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

[Stephen Tashi]

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?

 

[Dale]

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.

 

[Stephen Tashi]

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]

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.

 

[hutchphd]

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.

 

[Stephen Tashi]

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.

 

[Stephen Tashi]

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]

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?

 

[Stephen Tashi]

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.)

 

[hutchphd]

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

 

[Stephen Tashi]

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" ?

 

[hutchphd]

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)

 

[Stephen Tashi]

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.

 

[hutchphd]

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 $\omega =ck$

 

[hutchphd]

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?"

 

[Stephen Tashi]

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".

 

[hutchphd]

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.

 

[Stephen Tashi]

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?

 

[hutchphd]

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

 

[Stephen Tashi]

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?

 

[hutchphd]

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

 

[Stephen Tashi]

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.

 

[hutchphd]

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]

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

 

[Stephen Tashi]

Unless you define what you are talking about nobody can help you.

That means noboby can help physics texts who speak of disturbances propagating through a medium. After all, it isn't just me that uses such terminology.

 

[hutchphd]

Typically they have already supplied or are commencing a description of the dynamics of the medium, usually describing a wavelike or a diffusive process. The statement by itself is not well defined.
You have historically pursued precise definition, this is not different.

 

[jasonRF]

That means noboby can help physics texts who speak of disturbances propagating through a medium. After all, it isn't just me that uses such terminology.

I think the last sentence of my post was ambiguous and a little out of line - please accept my apology. Also, for some reason I hadn’t seen posts 22 or 23 when I posted.

Anyway,I simply don’t understand what complex situations you are interested in. That is what I was trying to communicate. So I will bow out of the conversation.
Jason

 

[Dale]

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.

What’s wrong with using the standard terminology? Group velocity and phase velocity.

 

[Stephen Tashi]

What’s wrong with using the standard terminology? Group velocity and phase velocity.

Nothing's wrong that if the field is a familiar example of a wave - plane wave, spherical wave.

However, if we have a inhomogenous medium, I think we might need the idea of the propagation speed of a "disturbance" at a point in the medium.

 

[Dale]

Nothing's wrong that if the field is a familiar example of a wave - plane wave, spherical wave.

However, if we have a inhomogenous medium, I think we might need the idea of the propagation speed of a "disturbance" at a point in the medium.

Why? What aspect of a “disturbance” is not adequately captured in the standard terms?

 

[vanhees71]

What’s wrong with using the standard terminology? Group velocity and phase velocity.

What's also important in this context is also "front velocity", which is the velicity which must be $\leq c$. Both group and phase velocity can be $>c$ without violating relativistic causality. This was pointed out to Wien by Sommerfeld as early as 1907. It's one of the most beautiful applications of the theorem of residues used to do the Fourier integral from frequency to time representation.

 

[Stephen Tashi]

Why? What aspect of a “disturbance” is not adequately captured in the standard terms?

What is the context for the standard terms? How do they apply when, for example, the variation of the values in a field over time is not described by a wave?

(This is a genuine question, not a rhetorical remark asserting the standard terms don't apply. We might have answer the question "What is a wave?" in technical manner. )

 

[Dale]

How do they apply when, for example, the variation of the values in a field over time is not described by a wave?

They don’t. But I am not sure how that is relevant to your OP (and the thread title). Your OP at least was describing a wave. Are you now thinking about a scenario with some near field phenomenon that does not propagate?

 

[Stephen Tashi]

They don’t. But I am not sure how that is relevant to your OP (and the thread title).

I agree the relation isn't straightforward!

The discussion of various definitions of "velocity" for waves asserts that only one of these velocities is appropriate for measuring how fast a signal can be transmitted. So the supplementary question arises: How do we establish a speed limit for signals ( and/or disturbances) in examples where the field is not described by a wave?

We can't answer that question till we define the terms "signal" and"disturbance" in a context that doesn't depend on the notion of "wave". ( Without waves, are there no such things as signals?)There is a basic vocabulary difficulty with the word "field". For example, one may refer to "the gravitational field" as a single concept, or one may refer to specific or local cases like "the gravitational field of a unit sphere with mass M" or include time in the example, such as "the EM field of a capacitor discharging through a resistor". I'm not sure what context is needed to give a technical definition for "signal" or "disturbance".

