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Phase vs. Group Velocity

The precise relationship between angular frequency  $\omega$  and wavenumber  $k$  for deep-water waves is

$$\omega \; = \; \sqrt{ g \, k \over 2 }$$ (14.26)

where  $g$  has its usual meaning. Such a functional relationship  $\omega(k)$  between frequency and wavenumber is known as the DISPERSION RELATION for waves in the medium in question, for reasons that will be clear shortly.

If we have a simple traveling plane wave   $A(x,t) = A_{_0} \exp[i(kx - \omega t)]$, with no beginning and no end, the rate of propagation of a point of constant phase (known as the PHASE VELOCITY $v_{\rm ph}$) is still given by Eq. (6):

$$\mbox{ \fbox{ \rule[-1.0\baselineskip]{0pt}{2.5\baselineski . . . . . . aystyle v_{\rm ph} \; \equiv \; { \omega \over k } }$~ }}$$ (14.27)

However, by combining Eq. (27) with Eq. (26) we find that the phase velocity is higher for smaller  $k$  (longer $\lambda$):

$$v_{\rm ph} \; = \; \sqrt{ g \over 2 k } .$$ (14.28)

Moreover, such a wave carries no information. It has been passing by forever and will continue to do so forever; it is the same amplitude everywhere; and so on. Obviously our PLANE WAVE is a bit of an oversimplification. If we want to send a signal with a wave, we have to turn it on and off in some pattern; we have to make wave pulses (or, anticipating the terminology of QUANTUM MECHANICS, "WAVE PACKETS"). And when we do that with water waves, we notice something odd: the wave packets propagate slower than the "wavelets" in them!

Figure: A WAVE PACKET moving at  $v_{\rm g}$  with "wavelets" moving through it at  $v_{\rm ph}$.

Such a packet is a superposition of waves with different wavelengths; the $k$-dependence of  $v_{\rm ph}$ causes a phenomenon known as DISPERSION, in which waves of different wavelength, initially moving together in phase, will drift apart as the packet propagates, making it "broader" in both space and time. (Obviously such a DISPERSIVE MEDIUM is undesirable for the transmission of information!) But how do we determine the effective speed of transmission of said information - i.e. the propagation velocity of the packet itself, called the GROUP VELOCITY  $v_{\rm g}$?

Allow me to defer an explanation of the following result until a later section. The general definition of the group velocity (the speed of transmission of information and/or energy in a wave packet) is

$$\mbox{ \fbox{ \rule[-1.0\baselineskip]{0pt}{2.5\baselineski . . . . . . \; \equiv \; { \partial \omega \over \partial k } }$~ }. }$$ (14.29)

For the particular case of deep-water waves, Eq. (29) combined with Eq. (26) gives

$$v_{\rm g} \; = \; {1\over2} \sqrt{ g \over 2 k } .$$ (14.30)

That is, the packet propagates at half the speed of the "wavelets" within it. This behaviour can actually be observed in the wake of a large vessel on the ocean, seen from high above (e.g. from an airliner).

Such exotic-seeming wave phenomena are ubiquitous in all dispersive media, which are anything but rare. However, in the following chapters we will restrict ourselves to waves propagating through simple non-dispersive media, for which the DISPERSION RELATION is just   $\omega = c \, k$  with  $c$  constant, for which   $v_{\rm ph} = v_{\rm g} = c$.