# Must the limits on the propagation speed of waves refer to a media?

## Summary:

Must a limit to the propagation speed of a wave refer to properties of a medium? If not, how can "wave" and "propagation speed" be defined mathematically so they rule-out illusions?
An example (I think) of creating a phenomena that appears to propagate faster than the speed of light would be to have a line of people holding flashlights and giving each person a schedule of when to blink his light. With proper schedule we could create the illusion that point of light is moving along the line "faster than light". Of course this would not involve any single physical object actually doing so.

Typically waves in media do not involve material parts of the media moving great distances. So if we wish to impose some speed limit on how fast a wave travels, must we do it by limiting how fast the parts of the media can make their small motions? If we don't take that approach, how do we avoid the confusion between a "real" wave and phenomena like the line of people blinking flashlights?

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Dale
Mentor
What you describe is the difference between phase velocity and group velocity. Note that the superluminal wave you describe carries no information. The flashlights just turn on and off according to a preset schedule. To transmit information would require altering the schedule, which could only be done at c or less.

etotheipi, sophiecentaur, jasonRF and 1 other person
What you describe is the difference between phase velocity and group velocity.
That distinction is clear only when we have a mathematical description of the phenomena. The line of flashlights is an example of physically implementing a field that changes in time. So we need something in our theory to say that we are not permitted to write equations for the changing field due to a line of flashlights in the same way we write equations for changes in the field due to other phenomena.

Note that the superluminal wave you describe carries no information.
That's intuitively clear - using an intuitive concept of "information".

The flashlights just turn on and off according to a preset schedule. To transmit information would require altering the schedule, which could only be done at c or less.
Ok, but are you saying that in general ( for sound waves, ocean waves, etc.) we limit the speed of waves by stating a limit on how fast information can be propagated? Is that the starting assumption? Or do we begin with different assumptions and then prove (with some appropriate technical definition of "information") that information can only be propagated at some finite speed?

jbriggs444
Homework Helper
2019 Award
Groping for a way to express it...

Do your wave equations have any consistency requirements? Or are they simply a rule that specifies the field values at all positions and all times? e.g. a movie recorded on DVD.

If there are consistency requirements and if you have a solution for all space and time...

1. Can there be a local perturbation for which there remains a globally consistent solution? That is to say that you pick a time and time and change some field values locally (in space) while leaving remote (in space) field values unchanged and see whether there is still a global solution consistent with those changes.

2. How do the required "changes", if any, from such a local perturbation ripple forward (and backward) in time. Can we place a bound on this -- a kind of "change cone" in either direction?

There are some language problems here. More formal phraseology would avoid calling these "changes" and would instead talk about regions where discrepancies exist between the unperturbed and the perturbed solution.

Obviously, our usual physical laws have some pretty tight consistency requirements and some pretty clear speed limits.

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jasonRF
So we need something in our theory to say that we are not permitted to write equations for the changing field due to a line of flashlights in the same way we write equations for changes in the field due to other phenomena.
Why? If we restrict ourselves to a finite region, this seems an odd statement to me.
The long line of terra cotta soldiers with pre-timed flashers offends neither me nor James Clerk Maxwell. This will not look like a propagating light wave. It will be what it is.
So why do we need an addition to Mr Maxwell's theory?

Ok, but are you saying that in general ( for sound waves, ocean waves, etc.) we limit the speed of waves by stating a limit on how fast information can be propagated? Is that the starting assumption?
In general the group velocity ##v_g=\frac {d\omega}{dk}## gives the maximum rate of information transferrable over any appreciable time and is derived from the dispersion relation for the system in question based on the appropriate constituative equations. Not from any arbitrary fiat.
Similarly for the speed of light which follows from Maxwell's equations. There is nothing in the equations which indicate that c is universal ....it is an experimental parameter.

The long line of terra cotta soldiers with pre-timed flashers offends neither me nor James Clerk Maxwell. This will not look like a propagating light wave.
If that's true, it would be interesting to see a proof. More generally, its it impossible to arrange a collection of "external" disturbances of a field whose net result can be described mathematically as a wave?

Another way of putting it is: Given the value of a field on some region of space over a time interval, is it possible to distinguish whether the field is being affected by something that adds energy (or changes its value in some way) at a single point or small region versus whether a set of things is adding energy to the field over the entire region?

When I conceive of the velocity of a wave , I think of some localized "external" disturbance that occurs. When there is finite propagation speed, the field at distant places is not immediately affected by the disturbance, but only varies because according to equations that describe how the value of the field at one place at space and time relates to other nearby values of the field where there are no "external" disturbances.

If someone arranged a line of disturbances who net result was mathematically a wave, I'd say the velocity that wave is different that my definition of "the velocity of a wave" because in my definition, I'm thinking of a different experiment.

Groping for a way to express it...

