The nonlinear evolution of linearly polarized, hydromagnetic waves propagating along the initial radial magnetic field in the solar wind is considered. Numerical solutions of the dissipationless, time-dependent equations of motion are used to study the steepening of the waves into hydromagnetic shocks and the subsequent propagation of the shocks. If the shocks propagate into a region in which the plasma beta (ratio of thermal to magnetic pressure) is greater than one, each shock splits into a second shock and a rotational discontinuity. It is shown that these secondary hydromagnetic shocks readily propagate through the rotational discontinuities without altering their structure. With increasing time the secondary hydromagnetic shocks propagate out ahead of the rotational discontinuities and become hydrodynamic shocks. Hence an initial packet of linearly polarized hydromagnetic waves ultimately evolves into a packet of rotational discontinuities preceeded by a packet of hydrodynamic shocks. The conditions for which the waves steepen into shocks, the time required for shock formation, and further implications of the results in the solar wind are discussed.