What is the meaning of constant speed of light?

[thaiqi]

TL;DR

What is the meaning here for speed?

Which one does it mean: "phase velocity" or "group velocity" or "speed of the wave front"? In the postulate of constant speed of light .

 

[Dale]

The postulate refers to the speed of light in vacuum, so the phase velocity and the group velocity are the same.

However, in modern thought the central concept is the “invariant speed”. We call it the speed of light for historical reasons (light was the first thing we discovered that moves at the invariant speed). But it is not the characteristics of light (phase or group velocity) that is important but rather the characteristics of spacetime (invariant speed) that is of interest.

 

[vanhees71]

It's the phase velocity. In the vacuum it's also the group velocity:
$$\vec{v}_\text{g}=\frac{\partial \omega}{\partial \vec{k}}=\partial_{\vec{k}} c |\vec{k}|=c \hat{k}.$$
It's also the "speed of the wave front" in simple ("Drude like") models of dielectrica.

For a very deep understanding of the latter, see the now famous papers by Sommerfeld and Brillouin (I'm sure there are English translations of those):

A. Sommerfeld, Über die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys. (Leipzig) 349 (1914) 177.
https://dx.doi.org/10.1002/andp.19143491002

L. Brillouin, Über die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys. (Leipzig) 349 (1914) 203.
https://dx.doi.org/10.1002/andp.19143491003

You also find it in Sommerfeld's lectures (vol. IV) as well as in Jackson Classical electrodynamics.

 

[thaiqi]

It's the phase velocity. In the vacuum it's also the group velocity:
$$\vec{v}_\text{g}=\frac{\partial \omega}{\partial \vec{k}}=\partial_{\vec{k}} c |\vec{k}|=c \hat{k}.$$
It's also the "speed of the wave front" in simple ("Drude like") models of dielectrica.

If so, what is the difference between constant speed of light c and that of sound (340m/s) ? Won't all moving observers for a sound source watch the same phase velocity ?

 

[PeroK]

If so, what is the difference between constant speed of light c and that of sound (340m/s) ? Won't all moving observers for a sound source watch the same phase velocity ?

The speed of light in vacuum is invariant across all inertial reference frames. That means that even if two observers are moving with respect to each other they measure the same speed for all light signals. You can see this from the relativistic velocity addition formula, which highlights that the speed of light in vacuum, $c$, is an invariant speed.

The speed of sound in air is $340m/s$ relative to the air. If you are moving relative to the air, then the speed of sound relative to you will be more or less than that.

 

[thaiqi]

The speed of sound in air is $340m/s$ relative to the air. If you are moving relative to the air, then the speed of sound relative to you will be more or less than that.

The wave front speed will be more or less. But I think the phase velocity calculated will not alter, will it?

 

[PeroK]

The wave front speed will be more or less. But I think the phase velocity calculated will not alter, will it?

It's not particularly relevant that light is (classically) an EM wave. What is relevant to this discussion is the speed of propagation through a vacuum or through a medium.

Imagine you are an observer in space, looking at the Earth. You observer a sound wave in the atmosphere. The atmosphere is moving at about $250m/s$ relative to you; and the sound wave, therefore, is moving at $250m/s \pm 340 m/s$ relative to you.

You can say that the sound is moving at $340m/s$ relative to the air. But you cannot say that the sound is moving at $340m/s$ relative to you. The speed of sound is, therefore, not invariant.

PS both phase and group velocities may be constant relative to the medium (air), but again neither can possibly be invariant. I.e. they can't possibly be the same for an observer at rest in the air and an observer moving with respect to the air.

 

[A.T.]

Won't all moving observers for a sound source watch the same phase velocity ?

No.

 

[thaiqi]

No.

Why? Won't they all watch the same geometric wave form (concentric circles) and do the same calculation?

 

[PeroK]

Why? Won't they all watch the same geometric wave form (concentric circles) and do the same calculation?

You, like a lot of people, are struggling with the concept of relative speed. Imagine a car is traveling at $30m/s$ relative to the road. Imagine you are in a car traveling in the opposite direction. Sure, you can also calculate the speed of the car relative to the road and get $30m/s$. But, the car relative to you is traveling at perhaps $60m/s$.

You have to try to imagine that the car does not have an absolute speed of $30m/s$ that everyone will measure. The speed of the car relative to you depends on your motion relative to the road.

