And, when this equation is multiplied by r the result is

r²(∂²E/∂t²) = C²[2r(∂E/∂r) + r²(∂²E/∂r²) ]
the same as the above
wave equation

The equation

(∂²(rE)/∂t²) = C²(∂²(rE)/∂r²)

is a one dimensional wave equation in rE. The d'Alembert solution therefore applies; i.e.,

rE(r, t) = ½(r-Ct)E(r-Ct, 0) + ½(r+Ct)E(r+Ct, 0)

In this solution one half of the field shoots away toward the horizon at the speed of light. The
other half moves through the center of the field at the speed of light. It would seem that this other
solution is not physically relevant. The physically relevant solution is then

rE(r, t) = (r-Ct)E(r-Ct, 0)

Consider a field generated by a spherical charge of radius R.

E(r, 0) = K/r² for r>R
E(r, 0) = 0 for r<R

If the charge were to disappear the hole in the field would expand from a radius of R at t=0 at the speed of light, as depicted below.

Free fields not generated by a charged particle disappear toward the horizon at the speed of light. There can be no static free-fields.
The goal of Quantum Field Theory of replacing charged particles with autonomous free fields is thus grossly misguided.