 

[sophiecentaur]

The discussion of various definitions of "velocity" for waves asserts that only one of these velocities is appropriate for measuring how fast a signal can be transmitted.

What do you mean by "a signal"? I think that could be the problem because you would need to specify how much 'information' constitutes a signal. If the information is just that the transmitter has been turned on then how long would you need to wait with your receiver to decide that it definitely has been switched on, rather than we just experience a slow noise spike. Afaics, the definitions used to describe the various wave speeds may not include that.

∂φ/∂k defines group velocity and that tells you how fast a modulated wave carries the information. It only includes a 'medium' by implication, and describes the point to point transfer.

I have a suspicion that you may, at this stage, just be trying to find something inadequate in received Science, If you want to do that then you will need, at least, to come up with a very non-vague question for PF.

 

[Stephen Tashi]

What do you mean by "a signal"?

Since I'm not the only person who uses the terminology "signal", I'm willing to accept a technical definition of what others mean by it. My questions are: What is that technical definition? Does it apply in contexts where there are no waves? If so, how is that definition stated without reference to a wave?

I think that could be the problem because you would need to specify how much 'information' constitutes a signal.

Perhaps - and we'd have to give a technical definition for "information".

I have a suspicion that you may, at this stage, just be trying to find something inadequate in received Science.

I don't understand exactly what you mean.

If you are saying that I'm trying to point out something inadequate and paradoxical about known science, that isn't the case. I genuinely don't know if conventional science has a way of giving a technical defintion for the intuitive notion of the propagation speed of a disturbance without referring to the concept of waves.

 

[hutchphd]

We (you) can define it any way we like ( I will resist the quotation from Humpty Dumpty ). I would define information as a single bit...all else is generalization. Try to transmit it efficiently.
The separation into near and f ar field will fall out naturally I think.
Also what disturbances cannot be described as waves? There is a place to start. I believe the answer is effectively none but am open to refutation...

 

[Dale]

The discussion of various definitions of "velocity" for waves asserts that only one of these velocities is appropriate for measuring how fast a signal can be transmitted. So the supplementary question arises: How do we establish a speed limit for signals ( and/or disturbances) in examples where the field is not described by a wave?

I don’t think that question actually arises. If the “disturbance” propagates then phase and group velocity is appropriate. If the signal doesn’t propagate then it is stuck in the near field and it isn’t really a signal or at least it can’t be transmitted.

Anyway, I find your description very vague and I am doubtful that what you are talking about even exists. I think you need to do a bit less speculation and a bit more research.

 

[Dale]

@Stephen Tashi sorry about the delay on this. Ireally cannot figure out what you are talking about. From my understanding what you describe in your OP is simply a group velocity. You seem to think that it is not. So if you think it is not a group velocity then you need to find something that describes what your real question is.

The literature clearly explains phase velocities and group velocities (which is still what I think your question is about), but without some guidance from a professional scientific source we cannot blindly discuss "disturbances".

This thread is reopened, but needs to be clearly defined by you. Either to discuss group velocity or if you can provide a scientific reference that defines what you mean by “disturbances” (even if they use different wording).

 

[Stephen Tashi]

I'll pretend to answer some of my own questions in an authoritative manner and we'll see how that goes.

===========
In the theory of fields, Is there a technical definition for a "signal" or a "disturbance" in the field that does not rely on the specific case of waves? Is there a definition for the speed of propagation of a signal or disturbance in a field that does not refer to the concept of waves?
============

The concepts of a signal or disturbance have no standard technical definitions in the theory of fields. They are intuitive concepts and when they are used they are always used in the context of waves. In that context, a wave is regarded as modeling a shape that moves through the field as time passes. The shape is the signal or disturbance. However, there are no formal definitions that say what a "shape" is. For example, in the case of a standing wave, which part of the wave's shape is the "disturbance" or the "signal" ? People can express different opinions. There are no technical definitions that answer the question.

