= Outgoing radiation pulse
{title2=$\overline S_r(r,t)=P_{\rm emit}(t-r/c)/(4\pi r^2)$}
An <outgoing radiation pulse> carries a finite amount of <electromagnetic energy> through successively larger enclosing <spheres>. In vacuum, far from a localized source and without absorption or additional reflections, let $P_{\rm emit}(t)$ be the emitted <power>. The <retarded time> at <radius> $r$ is $s=t-r/c$. Define $\overline S_r$ as the angular average of the radial <Poynting vector> over the <sphere>, not a cycle average:
$$
\overline S_r(r,t)=\frac{P_{\rm emit}(t-r/c)}{4\pi r^2},\qquad
\overline u(r,t)=\frac{\overline S_r(r,t)}{c},\qquad
\frac{dU}{dr}=4\pi r^2\overline u=\frac{P_{\rm emit}(t-r/c)}c.
$$
Here $\overline u$ is the angularly averaged <energy density> in the leading <radiation zone> approximation. Multiplying by shell <area> removes geometric dilution: the <energy> per unit radial thickness translates outward without losing its shape. These angular averages do not assume that an actual <electric dipole> emits equally in all directions. Nor do they specify an exact spherically symmetric transverse solution of the <Maxwell equations>.
For an explicitly stipulated model, emit
$$
P_{\rm emit}(s)=P_0g(s/\tau),\qquad
g(q)=\begin{cases}\sin^2(\pi q),&0\le q\le1,\\0,&\text{otherwise},\end{cases}
\qquad U=\int P_{\rm emit}(s)\,ds=\frac{P_0\tau}{2}.
$$
After emission ends, $t>\tau$, the pulse lies between $r=c(t-\tau)$ and $r=ct$, provided this shell is in the <radiation zone>. Its radial thickness remains $c\tau$. Substituting $s=t-r/c$ in the shell <integral> $\int4\pi r^2\overline u\,dr$ gives $U=P_0\tau/2$ at every such time. The same substitution in the time <integral> of $4\pi r^2\overline S_r$ at any enclosing <sphere> gives that same total crossing <energy>. This is <conservation of energy> in two different views.
\Image[/outgoing-radiation-pulse-spherical-spreading.png]
{title=Outgoing pulse: increasing shell <area> dilutes local <energy flux>}
{description=Original calculated model with $L=c\tau$ and $P_{\rm emit}=P_0\sin^2(\pi s/\tau)$ during emission. The circle diagrams show shell support, not equally strong emission in all directions. The lower-left curves show angularly integrated <energy> per radius and have equal areas. The lower-right curves compare <electric field> and <energy flux> ratios at corresponding <retarded times> along a fixed direction. All values are model predictions, not historical measurements.}
Along any fixed direction with nonzero outgoing <power>, compare the same part of the pulse at two <radii>, hence at different arrival times. The leading <electric field> amplitude falls as $1/r$, while its squared <wave intensity> falls as $1/r^2$. At $2r$ the field is one half and the <energy flux> one quarter; the signal arrives $r/c$ later. A small fixed <area> receiver therefore intercepts less <energy> at greater distance even though every enclosing <sphere> receives the same total pulse <energy>. The <inverse-square law> expresses spreading over <area>, rather than destruction of <energy>. Close to the source, the neglected <near and far field> terms prevent extrapolating these formulas down to $r=0$.
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