A light elastic string is stretched to tension T0 between two fixed points A and B a distance 3a apart, and two particles of mass m are attached to the string at equally spaced intervals.
The strength of each of the three sections of the string is α. The system performs small oscillations in a plane through AB.
Without making any prior assumptions, prove that the particles oscillate longitudinally in two of the normal modes and transversely in the other two. Find the four normal frequencies.