The solution of the wave equation for a plane stratified medium is obtained by using a fourth‐order accurate numerical integration technique. The general solution thus obtained is described in terms of four characteristic waves, two upgoing and two downgoing. A compact expression for the coupling coefficients Γij(i≠j), which is suitable for numerical calculation, is also derived. These techniques are then used in a quantitative study of transmission coefficients, coupling factors, and wave structures through the crossover region where the proton whistlers are generated. The negligible influence of collisions upon the transmitted energies in the two upgoing modes is discussed. The results are then compared with previous works in which the thin‐slab full‐wave method