillation is zero, i.e. displacement is minimum from the mean.

EXPLANATION:

The gap between a consecutive node and anti-node in a stationary wave is λ/4. So option 4 is correct.

• Standing Wave: In standing wave or stationary wave, i.e. a combination of two waves moves in the opposite direction with the same amplitude and frequency get superimposed and form nodes and anti-nodes.
  • An antinode is a point on the wave where the amplitude is maximum and hence it is the wave crest.
  • Node is a point where the amplitude of oscillation is zero, i.e. displacement is minimum from the mean.

• Organ pipe: The sound-producing device which resonates at a particular pitch is called an Organ pipe.
  • Closed organ pipe: when one end of an organ pipe is closed and the other end is open then it is called a closed organ pipe.  
  • Open organ pipe: when both the ends of the organ pipe are open then it is called open organ pipe.

Where n = positive integer, v = velocity of sound, and l = length of the organ pipe.
EXPLANATION:

In an open pipe, there will be 1 node and 2 anti-nodes for first harmonic as shown in the figure. So option 3 is correct.

• Stationary wave is also known as the standing wave.
  • It is a combination of two waves, with the same amplitude and the same frequency, moving in the opposite direction.
    • It is a result of interference.

• Harmonics of an instrument: A musical instrument has a set of natural frequencies at which it vibrates when a disturbance is introduced into it.
  • These set of natural frequencies are known as the harmonics of the instrument.

The frequency (f) in fundamental mode is given by:
f = v/2L
where v is the speed of sound, L is the length string.
So option 4 is correct.

• Stationary wave is also known as the standing wave.
  • It is a combination of two waves, with the same amplitude and the same frequency, moving in the opposite direction.
    • It is a result of interference.

• Harmonics of an instrument: A musical instrument has a set of natural frequencies at which it vibrates when a disturbance is introduced into it.
• These set of natural frequencies are known as the harmonics of the instrument.

The Frequency of nth harmonic in the standing wave is given by:

where n is nth harmonic, v is the speed of sound, L is the length string.
CALCULATION:
For second harmonic n = 2
Frequency of 2nd harmonic in standing wave

So the correct answer is option 3.

Standing Wave: 
It is a combination of two waves that move in the opposite direction with the same amplitude and frequency get superimposed and form nodes and antinodes.
Nodes (N): 

• The points where the amplitude is minimum are called nodes. 
• The distance between two successive nodes is λ/2.
• Nodes are at permanent rest.
• At nodes air pressure and density both are high.

Antinodes (A):

• The points of maximum amplitudes are called antinodes.
• The distance between two successive nodes is λ/2.
• At nodes air pressure and density both are low.
• The distance between a node (N) and adjoining antinode (A) is λ/4. 

Explanation:
Given - Length of the string (l) = 1.25 meters​

It is given that three nodes and two antinodes are produced in the string and as we know that the distance between two successive nodes is λ/2 and the distance between a node (N) and adjoining antinode (A) is λ/4. 
Therefore the wavelength of the wave is 1.25 meters.

CONCEPT:

• Stationary wave is also known as the standing wave.
• It is a combination of two waves, with the same amplitude and the same frequency, moving in the opposite direction.
• A wave that travels in a medium with the same amplitude in the same direction is called a traveling wave or a progressive wave.

EXPLANATION:

• Difference between Progressive wave and standing stationary wave: