= Solution
Let the positive real material parameters depend on $x$ but not on time. Define the <flux-normalized elastic characteristic amplitudes> and logarithmic <seismic impedance> <gradient>
$$
\phi_\pm=\frac12\left(\sqrt Z\,v\mp\frac{\sigma}{\sqrt Z}\right),\qquad g=\frac12\frac{d\log Z}{dx}.
$$
Write $P=\phi_++\phi_-$ and $Q=\phi_+-\phi_-$. Then $v=Z^{-1/2}P$, $\sigma=-Z^{1/2}Q$, and direct substitution gives
$$
P_t=-\beta(Q_x+gQ),\qquad Q_t=-\beta(P_x-gP).
$$
For example, the first relation uses $(Z^{1/2})'=gZ^{1/2}$ and $Z/\rho=\beta$; the second uses $\mu/Z=\beta$. Adding and subtracting now proves
$$
\boxed{\partial_x\phi_++\beta^{-1}\partial_t\phi_+=g\phi_-,\qquad \partial_x\phi_--\beta^{-1}\partial_t\phi_-=g\phi_+.}
$$
The 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.
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