A long taut spring of mass m, stiffness k and length L is stretched to length L + s and released. When longitudinal waves propagate along the spring, the equation of motion of a length δx may be...

A long taut spring of mass m, stiffness k and length L is stretched to length L + s and released. When longitudinal waves propagate along the spring, the equation of motion of a length δx may be written as mδx × ∂2u ∂t 2 = ∂F ∂x δx where u is the longitudinal displacement and F is the restoring force. Derive the equation of longitudinal motion of the spring and show that waves propagate along the length of the spring in both directions. Hence, derive an expression for the velocity of wave propagation along the spring.