Simplified Impedance Analysis

The above results are quickly derived from the general reflection-coefficient for force waves (or voltage waves, pressure waves, etc.):

$$\zbox{\rho = \frac{R_2-R_1}{R_2+R_1} = \frac{\mbox{Impedance Step}}{\mbox{Impedance Sum}}}$$

where $\rho =$ reflection coefficient of impedance $R_2$ as ``seen'' from impedance $R_1$

When a force wave crosses from impedance $R_1$ to $R_2$ it splits into

1. a reflected wave $f^{{-}}=\rho f^{{+}}$ in $R_1$, and
2. a transmitted wave $(1+\rho)f^{{+}}$ in $R_2$

Therefore, a velocity wave $v^{+}$ splits into

1. reflected wave $v^{-}= - \rho v^{+}$ and
2. transmitted wave $(1-\rho)v^{+}$

These relations are of course unchanged in the Laplace domain, by linearity of the Laplace transform.