We prove the existence and find necessary and sufficient conditions for the uniqueness of the time-periodic solution to the equations $u_{tt}-{\Delta }_{x}u\pm F\left(u\right)=g(x,t)$ for an arbitrary (sufficiently smooth) periodic right-hand side $g$, where ${\Delta }_x$ denotes the Laplace operator with respect to $x\in \Omega \subset R^N,N\ge 1$, and $F$ is the Ishlinskii hysteresis operator. For $N=2$ this equation describes e.g. the vibrations of an elastic membrane in an elastico-plastic medium.