You have the same problem with sound waves. You are failing to imagine a measurement other than relative to the air. You somehow have to picture yourself moving relative to the air and measuring a speed different from the speed relative to the air.

Every calculation you try to do, you fall back to the reference frame of the air. And fail to take into account the observer's motion relative to the air.

 

[A.T.]

Why? Won't they all watch the same geometric wave form (concentric circles) and do the same calculation?

If the center of the growing concentric circles is moving, the points on the concentric circles are moving at different speeds.

You seem to confuse radius increase per time with the actual speed of different parts of the growing circle. In a frame where the circle center is moving these two are not the same.

 

[thaiqi]

I think the speed of the sound relative to an observer is composed of three factors: its phase velocity, the speed of the medium it is in , and the speed of the observer(relative to the medium), isn't it?

 

[A.T.]

I think the speed of the sound relative to an observer is composed of three factors: its phase velocity, the speed of the medium it is in , and the speed of the observer(relative to the medium), isn't it?

For arbitrary directions you have to use vectors, then the magnitude at the end will give you the speed. For sound Galilean Transformation is sufficient:

velocity_of_sound = velocity_of_sound_wrt_medium + velocity_of_medium

It doesn't matter if you use phase velocity or group velocity, they both will vary in a frame where the medium is not at rest.

 

[thaiqi]

I think the speed of the sound wave front relative to an observer is composed of three factors: its phase velocity, the speed of the medium it is in , and the speed of the observer(relative to the medium).

 

[A.T.]

I think the speed of the sound wave front relative to an observer is composed of three factors: its phase velocity, the speed of the medium it is in , and the speed of the observer(relative to the medium).

"Composed" as shown in post #13 .

 

[vanhees71]

If so, what is the difference between constant speed of light c and that of sound (340m/s) ? Won't all moving observers for a sound source watch the same phase velocity ?

For sound you have a preferred reference frame, i.e., the medium whose vibrations are the sound waves. That's why the speed of sound is of course frame dependent. Its quoted value is usually defined in the (local) rest frame of the medium, as any intrinsic quantity charcterizing a medium in relativistic many-body physics. Everything else leads to a lot of confusion!

 

[vanhees71]

I think the speed of the sound wave front relative to an observer is composed of three factors: its phase velocity, the speed of the medium it is in , and the speed of the observer(relative to the medium).

One should note that in general phase, group, and "front velocity" are all different quantities, which have to be defined and physically interpreted with some care!

E.g., it was well known that the phase and group velocity for light in a medium close to a resonance frequency shows "anomalous dispersion" leading to faster-than-light values for both quantities. It was Sommerfeld who showed as early as 1907 in one line (arguing elegantly with the residuum theorem of complex-function theory applied to Fourier transformations), when this objection against the Special Theory of Relativity was made by Wien, that this is no contradiction to the relativistic causality structure nor concerns this quantities of the dimension of velocity which are not allowed to get larger than $c$ in relativity. This was worked out by Sommerfeld himself as well as Brillouin in the above quoted Ann. Phys. papers in 1913.

 

[Dale]

The wave front speed will be more or less. But I think the phase velocity calculated will not alter, will it?

Yes, it will alter. The phase velocity of sound is not invariant. All of those velocities are measured relative to the air.

There is one and only one invariant speed. That is c which is called the speed of light for historical reasons. There is not a different invariant speed for different waves because the invariant speed is a feature of spacetime, not a particular wave or type of wave velocity.

 

[thaiqi]

Yes, it will alter. The phase velocity of sound is not invariant.

Imagine one sound source, and an observer, all are at rest in the very beginning. The man calculates the phase velocity, that is, 340m/s. Then the observer moves at e.g. 10m/s. Surely the wave front speed measured differently, but I think the phase velocity calculated will be the same as before (because the wavelength and the period do not vary). Why do you say the phase velocity will differ?

 

[jbriggs444]

Imagine one sound source, and an observer, all are at rest in the very beginning. The man calculates the phase velocity, that is, 340m/s. Then the observer moves at e.g. 10m/s. Surely the wave front speed measured differently, but I think the phase velocity calculated will be the same as before (because the wavelength and the period do not vary). Why do you say the phase velocity will differ?

You should define the phase velocity before you attempt to measure and calculate it.

