 Solution

For opposite propagation directions on a uniform string, write the equal-frequency transverse waves asThe cosine addition identity givesThis 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 $\pi /k$, half a wavelength; antinodes have displacement amplitude $2A$. 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 givesFor 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 $k,\omega >0$ on the same uniform string.