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Differential equations of coupled longitudinal-flexural vibrations of a rod have been developed. Such a motion is excited by the longitudinal impact of a rigid body. The material of the rod may be ideally elastic or exhibit linear strain-hardening. The conditions of the propagation of longitudinal-flexural waves have been investigated for a homogeneous compression in the excitation zone. It has been shown that to each compression, i.e. to each impact velocity corresponds a certain wavelength of bending, for which the phase velocity decreases to zero and the group velocity increases infinitely. That is characteristic for instability in the dynamic sense. A confrontation of the formulas for the buckling wavelenegth with the test results described in [9], [11] shows a good agreement in the range of both elastic and plastic waves. In this second case a better agreement with the experiment is obtained by means of formula (4.5), in which instead of the strain-hardening modulus, the average tangent modulus is substituted.