Custom Problem 7

Find the normal mode configurations and frequencies for the first five modes of transverse vibration of a continuous string with tension T, mass density ρ, and length L given the boundary conditions that both ends are free. (They slide on frictionless rods that pass through massless rings at each end of the string.) These boundary conditions correspond to the condition that the first derivative of the displacement at the ends must be zero. Show that the lowest mode has the peculiar property of having infinite wavelength and zero frequency. In this mode, the string translates with constant velocity.