Is the Definition of a Standing Wave Accurate?

[kasheee]

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.]

 

[BvU]

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.

$\$

 

[jtbell]

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.

 

[kuruman]

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.

 

[Orthoceras]

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?

Image from wikipedia. Travelling waves between the nodal lines.

 

[Lord Jestocost]

"In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space."

from: https://en.wikipedia.org/wiki/Standing_wave

 

[Orthoceras]

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?

 

[vanhees71]

It only demonstrates once more that a definition in words is usually ambiguous. Only with help of mathematics we can express physics accurately. @kuruman 's definition is clear and accurate. Also Wikipedia comes, after many words, to a clear mathematical description:

https://en.wikipedia.org/wiki/Standing_wave#Mathematical_description

 

[Orthoceras]

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.)

 

[sophiecentaur]

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.

 

[sophiecentaur]

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.

 

[jtbell]

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.

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)."

 

[hutchphd]

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.

 

[vanhees71]

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?

 

[hutchphd]

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.

 

[jtbell]

"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:

"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.

 

[vanhees71]

It's a solution of the wave equation of the form
$$f(t,\vec{x})=A(t) B(\vec{x}).$$

 

[sophiecentaur]

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.

 

[kuruman]

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.

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.

 

[Orthoceras]

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 $f(t,\vec{x})=A(t) B(\vec{x})$.