In this paper we study the propagation of longitudinal magnetoelastic waves in a rod with damage. It is shown that for a stationary magnetic field the system of equations of magnetoelasticity can be reduced to one evolution equation with respect to the function of longitudinal deformation. The equation comprises variants of generalized unperturbed Burgers equations, when the medium does not have conductivity. For these equations, solutions have been found in the form of stationary shock waves. The connection between the main parameters (amplitude, width of the front) of the shock wave and the parameters of the system have been established. The influence of the damage parameters and the elastic nonlinearity of the material on the width of the front of the shock wave is determined. The evolutionary equation of magnetoelasticity has been investigated by an approximate method, when the medium is conductive. The influence of the conductivity parameters of the medium and material damage on the amplitudes of the first and second harmonics of the decomposition has been analyzed.