In this communication we describe an original approach to numerical modeling for acoustic waves in a solid structure with a crack. The method allows us simulating the conditions of a real acoustic experiment for non-destructive testing and damage visualization. The presence of damage, for instance, in the form of a crack, means that it is necessary to set at the crack faces an additional boundary condition corres ponding to the physics of contact interaction. A most pertinent contact phenomena in this case is friction that typically causes hysteretic response. This problem is solved by generalizing the well-known Hertz-Mindlin solution for two effective bodies with axial symmetry that provide the same response as fragments of rough surfaces of a real crack. The solution can be written in a semi-analytical form valid for any variable external mechanical action, in particular, for an acoustic signal. Previously, this method was applied for solving several demonstration problems related to wave propagation in a medium with a crack and the corresponding non-linear generation. At the current stage, we turn to modeling for a real experiment on non-destructive testing, in which a standing wave is formed with a nonlinear component generated by an internal nonlinear contact present in the sample. The calculated distribution of higher harmonics on the sample surface clearly indicates the position and extent of damage located under the surface. It is expected that the developed numerical approach in the future will make it possible to reconstruct damage parameters more accurately in comparison to purely experimental methods. Indeed, a hypothesis on some certain geometric structure of damage can be accepted or rejected by comparing the calculated and experimental data. In addition, a vector character of the displacement field on the surface can carry information about the crack orientation. Moreover, an accurate reconstruction of the damage offers an opportunity of predicting its future evolution by using appropriate fracture mechanics techniques. Correspondingly, the ultimate goal of the study is the creation of an innovative computer-assisted diagnostics-prognostics method based on existing non-destructive testing technologies.