When the second order differential equation governing transmission of waves through a potential barrier is solved approximately, two approximate solutions arise within the barrier, one exponentially large and one exponentially small. When a linear combination of these solutions is considered, the error involved in the exponentially large solution is much larger than the actual smaller solution, leading to conceptual difficulties as to how this small solution is to be interpreted within the linear combination. Here, a new approach is adopted, whereby linear combinations of approximate solutions are avoided. The reflection coefficient of the barrier is derived and the series expansion of its modulus is obtained before approximations are introduced. The analysis is so arranged that ratios rather than linear combinations enter this modulus, and error analysis then shows exactly why the error consists of certain unexpected exponentially small terms rather than expected terms of larger order of magnitude.