Reflection from a finite string segment with a fixed far end ($R(q)=(r-q)/(1-rq)$)

At a lossless string junction with displacement reflection coefficient $r$ from the incident side, reverse incidence has coefficient $-r$ and the product of transmission coefficients is $1-r^2$. Reflection from a fixed far end contributes $-1$. Summing all round trips with phase $q=e^{2ikL}$ gives $R=r-(1-r^2)q/(1-rq)=(r-q)/(1-rq)$. For real $r$ with $|r|<1$ and $|q|=1$, numerator and denominator have equal absolute value, so $|R|=1$: the finite segment returns all incoming energy while adding a frequency-dependent phase. Short, nonoverlapping pulses can instead be counted as a geometric series of returning pulse energies; overlapping echoes must be added as amplitudes before their energy is evaluated.