For opposite propagation directions on a uniform string, write the equal-frequency transverse waves as

y_{1} = A cos (k x - ω t + α), y_{2} = A cos (k x + ω t + β) .

The cosine addition identity gives

y_{1} + y_{2} = 2 A cos (k x + \frac{α + β}{2}) cos (ω t + \frac{β - α}{2}) .

(2)

This is a standing wave: the spatial pattern is fixed while its amplitude oscillates in time. Nodes occur at the zeros of the first cosine and are separated by

π / k

, half a wavelength; antinodes have displacement amplitude

2 A

. The two equal opposite travelling waves carry equal opposite average energy flux, so the net average transported power is zero.

For the same propagation direction, the same identity instead gives

A cos (k x - ω t + α) + A cos (k x - ω t + β) = 2 A cos \frac{α - β}{2} cos (k x - ω t + \frac{α + β}{2}) .

(3)

For a nonzero resultant this is another travelling wave, with fixed amplitude and moving constant-phase points, not a nontrivial standing wave. If the prefactor vanishes, the waves cancel identically; the zero disturbance is a trivial exception rather than a standing oscillation. The argument assumes propagating waves with