Solution
= Solution
At the zero-slope boundary, the rightgoing and leftgoing slopes cancel, so $A_r=A_i$. Consequently
$$
\boxed{r_\psi=+1,\qquad P_r=P_i,\qquad P_t=0}.
$$
This <free-end reflection of a transverse string wave> uses a <Neumann boundary condition>: the displacement has an antinode and is not forced to vanish. The instantaneous flux $-T\psi_x\psi_t$ through the boundary is zero, confirming that it transmits no <power>.