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We derive a corrected analytical solution for the propagation and enhancedphase mixing of torsional Alfv\'en waves, in a potential magnetic field withexponentially divergent field lines, embedded in a stratified solar corona.Further we develop a code named TAWAS which calculates the analytic solutiondescribing torsional Alfv\'en waves using IDL software language. We then useTAWAS to demonstrate that both our correction to the analytic solution and theinclusion of wave reflection have a significant impact on Alfv\'en wavedamping. We continue to utilise TAWAS by performing a parameter study in orderto identify the conditions under which enhanced phase mixing is strongest. Wefind that phase mixing is the strongest for high frequency Alfv\'en waves inmagnetic fields with highly divergent field lines and without densitystratification. We then present a finite difference solver, Wigglewave, whichsolves the linearised evolution equations for the system directly. Comparingsolutions from TAWAS and Wigglewave we see that our analytical solution isaccurate within the limits of the WKB approximation but under-reports the wavedamping, caused by enhanced phase mixing, beyond the WKB limit. Both TAWAS andWigglewave solve the linearised governing equations and not the completenonlinear MHD equations. Paper II will consider simulations that solve the fullMHD equations including important nonlinear effects.

Enhanced Phase Mixing of Torsional Alfv\'en Waves in Stratified and Divergent Solar Coronal Structures, Paper I: Linear Solutions