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

[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 definition 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.

 

[vector222]

You say "with a proper schedule" the flash light turn on in sequence. Turning on the flashlights quickly in sequence should give the illusion of something happening faster than light speed

To do so would require a way to switch on the flashlights in sequence and this would require a timing device.

If we have 2 flashlights 2 light years apart and we switch them on at the same time by a series connected circuit the bulbs should light at the same time. Does this prove that something can happen in no time?. There is only one event because of the series connected circuit. If we have a line of billions of flashlight bulbs , 2 light years apart, (each bulb one wavelength apart)connected to a timing device then we might expect it would take a minimum of 2 years to switch on each bulb in sequence because the timing device speed is limited to C

 

[Stephen Tashi]

Does this prove that something can happen in no time?.

That isn't the question asked in the original post. The technical question is whether the function that specifices the field caused by line of flashlights is a wave function that has a component which propagates along the line of flashlights.

 

[vector222]

I agree

i was comparing 2 flashlights to billions of flashlights suggesting the flashlights need to be one wavelength apart to be fair. If the lights are not that close together then nothing can be proved. Even with a timing device, 3 lights, 2 light years apart proves nothing as far as a real light speed propagation is concerned.

Im talking about synchronous timing device. You could also switch the lights as a real propagation like a domino effect but I thinks no matter how its done, the information transfer is limited by the speed of light.

Relating that to a wave function should be doable i have no idea how to do it.

 

[sophiecentaur]

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

'Throughout all time' implies a very very long propagation time. This is why the quantity Group Delay is so useful and I still don't see why you are not happy with it; it's well defined and of great practical use. Transmitting and Receiving Information is tied in with signal to noise ratio. The amount of information you want to transmit will depend on the time and the channel noise. (I'm not talking about bit rate for a simple digital signal because that performs well below the limit imposed by Shannon). Your "disturbance" is a very vague term and should include the "throughout all time" qualification, as you say. In your head, you can see a wiggle on a time graph of the received sound / radio signal etc but how much information does your wiggle convey? When is there enough? Your 'disturbance' never ends. It's much too easy to think of this in terms of a sine wave (phase velocity) and ignore the fact that information is only transmitted when the sine wave is modulated in some way, to mark a 'point in time and space'. So here comes Group Delay.

The point of using Group Delay in communications is that the group delay can be specified over the bandwidth of interest (it will always vary across the band of interest when channel filters or transducers are used,) It will give you a proper quantified answer to your original question because it will specifically deal with the truncation of the bandwidth in the receiver and, in the end, will help tp tell you what you need to know about the actual error in your signal. Anything else tends just to be arm waving.

 

[sophiecentaur]

but I have not seen a definition of "group velocity" that defines "group" first and then defines its velocity

You almost certainly have - it's just that you did not recognise what was written. Group Velocity is not a single value for a particular channel. The definition doesn't start with word "group"; it uses the word Group to describe a significant part of the signal that arrives at the other end. I think you are mixing the order of the definition and the term that's used.

 

[Dale]

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.

And thus we will close the discussion. We cannot discuss concepts that are deliberately outside standard science. I am saddened that you have chosen to respond thus.

Is there a standard definition for the "group of a wave"?

For a narrow band signal, yes: $$\int _{-\infty}^{\infty} A(k) e^{i (k-k_0)(x-\omega’_0 t)}dk $$