Group velocity Definition and 123 Threads

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...

I have a simple question but I'm putting down the whole derivation as it is relevant. There is a point that I don't understand, or seems wrong to me. This is a derivation of Group Velocity followed by simplifying(approximating it) for long wavelength waves in shallow water. This appears in a...

In my notes on waves (specifically water waves) there is a derivation of Group Velocity. They consider two waveforms with the same amplitude, that differ slightly in wavelength and frequency, which are then superimposed to give wave groups. kis wavenumber, \delta k is how much the wavenumbers...

Is the relation v_{\varphi }v_{g}=v^{2}=\frac{1}{\mu \varepsilon } always true in a plasma ? Where v_{\varphi }, v_{g} are respectively the phase and group velocity of the electromagnetic wave that is propagating in the plasma.

While studying the brillouin zone I came across the dispersion relation and the group velocity. The group velocity is given by v=dω/dκ, I understand this in the sense of beats where it is Δω/Δκ and I understand that the group velocity is the propagation speed of the envelope function. However...

Can the group refractive index ng be 1>ng>0 ?

Does group velocity effect long linear waves generated by a paddle generating waves in deep water? I have developed a numerical wave tank in CFD at full scale, using a bottom hinged flap paddle that oscillates to produce regular waves, the domain is roughly three wavelengths long, and a beach...

I am working on an assignment here; A linear chain with a two-atom primitive basis, both atoms of the same mass but different nearest neighbor separation and thus different force constants. I have made a rigorous calculation in order to find the dispersion relation ω(k), with extensive...

There are various methods to slow and even stop visible light. Most of these methods appear to be slowing the group velocity of light, not the phase velocity. Can you slow the velocity of propagation of electricity? I have seen estimates that electricity, in a wire, propagates at 2/3 the...

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...

Infrared radiation, of wavelength λair = 1um in air, travels through a dispersive medium with refractive index n = 1.4505 and with dn/dλair = -0.01 per um at this wavelength. Calculate the speed at which the radiation carries information. So know that c/n = λf radiation carries...

Homework Statement see: http://www.flickr.com/photos/55153239@N03/6371848305/ Homework Equations vp = w/k vg = dw/dk The Attempt at a Solution I know the relationship of: vg = vp(1+lambda/n * (dn/dlambda)) but i don't know where to go from here..

Homework Statement In a lab experiment about Gravity Waves and Dispersion, one of the preliminary questions is: Show that for gravity waves the group velocity is: C_{g} = \frac{C_{p}}{2}\sqrt{1 + \frac{2kh}{sinh(2kh)}} Homework Equations C_{g} = dω/dk ω = \sqrt{gktanh(kh)}...

Hello there PF readers, The group velocity for example of electron waves is given by the derivative of the dispersion relation: \frac{dE}{dp}=v (this is for free electrons) ^{1}. Now the Heisenberg's uncertainty principle has two forms, one for position and momentum and the other for energy...

Homework Statement A wave group is generated as a superposition of harmonic waves of average wavelength lambda. Show that V_{g}=V_{p}-\lambda \frac{d V_{p}}{d\lambda } The Attempt at a Solution All I know is that Vg = dw/dk and Vf = w/k. I am not sure what I need to do.

Homework Statement Surface waves generated by a mid-Atlantic storm arrive at the British coast with period 15 seconds. A day later the period of the waves has dropped to 12.5 seconds. Roughly how far away did the storm occur? Homework Equations Group velocity c = \frac{\omega}{2k} \omega =...

I am doing my extended essay on the maths involved in GPS, and I am trying to understand the concept of the group velocity of a wave packet, http://www.mathpages.com/home/kmath210/kmath210.htm" they explain it more or less, but I still don't get how do they differenciate dw/dk to get...

Homework Statement a wave function \Psi=Aei(px-Et)/h where p is momentum and E=p2/2m calc group and wave velocityHomework Equations group vel=dw/dk wave v=w/kThe Attempt at a Solution I just have no idea what they did here to solve it. i(px-Et)/h=i(kx-wt) than somehow got k=p/h than...

