Standing or Stationary Wave: Exploring the Definition and Interpretation

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In summary: That would be more accurate. Different points on a standing or stationary wave reach their respective maximum displacements at the same time, but the actual displacement varies with time.
  • #1
kasheee
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Homework Statement:: Definition
Relevant Equations:: Definition interpretation

I saw the definition below for a standing or stationary wave. Is this definition correct, as my definition of a 'fixed displacement' for this type of wave applies only to anti-nodes on a stationary wave? Thanks.

A standing wave or a stationary wave, is a wave that has fixed displacement for all points along its length.

[Moderator's note: moved from a homework forum.]
 
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  • #2
kasheee said:
I saw the definition below
And where was that ? On a billboard ?

[edit] You are posting in the advanced physics forum -- we expect you to provide references that are better than just 'I saw'.

Electromagnetic waves have no displacement.

##\ ##
 
  • #3
I assume we're talking about mechanical waves (strings, water waves, sound waves, etc.) not electromagnetic waves.

Displacement usually means the distance of a point on the wave from its equilibrium point, at any specific time. It varies with time, as the wave oscillates.

I would say that "A standing wave or a stationary wave, is a wave on which each point has a fixed amplitude of oscillation." The amplitude of oscillation of a point on the wave is its maximum displacement, during the course of its oscillations about its equilibrium point. Different points on a standing or stationary have different amplitudes of oscillation.
 
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  • #4
I think the definition is trying to convey that the waveform of a standing wave can be written as ##y(x,t)=f(x)g(t)## instead of ##y(x,t)=f(x\pm vt)##. In other words, there is no constant-phase propagation (no constant-phase "displacement") in a time interval Δt. The definition quoted by the OP is confusing, I agree.
 
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  • #5
Is the wave behind a double slit a standing wave? The definition by jtbell implies it is a standing wave, whereas the definition by kuruman implies it is not. Or is the concept of a standing wave in this case just a matter of taste?

double_slit.gif

Image from wikipedia. Travelling waves between the nodal lines.
 
  • #6
  • #7
Ok, so the definitions by wikipedia and kuruman are contradictory. Do you mean that the definition by wikipedia is better, or is your post inconclusive?
 
  • #9
Ok, so what is your conclusion after having read the mathematical section in wikipedia: is the wave behind a double slit a standing wave?

(It seems to me that the section about the 2D standing wave with a rectangular boundary only discusses a boundary with a fixed end, where the displacement must be zero. That is more like a Cladni pattern than the wave behind a double slit.)
 
  • #10
BvU said:
Electromagnetic waves have no displacement.
Strictly true but, the Maths of EM waves is identical to that of other waves, which may or may not have actual displacement. You could claim that a 'pressure' wave has no displacement is all you are doing is looking at the pressure. It's probably best, if one feels it's confusing, to say that the H or E fields in EM waves are 'equivalent to' the displacement that we use for other types of wave.
Stressing the common concepts in Science is at least as important as pointing out differences.
 
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  • #11
Orthoceras said:
Ok, so what is your conclusion after having read the mathematical section in wikipedia: is the wave behind a double slit a standing wave?
This is the sort of problem that makes itself a problem. In fact, interference is caused by multiple waves, traveling along different paths - either from two slits or from the two ends of a resonant tube. In either case, the resultant amplitude is caused by different path lengths for all the components at different points in space.
Best not to cause angst trying to resolve a non-existent distinction. There are enough real things with which to scare students.
 
  • #12
jtbell said:
I would say that "A standing wave or a stationary wave, is a wave on which each point has a fixed amplitude of oscillation." The amplitude of oscillation of a point on the wave is its maximum displacement, during the course of its oscillations about its equilibrium point. Different points on a standing or stationary have different amplitudes of oscillation.

Orthoceras said:
Is the wave behind a double slit a standing wave? The definition by jtbell implies it is a standing wave,

I was thinking of simple cases of a one-dimensional wave such as a vibrating string, that you see in introductory textbooks. To cover cases like yours, how about if we add, "Different points on a standing or stationary wave reach their respective maximum displacements at the same time(s)."
 
  • #13
Orthoceras said:
Ok, so what is your conclusion after having read the mathematical section in wikipedia: is the wave behind a double slit a standing wave?
It really does not matter. A plane standing wave can be (rigorously) thought of as the equal sum of a rightward and a leftward traveling wave. At the slits only the rightward part of the wave produces the sources in this picture.
In fact the exact nature of the waves very close to each slit is quite complicated but need not concern us because it does not affect the radiation pattern far away.
 
  • #14
Orthoceras said:
Ok, so what is your conclusion after having read the mathematical section in wikipedia: is the wave behind a double slit a standing wave?

