Exact Solutions for Transient Wave Propagation in a Two-Dimensional Array of Particles

Abstract

We derive exact analytical solutions for transient waves propagating in an array of particles occupying the half-plane. x ≥0, forced by general external excitations applied at the top horizontal row x = 0. We consider only nearest neighbour interactions between the particles, resulting in four-particle interactions throughout the array. First, we assume spatially periodic forces and internal displacements, and apply Fourier and Z-transforms to obtain closed form transformed solutions. Next, we relax the spatial periodicity assumption by resorting to the following limiting analysis: We let the wavelength of the force distribution tend to infinity, and employ continuity arguments to obtain the exact transformed wave responses. These transformed solutions are Fourier-inverted in closed form by employing previous results of WANG and LEE [10]. We perform numerical simulations using the exact analytical formulas, and compute transient waves in the array due to a single transient force of trapezoidal form applied at the top row. Assuming spatially harmonic force we compute propagation and attenuation zones of the array. Finally, we consider a second limiting process by allowing the horizontal distances between particles to tend to zero. Performing rescalings and limiting calculations we derive an additional closed form solution for transient waves propagating in a half-plane continuum of borizontal strings transversely coupled by distributed elasticities, due to forces applied to the top string.