What is Phase velocity: Definition and 82 Discussions
The phase velocity of a wave is the rate at which the wave propagates in some medium. This is the velocity at which the phase of any one 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. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as
v
p
=
λ
T
.
{\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}
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) vp,
v
p
=
ω
k
.
{\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}.}
To understand where this equation comes from, consider a basic cosine 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 would have to propagate faster when higher frequency oscillations are distributed less densely in space unless the wave length is compensatorily shortened. Formally, Φ = kx − ωt is the phase, where
∂
x
∂
t
=
−
∂
Φ
/
∂
t
∂
Φ
/
∂
x
.
{\displaystyle {\frac {\partial x}{\partial t}}=-{\frac {\partial \Phi /\partial t}{\partial \Phi /\partial x}}.}
Since ω = −dΦ/dt and k = +dΦ/dx, the wave velocity is v = dx/dt = ω/k.
The current explanation in Wikipedia for the slowing of the light wave phase velocity in a transparent medium considers:
From another discussion I learned that the waves radiated by the "shaken" charges are real waves (not virtual), so they would also leave the medium, as the original wave. In...
What exactly is a signal in wave physics? Is any wave considered a signal? Like, consider a superposition of harmonic plane waves, is the signals it carries considered the envelope(that travels at the group velocity) or the individual rippes that travel at a the phase velocity?
The relationships for matter waves are (see e.g. https://en.wikipedia.org/wiki/Matter_wave):
λ = h / p and E = h f, where E = m c2
From this the phase velocity can be derived and we get vph = c2 / v.
v is the group velocity, which is also the velocity of the particle.
If I consider these...
Hi,
I saw that the group velocity for an electromagnetic wave can be calculate with the following formula
##v_g = v_p + k \frac{d v_p}{dk}##
Thus, since ##v_p = \frac{c}{n} = \frac{\omega}{k}##
Is it correct to say that ##v_g = \frac{c}{n} + k(- \frac{\omega}{k^2})## where ##k =...
Refractive index is a function of velocity in the medium. How is this related to deviation angle inside the medium? I am not asking for the known formula, but for the mechanism behind it.
Hello,
Regarding the wave oblique angle propagation and based on Balanis "Advanced engineering Electromagnetic" book on page 136 ( it has been attached) I need to know why the phase velocity in x direction is not important to keep in step with a constant phase plane( Just equation 4-23).
I...
Hey everyone,
I'm trying to measure the stray capacitance in a circuit comprised of an rf signal generator, an oscilloscope, a coaxial cable (short-circuited) and some capacitors. I measured the resonant frequencies of the coaxil cable for varying values of capacity (0pF to 850pF) between 4MHz...
I am studying phase and group velocity in non-dispersive and dispersive media. My question is the following: Is there any reason why a dispersive medium simply cannot be modeled as a type of field?
Homework Statement
An electromagnetic wave with the frequency f = 1.4 GHz is propagating in the z-direction from vacuum in a plasma with the magnetic field B_0 = 0.1T z^. The plasma density is 1.0*10^17 particles/m^3. The wave is the separated into a R- and L-wave in the plasma.
Homework...
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?
Given a massive particle traveling freely. Also assume that it is in a momentum eigenstate - a pure unmodulated sine wave in position space. In non-relativistic quantum mechanics, the phase velocity for that particle would be greater than the velocity of light.
Does the phase velocity for...
Homework Statement
Find the group velocity for a shallow water wave: ##\nu = \sqrt{\frac{2\pi\gamma}{\rho\lambda^3}}##
Homework Equations
Phase velocity: ##v_p = \nu\lambda##
group velocity: ##v_g = \frac{d\omega}{dk}##
##k=\frac{2\pi}{\lambda}##
##\omega = 2\pi \nu##The Attempt at a Solution...
Hi all,
I have a difficulty to understand the concept of phase velocity of a wave packet. I read some materials online and my text, I didn't find a very straightforward explanation on this concept except the math definition. I also read some threads from this forum, but still have some...
Could you please explain the derivation of
group velocity = dw/dk
I read ut here https://en.m.wikipedia.org/wiki/Group_velocity
Is it approximation, if so under what circumstances
Hi! I am having problem in understanding the difference between phase and group velocity clearly. In my textbook phase velocity is given by ω/κ while group velocity is by dω/dκ. What is the difference between these two terms?
Thank you!
I just read some words from the book Optical Waveguide Theory by Snyder, Allan W., p226. It says that
"However, although it is possible for the phase velocity of a mode to exceed the maximum speed of light in the cladding (of waveguide), c/n_cl, this cannot occur without losing power to...
As I'm seeing things, there are these choices for a definition of relativistic phase velocity u;
a) u = (mc^2)/ p
b) u = E / p
Now I like choice a) because it leads to a correct-looking index of refraction via n = c / u. It leads to a correct-looking solution to the ray equation. However...
If i am given the de broglie wavelength
Of any particle then its sure that i can find the velocity of tat particle if its mass is given.
λ= h/mv
But the velocity which i found is the group velocity or phase velocity?If it's not the group velocity
How can i find it?
Homework Statement
how to visualize group velocity and phase velocity?
I tried to to visualize it
The velocity of up and down vibration
Of the wave as phase velocity.
And
The velocity of the whole wave in propagating direction as group velocity.
AM I CORRECT OR WRONG?
Homework EquationsThe...
Hi All,
I've heard it said that the superluminal phase velocity of the KG eqn is not a problem for relativistic causality because signals travel at the packet/group velocity, which is the inverse of the phase velocity (c being 1). I'm a bit skeptical of this.
We can strip away all the quantum...