Homework Statement so I am asked to derive the group velocity of an equation in terms of its frequency and wavelength, but I am stuck at a simple derivative, which is df/d(1/n) where is the frequency and n is the wavelenght. Homework Equations the group velocity is dw/dk The...

I know the phase vs group question has been asked before, but I really don't quite understand it. I've read in various places that the phase velocity of a light wave has been made to go faster than C, and in others that the group velocity has been made to go faster than C, and even made negative...

Superluminal Group Velocity? An electron has a De Broglie wavelength L of 2*10^(-12) m. Calculate its kinetic energy, and phase and group velocities. So I used E = p^2 / 2m, p = h / L. Phase velocity is w / k and group velocity is dw/dk. This gave me expressions E/p and p/m...

given: E(k) = 2.5 x 10-28 k2 Joules k = 5.8 x 109 m-1 Find the group velocity. i know that the equation for vg = 2pi/h (dE/dk) dE/dk = 2(2.5 x 10-28) = 5.0 x 10-28 k plugging in k, i get dE/dk = 2.9 x 10-18 J/m vg = 2.75 x 1016 m-1 s-1 what i am more concerned about here is the...

understanding for group velocity and phase velocity?? Hi, everyone. I learned physics for 3 years, but up to now I still cannot understand clearly the difference between the group & phase velocity. What is exatly "phase"? and what is exactly "group". From book I read the Vph corresponds to the...

Homework Statement Ok hello, The problem is there is a system in the state characterized by \Psi(x,t) = \int dk e^(ikx)e^(-iwt)f(k), which is the time dependent representation of a wave packet, where individual waves travel with velocity vp = w/k and f(k) =...

hi, I am trying to derive the group velocity equation (actually its inverse): \frac{1}{v_g} = \frac{1}{c}(n(\lambda) - \lambda\frac{dn}{d\lambda}) where n is the refractive index and depends on the light's wavelength. I have started by saying that if v_g = \frac{\partial\omega}{\partial...

Homework Statement In an electron wave, show that the group velocity = particle velocity of the electron Homework Equations E = p^{2}c^{2} + m^{2}c^{4} The Attempt at a Solution V_{g} = \frac{d\omega}{dk} = \frac{dE}{dp} = \frac{d}{dp} \sqrt{(p^{2}c^{2} + m^{2}c^{4})} =\frac{1}{2}...

Hey ppl, I have a question for you. For the case of light propagation in the air the the group-velocity dispersion parameter D is infinity because the derivative of the group velocity should be a zero (group velocity is constant ; w=ck and Vgroup=dw/dk). Who can explain me this? dispersion...

"Group velocity (equal to an electron's speed) should not be confused with phase velocity (equal to the product of the electron's frequency multiplied by its wavelength)." Above statement is from wikipedia, and I still don't quite understand it. As one of the postulate of QM, group velocity...

Hi The group velocity of an electron wavepacket in a homogeneous lattice is vgroup(k) = ∇kEk, where Ek is the dispersion. I have just read an article, where they use this to find the group velocity of a wavepacket in an inhomogeneous lattice, but they use the homogeneous dispersion. I...

Hi there So I was looking into group velocity and related matters and found myself quite confused. So now I have a few questions which I feel I need to understand (primarily the first one). Any help with these would be awesome and I would be very grateful... 1) Why is the group...

Homework Statement Ok so I'm confused by an apparent inconsistency in my notes.. On the one hand, we are told that in a non-dispersive medium, all frequencies propagate at the same speed, hence phase velocity of individual waves in superposition = group velocity.. so the idea of group...

Homework Statement Ok so we have different expressions for group velocity, g... The most obvious being g = dw/dk But then the text I am using says sinve w=vk we can write g = v+k dv/dk alongside other expressions, including k etc.. My question is - what is v in the equation...

*for a group of waves, the energy is transmitted with the group velocity. what does that mean? *what is meant, as in actually meant, by dispersion of a wave in a medium? whatever i looked up says phase velocity depends on frequency for a dispersive medium and vice versa. Why? * wave...

