The temporal evolution of weakly nonlinear, plane, linearly polarized Alfvén pulses in a cold homogeneous plasma is investigated. A static initial pulse-like disturbance in transverse velocity produces two Alfvén pulses that travel in opposite directions along the magnetic field. The ponderomotive force of the two pulses produces a static shock in longitudinal velocity at the starting position. The travelling pulses form a shock front that is governed by the scalar Cohen–Kulsrud equation. We find good agreement between the analytical solutions we derive and the results from a fully nonlinear numerical MHD code.