In the Laser Wakefield Accelerator (LWFA), an intense picosecond laser pulse propagates through a plasma and nonresonantly generates large amplitude axial and transverse wakefields. The intense axial electric field associated with these wake plasma waves could be used for accelerating electron beams to very high energies. The propagation of such short laser pulses through a plasma and the generation of the wakefields are studied with numerical simulations. The formulation is a three-dimensional cold fluid model with self-consistent density perturbation and relativistic focusing effects. The laser pulse shape is found to be modified which in turn affects the wakefields.