Do your wave equations have any consistency requirements? Or are they simply a rule that specifies the field values at all positions and all times? e.g. a movie recorded on DVD.

If there are consistency requirements and if you have a solution for all space and time...

1. Can there be a local perturbation for which there remains a globally consistent solution? That is to say that you pick a time and time and change some field values locally (in space) while leaving remote (in space) field values unchanged and see whether there is still a global solution consistent with those changes.
That agrees with my notion of how to define the velocity of a wave -i.e. it restricts the type of mathematical wave function we analyze to one generated by local perturbations.

jasonRF
Gold Member
If that's true, it would be interesting to see a proof. More generally, its it impossible to arrange a collection of "external" disturbances of a field whose net result can be described mathematically as a wave?
When you say "described mathematically as a wave", what do you mean?

hutchphd
When you say "described mathematically as a wave", what do you mean?
I'm thinking of a field that is described by a plane wave - because I can understand what "velocity" of the wave means in that case. Your question forces me to ponder what "velocity" and "max propagation speed" would mean in more general cases.

For the general case, suppose we have a family of time varying fields in 3D given by the 7 dimensional vectors of the form ##(f_x, f_y, f_z, x,y,z,t)##. How do we define the "maximum propagation" speed or "wave velocity" in such a general case? (I'm not assuming the members of the family are each solutions to some differential equation. There might not be any plane waves in the family.)

What do you mean by "general case"? Clearly an arbitrary field can do whatever you want. I do not understand.

What do you mean by "general case"? Clearly an arbitrary field can do whatever you want. I do not understand.
Yes, I'm speaking in mathematical terms - an arbitrary family of fields.

I'll try to refine the question in the OP. Suppose I have a family fields that describe the possible flows of a fluid. Intuitively, I have no trouble imagining that for a particular member of the family and for a particular location in space, there is an instantaneous velocity of the fluid at that point. If I look at all members of the family and all points in space, I can imagine that there is a maximum magnitude for these instantaneous velocities.

If, instead of that, I think about a family of fields that doesn't describe the flows of a fluid, perhaps I can pretend it does and apply the same mathematics to define an instantaneous velocity and still get a maximum magnitude for such velocities.

The above approach seems different than the approach used to define the maximum propagation speed for waves in electromagnetic fields or sound waves, ocean waves etc. That approach, as presented in elementary texts, is to consider a field that has a nice mathematical expression ( plane wave, spherical wave etc). The equations for these fields have a term of the form "##-ct##" in them. The intuitive interpretation of the mathematical expression is that some "disturbance" is propagating along within the field with a velocity of ##c##.

Consider an inhomogeneous medium. (I think) the solutions to Maxwell's equations (for various boundary conditions) all contain the term ## - ct## even though there need not be a solution that is a simple plane wave or spherical wave.

In regard to your earlier post about a line of terra cotta soldiers with flashlights not escaping Maxwell's theory. I interpret that to mean that even if the boundary conditions for Maxwell's equations include some time varying external disturbances applied to the EM field, the solutions to Maxwell's equations still depend on the "##-ct##" term. Is that correct?

Does a similar statement apply for sound waves in inhomgeneous media?

I can see why ##c## can be interpreted as the "speed of propagation" from a purely mathematical point of view. I don't understand if there is a more detailed way to think (rigorously) about it in terms of physics. In terms of physics, the intuitive concept is that a "disturbance" "propagates" through space. However, I don't see how to rigorously define a "disturbance" and "propagates through space" except by a set of examples - simple cases like plane waves, spherical waves, etc.

As you said, an arbitrary field can do whatever it wants. so it would not be possible to do this for an arbitrary field. Is it possible to define "disturbance" and "propagates" for some particular fields that are not describe by equations with a "##-ct##" in them. Is there a physical definition for "disturbance" and "propagates" that is different than "look for a ##-ct## term in the equation" ?

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But for media the structured lumps that have some persistence propogate at the group velocity associated with a mean wavelength. This group velocity doe not appear per se in the equations but is a vestige of applying stationary phase arguments to the fourier synthesized solutions.
To the degree that an arbitrary disturbance is subject to fourier analysis this has all been done it seems to me. I would also direct you to wavelet transforms which are similar in spirit but I do not pretend to understand. Perhaps I will give them a look (I never get far for some reason)

To the degree that an arbitrary disturbance is subject to fourier analysis this has all been done it seems to me.
Considering "an arbitrary disturbance" to be the entire field, I interpret your remark to mean that a reasonable time-varying field can be represented as a superposition of simple waves of some type (like spherical waves) and each of the simple waves can be interpreted as a disturbance moving through the field. Furthermore the disturbances in the simple waves all move with the same velocity.

