Imagine a string of length L1 = L and mass per unit length, µ1 = µ is connected to a thick rope of length of length L2 = 2L/3 and mass per unit length, µ2 = 4µ (Fig. 1).
a) If the end of the string is oscillated, generating a sine wave with frequency ω1 moving to the left with speed, v1, what frequency wave moves through the thick rope? Does it have the same speed?
b) Show that in this case the wave moving through the rope has wavelength, λ2 = λ1/2 1
c) Imagine that the ends of both the rope and the string are held firmly in place. If the string has length, L1 = 2m, and is under tension, τ1 = 1 N, find the lowest frequency that generates a standing wave throughout the whole system (assume the mass/Length of the string is µ1 = 1/16 kg)
d) How many nodes are observed at this frequency? (sketch the standing wave at a fixed moment in time)