What is Group velocity: Definition and 123 Discussions
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.
For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave group, within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group.
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?
hello everyone!
Recently,i'm reading a paper about slow light,that's really a famous work published in Nature.[Light speed reduction to 17 metrespersecond in an ultracold atomicgas].
But I'm trouble with some calculation about the velocity of slow light.here are below:
i try to use the...
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 =...
$$\tau _{01} = 10 \tau _{01}$$
If I calculate ##\frac{\tau_{p1}}{\tau_{p1}}## and set z=d=1cm I do not know how to continue from there as I can't solve the equation without knowledge of τ0 for D.
$$\frac{\tau_{p1}}{\tau_{p1}} = \frac{\tau_{02} \cdot 10}{\tau_{02}} \sqrt{\frac{1+\frac{d^2 \cdot 4...
After noting w=vk and differentiating with respect to k, and lots of simplifying, I get:
Vg = c/n +(2*pi*0.6)/(k*n)
This doesn't correspond to any numerical value though...
I'm trying to wrap my head around the dispersion relation ##\omega(k)##. I understand how you can construct a wavepacket by combining multiple traveling waves of different wavelengths. I can then calculate the phase and group velocities of this wavepacket:
\begin{align*}
v_p &=...
I just confused about it.Why can't we discribe a particle just one wave function instead of wave packet(group of waves with different phase velocities)?
Homework Statement
My question is, how do I show that speed is equal to group velocity? More information at https://imgur.com/a/m6FwNaG
Homework Equations
v_g = dw/dk
The Attempt at a Solution
Part a is substitution, part b uses v_g = dw/dk, part c is multiplication by h-bar, but I am stuck...
Homework Statement
I know that for a dispersive wave packet, the group velocity equals the phase velocity, which is given by v=w/k. But how do I calculate the group velocity of a non-dispersive wave packet? I'm supposed to be giving an example with any functional form.
Homework Equations...
In Quantum Mechanics Concepts and Applications by Zettili the following formulas are used
for phase and group velocities.
{\rm{ }}{v_{ph}} = \frac{w}{k} = \frac{{E\left( p \right)}}{{p}}{\rm{ }}\\
{\rm{ }}{v_g} = \frac{{dw}}{{dk}}{\rm{ = }}\frac{{dE\left( p \right)}}{{dp}}{\rm{ }}
In...
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?
Hi!
Dealing about wave propagation in a medium and dispersion, wavenumber k can be considered as a function of \omega (as done in Optics) or vice-versa (as maybe done more often in Quantum Mechanics). In the first case,
k (\omega) \simeq k(\omega_0) + (\omega - \omega_0) \displaystyle \left...
If general relativity in the formal sense constrains all velocities to the speed of light as a maximum, how would superluminal group velocities exceeding speeds of light (at their superpositions) be evaluated in mainstream physics? Would this be a case of General Relativity and Physics...
Hello!
Starting from a gaussian waveform propagating in a dispersive medium, is it possible to obtain an expression for the waveform at a generic time t, when the dispersion is not negligible?
I know that a generic gaussian pulse (considered as an envelope of a carrier at frequency k_c) can be...
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...
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
i get the differentiation
the halves cancel the 2 the h bar cancels the h bar square
and to get rid of the root in the denominator the entire thing is squared
But I cannot understand where the huge square root came from at the end where I circled.
Can someone help me here
In the propagation of non-monochromatic waves, the group velocity is defined as
v_g = \displaystyle \frac{d \omega}{d k}
It seems here that \omega is considered a function of k and not viceversa.
But in the presence of a signal source, like an antenna in the case of electro-magnetic wave or a...
In class I learn that we can get the dispersion relation for particles by using E=hbar*w and p=hbar*k. The calculated phase velocity is w/k = hbar*k/2m, while the group velocity is dw/dk=hbar*k/m. All these make sense to me, except one thing: I always thought that E=hbar*w=hf is only applicable...
Homework Statement
Consider a propagating wavepacket with initial length ## L_{0}##. Use the bandwidth theorem to show that the minimum range of angular frequencies present in the wavepacket is approximately:
$$ \Delta{\omega}\approx \frac{v_{g}}{L_{0}} $$
Homework Equations
Bandwidth theorem...
