  In this paper, a self-consistent transverse beam dynamics framework is
demonstrated, that incorporates acceleration into the transverse beam dynamics
studies for a proton linac machine. Two focusing schemes are developed and
discussed; the FODO-like scheme, and the minimum aperture scheme. The FODO-like
scheme is a simple scheme, requiring only one quadrupole per cavity. The scheme
is analytically solved to minimise the beam size at the cavity entrance/exit
and ensures a constant beam size along the lattice, with respect to adiabatic
damping due to longitudinally accelerating rf cavities. The minimum aperture
scheme describes the regime that matches the beam ellipse to the acceptance
ellipse of a cavity, allowing for the smallest possible aperture, for a given
cavity length. A simple approximation of an rf cavity map is determined to
allow changes in particle energy along a lattice, and acceleration is assumed
only in the longitudinal direction.
