  The suppression of transverse wakefield effects using transversely elliptical
drive beams in a planar structure is studied with a simple analytical model
that unveils the geometric nature of this phenomenon. By analyzing the
suggested model we derive scaling laws for the amplitude of the longitudinal
and transverse wake potentials as a function of the Gaussian beam ellipticity -
$\sigma_x/a$. We explicitly show that in a wakefield accelerator application it
is beneficial to use highly elliptical beams for mitigating transverse forces
while maintaining the accelerating field. We consider two scaling strategies:
1) aperture scaling, where we keep a constant charge to have the same
accelerating gradient as in a cylindrical structure and 2) charge scaling,
where aperture is the same as in the cylindrical structure and charge is
increased to match the gradient.
