 Transverse momentum current of an elastic string

For $\mathcal{L}=\rho ({\dot{\psi }}_y^2+{\dot{\psi }}_z^2)/2-F(({\psi }_y^{\prime }{)}^2+({\psi }_z^{\prime }{)}^2)/2$, constant displacement symmetry in $y$ gives $j^0=\rho {\dot{\psi }}_y$ and $j^x=-F{\psi }_y^{\prime }$. The wave equation implies ${\partial }_t{j}^0+{\partial }_x{j}^x=0$. Its integrated charge is physical transverse momentum; endpoint forces can change that charge on a finite segment.