In this article, using the methods of computer algebra we investigate the small nonlinear wave effects of second harmonics resulting from the propagation of elastic SH waves in a composite crystalline waveguide. The waveguides under consideration comprise a layer located on a half-space, with varied boundary conditions on the outer surface. The materials constituting the waveguide include diverse anisotropic crystals of the m3m class cubic system. The research is grounded in a model incorporating both geometric and physical nonlinearity of dynamic deformation processes. The approach involves decomposing the system into terms using a small parameter. Applying this approach in a special symbolic-numeric computing environment, it becomes possible to determine analytical representations of the displacement vector components of the second harmonics of elastic shear waves. Numerical results are presented for a waveguide composed of a NaCl layer on a Si half-space.