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In an ideal matched-port two-port network for an antenna pair, the incident amplitude traverses the link, reflects from the receiver's load with amplitude coefficient ${\Gamma }_L$, then traverses the reciprocal link again. If each one-way power transfer factor is $\mathcal{F}$, the returned power fraction is $|{\Gamma }_L{|}^2{\mathcal{F}}^2$. With fixed gains, $\mathcal{F}\propto {\lambda }^2$ and this echo scales as ${\lambda }^4$. Actual antenna structural scattering or an existing input mismatch can add a coherent return not specified by the Friis transmission equation alone.

Distinguish reflection at the input terminal, reflection at the receiving load, and a returned signal that traverses the free-space link twice. The input antenna is initially matched, so its direct terminal reflection is zero.

In the usual weak-coupling, matched-port two-port network model for the antenna pair, let the forward and reverse wave amplitude transfer coefficients be $t_{21}$ and $t_{12}$. Electromagnetic reciprocity gives equal power transfer factors $|t_{21}{|}^2=|t_{12}{|}^2=\mathcal{F}$, where $\mathcal{F}=(G_a\lambda /(4\pi d){)}^2$ from part (a). The receiving-load reflection returns wave amplitude $t_{12}{\Gamma }_L{t}_{21}$ to the original input port. The load-mismatch contribution to the returned power is thereforeFor fixed gains and impedances this returned power scales as ${\lambda }^4$: it contains two link traversals. By contrast, the reflected power at the receiving load before that return traversal is $P_i\mathcal{F}|{\Gamma }_L{|}^2\propto {\lambda }^2$. Calling that local reflected power the power already back in the input line would omit one propagation loss.

The original wording does not specify an antenna scattering model. The boxed result is the load-dependent echo in an ideal matched-port link, with structural scattering and residual input coupling neglected. The actual total input return can include coherent scattering from the receiving antenna even when ${\Gamma }_L=0$; gains and effective areas alone do not specify that background. If “reflected back” means only the direct reflection at the stated matched transmitting antenna, its value is $0$. These meanings should not be conflated; the ${\lambda }^4$ result requires the two-pass echo interpretation and its stated model.

A two-port network relates voltage and current, or incoming and outgoing travelling-wave amplitudes, at two pairs of electrical terminals. Power-normalized scattering parameters obey $b_i={\sum }_j{S}_{ij}{a}_j$. In a reciprocal network with equal reference impedances, $S_{12}=S_{21}$. A load reflection at port 2 can return through both transmission coefficients to port 1.