## Meaning of Sommerfeld radiation conditions

Hello!
Promising that I will not make other new questions in the next days , I have a doubt about the meaning of a pair of expressions.
Sommerfeld's conditions for an electromagnetic field produced by a finite source bounded by a finite volume are:

$\lim_{r \to +\infty} r|\mathbf{E}| < q\\ \lim_{r \to +\infty} r|\mathbf{H}| < q\\ \lim_{r \to +\infty} r \left[\mathbf{E} - \eta \mathbf{H} \times \mathbf{\hat{u}}_k \right] = 0\\ \lim_{r \to +\infty} r \left[\mathbf{H} - \displaystyle \frac{\mathbf{\hat{u}}_k \times \mathbf{E}}{\eta} \right] = 0$

where $q$ are finite quantities, $\eta$ is the wave impedance in the considered medium (for example the free space), $\mathbf{\hat{u}}_k$ is the direction of propagation and $r$ is the distance from the source.
The first two state that the fields' module must decrease at least as $1/r$.
The last two state that the fields must be similar to those of a plane wave: they must be mutually orthogonal and also both orhogonal to the direction of propagation. Moreover, the "part" of $\mathbf{E}$ (in the first) and $\mathbf{H}$ (in the second) which does not contribute to that plane wave must decrease at least as $1/r^2$.

Why these last two conditions are called "radiation conditions"? As a matter of fact, in the electric dipole non-radiative field components decrease as $1/r^2$ or $1/r^3$. But why this is a necessary requirement to build a "radiation"? Couldn't we have a not-radiating component which decreases as $1/r$? Why?
Thank you, again, for having read.

Emily
 Mentor Every component which decreases with 1/r looks like radiation, and therefore it is called radiation.