Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 11 / a

For two equal-amplitude waves with the same frequency and wavenumber, choose their phases so thatThe spatial amplitude and temporal oscillation separate. Positions with $\cos kx=0$ are fixed nodes of a standing wave, while positions with $|\cos kx|=1$ are antinodes. Different initial phases merely shift this pattern in space and time.

A travelling wave has a phase profile that moves at its phase velocity; a standing wave has stationary nodes and each point oscillates with its fixed local amplitude. The two counterpropagating waves carry equal opposite time-averaged energy flux, so their superposition has zero net time-averaged energy transport. Instantaneous energy can still move within the standing pattern.