Not exactly.
For a localized disturbance,the fourrier synthesis is general enough, but the details of the lump dynamics depend upon the dispersion relation ω=ω(k).
For light in vacuo of course ##\omega =ck ##

I can see why c can be interpreted as the "speed of propagation" from a purely mathematical point of view.
I don't know what prompted this thought but I believe this discussion has, at its heart, the "Lighthouse Paradox". The laser spot on the moon can be made to move faster than c. To grossly paraphrase @Dale, "so what?"

Dale and vanhees71
I don't know what prompted this thought but I believe this discussion has, at its heart, the "Lighthouse Paradox". The laser spot on the moon can be made to move faster than c. To grossly paraphrase @Dale, "so what?"
A "Lighthouse Paradox" isn't a situation where information can be transmitted at arbirarily great speeds. So a person whose is focus is a speed limit on transmitting information can look at a "Lighthouse Paradox" and say "So what?". I assume that's what you mean.

However, my interest is in a precise definition for "the speed of propagation of a disturbance" in a field. Presumably, appropriate definitions of "disturbance" and "speed of propagation" would agree with the conclusion that what happens in a "Lighthouse Paradox" is not an example of propagating a disturbance through a field at arbitrary speeds.

Refuting a "Lighthouse Paradox" as a claim of transmitting signals faster than light is a familiar exercise and I understand why that would be a common reaction to the original post. However, stating that a "Lighthouse Paradox" is not an example of propagating a disturbance through a medium at an arbitrary speed doesn't, by itself, provide a definition of "disturbance" and "speed of propagation". It merely uses those words as being synonymous with "you can't transmit information at arbitrary speeds".

Presumably, appropriate definitions of "disturbance" and "speed of propagation" would agree with the conclusion that what happens in a "Lighthouse Paradox" is not an example of propagating a disturbance through a field at arbitrary speeds.
And my point is that any such attempt will be a "spot on the moon".
So you asked the question. Please provide a definition, otherwise we are blind men with an elephant! "Information" propagation has to do only with the boundary values.

And my point is that any such attempt will be a "spot on the moon".
I don't understand what you mean by that.

Are you saying that any attempt to precisely define "disturbance" and "speed of propagation" will result in the spot on the moon being an example of the propagation of a disturbance?

Yes. If you do so, we shall see. Otherwise blind men.

Yes. If you do so, we shall see. Otherwise blind men.

Bind men? Your metaphors are confusing.

My question is about finding appropriate definitions for "disturbance" and "speed of propagation" in field. Are you implying that I must define "distrubance" and "speed of propagation" myself? - that it's just a matter of personal opinion? Are you implying that it's futile to seek specific definitions for the terminology?

So you asked the question. Please provide a definition, otherwise we are blind men with an elephant!
Perhaps you don't know the parable.
Yes, at least you need to somehow rigorously specify your question

Yes, at least you need to somehow rigorously specify your question
Well, I'm trying. If a question is how to define something, then it seems a bit unfair to ask the questioner to begin by defining the something he is asking about!

Physics texts commonly speak of "the speed of propagation" of waves. For waves with simple mathematical expressions, I don't think the concept of a "disturbance" "propagating" needs to be made precise in order to understand such examples. The question is how to use those definitions in more complex situations.

One attempt is to define a "disturbance" ##D_{s,t}## to be the set of values of a field on a particular connected subset ##s## of space at a particular time ##t##. To say that ##D## propagates along a line from times ##t = t_a## to ##t=t_b## means that the values of the field along that line at intermediate times can be found by translating the values in ##D_{s,t_0}## along that line. That's not a completely precise mathematical definition, but I think the idea can be made mathematically precise.

That definition does not deal with curvy paths or inhomogenous media where we need the concepts of varying speeds of propagation and instantaneous speeds. However, if we can define things for averages over time, we can probably use the ideas of calculus to define them for instants.

That approach doesn't deal with concept of a distrubance that is propagating with a change in its amplitude. It also doesn't say how to define the global properties of fields - e.g. how to use a definition for the propagation of particular disturbances to make a definition of the (single, unique) propagation speed for all disturbances in a medium.

I am not an expert on the abstractions of field theory but it seems to me one additionally needs some kind of constituative rules to define the fields (like maxwell eqn).
Asking the question is always the hard part!

jasonRF
Gold Member
I'm thinking of a field that is described by a plane wave - because I can understand what "velocity" of the wave means in that case. Your question forces me to ponder what "velocity" and "max propagation speed" would mean in more general cases.
Even for plane waves it can be complicated and there are multiple velocities to consider. For example, see prof. Fitzpatrick ‘s notes on electromagnetic plane waves in linear isotropic dispersive media.

http://farside.ph.utexas.edu/teaching/jk1/lectures/node75.html

Saying a field is “described by a plane wave” doesn’t force us into a simple situation.

I will agree with everyone else. Unless you define what you are talking about nobody can help you.

Jason

vanhees71