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I am working thru a derivation of the group velocity formula and I get to this stage: $$y=2A\cos(x\frac{\Delta K}{2} -t\frac{\Delta \omega}{2})\sin( \bar k x-\bar \omega t)$$ Then all the derivations I have seen say that $\frac{\Delta \omega}{\Delta K} $ is the group velocity. I know mathematically why this is a velocity but what I don't get is why do we know that this is the group velocity rather then the phase velocity and that $\frac{\bar \omega}{\bar k}$ is the phase velocity and not the group velocity?

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Consider a wave

$$A = \int_{-\infty}^{\infty} a(k) e^{i(kx-\omega t)} \ dk,$$

where $a(k)$ is the amplitude of the kth wavenumber, and $\omega=\omega(k)$ is the frequency, related to $k$ via a dispersion relation. Note, if we wanted to track a wave, with wavenumber $k$, with constant phase, we would see that this occurs when $kx=\omega t$, i.e. $x/t = \omega/k = c$, with $c$ the $\textbf{phase}$ velocity.

We would like to know the speed at which the envelope $|A|$ is traveling.

For $\textbf{narrow banded}$ waves, the angular frequency $\omega$ can be approximated via the taylor expansion around a central wavenumber $k_o$, i.e.

$$\omega(k) = \omega(k_o) + \frac{\partial \omega}{\partial k} (k-k_o) + \mathcal{O}((k-k_o)^2),$$

where the scale of the bandwidth is quantified by the small parameter $(k-k_o)$. Therefore, we can rewrite $A$ as

$$A \approx e^{-i(\omega(k_o)t-k_o\frac{\partial \omega}{\partial k}t)} \int_{-\infty}^{\infty} a(k) e^{ik(x-\frac{\partial \omega}{\partial k} t)} \ dk.$$

Therefore

$$|A| = \int_{-\infty}^{\infty} a(k) e^{ik(x-\frac{\partial \omega}{\partial k} t)} \ dk,$$

which says that the envelope, $|A|$, travels at speed $\frac{\partial \omega}{\partial k}$, i.e. $$|A(x,t)| = |A(x-c_g t,0)|,$$ where we have defined $$c_g \equiv \frac{\partial \omega}{\partial k}.$$

The group velocity has dynamical significance, as it is the velocity at which the energy travels.

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Definitions

Before we begin, we should define some terms and parameters/functions that will be used later:

Wave Number: $\equiv$ effectively the number of wave crests (i.e., anti-node of local maximum) per unit length $\leftrightharpoons$ ``density'' of waves $\rightarrow$ $\boldsymbol{\kappa}$ $=$ $\boldsymbol{\kappa}\left(\omega,\textbf{x},t\right)$ in general

Wave Frequency: $\equiv$ effectively the number of wave crests crossing position $\mathbf{x}$ per unit time $\leftrightharpoons$ ``flux'' of waves $\rightarrow$ $\omega$ $=$ $\omega\left(\boldsymbol{\kappa},\textbf{x},t\right)$ in general

Wave Phase: $\equiv$ position on a wave cycle between a crest and a trough (i.e., anti-node of local minimum) $\rightarrow$ $\phi$ $=$ $\phi\left(\textbf{x},t\right)$ in general

Phase and Continuity

Then, we can define an elementary solution to periodic wave equations as: $$ \psi\left( \mathbf{x}, t \right) = \mathcal{A} \ e^{ {\displaystyle i\left( \boldsymbol{\kappa} \cdot \mathbf{x} - \omega t \right) } } $$ where $\mathcal{A}$ is the wave amplitude and, in general, can be a function of $\boldsymbol{\kappa}$ and/or $\omega$, but we will assume constant for now. Let us assume that a dispersion relation, $\omega$ $=$ $\mathcal{W}\left( \boldsymbol{\kappa}, \textbf{x}, t \right)$, exists and may be solved for positive real roots. In general, there will be multiple solutions to the dispersion relation, where each solution is referred to as different modes. The term in the exponent is known as the wave phase, given by: $$ \phi\left( \mathbf{x}, t \right) = \boldsymbol{\kappa}\left( \omega, \mathbf{x}, t \right) \cdot \mathbf{x} - \omega\left( \boldsymbol{\kappa}, \mathbf{x}, t \right) \ t + \phi{\scriptstyle_{o}} $$ Because $\phi\left(\textbf{x},t\right)$ results from solutions of the wave equation, its derivatives must satisfy the dispersion relation through the following: $$ - \frac{ \partial \phi\left( \mathbf{x}, t \right) }{ \partial t } = \mathcal{W}\left( \frac{ \partial \phi\left( \mathbf{x}, t \right) }{ \partial \mathbf{x} }, \mathbf{x}, t \right) $$ and we can see from the equation for $\phi\left(\textbf{x},t\right)$ that the following is true: $$ \begin{align} \boldsymbol{\kappa} & = \frac{ \partial \phi\left( \mathbf{x}, t \right) }{ \partial \mathbf{x} } \\ \omega & = - \frac{ \partial \phi\left( \mathbf{x}, t \right) }{ \partial t } \end{align} $$ We also know that $\partial^{2} \phi$/$\partial \mathbf{x} \partial t$ $=$ $\partial^{2} \phi$/$\partial t \partial \mathbf{x}$, therefore: $$ \begin{align} \frac{ \partial^{2} \phi }{ \partial t \partial \mathbf{x} } - \frac{ \partial^{2} \phi }{ \partial \mathbf{x} \partial t } & = 0 \\ & = \frac{ \partial \boldsymbol{\kappa} }{ \partial t } - \frac{ - \partial \omega }{ \partial \mathbf{x} } = 0 \\ & = \frac{ \partial \boldsymbol{\kappa} }{ \partial t } + \frac{ \partial \omega }{ \partial \mathbf{x} } = 0 \\ & = \frac{ \partial \boldsymbol{\kappa} }{ \partial t } + \nabla \omega = 0 \end{align} $$ One can see that this final form looks similar to a continuity equation, so long as $\boldsymbol{\kappa}$ $\leftrightharpoons$ density of the waves, and $\omega$ $\leftrightharpoons$ flux of the waves.