In other fields of study (for example, applications of field theory to comunications) "signal" may have a technical definition.

According to wikipedia (https://en.wikipedia.org/wiki/Wave_equation ):

The inhomogeneous wave equation in one dimension is the following:
$u_{tt}(x,t) - c^2 u_{xx}(x,ty) = s(x,t)$

with initial conditions given by
$u(x,0) = f(x)$
$u_t(x,0) = g(x)$

The function $s(x,t)$ is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism.

So in that specific context, you could regard $s(x,t)$ as a disturbance or signal. I don't know whether the concept of source functions generalizes to higher dimensions.

==========

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

==========

A field need not be defined in terms of the properties of a media. However, it may be meaningful to discuss the properties of a field at a point in space without talking about any physical substance that causes these properties.

For example, in the context of a plane wave of constant amplitude propagating through "free space" or in some unspecified homogenous medium, it is clear how to define the speed of propagation of the wave - just as it is clear how to define the velocity of an object that is moving at a constant velocity. We can say that the propagation speed is property of the field without invoking any specific medium to explain this property. If the wave propagates over several regions of the field and has a different constant velocity in each region, we can compute the average velocity of the wave over all the regions.

From average quantities, we can often define useful instanteous quantities. For example, we can define instantaneous velocity of a moving object as a limit of average velocities taken over small distances. So it may be possible to define the instantaeous propagation speed of a wave at a point in time and space. It may be that at a specific point (x,y,x) in space, all waves or all waves of a certain type must have the same instantaneous speed. We can regard that speed as a location dependent property of the field.

It's an interesting question of semantics whether an approach that derives the theory of a field from assumed local properties is effectively reviving the concept of an aether - a medium which has those properties!

 

[Dale]

In that context, a wave is regarded as modeling a shape that moves through the field as time passes. The shape is the signal or disturbance.

How is that different from the group velocity? The group is the shape that moves.

 

[jasonRF]

According to wikipedia (https://en.wikipedia.org/wiki/Wave_equation ):
So in that specific context, you could regard s(x,t) as a disturbance or signal. I don't know whether the concept of source functions generalizes to higher dimensions.

I would call $u$ the disturbance and $s$ a source. Of course this generalizes to higher dimensions than your example. A light bulb is a 3D source of electromagnetic waves. A loudspeaker is a 3D source of acoustic waves.

 

[Stephen Tashi]

How is that different from the group velocity? The group is the shape that moves.

How is what different than the group velocity?

A velocity is a vector and a shape is ...well, intuitively it's more than a 3D vector. Group velocity has a formal definition which is, as far as I can tell, based on the looking at the mathematical expression for a wave. Is there a formal definition for "group"? - I mean a formal definition for "group" as a thing that can have a velocity (not the mathematical definition of "group" as an algebraic structure).

 

[jasonRF]

Must the limits on the propagation speed of waves refer to a media?
==========
A field need not be defined in terms of the properties of a media. However, it may be meaningful to discuss the properties of a field at a point in space without talking about any physical substance that causes these properties.

I agree that it may be meaningful. Once you write down a set of PDEs and/or integral equations and/or some other mathematical description of the evolution of your field (often including boundary conditions), then you have already included the physics of the phenomena and media that you presumably care about. If your idea is to forget the media after you have the description and simply understand the properties of that mathematical model, then that can be very useful and you may be learning the physics of multiple phenomena that happen to satisfy the same model. Of course most models have limits of validity (eg most linear models are only valid for small amplitudes for most phenomena in most media), but as long as you don't violate those limits the mathematical description should hold.

