A system of non-strictly hyperbolic conservation laws arising in elasticity theory

where r = q~(u, v). This system models the propagation of forward longitudinal and transverse waves in a stretched elastic string which moves in a plane. The wave propagation problem on an idealized nonlinear string, admitting both forward and backward waves, leads to a closely related system of four conservation laws which we also solve. The feature of interest in system (1) is that the equations are non-strictly hyperbolic in the following sense. Introduce vector notation U = (u, v), F = (~b u, r v); then the system (1) can be differentiated to produce