Past exam of the mathematical methods II course of the University of Oxford / 2017 / 10 / c

Each light-string round trip combines wall reflection $-1$ with junction reflection $-1/3$, multiplying the remaining pulse amplitude by $1/3$. The initial directly reflected heavy-string pulse has amplitude $A_i/3$. The subsequent outgoing heavy pulses have amplitudesThey all have width $w$ in the same heavy medium. For nonoverlapping pulses their energy fractions therefore sum toThe first transmission leaves energy fraction $8/9$ in the light segment; each return encounter leaves only $1/9$ of the previous stored energy there, so this residual tends to zero.

For reflection from a finite string segment with a fixed far end, if pulses overlap, squared amplitudes cannot simply be added because of interference. The same result follows frequency by frequency: with round-trip phase $q=e^{2i{k}_l{L}_l}$, $L_l=2\mathrm{m}$, the total amplitude coefficient isFor real frequencies $|q|=1$, so $|1/3-q|=|1-q/3|$ and $|R|=1$. Thus the ideal lossless fixed-ended segment returns all incident finite-pulse energy even when individual echoes overlap; their relative phases must be retained.