For a rightgoing wave packet, put

ξ = z - c t

and assume

f^{'}

decays sufficiently rapidly. Then

Π = - A ρ c f^{'} (ξ), H = A ρ c^{2} [f^{'} (ξ)]^{2} .

(1)

The shift charge becomes

Q_{1} = - A ρ c \int f^{'} (ξ) d ξ = - A ρ c [f (+ \infty) - f (- \infty)] .

(2)

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 = ξ + c t

Q_{2} = A ρ c \int (c t - z) [f^{'} (ξ)]^{2} d z = - A ρ c \int ξ [f^{'} (ξ)]^{2} d ξ .

(3)

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.