Nonlinear waves on a two-part stretched string are considered as a reduced model of nonlinear waves in sea ice near a sudden change in ice properties. Geometric nonlinearity is retained, and the waves are confined to a wave packet. The weakly nonlinear approach results in three coupled nonlinear amplitude equations, one for incident, reflected, and transmitted waves. The waves create a local mean displacement at the interface of the two sides, which provides the primary nonlinear effect. This interfacial mean requires a time-history approach, treated here with Laplace transforms. The results show that the mean is positive when the stiffness decreases across the interface and negative otherwise. The magnitude of the mean displacement near the interface is much larger when the stiffness increases, as it would where ice increases in thickness.