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...
Homework Statement
Our lecturer seemed to skip over how to get from the Group Velocity Dispersion to the actual temporal stretch of a pulse sent down an optical fibre, instead we were given just the two formula. I've been trying to work out where the temporal stretch comes from but can't work...
What is the relationship between transmission of information and group velocity of a wave packet?
I always keep hearing things like information always travels at the group velocity, it can't go faster than light etc. While I do understand (to an extent) about information not exceeding the...
On first reading, the description of ‘group velocity [vg]’ appears to be quite straightforward. However, I also found a number of speculative explanations as to ‘how’ and ‘why’ the group velocity may exceed the ‘phase velocity [vp]’. Therefore, in order to get a better intuitive understanding of...
Hi all,
I understand the concept of group velocity when applied to superimposed sine waves of the same amplitude, and even when applied to wave packets (in which case you get the well-known expression ∂ω/∂k).
My question is what happens when you add two sine waves of different amplitudes? So...
From my understanding, normal and anomalous dispersion are because the phase velocity is a function of k so it is different for different components of a group so the group will spread out over time.
So what's group velocity dispersion? Is it the same affect (dispersion/ spreading out)...
Homework Statement
Prove that the group velocity of a wave packet is equal to the particle’s velocity
for a relativistic free particle.
Homework Equations
vgroup = Δω/Δk = dω/dk
E = (h/2π)*ω = √(p2c2 + m2c4)
The Attempt at a Solution
I'll be honest..I have no idea where to...
You people know that group velocity of a wave packet is calculated with the formula v_g=\frac{d \omega}{d k} .But this gives an expression which,in general,is a function of k.My problem is,I can't think of an interpretation for it.What is that wave-number appearing in the expression for group...
In transitions in the crystals we always use conservation of wave vector of electron, not electron momentum conservation. For example in an indirect transition from top of valence band to bottom of conduction band, the group velocity of electron and hence its momentum would not change (it is...
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 =...
Homework Statement
The dielectric constant k of a gas is related to its index of refraction by the relation k = n^{2}.
a. Show that the group velocity for waves traveling in the gas may be expressed in terms of the dielectric constant by
\frac{c}{\sqrt{k}}(1 -...
I was reading the derivation on Wikipedia:
http://en.wikipedia.org/wiki/Group_velocity#Derivation
Why is the first part before the integral sign ignored when calculating the velocity? Surely it would also cause a phase shift in some time interval and make the waves move forward (or backward)?
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
Show that the group velocity
vg=dω/dk
can be written as
vg=v-λ*dv/dλ
where v = phase velocity
Homework Equations
n=n(k)=c/v
k=2∏/λ
ω=2∏f=kv
fλ=c
The Attempt at a Solution
dω/dk = d(kv)/dk= v+k(dv/dk)= v+ck(d(n^-1)/dk) =v-(ck/n^2)(dn/dk)...
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
Let's consider a single frequency signal of frequency say 'f'. If the wave is propagating through a medium (EM wave with a velocity of 'c') then what will be the phase and group velocity? I believe that we can't find out the phase velocity and that the group velocity should be equal to the...
ello everybody,
how can I calculate the group velocity of a wave package in an infinite square well?
I know only how it can be calculated with a free particle, the derivation of the dispersion relation at the expectation value of the moment.
But in the well, there are only discrete...
An infinitely long "mass-spring transmission line", consisting of masses (m) connected by springs (spring constant s) obeys the following dispersion relation:
ω = \sqrt{4s/m} sin(kd/2).
The group velocity is
dω/dk = d/2 \sqrt{4s/m} cos(kd/2).
What does zero group velocity "mean" for...
everyone knows, there exists the relation between the group velocity and energy dispersion.
a question is how to expression the relation between the velocity and momentum k?
it seems that the Dirac electron in graphene is massless.
I have difficulty understanding the exact concept of group velocity. Consider a wave packet as a linear combination of a number of eigenstates of a 1-D particle in box. The dispersion curve(\omegaversus k) is composed of discrete points located on a parabola. Well, for each point one can...