Using Integro-Differential Equations to Model the Propagation of Seismic Waves Through a Barrier with a Memory Effect

Introduction. The present paper considers the propagation of seismic waves through a barrier with a memory effect based on integro-differential equations. Conventional wave models, founded upon elastic equations, frequently neglect the viscoelastic characteristics of actual soils and seismic barriers, which possess the capacity to “remember” prior deformations. To achieve a more accurate description of the phenomenon, an integro-differential model with an exponential memory kernel is employed. This model allows for the modelling of a wide range of dissipative effects and the derivation of analytical solutions applicable to seismic protection problems.

Materials and methods. The model is predicated on integro-differential equations of motion, which take into account the deformation history and material relaxation. Direct and inverse Fourier and Laplace transforms are applied in order to obtain analytical solutions. Two forms of pulses are investigated: the delta function and the Gaussian pulse.

Results. In the context of a delta pulse, supplementary “tails” and bursts are formed within the medium that exhibits memory, with the kernel parameters (α and β) exerting an influence on the rate of “forgetting” and its intensity. In the case of a Gaussian pulse, the introduction of memory effects results in a more gradual blurring of the waveform, accompanied by the acquisition of additional distortions, particularly at high values of α and during slow memory decay β. It has been demonstrated that by manipulating the values of α and β, a substantial alteration in the nature of the interaction can be achieved, resulting in either a sharp local peak or a more uniform distribution, characterized by significant energy dissipation.

Conclusions. This study demonstrates the importance of taking into account the “memory effect” when modelling seismic barriers. Integro-differential equations with an exponential kernel facilitate a more precise description of the processes of attenuation, energy dissipation and transformation of seismic wave shape in real ground conditions. The analytical solutions obtained from this study form a foundation for the design of more efficient seismic barriers, capable of “tuning” to the required range of vibration frequencies.

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