To test if "back radiation" is a real phenomenon, we suggest the following experiment: On a night with moon-light so feeble that you can cannot read a newspaper, place yourself in front of a mirror letting the moonlight reflect from the newspaper to the mirror and back again, and check if you can now read. You will probably find that the paper is still unreadable, as if "back radiation" does not give more light.

To give this experiment theoretical support we consider the mathematics of wave propagation from a source at x=0 (Earth surface) to a receiver at x=1 (atmospheric layer) described by the wave equation (as a model of Maxwell's equations describing light as electromagnetic waves):

U_tt - U_xx = 0 for x in the interval (0,1)

with solution U(x,t) being a combination of waves traveling with velocity +1 and -1 along the x-axis, and with subindices indicating differentiation with respect to space x and time t. The boundary condition at the receiver may take the form

AU_t(1,t) + U_x(1,t) =0

with a positive coefficient A signifying:

- A = 0: soft reflection with U_x(1,t) = 0
- A large : hard reflection with U_t(1,t) = 0
- A = 1: no reflection: transparent absorption of all incoming waves at x = 1.

The basic energy balance is obtained by multiplying the wave equation by U_t and integrating

with respect to x to give:

E_t + AU_t(1,t)^2 = -U_x(0,t)U_t(0,t) = Input Energy.

where E(t) is the energy of the wave over the interval (0,1). Assuming that E(t) stays constant so that energy is no accumulating in the interval (0,1), we have that

Output Energy = A U_t(1,t)^2 = Input Energy.

In particular, with soft reflection with A = 0, the Input Energy is also zero. We learn that it is not possible to "pump the system" by reflection at x = 1: If you change from transparency with A = 1 to reflection with A = 0, the system reacts by refusing to accept Input Energy.

Ergo: Reflection/back radiation cannot increase the insolation to the Earth surface.

(Back radiation seeks support in a description of light as a stream of particles proposed by Newton, which was replaced by Maxwell's wave theory in the late 19th century).