The techniques used in optics and microwave and radio physics for the easy and efficient calculation of reflection and transmission by stratified media are adapted to acoustic compressional waves. The method involves taking the product of N 2×2 matrices when the stratification is approximated by N layers. These layers can be chosen to have linear variation in the acoustic parameters to best represent the actual stratification without undue complexity in the resulting matrix elements. It is possible to guarantee unimodularity of the matrices, thus making sure that energy conservation and a reciprocity law are automatically satisfied. Accuracy is tested against an exactly solvable model stratification, in which the density and speed of sound both vary exponentially with depth.
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John Lekner (1990) studied this question.