(It seems to me that the section about the 2D standing wave with a rectangular boundary only discusses a boundary with a fixed end, where the displacement must be zero. That is more like a Cladni pattern than the wave behind a double slit.)
No, behind a double slit the wave is moving. Why should it be standing?
 
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  • #15
vanhees71 said:
No, behind a double slit the wave is moving. Why should it be standing?
For a big reflective wall in steady state it would likely be more like a standing wave. But it is not salient.
 
  • #16
jtbell said:
"A standing wave or a stationary wave, is a wave on which each point has a fixed amplitude of oscillation."
Now that I think of it, this also applies to simple traveling waves. So I would now replace this with my newer statement:
jtbell said:
"Different points on a standing or stationary wave reach their respective maximum displacements at the same time(s)."
... whereas on a traveling wave, different points reach their respective maximum displacemments at different times.
 
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  • #17
It's a solution of the wave equation of the form
$$f(t,\vec{x})=A(t) B(\vec{x}).$$
 
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  • #18
hutchphd said:
For a big reflective wall in steady state it would likely be more like a standing wave. But it is not salient.
If there is any reflection then of course there will be interference. Even with non normal incidence there will be a region of standing wave.
But honestly, who gives a monkey’s what you call these things? You could say that any wave through a finite medium will have ‘standing’ components due to restrictions. You just may need to be inventive to recognise them.
 
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  • #19
A picture with math is worth 103 words.
Below is shown a collection of 41 independent vertical spring-mass systems oscillating up and down with no energy losses. Each of the masses, differs from its two neighbors by a constant phase.

1. Traveling Wave
##y(x,t)=\sin(kx-\omega t).##
If you concentrate on a single dot, you will see it execute harmonic motion; if you concentrate on a peak or valley (constant phase), you will see it move (displaced) to the right. All point masses have the same amplitude of 1.



2. Standing Wave or Two Waves traveling in opposite directions
##y(x,t)=\sin(kx-\omega t)+\sin(kx+\omega t).##
There is no displacement of constant phase. The point masses have position-dependent amplitudes varying from 2 to zero.



3. Two traveling waves close to destructive interference (in one dimension)
##y(x,t)=\sin(kx-\omega t)+\sin(kx-\omega t+\frac{15\pi}{16})##
This is a traveling wave with reduced amplitude and different initial value from the one above. All point masses have the same amplitude of about 0.2



4. Two traveling waves exactly at destructive interference (in one dimension)
When the phase difference is exactly ##\pi##, all masses lie on the x-axis. Not animated because there is no need. All amplitudes are zero at all points in the one-dimensional space available to the wave.

Destructive Interference.png


The point is that, in the two-dimensional double slit interference pattern shown in post #5, there are traveling waves everywhere. Along one of the bright propagating lines, one would have case 1 with amplitude 2; along one of the dark lines, one would have case 4 with amplitude zero; along any line between bright and dark lines, one would have case 3 with amplitude between 2 and zero.
 
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  • #20
From my point of view, the topic started in post #3 and #4 with contradictory definitions of a standing wave by jtbell and kuruman. My example in post #5 served to illustrate the contradiction. The contradiction has been solved in post #12 by jtbell adapting his definition, so that it is in agreement with the uncomplicated definition by kuruman in #3, and vanhees71 in #17, that a standing wave is a solution of the wave equation of the form ##f(t,\vec{x})=A(t) B(\vec{x})##.
 
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FAQ: Standing or Stationary Wave: Exploring the Definition and Interpretation

What is a standing or stationary wave?

A standing or stationary wave is a type of wave that forms when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This creates a pattern of nodes (points of no displacement) and antinodes (points of maximum displacement) that do not move or propagate through space.

How is a standing wave different from a traveling wave?

A standing wave does not travel through space like a traveling wave does. Instead, it appears to be standing still because the energy of the wave is constantly being reflected back and forth between the two interfering waves. This results in a pattern of nodes and antinodes that do not move.

What factors affect the formation of standing waves?

The formation of standing waves is affected by the frequency, wavelength, and amplitude of the two interfering waves. The distance between the two waves, as well as the medium through which the waves are traveling, can also impact the formation of standing waves.

What are some real-life examples of standing waves?

Standing waves can be found in various natural and man-made systems. Some examples include vibrating strings, musical instruments (such as a guitar or violin), sound waves in a pipe or organ, and electromagnetic waves between two parallel conductors.

How are standing waves used in scientific research or technology?

Standing waves have many practical applications in science and technology. They are used in fields such as acoustics, optics, and electronics for tasks such as measuring properties of materials, creating resonant frequencies, and producing stable signals. In addition, standing waves are also used in medical imaging techniques such as ultrasound and MRI.

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