= Solution
Choose both <waves> as <sinusoids> with a common zero phase and amplitude $A$. The trigonometric sum identity gives
$$
A\cos[(k+\delta)x-(\omega+\epsilon)t]+A\cos[(k-\delta)x-(\omega-\epsilon)t]=2A\cos(\delta x-\epsilon t)\cos(kx-\omega t).
$$
The slowly varying <wave envelope> has constant phase when $\delta x-\epsilon t$ is constant. Its speed is $\epsilon/\delta$, and in the close-frequency limit the <group velocity> is
$$
\boxed{v_g=\frac{d\omega}{dk}.}
$$
It 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 $v_g=v=v_p$.
Back to article page
