= Dilatation charge of a one-dimensional elastic wave
For constant $A,\rho,K$ on all space and time, the active waveform transformation $\psi_b(z,t)=\psi(bz,bt)$ leaves the <action> of $\mathcal L=A(\rho\dot\psi^2-K\psi'^2)/2$ invariant: the density gains $b^2$ and the integration measure gains $b^{-2}$. With <canonical momentum> $\Pi=A\rho\dot\psi$, its <Noether current> can be chosen as
$$
j_D^0=t\mathcal H+z\Pi\psi',\qquad
j_D^z=-tAK\dot\psi\psi'-z\mathcal H.
$$
The integrated charge $D=\int j_D^0dz$ is conserved if its boundary flux vanishes and the weighted integral exists. For a rightgoing <wave packet> $\psi=f(z-ct)$, $c=\sqrt{K/\rho}$, it becomes $D=-A\rho c\int\xi[f'(\xi)]^2d\xi$. A fixed finite interval generally breaks this scale symmetry unless its boundaries and boundary data are transformed too.
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