Conceptual quest on standing wave

In summary, standing waves do not require string lengths to be integral multiples of wavelength and can occur for any wavelength. However, in the special case of a string fixed at both ends, the length of the string must be an integral multiple of half the wavelength of the original wave for a standing wave to occur. This is because both ends must be zero amplitude nodes in order to maintain the standing wave pattern.
  • #1
PrakashPhy
35
0
How can we predict whether, a wave ( say a one dimensional string wave) traveling in any direction (say +x) after reflection from the fixed end, forms a standing wave in the medium or not (here string)?
I do not understand why the length of stirng should be integral multiple of wavelength( lymbda) for the formation of standing wave.
Please help me.
 
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  • #2
Hi PrakashPhy! :smile:

(have a lambda: λ :wink:)
PrakashPhy said:
I do not understand why the length of stirng should be integral multiple of wavelength( lymbda) for the formation of standing wave.

Draw a wave, cut it at any point, and turn one of the bits round to fit on top of the other bit (with or without reflection) …

you'll quickly find that you can't match the two bits unless you make the cut at a top middle or bottom position. :wink:
 
  • #3
PrakashPhy said:
I do not understand why the length of string should be integral multiple of wavelength for the formation of standing wave.
Please help me.

Your intuition is correct. Standing waves, in general, do not require string lengths to be integral multiples of wavelength.

When any sinusoidal wave is reflected back on itself from a single reflecting surface or point the result is two superimposed waves of equal wavelength traveling in opposite directions. The two superimposed waves will always sum to give a standing wave.

The amplitude of the sum of the two waves at the exact reflecting point is zero.
The amplitude of the sum of the two waves rises up to a maximum at one quarter wavelength from the reflecting point, drops down to zero again at half a wavelength from the reflecting point ... and so on.
The standing wave pattern extends for whatever overall distance the two waves continue to overlap, with zero amplitude nodes at regular half wavelength intervals.
This will occur for any wavelength.

In the special case of a piece of string fixed at each end the thing that makes a difference is that the string is fixed at two points, not just one point. For a standing wave to be maintained with two fixed points both those points must be zero amplitude nodes.

As described above, the distance from node to node in a standing wave is half the wavelength of the original sinusoidal wave. Therefore if the string is fixed at each end the length of the string must be an integral multiple of that distance.
 
  • #4
AndyNewton said:
In the special case of a piece of string fixed at each end the thing that makes a difference is that the string is fixed at two points, not just one point. For a standing wave to be maintained with two fixed points both those points must be zero amplitude nodes.

What happens if two ends are at max amplitude (sum of either amplitudes) rather than zero. I mean why should the ends be at zero amplitude to "maintain" the standing wave pattern?
 
  • #5
The same as if there were nodes at each end. We still have a double constraint and the possible wavelengths of standing waves are 2L/n in which L is the string length and n = 1, 2, 3... . This what you get for sound waves in a pipe open at each end - a so-called 'open pipe'! If there'e a node at one end and an antinode at the other (as for sound in a pipe closed at one end and open at the other - a 'closed pipe'), it's easy to see that the possible wavelengths are 4L, 4L/3, 4L/5 ... .
 
  • #6
PrakashPhy said:
What happens if two ends are at max amplitude (sum of either amplitudes) rather than zero. I mean why should the ends be at zero amplitude to "maintain" the standing wave pattern?

Because the ends are "fixed." Meaning whatever else happens to the string, the ends stay at their original positions, hence the wave amplitude at the ends has to work out to be zero.
 

1. What is a standing wave?

A standing wave is a type of wave that forms when two waves with the same frequency and amplitude travel in opposite directions and intersect with each other. The points where the waves intersect experience no net movement, resulting in a pattern of stationary nodes and antinodes.

2. How is a standing wave different from a traveling wave?

A standing wave remains in a fixed position while a traveling wave moves through a medium. Additionally, a standing wave is formed by the interference of two waves, while a traveling wave is created by a single source.

3. What factors affect the formation of a standing wave?

The formation of a standing wave is affected by the frequency and amplitude of the waves, as well as the properties of the medium through which the waves are traveling. The length of the medium and the boundary conditions also play a role in the formation of a standing wave.

4. What are the applications of standing waves?

Standing waves have a variety of applications in science and technology. They are used in musical instruments to produce various notes, in seismic studies to analyze the properties of the Earth's interior, and in optical fibers to transmit signals. Standing waves are also used in medical imaging techniques such as ultrasound and MRI.

5. Can standing waves interfere with each other?

Yes, standing waves can interfere with each other. When two standing waves with different frequencies are present, they can combine to create a new pattern of nodes and antinodes. This phenomenon is known as beats and is used in music to tune instruments.

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