 Flux-normalized elastic characteristic amplitudes

These characteristic wave amplitudes split a one-dimensional elastic wave into forward and backward components. Their instantaneous net elastic-wave energy flux is ${\varphi }_{+}^2-{\varphi }_{-}^2$. For real-part harmonic phasors the mean flux is $(|{\varphi }_{+}{|}^2-|{\varphi }_{-}{|}^2)/2$. In a static graded medium with real positive seismic impedance, put $g=(\log Z{)}^{\prime }/2$; then ${\partial }_x{\varphi }_{\pm }\pm {\beta }^{-1}{\partial }_t{\varphi }_{\pm }=g{\varphi }_{\mp }$. Seismic impedance gradients couple the two directions while preserving net flux.