How exactly are standing waves formed?

In summary: This is why a fixed end reflects a transverse wave upside down.In summary, a standing wave is formed when two coherent waves interfere and undergo constructive and destructive superposition. The incident and reflected waves must have a phase difference that satisfies the boundary condition of being independent of time. This can be found by summing the equations for the two waves. Understanding the sinusoidal wave equation and knowing how to obtain the phase difference are necessary for understanding standing waves. The reflected wave is upside down because the fixed end of the medium does not move, causing the reflected wave to be in complete anti phase with the incident wave.
  • #1
PFuser1232
479
20
I understand the equations for the nth harmonic and so on, but I really don't get how a standing wave is formed. My physics teacher said that an incident wave and an identical reflected wave interfering would cause a stationary wave, but I still don't understand how this happens. For example, what should be the phase difference between the two waves, in order for a standing wave to form?
 
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  • #2
You can figure this out. Write down the equations for the incident wave and the reflected wave. Sum them. The sum must be independent of time. That gives you the condition for the phase difference.
 
  • #3
voko said:
You can figure this out. Write down the equations for the incident wave and the reflected wave. Sum them. The sum must be independent of time. That gives you the condition for the phase difference.


But this is beyond the scope of my syllabus
 
  • #4
MohammedRady97 said:
But this is beyond the scope of my syllabus

Have you taken into consideration the principle of superposition?
 
  • #5
MohammedRady97 said:
But this is beyond the scope of my syllabus

What is beyond the scope of your syllabus? Knowing the sinusoidal wave equation or being able to do simple trigonometry?
 
  • #6
MohammedRady97 said:
But this is beyond the scope of my syllabus

Why is that a problem? That doesn't mean it's too hard for you to understand!

This might help (don't be put off by the cheesy introduction to the real video).
https://www.youtube.com/watch?v=DovunOxlY1k
 
  • #7
I think there was just a thread discussing this topic. :wink:
 
  • #8
voko said:
What is beyond the scope of your syllabus? Knowing the sinusoidal wave equation or being able to do simple trigonometry?

Knowing the sinusoidal wave equation and what it actually means and how to obtain the phase difference given two sinusoidal wave equations.
The math is easy, I am well trained in trigonometry, calculus and algebra. I study A level Mathematics and A level physics.
 
  • #9
AlephZero said:
Why is that a problem? That doesn't mean it's too hard for you to understand!

This might help (don't be put off by the cheesy introduction to the real video).
https://www.youtube.com/watch?v=DovunOxlY1k

Why is a wave reflected upside down if it's a rigid end?
 
  • #10
Why is it reflected upside down?
 
  • #12
In simple terms, Two coherent waves (or one wave that has been reflected back upon its self), interact and under go constructive and destructive superposition. At points where the amplitudes are completely opposite (http://leiferlingsson.files.lege.net/2008/07/standing_waves_400x300.jpg) the amplitudes cancel and form a node. Where the amplitudes rare both on the same side of the x-axis they reinforce each other, The standing wave is merely the resultant of two wave forms.
 
  • #13
MohammedRady97 said:
Why is it reflected upside down?
Here is an interesting way of seeing why:

Imagine two identical waves at opposite ends of a long string. They meet at "x= 0". If the two waves are on the same side of the string when they meet they add so that the height at x= 0 will be twice as high as the two separately. If they are on opposite sides, they will "subtract" and the point at x= 0 will not move.

Since we can think of the reflected wave as a wave coming from "beyond" the end, if the end cannot move, then the reflected wave must be upside down.
 
  • #14
MohammedRady97 said:
Why is it reflected upside down?

Another way to look at it:

The idea of Boundary Conditions is a useful one to get familiar with and the boundary condition is that there is no net displacement at the end. Take the example of a string with a transverse wave. The instantaneous force on the fixed end point, due to the incident wave is in one direction and with a certain value. That point is in equilibrium (is doesn't move) so there must be an equal and opposite force 'from' the fixing. This force will be generating a reflected wave with an equal and opposite force at all times. This corresponds to launching a wave that is in complete anti phase with the incident wave.
 

FAQ: How exactly are standing waves formed?

1. How do standing waves form?

Standing waves are formed when two waves with the same frequency and amplitude traveling in opposite directions interfere with each other. This creates a pattern of nodes (points of maximum constructive interference) and antinodes (points of maximum destructive interference) that do not appear to move.

2. What are the necessary conditions for standing waves to form?

In order for standing waves to form, there must be a fixed boundary at either end of the medium that the waves are traveling through. The medium must also be able to support the specific frequency of the waves. Additionally, the two waves must have the same amplitude and frequency and be traveling in opposite directions.

3. How is the wavelength of a standing wave determined?

The wavelength of a standing wave is determined by the distance between two consecutive nodes or antinodes. This distance is equal to half of the wavelength of the individual waves that are interfering to create the standing wave.

4. Can standing waves form in any medium?

Standing waves can form in any medium that meets the necessary conditions, such as a string, air column, or water surface. However, the specific frequency of the standing wave will vary depending on the properties of the medium.

5. What is the difference between standing waves and traveling waves?

The main difference between standing waves and traveling waves is that standing waves do not appear to move, while traveling waves do. Standing waves are the result of two waves interfering with each other, while traveling waves are the result of a disturbance traveling through a medium. Additionally, standing waves have nodes and antinodes, while traveling waves do not.

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