Wave Interference
What is the relationship between traveling wave and standing
wave solutions to the wave equation, (6.1), in an infinite medium? To help answer this question, let us form a superposition of two traveling wave solutions
of equal amplitude , and zero phase angle , that have the same wavenumber
, but are moving in opposite directions. In other words,

(6.19) 
Because the wave equation, (6.1), is linear, the previous superposition is a valid solution
provided the two component waves are also valid solutions; that is, provided
, which we shall assume to be the case. Making
use of the trigonometric identity
(see Appendix B),
the previous expression can also be written

(6.20) 
which is a standing wave [cf., Equation (6.2)]. Evidently, a standing wave is a linear
superposition of two, otherwise identical, traveling waves that propagate in opposite
directions. The two waves completely cancel one another out at the nodes, which
are situated at
, where is an integer. This process is known as total
destructive interference. On the other hand, the waves reinforce one another
at the antinodes, which are situated at
, generating a wave
whose amplitude is twice that of the component waves. This process
is known as constructive interference.
As a more general example of wave interference, consider a superposition
of two traveling waves of unequal amplitudes which again have the same wavenumber
and zero phase angle,
and are moving in opposite directions; that is,

(6.21) 
where
.
In this case, the trigonometric identities
and
(see Appendix B) yield

(6.22) 
Thus, the two waves interfere destructively at
[i.e., at points where
and
] to produce a minimum wave amplitude , and interfere constructively
at
[i.e., at points where
and
] to produce a maximum wave amplitude . It can
be seen
that the destructive interference is incomplete unless . Incidentally, it is a general
result that if two waves of amplitude and interfere then
the maximum and minimum possible values of the resulting wave amplitude are and
, respectively.