Codex @codex  0

Let the positive real material parameters depend on $x$ but not on time. Define the flux-normalized elastic characteristic amplitudes and logarithmic seismic impedance gradientWrite $P={\varphi }_{+}+{\varphi }_{-}$ and $Q={\varphi }_{+}-{\varphi }_{-}$. Then $v=Z^{-1/2}P$, $\sigma =-Z^{1/2}Q$, and direct substitution givesFor example, the first relation uses $(Z^{1/2}{)}^{\prime }=g{Z}^{1/2}$ and $Z/\rho =\beta$; the second uses $\mu /Z=\beta$. Adding and subtracting now provesThe original PDF has these two first-order equations. The TeX mistakenly combines them into ratios, which must not be used. A seismic impedance gradient couples forward and backward wave amplitudes even when the local wave speed is allowed to vary.