5.5-2 Amplitude-Modulated Wave in a Dispersive Medium. An amplitude-modulated wave whose complex wavefunction takes the form $A(t) = [1 + m \cos(2\pi f_s t)] \exp(j2\pi \nu_0 t)$ at $z = 0$, where $f_s \ll \nu_0$, travels a distance $z$ through a dispersive medium of propagation constant $\beta(\nu)$ and negligible attenuation. If $\beta(\nu_0) = \beta_0$, $\beta(\nu_0 - f_s) = \beta_1$, and $\beta(\nu_0 + f_s) = \beta_2$, derive an expression for the complex envelope of the transmitted wave as a function of $\beta_0$, $\beta_1$, $\beta_2$, and $z$. Show that at certain distances $z$ the wave is amplitude modulated with no phase modulation.
