Now, the intensity I (``brightness'' in the case of light) of
the pattern at the observation screen is a just a measure of the
amount of energy arriving at each point 
on the screen per
unit time. In a real experiment, the screen is a two-dimensional
surface. Thus, the precise measure we use for the intensity is the
average energy arriving per unit time per unit area, or, simply,
average power per unit area. To relate this to the solution for the
waves 
, we can generalize from the result we know for
power on a string,
It turns out that all phenomena (such as sound or light) which obey
the wave-equation obey a very similar formula. The primary difference
in going from waves on a string to any other type of wave is that the
constant 
(which measures the tension of the string) is replaced
by some other constant characterizing the medium of propagation for
the waves. If we consider traveling sinusoidal waves, then the
solution will have the general form 
.
Then, using Eq. 1 we will find a result of the following
form for the power. (Here, 
indicates the average
value of the quantity q.)
Note that, at any instant in time t, the power may be different: it
varies with the oscillations of 
. This is why we
define intensity as the time average of the arriving power.
Finally, because the time average of 
or 
is just 1/2, we
have
Note that in the last line we replaced the factor 
with a general proportionality constant 
. We do this because
for waves which are not waves on a string, we end up with
factors other than to describe the medium of propagation. What is
the same for all types of waves, the main point here, is that the
intensity is proportional to the square of the amplitude.
