Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 4 / b / 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 areThe wave equation gives ${\partial }_t{j}_y^0+{\partial }_x{j}_y^x=0$. ThereforeThis 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.