If
we
let
subscripts
denote
partial
derivatives,
then
the
simplest
wave
equation
for
rightward
propagating
waves
reads:
A_{t}
=
-cA_{x}
.
While
any
initial
condition
just
moves
to
the
right
with
speed
c
without
changing
form,
a
particularly
useful
form
of
the
solution
is
sinusoidal:
A
=
A_{0}sin(kx-wt)
where
lambda
(wave
length)
=2pi/k.
The
wavenumber, k,
tells
us
how
quickly
the
wave
oscillates *in
space*,
while is
the
frequency,
w,
tells
us
how
quickly
the
wave
oscillates *in
time*.
If
we
know
the
propagation
speed
in
the
equation
(say
c=5
units
per
second)
then
if
we
choose
the
wavelength,
and
hence
k,
we
*
must*
choose
w
so
that
(w=ck
make
a
new
equation
for
this).
For
this
simplest
of
cases
the
relation
is
linear.
In
a
more
general
physical
setting
(waves
generated
by
a
pebble
dropped
in
a
pond,
for
example)
we
do
not
expect
the
simple
linear
relation
to
hold,
and
indeed
for
the
simple
wave
equation
above
to
apply,
but
we
*
do*
still
expect
wave
solutions
(experience
tells
us
to)
but
perhaps
with
a
more
complicated
relation
between
w
and
k.
For
a
given
physical
situation
we
aim
to
find
some
function,
w(k),
called
a *dispersion
relation*,
that
determines
just
how
w
depends
on
k.
The
reason
for
this
name
will
become
clearer
as
we
go
on.

For
the
next
example
consider
a
traveling
wave
pulse
that
can
be
broken
up
into
two
simpler
component
waves,
y_{1}
and
y_{2},
where:

y_{1}
=
A
sin(k_{1}x-w_{1}t)

y_{2} =
A
sin(k_{2}x-w_{2}t)

We assume that the two wave numbers are nearly, but not exactly, the same. Similarly the two frequencies are assumed to be nearly, but not exactly, the same. Recall the addition formula for sines which allows us to combine these eqations into:

y =
y_{1}
+
y_{2}

=
A
sin(k_{av}x-w_{av}t)sin(k_{mod}x-w_{mod}t)

Where
k_{av}
and
w_{av} are
the
averages
of
wave
number
and
frequency,
respectively. w_{mod}
is
a
slower
version
called
the *modulation frequency*
and
k_{mod}
is
half
the
difference
of
the
wave
numbers
of
the
component
waves.

Graphing
the
wave
pulse
and
the
modulation
curve
we
see
something
familiar:
our
wave
pulse
resembles
a
kind
of
beat
phenomenon.
The
modulation
curve
is
an
“envelope”
for
all
of
the
little
wave
crests
that
make
up
the
wave
pulse.
This
kind
of
wave
pulse
is
commonly
called
a *wave
packet*.

Combining the equations for period and wavelength we can calculate:

v_{p}
=
w/k
,
which
is
the *phase
velocity *and
it
represents
the
speed
at
which
the
individual
component
waves
are
traveling.

We
can
calculate
the
speed
at
which
the
wave
packet
travels
by
using
the
phase
velocity
but
using k_{mod} and
w_{mod}.
Since k_{mod} and w_{mod} are
infact
differences
we
can
take
the
limit
as k_{} goes
to
zero
and
derive
that:

v_{g}
=
d/dk
w/k

Group velocity, along with the concept of phase velocity, is a very important idea in the theory of waves. All of these concepts carry over to cases where our traveling wave pulse is no longer the superposition (sum) of just two simpler component waves, but rather many or possibly infinitely many component waves. We will now use the ideas of phase and group velocity to analyze different dispersion relations.

The next simplest wave equation to consider is the Partial Differential Equation (PDE):

A_{t}
=
bA_{xxx} -cA_{x}

You
can
convince
yourself
that
only
odd
numbered
space
derivatives
can
appear
if
we
do
not
wish
our
waves
to
spontaneously
decay.
Assuming
solutions
of
the
form: A
=
A_{0}sin(kx-wt)

Substituting
and
solving
the
differential
equation
gives
the
dispersion
relation:
w=
bk^{3}+ck

Taking
the
values
of
c
and
b
to
be
1
and
0.1
respectively,
our
dispersion
relation
becomes:
w =
k+0.1k^{3}.
Which
looks
like:

Calculating the phase and group velocities we get:

v_{p}
=
1+0.1k^{2}
and
v_{g}
=
1+0.3k^{2}

Suppose
that
we
define
a
set
of *initial
conditions* so
that
our
wave
pulse
initially
looks
like
this:

And obeys the differential equation, Applying Fourier analysis, we can track the individual wave components that make up the wave pulse and determine what the pulse will look like at any point in the future. A little further ahead in time, the wave pulse looks like:

And further ahead:

We
can
now
make
an
important
observation:
Waves
of
shorter
wavelengths
travel
faster
than
waves
of
longer
wavelengths.
Because
waves
of
different
wavelengths
travel
at
different
velocities,
the
pulse
“spreads
out”
as
time
goes
on.
This
phenomenon
is
called *dispersion*,
and
hence
the
reason
why
the
function
is
called
a
*
dispersion
relation*.

So
far,
we
have
only
looked
at
our
wave
pulse
from
a
spatial
perspective,
looking
at
a
“snap-shot”
of
the
wave
pulse
in
time.
We
can
also
consider
a *contour
plot*,
which
is
a
set
of
level
curves
taken
in
the
x-t
plane.
(If
you
aren’t
familiar
with
level
curves,
thing
of
a
contour
plot
as
a
type
of
topographical
map,
such
as
those
found
in
a
world
atlas.
The
only
difference
is
that
we
are
measuring
wave
amplitude
instead
of
mountain
height.)

The contour plot can tell us a lot about our wave pulse. The colour gradient tells us the amplitude of the individual waves at a point in space-time. The width of the bands of colour, (in the x-direction), tell us the wavelengths of the individual component waves. As we noticed before, we can see that the waves of shorter wavelengths travel faster than waves of longer wavelengths. In general, the energy of the wave pulse travels in the same direction as the group velocity.

We
now
consider
what
happens
when
we
make
one
small
change
in
our
dispersion
relation.
This
time
we
keep
c
=
1,
but
we
now
let
b
=
−0.1.
The
dispersion
relation
now
becomes:
w
=
k -
0.1k^{3}.
A
graph
that
looks
like:

With
v_{p}
=
1-0.1k^{2}
and
v_{g}
=
1
-0.3k^{2}. Taking
the
same
initial
conditions
as
before
we
have
our
familiar
looking
wave
pulse:

After some time:

And,

We see again this time that the waves of shorter wavelengths are traveling faster than waves of longer wavelengths, but now, many of the component waves are traveling in a direction that is opposite that of the wave packet itself.

The contour plot gives us a better picture of what’s going on:

This confirms what we deduced earlier.

Therefore, we see that if we keep the same initial conditions, but change the dispersion relation ever so slightly, we end up with remarkably different results. We have also seen that there are many possible outcomes when considering traveling wave pulses. The component waves, and the wave packet and energy, may travel at different speeds and in the same or opposite directions. (In fact, for special kinds of 2D waves, called internal gravity waves, the phase and group velocities can be perpendicular to each other!)

One
last
thing
to
note,
is
that
when
the
phase
and
group
velocities
are *always* equal,
all
of
the
component
waves
and
the
wave
packet
itself
travel
at
the
same
speed.
Hence,
the
waves
are
non-dispersive.