Question #43566

Consider a flat circular disk of radius $R$ lying in the $x-y$ plane, centered at the origin. The entire disk acts as a coherent source of scalar waves, oscillating in phase at an angular frequency $\omega$. According to the Huygens-Fresnel principle, each infinitesimal area element $dA$ on the disk emits a spherical secondary wavelet of the form $d\psi = \frac{\psi_0}{r'} e^{i(kr' - \omega t)} dA$, where $r'$ is the distance from the element to the observation point, $k = \omega/c$, and $c$ is the speed of the wave in the medium. Ignore any obliquity factors. By integrating the contributions from the entire continuous distribution, determine the effective phase velocity $v_p$ of the resultant wave along the central $z$-axis as a function of the distance $z$ from the disk.