Homework Statement
a) Starting from Equation 5-1, show that the group velocity can also be expressed as: vg = vp - λ(dvp/dλ)
b) The phase velocity of each wavelength of white light moving through ordinary glass is a function of the wavelength; that is, glass is a dispersive medium. What is...
Homework Statement
This is Problem 7.6 from Electronic Properties of Engineering Materials by Livingston.
"Over a wide range of frequencies, the dielectric constant of a polymer is found to be proportional to the inverse square root of frequency. (a) How does the phase velocity of EM-waves...
The mean velocity of a wavepacket given by the general wavefunction:
\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int dk A(k)e^{i(k x - \omega(k) t)},
can be expressed in two ways.
First, we have that it's the time derivative of the mean position (i.e., its mean group velocity):
\frac{d \langle...
I am currently taking an undergrads course in atomic physics, and I have just grazed the concepts of matter wave functions, group waves, group velocity, and phase velocity. For the latter, it has been stated that phase velocity can exceed the speed of light. I don't fathom this. Could anyone...
Homework Statement
Say you have a small boat moving through water, and creating a short wave-group which is a superposition of waves in the range of 0.2m-2m. If the shore 50meters away, how long will it take the fastest of the wave-components to reach shore? {assume the depth is constantly very...
I have no idea how to do this or where to start. Can someone please help me?
Problem 4.4- Suppose n o and n e are given. In (a) you only need to find the magnitude of the group velocity. Problem #2 in HW 10 may be helpful. You can also directly use the definition of group velocity, i.e., v g =...
Hi, I've been reading a lot today about waves and how they have both a phase velocity and a group velocity. The group velocity makes perfect sense to me, I would phrase it as the speed at which the wave is propagating. The phase velocity on the other hand is quite confusing to me. What exactly...
Why must the charged particle that leads to Cherenkov radiation travel faster than the phase velocity of light not the group velocity of light?
One of the sides of the triangle that is used to define cosθ is v=c/n i.e. the phase velocity. I don't see why it's one rather than the other.
Thanks!
Hi. Today I sat my final first year Modern Physics exam. It went very well, however I got stuck in one question. It asked (i) to prove the following relation for the matter wave \omega^{2}=k^{2}c^{2}+m^{2}c^{4}/\hbar^{2} and (ii) to obtain the group velocity and phase velocity of a matter wave...
Homework Statement
What is the phase velocity of light in glass if n=1.5 at λ=500nm
Homework Equations
n=c/vp =λ0/λ
The Attempt at a Solution
1.5 = (3x10^8) / vp
vp = (3x10^8)/1.5
I'm not entirely sure if the equation I have used is the correct one. It does not use the...
Homework Statement
(i) What is the phase velocity of EM radiation in free space?
(ii) What is its group velocity?
Homework Equations
The Attempt at a Solution
Using the definition as per this page:
http://physicsdaily.com/physics/Phase_velocity
(i) Vp = (C^2)/Vg
(ii)...
The phase velocity of ocean waves is (gλ/2∏)1/2,where g is the acceleration of gravity.Find the group velocity of ocean waves.
Relevant equations: λ=h/γmv phase velocity= c2/v(velocity of particle) group velocity=v (velocity of particle).
thnxx in advance
So for the phase velocity of a massive particle we have
Vph = Vg/2 for non-relativistic case
Vph = c^2/Vg for the relativistic case
Vg is the group velocity or particle velocity
But there seems to be a contradiction in that for the non-relativistic case the phase velocity is...
According to wiki-
For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases. Then have shown this diagram
http://en.wikipedia.org/wiki/File:Wave_group.gif
How is particle velocity of a wave related with the phase velocity of the...
According to the wiki page, this diagram:
shows the phase velocity and group velocity moving in opposite directions. I think I understand what group velocity is, its the velocity of the overall wavepacket, in the diagram I can see a Gaussian wavepacket moving to the right so I'm guessing that's...
Homework Statement
show that the group velocity and phase velocity are related by
d vgroup = vphase - \lambda ( (d vphase) / (d \lambda) )
Homework Equations
vphase = \lambdaf
vgroup = d\omega/dkThe Attempt at a Solution
dw/dk = \lambdaf - \lambda(d\lambdaf/d\lambda)
dw/dk = \lambdaf -...
I know the physical meaning of phase as well as the group velocity.
I want to know the mathematical relationship between the phase velocity and group velocity.
Thanks a lot.
Homework Statement
A wave packet in a dispersive medium is given as :
y(t) =cos(x-5t)cos(.2x-.4t)cos(.1x-.2t)
Find group velocity and phase velocity for the wave packet. Hence plot w-k variation for the calculated values.
2. The attempt at a solution
We know that for the wave, Vg = dw/dk and...
In quantum mechanics, what is the difference between velocity and phase velocity of matter wave? How can it also be that phase velocity of matter wave always exceeds the speed of light?
Hi,
I am reading about phase velocity vs group velocity and while I get most of it, there is one thing I don't see. How can the phase velocity ever exceed c? At least in a vacuum, I think it never happens. Is that right? And if it does exceed c in a medium, what is the general physics behind...
Hello!
My book here states that for a medium where the index of refraction n increases with increasing frequency (or wavenumber), "the group velocity is less than the phase velocity". This is stated for a wave which is the sum of two waves with equal amplitude and differing frequency...
de Broglie relations say that E = hf and p = h/λ which implies E/p = λf = ω/κ = vp, or p = E/vp. It seems to me like the momentum of a particle described by a matter wave should relate to the group velocity not the phase velocity, because the group velocity is generally the actual velocity of...
Calculating wavelength in a media
Hi
Trying to get how to calculate the wavelength of an electromagnetic wave in a media and I am getting confused about the permittivity.
I know the formula:
vphase = c/(sqrt(εr*μr))
where vphase is the phase velocity.
c is the speed of light in a...