Magnetic field reconnection is a macroscopic energy-conversion and transport process which operates in current sheets. Here we investigate analytically a physico-mathematical model of reconnection in a 2-D configuration consisting of a current sheet separating two different plasma regions with antiparallel magnetic field orientations. Reconnection is initiated in a localized region of the current sheet known as the diffusion region. The disruption of the current sheet generates MHD waves, which propagate the local disturbances into the system at large. We apply the MHD equations to describe and analyse the macroscopic response of the plasma and magnetic field to an imposed reconnection rate, which is generally a function of time. The dissipative process leading to disruption is not specified, and instead an arbitrary reconnection rate is used as an initial-boundary condition for solving the MHD equations. Analysis is restricted to an incompressible plasma, and to the case of weak reconnection, which implies that the magnetic field component perpendicular to the current sheet remains small relative to the field strength specified in the initial configuration. Time dependency and the inclusion of different plasma parameter values and field strengths on opposite sides of the current sheet lead to new features which are not evident in Petschek's analysis of steady-state reconnection in a symmetric current sheet configuration. These features include an asymmetric evolution of the outflow region on opposite sides of the current sheet, and a shift of the reconnection line into the region of higher field strength. The outflow region is partially bounded by a tangential discontinuity as a result of the different propagation speeds of the shocks and the surface waves. Once the reconnection rate drops to zero, i.e. reconnection is no longer active, the outflow region splits at the reconnection line and propagates like two solitary waves moving in opposite directions along the current sheet. The previous shift of the reconnection line now corresponds to a displacement of the current sheet downstream of the outflow region. Although no more flux is reconnected at this stage, the outflow region continues to grow in size and gathers up an increasingly large volume of reconnected plasma. The growth is eventually confined to a stretching of the outflow region along the current sheet. With velocity shear there is an additional asymmetry in the evolution on opposite sides of the reconnection line, and the conditions for the appearance and outward propagation of MHD waves are shown to be related to the criterion for Kelvin-Helmholtz stability of the current sheet.