Jakes is a bit beyond me, but I'm fairly sure that the Bessel function $J_0$ emerges, because of the integral presentation $$ J_0(x)=\frac1{2\pi}\int_{-\pi}^{\pi}e^{ix\sin t}dt. $$ If you have two plane waves of the same frequency, the other reflected, so that the two copies have angular separation $\theta$. Then their correlation would vary like $e^{ix\cos\theta}$, because the the projection of the wavelength of the other wave along the direction of propagation of the other gets multiplied by $\cos\theta$. Now treat $\theta$ as a random variable uniformly distributed over the circle...

IOW, I think that the Bessel function emerges as a consequence of the underlying geometry as opposed to being a design parameter.

Jakes is a bit beyond me, but I'm fairly sure that the Bessel function $J_0$ emerges, because of the integral presentation $$ J_0(x)=\frac1{2\pi}\int_{-\pi}^{\pi}e^{ix\sin t}dt. $$ If you have two plane waves of the same frequency, the other reflected so that the two copies have angular separation $\theta$. Then their correlation would vary like $e^{ix\cos\theta}$, because the the projection of the wavelength of the other wave along the direction of propagation of the other gets multiplied by $\cos\theta$. Now treat $\theta$ as a random variable uniformly distributed over the circle and compute the average.

IOW, I think that the Bessel function emerges as a consequence of the underlying geometry as opposed to being a design parameter. Hopefully a more knowledgeable person can answer. This was just a bit too long to fit into a comment.