The waves described by the functions
and
can be added algebraically to produce the resultant wave
We can define new variables
and
through the equations
we can write
The resultant wave above is a sinusoidal wave of the same frequency,
with
and
For a standing wave
and
since the reflected wave required for a standing wave to be formed undergoes a phase change of
on reflection.
which is not defined. Consideration of the sign of the numerator implies we may take