The classic 1-D problem that folks such as Sommerfeld and Brillouin examined over a century ago was that of of a semi-finite media occupying $x \geq 0$, and consider the boundary value $u(x=0,t) = \sin(\omega t)$ for $t\geq 0$ and $u(x=0,t) = 0$ for t<0 with the initial condition $u(x,t=0) = 0$ for all $x\geq 0$. If your model includes all of the relevant physics, then a careful mathematical analysis of this or some similar problem should yield the fact that there is a maximum speed at which anything travels (the maximum speed at which $u\neq 0$ can expand). That maximum speed should be no greater than the vacuum speed of light, since we need to be consistent with special relativity. If your model indicates larger or even infinite speeds are possible, like you would find by analyzing the "classic" diffusion equation $\nabla^2 u = \frac{\partial u}{\partial t}$, then your model does not include the physics you need in order to answer the questions you are asking of it. In dispersive media, a good model should show you that there are different types of features that propagate at different velocities, including those usually called phase velocity, group velocity, and signal velocity. For the case of electromagnetic waves, there also will be "precursors" that propagate at the vacuum speed of light even when the dielectric constant is $> 1$; the physical interpretation is that the finite inertia of the electrons (bound or free) allows the initial disturbance to propagate past each electron before it has time to respond.

Specifically, for the 1D case if $\omega$ is the frequency and $k$ is the wavenumber, in a dispersive media a given wave mode will satisfy some relation $\omega = \omega_\alpha(k)$ for some function $\omega_\alpha$. Then the phase velocity is $\omega/k$ and the group velocity is $\frac{\partial \omega_\alpha}{\partial k}$, both evaluated at a given wavenumber and frequency of interest. Fourier analysis then allows you to synthesize any shape waveform.

Is this the approach you are thinking of, or are you trying to completely divorce the field from the media to the point that you cannot write down the mathematics that describe how the field should evolve? My interpretation of some of your posts (and my interpretation may be wrong!) is that you just want to think of some abstract "velocity" at each location without being precise about how that velocity parameter fits into a mathematical description (is it a coefficient in a particular PDE or class of PDEs? something else?...).

By the way, I know I wrote that I would leave the conversation, but I couldn't resist!

Jason

 

[Stephen Tashi]

I would call $u$ the disturbance and $s$ a source.

I think of $u$ as the information for the entire field, what happens everywhere in it throughout all time.

The intuitive idea of a disturbance (to me) is that it is a local phenomena that spreads out in time. I don't know any formal definition that captures that intuition.

Of course this generalizes to higher dimensions than your example. A light bulb is a 3D source of electromagnetic waves. A loudspeaker is a 3D source of acoustic waves.

Yes, those are physical examples of what people call sources. Mathematically, what function is a source? For that matter, mathematically what is the inhomogeneous wave equation in higher dimensions. - and does "inhomogeneous" merely mean we have a differential equation that is not set equal to zero or do inhomogeneous wave equations also have something to do with the transmission of waves through inhomogeneous media?

 

[Stephen Tashi]

If your model includes all of the relevant physics, then a careful mathematical analysis of this or some similar problem should yield the fact that there is a maximum speed at which anything travels (the maximum speed at which $u\neq 0$ can expand). That maximum speed should be no greater than the vacuum speed of light, since we need to be consistent with special relativity.

Emphasizing one of my questions, when you say "maximum speed at which anything travels" , are you implying that a speed can be assigned to phenomena other than waves? Or do you mean "maximum speed at which any wave travels"?

In dispersive media, a good model should show you that there are different types of features that propagate at different velocities, including those usually called phase velocity, group velocity, and signal velocity.

If signal velocity is defined, is there a definition for "signal"? All the features you mentioned are properties of waves. so am I still safe in saying propagation speeds are only defined in the context of waves?

For the case of electromagnetic waves, there also will be "precursors" that propagate at the vacuum speed of light even when the dielectric constant is $> 1$; the physical interpretation is that the finite inertia of the electrons (bound or free) allows the initial disturbance to propagate past each electron before it has time to respond.

The way I visualize such analysis is that the field $\vec{u}(\vec{x},t)$ is written as a superposition of waves. If $\vec{u}(\vec{x},t)$ is defined by linear differential equations, each component wave is a solution. When that is the case it seems safe to think of each component as a real physical phenomena. What do we do if the differential equations satisfied by $\vec{u}(\vec{x},t)$ are nonlinear? - or does that case rarely occur?

