Phase velocity and wave velocity

[Pushoam]

Numerically, speed of wave propagation(defined as wave velocity) = ω/k = phase velocity
But, conceptually is there any difference between phase velocity and wave velocity?

 

[sophiecentaur]

Did you try Google? This wiki link tells you what you need to know. It was the first hit on my Google search. Do you have a further question, based on what you can find out on wiki?

 

[Pushoam]

Equivalently, in terms of the wave's angular frequency ω, which specifies angular change per unit of time, and wavenumber (or angular wave number) k, which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) ν_p,

To understand where this equation comes from, consider a basic sine wave, A cos (kx−ωt). After time t, the source has produced ωt/2π = ft oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance x to fit the same number of oscillations, kx = ωt.

Thus the propagation velocity v is v = x/t = ω/k. The wave propagates faster when higher frequency oscillations are distributed less densely in space.[2] Formally, Φ = kx−ωt is the phase. Since ω = −dΦ/dt and k = +dΦ/dx, the wave velocity is v = dx/dt = ω/k.



What Wikipedia is saying is:

and Wikipedia derives v = dx/dt = ω/k

so, what I can conclude from it is :
Numerically, speed of wave propagation(defined as wave velocity) = ω/k = phase velocity
which I had already said.
What I want to know is
Are wave velocity and phase velocity two different names for the same quantity i.e.ω/k?
Do wave velocity and phase velocity have the same meaning or they have different meanings,but their magnitudes are equal?

 

[nasu]

A wave can be described by several velocities (phase velocity, group velocity. signal velocity) . They are all "wave velocities" as they describe something about the wave. It's like asking is the difference between the "velocity of a car" and the "average velocity". You can talk about average velocity, instantaneous velocity bit they are both "car velocities".

 

[sophiecentaur]

That wiki article defines all three velocities. It really is worth reading as it talks sense. The phase velocity (ω/k) can exceed c but there is no 'relativity' problem because it doesn't mean that any information is carried faster than c. Group velocity (dω/dk) is the speed at which any information is carried by the wave (due to modulating the em wave) group velocity is always less than c.

 

[nasu]

Usually the group velocity is the same as signal velocity, which is the speed of the carried information.
However they are not the same and there are some (not very common) cases when they are different.
Interesting that this issue was discussed by Brillouin in the 1940's, and if I understand correctly, around that time they realized that even the group velocity may exceed c in some cases but the signal velocity does not.

The book is "Waves in periodic Structures", by Leon Brillouin.

 

[Pushoam]

Thank you ,all.

 

[Khashishi]

Group velocity can also be greater than c. But signal velocity is always less than c.

 

[nasu]

Did I say otherwise?

 

[Khashishi]

Sorry, I was replying to sophiecentaur and didn't realize that you said it already.

 

[davenn]

Sorry, I was replying to sophiecentaur and didn't realize that you said it already.

that's why it is always good to respond with quotes, so that there is no misunderstanding

 

[Pushoam]

conceptually is there any difference between phase velocity and wave velocity?

No, there is no difference.
For my sake,
University Physics, 13_th
The wave speed is the speed with which we have to move along with the wave
to keep alongside a point of a given phase, such as a particular crest of a wave on
a string. For a wave traveling in the that means kx -ωt constant.
Taking the derivative with respect to t, we find dx/dt = ω/k
Eq. (15.6): v = ω/k
Comparing this with Eq. (15.6), we see that dx/dt is equal to the speed v of the
wave. Because of this relationship, is sometimes called the phase velocity of
the wave. (Phase speed would be a better term.)

 

[nasu]

But if you have a short pulse in a dispersive medium, the pulse may not propagate with this speed. And you may not be able to follow a point of "given phase" as the pulse gets distorted./ This is the reason there is more than one "wave speed".

As long as you have an infinite (continuous wave) in a non-dispersive medium all the wave velocities are the same but this is not the most general case.

 

[Pushoam]

o.k.
Now what I have understood till now is
There are different kinds of wave velocities : group velocity , phase velocity, etc.
In case of infinite sinusoidal wave, group velocity is equal to phase velocity and so we identify phase velocity as wave velocity.
Is this o.k.?

 

[nasu]

I don't really understand what is the need to identify the two concepts.

It's like apple and fruit. Apple is a fruit. But there is more than one type of fruit.
If you have a special situation where there are only apples in the store, if you buy fruit you obviously buy apple. So id does not matter if you ask for fruit or for apples, you get the same thing. This does not mean that "fruit" should be identified as "apple". Of course, this has nothing to do with physics anyway.

 

[Pushoam]

Thank you , Nasu.
Now I got it.