In a rectangular box of width 
and height 
the modes which have
nodes at all boundaries are products of sinusoidal functions of the form
, where
and
.
Now the situation is a little more complicated, since
is a vector.
In fact, we call it the wavevector
instead of the wave number;
the wave number
is then given by
.
Why do we bother with the magnitude k
instead of sticking to the intrinsically multidimensional vector
?
Well, when we do kinematics we are often
concerned with the kinetic energy, which is a scalar
quantity depending only upon the magnitude of the momentum p
(and upon the effective mass, if any) of the particle in question.
Since we have discovered that photons (for example) are
in some sense particles which have energy
and momentum
,
we can conclude that 
(for massless particles only)
and so, if all we really care about is the energy
of a given mode, the only thing we need to know is its
wave number, k .
But we still need to count up how many modes have (approximately) the same wavenumber k . This is where we have to return to the two-dimensional picture and begin talking in terms of k-space.
for every 
and one allowed 
for every
, so there
is altogether 
allowed
for every "k-area" element
in two-dimensional k-space.
(Yes, this is getting a little weird. Pay close attention!)
Note that 
where
is the
actual physical area of the box in normal space.
This element of k-space contains exactly
allowed state,
so once again we may define the density of states in k-space,
or, for this two-dimensional (2D) case,
.
Note that the density of states in k-space is proportional to
the physical area of the region to which the waves are confined.
How many such states have (approximately) the same wavenumber k?
This is a crucial question in many problems.
To estimate the result we draw a ring in k-space
with radius k and width dk.
Recalling that only positive
values of 
and
are allowed
(standing waves and all that),
we only consider the upper right-hand quadrant of the circular ring;
its "k-area" is thus 
.
At a density of 
states per unit k-area, this gives
or 
states in that ring quadrant.
We can express this as a density of wavenumber magnitudes
in terms of the distribution function
which is defined as the number of allowed modes whose wavenumbers
are within dk of a given k.
Note that the number increases linearly with k,
unlike in the 1D case where it is independent of k.