Is this the approach you are thinking of, or are you trying to completely divorce the field from the media to the point that you cannot write down the mathematics that describe how the field should evolve?

No, I'm not interested in fields without any mathematical descripton!

I'm curious about what properties of fields are defined at a point in space. It's of some interested to discuss whether only knowing the distributions for such properties is sufficient to deduce the equations that define the global field, but I'd be happy just to know the definitions of some of the properties.

By the way, I know I wrote that I would leave the conversation, but I couldn't resist!

I'm glad you came back.

 

[jasonRF]

EDIT: we seem to be "leap-frogging" each others replies!

Mathematically, what function is a source? For that matter, mathematically what is the inhomogeneous wave equation in higher dimensions. - and does "inhomogeneous" merely mean we have a differential equation that is not set equal to zero or do inhomogeneous wave equations also have something to do with the transmission of waves through inhomogeneous media?

For a scalar field, in 3-D the simplest wave equation I can think of is $\nabla^2 u - \frac{1}{v^2} \frac{\partial^2 u}{\partial t^2} = f(\mathbf{r},t)$. This is a spatially uniform (homogeneous) media and does not include any dispersion, so does not apply for most phenomena in most media. All disturbances will propagate with a speed $v$. The right-hand side $f(\mathbf{r},t)$ is the source, when it is zero this is a homogeneous PDE, when it is not zero this is a nonhomogeneous (also called inhomogeneous) PDE. EDIT: If this vocabulary is unfamiliar, then you need to learn something about differential equations.

In a simple inhomogeneous medium, $v$ may be a function of position so the equation becomes $\nabla^2 u - \frac{1}{v(\mathbf{r})^2} \frac{\partial^2 u}{\partial t^2} = f(\mathbf{r},t)$.

A model of a dispersive media will be more complicated. Often it will involve more than one PDE, so would be called a system of PDEs. A simple example that describes the electric field of high-frequency transverse electromagnetic waves in an inhomogeneous, collisionless unmagnetized plasma would be
$$c^2 \nabla^2 \mathbf{E}(\mathbf{r},t) - \frac{\partial^2}{\partial t^2} \mathbf{E}(\mathbf{r},t) - \omega_p(\mathbf{r}) \mathbf{E}(\mathbf{r},t) = \frac{1}{\epsilon_0} \frac{\partial}{\partial t} \mathbf{J}(\mathbf{r},t)$$
where $\mathbf{J}(\mathbf{r},t)$ is a current source, $c$ is the vacuum speed of light, $\omega_p(\mathbf{r})$ is the spatially varying plasma frequency (proportional to the square root of the electron density), and $\epsilon_0$ is the permitivity of free space. This describes the evolution of the vector field $\mathbf{E}(\mathbf{r},t)$ in 3D space.

Jason

 

[Dale]

How is what different than the group velocity?

How is your vague concept of the velocity of the “disturbance”, which you later describe as “shape that moves”, different from the group velocity?

@Stephen Tashi I don’t appreciate your continued resistance to trying to clearly tie this discussion into the standard scientific literature. Nor do I appreciate you nitpicking my language that way when you have been so persistently vague this whole thread. It is now almost 50 posts long and you are still unclear on what you are asking about.

Out of respect for you I have tried to keep the thread open. You have three options for your next post:

1) a clear statement that you are discussing the group velocity
2) a clear statement that you are discussing the phase velocity
3) a reference from the professional scientific literature describing exactly the velocity that you are discussing

One of these must happen for this thread to continue.

 

[jasonRF]

Emphasizing one of my questions, when you say "maximum speed at which anything travels" , are you implying that a speed can be assigned to phenomena other than waves? Or do you mean "maximum speed at which any wave travels"?

I would call anything that propagates a "wave".

If signal velocity is defined, is there a definition for "signal"? All the features you mentioned are properties of waves. so am I still safe in saying propagation speeds are only defined in the context of waves?

