 Solution

Choose both waves as sinusoids with a common zero phase and amplitude $A$. The trigonometric sum identity givesThe slowly varying wave envelope has constant phase when $\delta x-ϵt$ is constant. Its speed is $ϵ/\delta$, and in the close-frequency limit the group velocity isIt describes the motion of a narrow-band envelope in a medium where the constituent waves retain a coherent packet interpretation; it is not an unconditional signal-speed theorem in strongly absorbing or anomalously dispersive media. The carrier phase velocity is $v_p=\omega /k$. In a nondispersive medium, $\omega =vk$ with constant $v$, so $v_g=v=v_p$.