Homework Statement Okay so I've been deriving various expressions for the group velocity starting with g=dw/dk then g=v+kdv/dk also g=v-lambda dv/dlambda (*) then finally g=c/u(1+lambda/u (du/dlambda)) Ok so finally I am asked to use equn (*) to derive the following expression...

Homework Statement The phase velocity of ocean waves is \sqrt{\frac{g\lambda}{2\pi}} , where g is 9.8 m/s2. What is the group velocity?2. Homework Equations v_{p} = \omega/k v_{g} = \partial\omega/\partial kThe Attempt at a SolutionSo I've been trying to find omega and k from the equation in...

Homework Statement The motion of short wavelength (about 1 cm or less) ripples on water is controlled by the surface tension S. The phase velocity of such ripples is given by; V_{p}^{2}=2\pi S/\rho \lambda where ρ is the water density, and λ is the wavelength. (a) Which of the formulae is...

How can we show that the group velocity is equal to the particle velocity?

Homework Statement For a particular Dieletric it is observed that over a range of frequencies, the group velocity varies exponetinally with wave number: v_{g}=ae^{bk} , where a,b are constants. *(PLease not that the superscript g on v on the LHS side is meant to be subscript, however it...

Homework Statement Electromagnetic wave, with the frequency \omega, travels through the medium for which: \epsilon=\epsilon_0\left(1+\frac{\omega_p^2}{\omega_0^2-\omega^2}\right) where \omega_p and \omega_0 are constants. How does the expression for the group velocity v_g(\omega) looks...

Homework Statement Starting from: vg=vp+k(dvp/dk) show that the group velocity can also be expressed by vg=vp-λ(dvp/dλ). Homework Equations I'm told that vg=vp+k(dvp/dk) with vg=group velocity and vp=phase velocity. I also know that k=(2π)/λ and ω=2πf. The Attempt at a Solution I...

Question on "Two theorems for the group velocity in dispersive media" In the paper (1), the authors show that the group velocity can be faster than light, slower than light, infinite, or even negative without contradicting the causality principle. I have no doubt about the validity of this...

It is well known that phase velocities can be FTL (faster than light). It is often said that group velocities cannot be FTL. But I doubt this last one. Starting from the Lienart-Wiechert delayed potential, what can be said for sure? For sure a response to a step signal cannot be FTL. But...

Hi, It is long time to come here sine i graduate University. Anyway, My question is whether exact form of Group velocity is \frac{dw}{dk} or \frac{dw}{dk}|_{\bar{k}} I want to know whether Group velocity is independent of K, propagation number Becasue Group velocity is 'proper'...

Homework Statement A particle in classical mechanics with a velocity v becomes a wave packet with a group velocity v_g in quantum mechanics. I have to show that \vec{v} = \vec{v}_g Homework Equations \vec{v}_g = \frac{\partial \omega(\vec{k})}{\partial \vec{k}} \omega(\vec{k})...

I know the definition of group and phase velocity but why? Why they are different and what's the physics behind it to define such velocities?

Homework Statement Find the group velocity of the wave f(x) = cos(b\sqrt{\omega}x - \omega t) 2. The attempt at a solution What? I thought group velocity was an interference phenomenon...The velocity of the beats produced when two waves with slightly different frequencies interfere...

Hi. We already know Group velocity is \frac{\partial w }{\partial k } But, liboff said Group velocity is \frac{\partial w }{\partial k }|_{k_{max}} where k_{max} is wave number which probablity this value can be measured is higher than other wave number. is it right? How can?

What is the relationship wavelength and group velocity in wave packet ? If a wave packet has big wavelength its group velocity bigger or not ?

I have a little trouble in demonstrating that the group velocity of an e.m. wave transform (in special relativity, by passing from an inertial frame to another) like the velocity of a particle with mass. So, \left( \frac{w}{c}, k_x, k_y, k_z\right) = \left( \frac{w}{c}, \vec k \ \right) is a...