The space-time evolution of beat wave generation is studied analytically and numerically. Electromagnetic cascading, collisional damping and relativistic frequency shift of the beat plasmon are taken into account in the model. In particular, detuning and dispersion effects are investigated. The achievable plasmon amplitude depends strongly on the collisional damping. At low electron temperatures, the induced beat wave follows the laser pulse and decays rapidly behind it. At high electron temperatures, amplitude modulation appears and an intense slowly decaying plasmon wake can be excited. The wake formation is controllable by varying the pulse length or by detuning the driver slightly off resonance. The amount of electromagnetic cascading is proportional to the plasmon amplitude and the propagation distance of the pulse. The EM spectra offer excellent diagnostics for beat wave experiments, because plasmon amplitude variations are directly reflected in them.