Axial angular-momentum current of an elastic string ($L_x=\int \rho ({\psi }_y{\dot{\psi }}_z-{\psi }_z{\dot{\psi }}_y)dx$)

Rotating the two transverse displacement components around the string axis gives density ${\ell }_x=\rho ({\psi }_y{\dot{\psi }}_z-{\psi }_z{\dot{\psi }}_y)$ and flux ${\mathcal{J}}_x=-F({\psi }_y{\psi }_z^{\prime }-{\psi }_z{\psi }_y^{\prime })$. The pair of wave equations yields ${\partial }_t{\ell }_x+{\partial }_x{\mathcal{J}}_x=0$. The total charge is physical angular momentum about the axis; it is conserved only when the net endpoint torque flux vanishes.