Hadamard compatibility condition implies existence of supersonic shock waves in bi-modular materials

It is known that the propagation of signals in linearly elastic rods cannot exceed the so called “rod” velocity; a similar restriction holds for nonlinearly elastic rods for which the rod velocity may depend upon strain. Herein, it is shown that the Hadamard compatibility condition applied to the propagation of acoustic waves in bi-modular rods implies (i) the appearance of discontinuities in strain, stress and rod velocity, known as strong shocks; and (ii) the instantaneous propagation velocity of these shocks can be unbounded, exceeding the maximal rod velocity. The existence of instantaneous supersonic velocities of strong shocks is demonstrated in the propagation of a harmonic acoustic wave.