Phase Velocity

From the above relations, we can see that on contours of constant $\phi\left(\textbf{x},t\right)$, we are sitting on local wave crests (i.e., phase fronts) where $\boldsymbol{\kappa}$ is orthogonal to these contours. These phase fronts move parallel to $\boldsymbol{\kappa}$ at a speed, $\mathbf{V}_{\phi}$, known as the phase velocity. The general form for this speed is given by: $$ \mathbf{V}_{\phi} \equiv \frac{ \mathcal{W}\left( \boldsymbol{\kappa}, \mathbf{x}, t \right) }{ \kappa } \boldsymbol{\hat{\kappa}} $$

Group Velocity

We can rearrange our continuity equation by multiplying by unity to get: $$ \begin{align} \frac{ \partial \boldsymbol{\kappa} }{ \partial t } + \frac{ \partial \omega }{ \partial \mathbf{x} } \cdot \frac{ \partial \boldsymbol{\kappa} }{ \partial \boldsymbol{\kappa} } & = 0 \\ \frac{ \partial \boldsymbol{\kappa} }{ \partial t } + \frac{ \partial \omega }{ \partial \boldsymbol{\kappa} } \cdot \frac{ \partial \boldsymbol{\kappa} }{ \partial \mathbf{x} } & = 0 \\ \frac{ \partial \boldsymbol{\kappa} }{ \partial t } + \left( \mathbf{V}_{g} \cdot \nabla \right) \boldsymbol{\kappa} & = 0 \end{align} $$ where $\mathbf{V}_{g}$ is called the group velocity, where we note that: $$ \frac{ \partial \omega }{ \partial \mathbf{x} } = \frac{ \partial \mathcal{W}\left( \boldsymbol{\kappa}, \mathbf{x}, t \right) }{ \partial \boldsymbol{\kappa} } \cdot \frac{ \partial \boldsymbol{\kappa} }{ \partial \mathbf{x} } + \frac{ \partial \mathcal{W}\left( \boldsymbol{\kappa}, \mathbf{x}, t \right) }{ \partial \mathbf{x} } $$ which shows that $\partial \mathcal{W}$/$\partial \boldsymbol{\kappa}$ $=$ $\left( \partial \omega / \partial \boldsymbol{\kappa} \right){\scriptstyle_{\textbf{x}}}$ $\Rightarrow$ different $\boldsymbol{\kappa}$'s propagate with velocity $\mathbf{V}_{g}$. In other words, $\mathbf{V}_{g}$ is the propagation velocity for $\kappa$ and $\mid$$\mathcal{A}$$\mid^{2}$ propagates with velocity $\mathbf{V}_{g}$.

Thus, an observer moving with the phase fronts (crests) moves at $\mathbf{V}_{\phi}$, but they observe the local wavenumber and frequency to change in time $\Rightarrow$ neighboring phase fronts (crests) move away from the observer in this frame. In contrast, for an observer moving with $\mathbf{V}_{g}$, they observe constant local wavenumber and frequency (with respect to time), but phase fronts (crests) continuously move past the observer in this frame.

References

Whitham, G. B. (1999), Linear and Nonlinear Waves, New York, NY: John Wiley & Sons, Inc.; ISBN:0-471-35942-4.

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  • $\begingroup$ +1 for any whitham reference i see on this site :-) @Joseph although this ray theory derivation is certainly legitimate, and interesting, it is kinematic. i think the dynamic approach also should be studied in some detail to understand the concepts at play. $\endgroup$ – Nick P Oct 29 '14 at 23:50
  • $\begingroup$ @NickP - I see you study water waves. I suppose I could have thrown in a Longuet-Higgins and Stewart reference as well ;) $\endgroup$ – honeste_vivere Oct 30 '14 at 11:51
  • $\begingroup$ now we're talking! $\endgroup$ – Nick P Oct 31 '14 at 3:33
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The frequency $\omega$ can be hight in a wave packet, but the envelope motion may be slow. The latter is determined with the $\cos(...)$; that is why they call it a group velocity. It is a velocity of displacement of the packet as a whole. There muct be applets on internet to show how a wave packet moves.

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  • $\begingroup$ How do we know it is determined by $cos(...)$ though?? $\endgroup$ – user43487 Oct 29 '14 at 14:28
  • $\begingroup$ @Joseph: Because the $\sin(...)$ is a carrier wave by definition. A time-dependent factor at it is a wave envelope, by definition too. $\endgroup$ – Vladimir Kalitvianski Oct 29 '14 at 14:48