In this thesis the propagation and reflection of waves on a nonlinear hyperelastic string, using a general strain-energy function, is investigated. The Riemann problem for the longitudinal motion is investigated and an iteractive Riemann solver is proposed. Upto shock formation characteristic methods are used to investigate breakdown and envelopes of characteristics for fairly general initial, boundary value problems. Therafter a numerical algorithm is required and Godunov's scheme combined with the solutions to the Riemann problems are used to investigate solutions after shock formation. A number of physical examples have been examined.