Let us find a definition.

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of anyone frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity.

Paraphrase: So if we follow the crest of a wave, the phase velocity is the velocity of that wave crest.

If we have an observer moving at 1000 m/s observing a wave whose crests are propagating at 340 m/s in the same direction then surely those crests are moving at 660 m/s in retrograde as measured by that observer.

 

[thaiqi]

Imagine one sound source, and an observer, all are at rest in the very beginning. The man calculates the phase velocity, that is, 340m/s. Then the observer moves at e.g. 10m/s. Surely the wave front speed measured differently, but I think the phase velocity calculated will be the same as before (because the wavelength and the period do not vary). Why do you say the phase velocity will differ?

Maybe I am wrong? The period will vary?

 

[Dale]

because the wavelength and the period do not vary

The period does vary. This is the Doppler shift.

 

[thaiqi]

It's the phase velocity. In the vacuum it's also the group velocity:
$$\vec{v}_\text{g}=\frac{\partial \omega}{\partial \vec{k}}=\partial_{\vec{k}} c |\vec{k}|=c \hat{k}.$$
It's also the "speed of the wave front" in simple ("Drude like") models of dielectrica.

For a very deep understanding of the latter, see the now famous papers by Sommerfeld and Brillouin (I'm sure there are English translations of those):

A. Sommerfeld, Über die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys. (Leipzig) 349 (1914) 177.
https://dx.doi.org/10.1002/andp.19143491002

L. Brillouin, Über die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys. (Leipzig) 349 (1914) 203.
https://dx.doi.org/10.1002/andp.19143491003

You also find it in Sommerfeld's lectures (vol. IV) as well as in Jackson Classical electrodynamics.

Thanks.
It is a pity that I cannot visit the two doi links given above.
I looked up Sommerfeld's book(vol.IV Optics) section 22(pp 114) titled "Phase velocity, signal velocity, Group velocity".
I looked up Jackson's book 3rd ed. section 7.8 titled "Superposition of waves in one dimension; Group velocity".

 

[PeroK]

Thanks.
It is a pity that I cannot visit the two doi links given above.
I looked up Sommerfeld's book(vol.IV Optics) section 22(pp 114) titled "Phase velocity, signal velocity, Group velocity".
I looked up Jackson's book 3rd ed. section 7.8 titled "Superposition of waves in one dimension; Group velocity".

What would be the speed of a wavefront as measured by someone moving along with it?

 

[vanhees71]

Thanks.
It is a pity that I cannot visit the two doi links given above.
I looked up Sommerfeld's book(vol.IV Optics) section 22(pp 114) titled "Phase velocity, signal velocity, Group velocity".

Yes, that's the right section.

I looked up Jackson's book 3rd ed. section 7.8 titled "Superposition of waves in one dimension; Group velocity".

Here it's particularly Sect. 7.10 ff.

 

[thaiqi]

What would be the speed of a wavefront as measured by someone moving along with it?

Sorry I don't catch your meaning of your question.

 

[thaiqi]

Yes, that's the right section.

Here it's particularly Sect. 7.10 ff.

I found in Sect 7.11 it is talked about:

 

[PeroK]

Sorry I don't catch your meaning of your question.

Your attempt to enforce an invariant phase velocity fails to an observer moving at that phase velocity. If an observer moves at the same velocity as the wave, then the wave is stationary in that reference frame. However you define phase velocity, it's still a velocity at which an observer can move.

The exception, of course, is light. You cannot move at the speed of light, which is a key point.

But, there is no problem moving at the speed of sound (phase or group, if indeed they differ).

 

[A.T.]

The period will vary?

If you mean periods measured by a receiver moving relative to the sound medium, then obviously yes.

 

[thaiqi]

Your attempt to enforce an invariant phase velocity fails to an observer moving at that phase velocity. If an observer moves at the same velocity as the wave, then the wave is stationary in that reference frame. However you define phase velocity, it's still a velocity at which an observer can move.

The exception, of course, is light. You cannot move at the speed of light, which is a key point.

But, there is no problem moving at the speed of sound (phase or group, if indeed they differ).

What is the meaning of "then the wave is stationary in that reference frame"? I think that though the wave front speed is 0 relative to the observer, he can still watch the concentric circles of the propagating wave extended in space and then he can still calculate the phase velocity.