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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):

$$\mbox{ \fbox{ \rule[-1.0\baselineskip]{0pt}{2.5\baselineskip . . . . . . style v_{\rm ph} \; \equiv \; { \omega \over k } }$ ~ } }$$ (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 the next chapter. The general definition of the group velocity (the speed of transmission of information and/or energy in a wave packet) is

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

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

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