  In this paper, wakefields driven by a relativistic electron beam in a cold
homogeneous plasma is studied using 2-D fluid simulation techniques. It has
been shown that in the limit when the transverse size of a rigid beam is
greater than the longitudinal extension, the wake wave acquires purely an
electrostatic form and the simulation results show a good agreement with the
1-D results given by Ratan et al. [Phys. Plasmas, 22, 073109 (2015)]. In the
other limit, when the transverse dimensions are equal or smaller than the
longitudinal extension, the wake waves are electromagnetic in nature.
Furthermore, a linear theoretical analysis of 2-D wakefields for a rigid
bi-parabolic beam has also been done and compared with the simulations. It has
also been shown that the transformer ratio which a key parameter that measures
the efficiency in the process of acceleration, becomes higher for a 2-D system
(i.e. for a beam having a smaller transverse extension compared to longitudinal
length) than the 1-D system (beam having larger transverse extension compared
to longitudinal length). Furthermore, including the self-consistent evolution
of the driver beam in the simulation, we have seen that the beam propagating
inside the plasma undergoes the transverse pinching which occurs much earlier
than the longitudinal modification. Due to the presence of transverse
dimensions in the system the 1-D rigidity limit given by Tsiklauri et al.
[Phys. Plasmas, 25, 032114 (2018)] gets modified. We have also demonstrated the
modified rigidity limit for the driver beam in a 2-D beam-plasma system.
