PX275 - I3 - point sources in 3D

the 3D wave equation:

\frac{\partial^{2} u}{\partial t^{2}} = c^{2} \nabla^{2} u

in spherical coordinates:

\begin{matrix} x = r \sin θ \cos ϕ \\ y = r \sin θ \sin ϕ \\ z = r \cos θ \\ \nabla^{2} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ϕ^{2}} \end{matrix}

since the point sources are spherically symmetric, there is no $θ$ or $ϕ$ dependence, only $r$ , ie: $u (\vec{r}, t) = u (r, t)$
this simplifies $\nabla^{2}$ and the radial wave equation is obtained:

\begin{matrix} (\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}) u (r, t) = 0 \\ \frac{2}{r} \frac{\partial u}{\partial r} + \frac{\partial^{2} u}{\partial r^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}} = 0 \end{matrix}

this is the 1D wave equation with an additional term in the beginning

\begin{matrix} \frac{2}{r} \frac{\partial}{\partial r} (\frac{v}{r}) + \frac{\partial}{\partial r} \frac{\partial}{\partial r} \frac{v}{r} - \frac{1}{c^{2}} \frac{\partial}{\partial t} \frac{\partial}{\partial t} (\frac{v}{r}) = 0 \\ = \dots = \frac{1}{r} \frac{\partial^{2} v}{\partial r^{2}} - \frac{1}{c^{2} r} \frac{\partial^{2} v}{\partial r^{2}} = 0 \\ ⟹ \frac{\partial^{2} v}{\partial r^{2}} = \frac{1}{c^{2}} \frac{\partial^{2} v}{\partial t^{2}} \end{matrix}

this is the 1D wave equation, but for $v$ , the solutions for which is:

v (r, t) = A \exp (i k (r - c t)) + B \exp (i k (r + c t))

for sources of light, outward propagating solutions are needed
so, without loss of generality:

u (r, t) = \frac{A}{r} \exp (i k (r - c t))

this is the solution for a point source
separating it into radial and temporal components:

u (r, t) = R (r) T (t) = \frac{A}{r} \exp (i k r) \exp (- i k c t)

for a stationary source, the time dependence is uniform
so the intensity patterns have a common time dependence
it is not possible to calculate the intensity patterns at given locations and for different shape apertures