= Solution
Put $Z=\rho\beta=\sqrt{\rho\mu}>0$, the <seismic impedance>. In a uniform medium, define $w_+=v-\sigma/Z$ and $w_-=v+\sigma/Z$. The field equations give
$$
(\partial_t+\beta\partial_x)w_+=0,\qquad(\partial_t-\beta\partial_x)w_-=0.
$$
Thus $w_+=2F(x-\beta t)$ and $w_-=2G(x+\beta t)$, with arbitrary independent profiles $F,G$. Reconstruction gives
$$
\boxed{v=F+G,\qquad\sigma=-ZF+ZG.}
$$
The two characteristic components therefore have $v_+=F$, $\sigma_+=-Zv_+$ and $v_-=G$, $\sigma_-=Zv_-$.
The kinetic plus <elastic energy> <mass density> is $e=\rho v^2/2+\sigma^2/(2\mu)$. Multiplying the equations by $v$ and $\sigma/\mu$ yields $e_t=(v\sigma)_x$. Hence the <elastic-wave energy flux> is $S=-v\sigma$. For a pure forward wave $S_+=Zv_+^2\geq0$; for a pure backward wave $S_-=-Zv_-^2\leq0$. In the mixture the cross terms cancel, giving \b[$S=Z(v_+^2-v_-^2)$]. Strict sign holds for nonzero components.
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