The definition of signal velocity, or wave-front velocity, is somewhat arbitrary to some extent. Some online notes that cover this can be found at
http://farside.ph.utexas.edu/teaching/jk1/jk1.html
Look at the sections starting with "wave propagation in dispersive media". Note these are graduate level notes, as this is a graduate-level topic. Along those lines, what is your background?

Regarding the term "signal", I would call anything that isn't a space-filling periodic feature in the field a "signal". At least that is my off-the-cuff definition.

The way I visualize such analysis is that the field $\vec{u}(\vec{x},t)$ is written as a superposition of waves. If $\vec{u}(\vec{x},t)$ is defined by linear differential equations, each component wave is a solution. When that is the case it seems safe to think of each component as a real physical phenomena. What do we do if the differential equations satisfied by $\vec{u}(\vec{x},t)$ are nonlinear? - or does that case rarely occur?

The general case is that the equations describing media are nonlinear, but for many situations we can linearize the equations and obtain a mathematical description that describes practical phenomena. The linear theory of dielectrics is an example, and many high-precision optical systems are based on it. However, if the wave electric field is large enough then nonlinear effects apply - indeed, if it is very large you will have arcing and destroy the material! Likewise, the example I gave for waves in the simplest plasma has been linearized. The issue with nonlinear waves is that superposition does not apply, so we don't have nice general techniques to solve the equations. My dissertation was on a type of plasma wave that only exists in a model that includes nonlinearities, but I was doing experimental work so didn't spend as much time on the mathematics as a theorist would. The short-course I took on methods of nonlinear plasma physics was basically a collection of mostly unrelated techniques that have proved useful for examining various types of wave phenomena, but again there are was no universal approach like superposition that we have for linear problems.

Jason

 

[Stephen Tashi]

@Stephen Tashi I don’t appreciate your continued resistance to trying to tie this discussion into the scientific literature. It is now almost 50 posts long and we still don’t know what you are asking about.

Other posters have answered some of my questions. So "we still don't know what you are asking about" can't refer to everyone.

How is your vague concept of the velocity of the “disturbance”, which you later describe as “shape that moves”, different from the group velocity?

Let's be clear that I have not insisted that "disturbance' must have a standard definition. In fact, in my answers to my questions (post #40), I said it didn't. So I see no point in being futher criticized for having only vague inuitive ideas about what a "disturbance" is. My question was whether any vague intuitive idea of "disturbance" has been made precise by some formal definition. Apparently the answer is No. So my question has been answered.

As far as vague concepts go, my vague concept of "group" as a thing that has a velocity is a specific case of my vague concept of shape that moves .

The phrase "group velocity" has a technical definition, but I have not seen a definition of "group velocity" that defines "group" first and then defines its velocity. So my concept of "group" as a thing that moves is vague. ( I'm not saying that "group velocity" as phrase is a vague concept. )

1) a clear statement that you are discussing the group velocity

I have asked the question whether the "group" of a wave has a standard definition. I understand that it is possible to define the meaning of phrases in technical terms without defining the individual words in the phrases. It is a fair question whether "group velocity" can be defined by defining a "group" and then defining its velocity. The alternative is that "group velocity" has technical definition, but "group" does not.

This a clear and specific question: Is there a standard definition for the "group of a wave"? I'm not insisting that there must be, I'm just asking whether there is.

For example if $u(x,t)$ is a solution to a wave equation, what is the "group of $u(x,t)$"? Is it a subset of the function $u$ of the form $\{u(x,t)| 0 < t < t1\}$ which has certain properties?

3) a reference from the professional scientific literature describing exactly the velocity that you are discussing

I am discussing whether any of the velocities, group, phase, or signal have standard definitions outside the context of waves. In my own answers to my questions (post #40) I said that they do not. As of this post, nobody has explicitly contradicted my answer. Do you agree that those things are only defined with respect to waves?

To contradict my own answer using scientific literature, I'd have to find an article where a velocity is defined for a phenomena in a field that is not a wave. That would be interesting, but I don't know of such an article.