= Solution
For a rightgoing <wave packet>, put $\xi=z-ct$ and assume $f'$ decays sufficiently rapidly. Then
$$
\Pi=-A\rho c f'(\xi),\qquad \mathcal H=A\rho c^2[f'(\xi)]^2.
$$
The shift charge becomes
$$
\boxed{Q_1=-A\rho c\int f'(\xi)d\xi=-A\rho c[f(+\infty)-f(-\infty)]}.
$$
It is time independent. It vanishes for a localized displacement packet that returns to the same displacement at both ends, but need not vanish for a localized strain pulse joining different asymptotic displacements.
For the dilation charge, substitute $z=\xi+ct$:
$$
Q_2=A\rho c\int(ct-z)[f'(\xi)]^2dz
=\boxed{-A\rho c\int\xi[f'(\xi)]^2d\xi}.
$$
The change of variables removes all time dependence. This charge also need not vanish; choosing an origin at the packet's <energy> centroid can make it zero. Conservation assumes the integral exists and there is no boundary flux, rather than merely assuming that the <energy> is finite.
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