BELIEVE ME NOT!   A SKEPTIC's GUIDE
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The precise relationship between angular frequency
and wavenumber for deepwater waves is

(14.26) 
where has its usual meaning.
Such a functional relationship
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
,
with no beginning and no end, the rate of propagation of a
point of constant phase (known as the PHASE VELOCITY
)
is still given by Eq. (6):

(14.27) 
However, by combining Eq. (27) with Eq. (26)
we find that the phase velocity is higher
for smaller (longer ):

(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
with "wavelets" moving through it at .

Such a packet is a superposition of waves with different wavelengths;
the dependence of 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 ?
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

(14.29) 
For the particular case of deepwater waves,
Eq. (29) combined with Eq. (26) gives

(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 exoticseeming 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 nondispersive media,
for which the DISPERSION RELATION is just
with constant, for which
.
Next: Sound Waves
Up: Water Waves
Previous: Water Waves
Jess H. Brewer 
Last modified: Sun Nov 15 18:05:46 PST 2015