Layers between two halfspaces

Let a stack of layers be sandwiched in between two halfspaces Figure 9.

 

8-9
Figure 9 Waves incident, reflected, and transmitted from a stack of layers between two half-spaces.

An impulse is incident from below. The backscattered wave is called C(Z) and the transmitted wave is called T(Z). We take these waves and load them into the multilayer equation (20).  

$$\left[ \begin{array} {c} 1 \\ C(Z) \end{array} \right] \e... ...t] \; \left[ \begin{array} {c} T(Z) \\ 0 \end{array} \right]$$ (21)

We may solve the first of (21) for the transmitted wave T(Z)  

$$T(Z) \eq {\sqrt{Z}^k \Pi t \over F(Z)}$$ (22)

and introduce the result back into the second of (21) to obtain the backscattered wave  

$$C(Z) \eq {G(Z)T(Z) \over \sqrt{Z}^k \Pi t} \eq {G(Z) \over F(Z)}$$ (23)

Let us be clear we understand the meaning of the solutions T(Z) and C(Z). They can be regarded two ways, either as functions of frequency $Z=e^{i\omega}$,or the polynomial coefficients (after dividing out the denominator F(Z)) as waveforms, as functions of time. Physical interpretations of F(Z) and G(Z) are on the right side of Figure 9.