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Note that if we plot the field for a fixed time, for example, at t = 0, the field
takes the shape of a sinusoidal function in space:

E(z, t = 0) = E cos[kz] (6.74)
0
From the above equation, one deduces that the wavenumber k = 2π/λ, where
λ is the wavelength of the wave (i.e., the length after which the wave shape
reproduces itself).
Now let us look at the field when an observer, located at z = 0, would mea-
sure it as a function of time. Then:

E(z = 0, t) = E cos[ωt] (6.75)
0

The temporal period, that is, the time after which the wave shape reproduces
2π
itself, is T = , where ω is the angular frequency of the wave.
ω
Next, we want to relate the wavenumber to the angular frequency. To do
that, consider an observer located at z = 0. The observer measures the field at
t = 0 to be E . At time ∆t later, he should measure the same field, whether he
0
uses Eq. (6.74) or (6.75) if he takes ∆z = c∆t, the distance that the wave crest
has moved, and where c is the speed of propagation of the wave. From this,
one deduces that the wavenumber and the angular frequency are related by
kc = ω. This relation holds true for all electromagnetic waves; that is, as the
frequency increases, the wavelength decreases.
If two traveling waves have the same amplitude and frequency, but one is
traveling to the right while the other is traveling to the left, the result is a
standing wave. The following program permits visualization of this standing
wave.

x=0:0.01:5;
a=1;
k=2*pi;
w=2*pi;
t=0:0.05:2;
M=moviein(41);
for m=1:41;
z1=cos(k*x-w*t(m));
z2=cos(k*x+w*t(m));
z=z1+z2;
plot(x,z,'r');
axis([0 5 -3 3]);

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