Halliday, Resnick & Walker — Waves–I: Problem Problem_93

Rock-climbing rescue. A stranded rock climber has hooked himself onto the bottom of a rope lowered from a cliff edge by a rescuer. The rope consists of two sections connected by a knot: The lower section has length \(L_1\) and linear density \(\mu_1\) and the upper section has length \(L_2 = 2L_1\) and linear density \(\mu_2 = 4\mu_1\). The climber happens to pluck the bottom end of the rope (as a "ready" signal) at the same time the rescuer plucks the top end. The mass of the rope sections is negligible compared to the mass of the climber. (a) What is the speed \(v_1\) of the pulse in section 1 in terms of the speed \(v_2\) of the pulse in section 2? (b) In terms of \(L_2\), at what distance below the rescuer do the two pulses pass through each other?