Abstract <p>The investigation of waves propagating in intricate structures encompassing diverse materials and diverse interfacial conditions holds immense significance in multiple fields, including geophysics, nondestructive testing, and sensor technology. Shear horizontal waves propagate out of plane, which is described by the direction of wave propagation and normal to the surface of the medium. The behavior of waves is influenced by material properties, the nature of bonding, and boundary conditions. The interior of the Earth is characterized by heterogeneity, stresses, and imperfect bonding between layers. Thus, the current study delves into examining the behavior of antiplane shear horizontal waves in an intricate geometrical structure comprising a heterogeneous layer lying on the initially stressed substrate. Perfect contact between materials with different properties is nearly impossible to achieve, thus bonding between the layer and the substrate was assumed to be imperfect. The imperfectness is modeled through various conditions including dislocation-like, force-like and spring-like conditions at the interfacial surface. These interfacial conditions are analyzed along with traction-free and rigidly fixed boundary conditions at the free surface of the layer. Dispersion relations are analytically derived under each scenario. Graphical representations are plotted to illustrate the impacts of various parameters, such as heterogeneity, initial stress, layer thickness, imperfectness, and jumping coefficients, on the propagation of antiplane shear horizontal waves.</p>