We report results of our study for an explosive behavior of nonlinear ultrasound occurring when acoustic nonlinearity is efficiently modulated by another physical process. In the considered example, a Lamb wave propagates in an antiferromagnetic plate in the presence of a harmonically oscillating magnetic field and a shear resonance. In this situation, the magnetic pumping induces a backward phase conjugate Lamb wave. Thus the system supports three phonons: two of the Lamb waves and one of the shear resonant mode.

The appropriate classical Hamiltonian contains a term describing the interaction of three waves with complex conjugate amplitudes. Correspondingly, the quantum counterpart of the Hamiltonian includes the product of three phonon creation operators. Therefore, it can be expected that the Lamb waves together with the shear resonant mode will be amplified with a huge factor. The amplification occurs in a two-phonon system as well i.e. when the modulation affects not the nonlinear parameter but the linear modulus. However, the amplitudes growth in the two-phonon system is exponential in time as it should be for a classical parametric amplification process, whereas the reported three-phonon system experiences an explosive instability. The explosive behavior means that in the beginning of the process the amplitudes grow exponentially and, soon after that, the positive feedback from the shear resonance induces a mathematical singularity in the time dependency of amplitudes. In other words, if the magnetic pumping is fixed and can not be depleted, the amplitudes become theoretically infinite at some finite moment of time after switching on the pumping. The corresponding synchronism condition is required which equates the magnetic pump modulation frequency to the sum of the double frequency of the Lamb wave and the shear resonance frequency. It is of interest to note that the effect of the additional resonance pumping is analogous to the action of the Feshbach resonance observed in ultra-cold gazes.

In this study, we write out the equations of motions for the considered exemplar system using the Hamiltonian formalism which is most suitable for the demonstration of the quantum analogy and of the Feshbach-type resonance effect. Then we theoretically analyze the stationary solution and formulate the conditions for the instability that occurs when the stationary solution is not possible. Finally, we numerically solve the equations of motion and illustrate the explosive dynamics with a number of examples. It is shown that the explosive scenario can occur with a very low signal level i.e. Lamb waves amplitudes comparable to the spontaneous noise in the system. From the practical point, we propose an extremely effective channel for converting magnetic energy into mechanical energy. The considered nonlinearity modulation mechanism is possible to extend onto systems of different physical nature and to apply in acousto-electronics, electro- and hydrodynamics and in microsystems designing.