  The interaction of a weakly nonlinear Alfv\'enic pulse with an Alfv\'en speed
inhomogeneity in the direction perpendicular to the magnetic field is
investigated. Identically to the phase mixing experienced by a harmonic
Alfv\'en wave, sharp transverse gradients are generated in the pulse by the
inhomogeneity. In the initial stage of the evolution of an initially plane
Alfv\'enic pulse, the transverse gradients efficiently generate transversely
propagating fast magnetoacoustic waves. However, high resolution full MHD
numerical simulations of the developed stage of the pulse evolution show that
the generation saturates due to destructive wave interference. It is shown that
the weakly non-linear description of the generated fast magnetoacoustic wave is
well described by the driven wave equation proposed in Nakariakov, Roberts &
Murawski (1997), and a simple numerical code (2D MacCromack), which solves it
with minimal CPU resources produces identical results to those obtained from
the full MHD code ({\it Lare2d}, Arber et al. 2001). A parametric study of the
phenomenon is undertaken, showing that, contrary to one's expectations, steeper
inhomogenities of the Alfv\'en speed do not produce higher saturation levels of
the fast wave generation. There is a certain optimal gradient of the
inhomogeneity that ensures the maximal efficiency of the fast wave generation.
