We start off modestly looking at the
interference between waves moving along a common direction in space with the same frequency. If the direction of propagation is
the same the resultant wave is just another wave propagating in the
same direction:
A good trick to figure out the result is to use complex numbers. The 
type functions can
be represented as the projection on the y axis of a vector forming the
angle 
with the x axis. (draw vector diagram) from the
diagram we see that
and perhaps less interesting is the phase :
We see that for
we have constructive interference that is the amplitude of the
resulting wave is greater than that of its constituents. However if
we have destructive interference and the resulting wave is weaker
then the strongest of the two waves. In the special case where the
amplitudes of the constituent waves are identical the two waves can
actually wipe each other out completely. So no wave can be thought
of as two waves propagating with a phase shift of 
. This
highlight the importance of adding amplitudes before determining
the amount of energy transported!