= Solution
Adding a constant to $\psi_y$ changes neither its time nor space derivatives, so the <action> is invariant. The associated Noether density and flux are
$$
j^0_y=\rho\dot\psi_y,\qquad j^x_y=-F\psi_y'.
$$
The <wave equation> gives $\partial_tj^0_y+\partial_xj^x_y=0$. Therefore
$$
\boxed{P_y=\int\rho\dot\psi_y\,dx,\qquad
\frac{dP_y}{dt}=F[\psi_y']_{\rm left}^{\rm right}.}
$$
This is total physical transverse momentum. It is conserved for a whole isolated infinite string with vanishing boundary flux, periodic <boundary conditions>, or endpoints exerting no net transverse force. For a finite subsystem momentum can flow through its ends, and fixed supports can exchange momentum with the string. Thus the local <conservation law> always holds in this model, while constancy of the integrated charge requires the stated <boundary condition>.
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