The interactions between linear elastic waves and a nonlinear crack with finite compressibility are studied in the one-dimensional context. Numerical studies on a hyperbolic model of contact with sinusoidal forcing have shown that the mean values of the scattered elastic displacements are discontinuous across the crack. The mean dilatation of the crack also increases with the amplitude of the forcing levels. The aim of the present theoretical study is to analyse these nonlinear processes under a larger range of nonlinear jump conditions. For this purpose, the problem is reduced to a nonlinear differential equation. The dependence of the periodic solution on the forcing amplitude is quantified under sinusoidal forcing conditions. Bounds for the mean, maximum and minimum values of the solution are presented. Lastly, periodic forcing with a null mean value is addressed. In that case, only a result about the mean dilatation of the crack is obtained.