Propagation of modal solutions in a smoothly inhomogeneous Timoshenko beam is considered. An asymptotic description is given to the interaction of high-frequency bending and shear modes that occurs if their phase velocities intersect at some point at a nonzero angle. A similar problem on the propagation of discontinuities is also considered. The results are presented in the generalized form, so that they have applications to problems on the interaction of other types of waves, e.g., water waves, electromagnetic waves, etc.