Past exam of the mathematical methods II course of the University of Oxford / 2021 / 11 / iv

For equal complex amplitudes, the superposition principle and the complex exponential formula for the cosine give the exact identityThe cosine is a real periodic modulation of the rapidly varying carrier. It varies slowly when $|\delta k|\ll |k_0|$ and $|\delta \omega |\ll |{\omega }_0|$. Its signed wave envelope travels with velocity $\delta \omega /\delta k$; its magnitude gives the visible amplitude envelope.

Equal magnitudes need not mean equal initial wave phases. If the two complex amplitudes are $a{e}^{i{\varphi }_{+}}$ and $a{e}^{i{\varphi }_{-}}$, write $\bar{\varphi }=({\varphi }_{+}+{\varphi }_{-})/2$ and $\delta \varphi =({\varphi }_{+}-{\varphi }_{-})/2$. The same result becomes $2a{e}^{i\bar{\varphi }}\cos (\delta kx-\delta \omega t+\delta \varphi )e^{i(k_{0}x-{\omega }_{0}t)}$, still with a real slowly varying modulation.