Exact mode shapes for a continuous tether

The vibrational problem of a tether modeled as a cable terminated by two end masses is considered (dumbbell configuration). For an inextensible cable the mode shapes are combinations of Legendre functions. The spacing of the frequencies depends on the mass ratios of the end masses and the cable mass. For the extensible cable, the exact vibrational problem is set up. The conditions under which approximations by more simple models are valid are derived.