  We derive a corrected analytical solution for the propagation and enhanced
phase mixing of torsional Alfv\'en waves, in a potential magnetic field with
exponentially divergent field lines, embedded in a stratified solar corona.
Further we develop a code named TAWAS which calculates the analytic solution
describing torsional Alfv\'en waves using IDL software language. We then use
TAWAS to demonstrate that both our correction to the analytic solution and the
inclusion of wave reflection have a significant impact on Alfv\'en wave
damping. We continue to utilise TAWAS by performing a parameter study in order
to identify the conditions under which enhanced phase mixing is strongest. We
find that phase mixing is the strongest for high frequency Alfv\'en waves in
magnetic fields with highly divergent field lines and without density
stratification. We then present a finite difference solver, Wigglewave, which
solves the linearised evolution equations for the system directly. Comparing
solutions from TAWAS and Wigglewave we see that our analytical solution is
accurate within the limits of the WKB approximation but under-reports the wave
damping, caused by enhanced phase mixing, beyond the WKB limit. Both TAWAS and
Wigglewave solve the linearised governing equations and not the complete
nonlinear MHD equations. Paper II will consider simulations that solve the full
MHD equations including important nonlinear effects.
