GETTING THE WAVES FROM THE REFLECTION COEFFICIENTS

An important result of the last section was the development of a ``layer matrix" (14) that is, a matrix which can be used to extrapolate waves observed in one layer to the waves observed in the next layer. This process may be continued indefinitely. To see how to extrapolate from layer 1 to layer 3 substitute (14) with k = 1 into (14) with k = 2, obtaining

$$\begin{eqnarray} \left[ \begin{array} {c} U \\ D \end{array} \right] _3 & = &... ...ight] \; \left[ \begin{array} {c} U \\ D \end{array} \right] _1\end{eqnarray}$$ (19)

Inspection of this product suggests the general form for a product of k layer matrices. We'll call it a multilayer matrix.  

$$\left[ \begin{array} {c} U \\ D \end{array} \right] _{k+1... ...ght] \left[ \begin{array} {c} U \\ D \end{array} \right] _1$$ (20)

Starting from computer programs that add and multiply polynomials by manipulating their coefficients, it is easy enough to write a computer code to recursively build up the F(Z) and G(Z) polynomials by adding a layer at a time starting from F(Z)=1 and G(Z)=c₁.