Periodic resistive loading of a transmission line

A shunt electrical resistance $R$ inserted between the conductors of a uniform lossless transmission line of characteristic impedance $Z$ has voltage reflection coefficient $r=-Z/(2R+Z)$ and transmission amplitude $t=1+r$. Equally spaced shunts produce coherent reflected waves. The round-trip phase between neighbours is $2kL$, so interference repeats in frequency with spacing $v_p/(2L)$ for nondispersive phase speed $v_p$.

Summing only paths that reflect at one shunt, with all previous shunts crossed once in each direction, gives a geometric series with ratio $t^2{e}^{2ikL}$. This is not the full multiple-reflection solution: it excludes paths that bounce between distinct shunts. For a semi-infinite array the exact amplitude instead obeys $R=r+t^2{e}^{2ikL}R/(1-r{e}^{2ikL}R)$. Near coherent reflection peaks the omitted paths need not be small, even if each shunt is weak.