ON BLOCK MATRICES ASSOCIATED WITH DISCRETE TRIGONOMETRIC TRANSFORMS AND THEIR USE IN THE THEORY OF WAVE PROPAGATION

Block matrices associated with discrete Trigonometric transforms （DTT＇s） arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators with the Method of Moments, the Boundary Element Method, and the Method of Auxiliary Sources. The DTT＇s are represented by the Fourier, Hartley, Cosine, and Sine matrices, which are unitary and offer simultaneous diagonalizations of specific matrix algebras. The main tool for the investigation of the aforementioned wave applications is the efficient inversion of such types of block matrices. To this direction, in this paper we develop an efficient algorithm for the inversion of matrices with U-diagonalizable blocks （U a fixed unitary matrix） by utilizing the U- diagonalization of each block and subsequently a similarity transformation procedure. We determine the developed method＇s computational complexity and point out its high efficiency compared to standard inversion techniques. An implementation of the algorithm in Matlab is given. Several numerical results are presented demonstrating the CPU-time efficiency and accuracy for ill-conditioned matrices of the method. The investigated matrices stem from real-world wave propagation applications.

Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory

This paper is concerned with the wave propagation behavior of rotating functionally graded（FG）temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field.Uniform,linear and nonlinear temperature distributions across the thickness are investigated.Thermo-elastic properties of FG beam change gradually according to the Mori–Tanaka distribution model in the spatial coordinate.The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function.The governing equations are derived by Hamilton’s principle as a function of axial force due to centrifugal stiffening and displacement.The solution of these equations is provided employing a Galerkin-based approach which has the potential to capture various boundary conditions.By applying an analytical solution and solving an eigenvalue problem,the dispersion relations of rotating FG nanobeam are obtained.Numerical results illustrate that various parameters including temperature change,angular velocity,nonlocality parameter,wave number and gradient index have significant effects on the wave dispersion characteristics of the nanobeam under study.The outcome of this study can provide beneficial information for the next-generation research and the exact design of nano-machines including nanoscale molecular bearings,nanogears,etc.

Physical mechanism of seismic attenuation in a two-phase medium

High-frequency seismic attenuation is conventionally attributed to anelastic absorption. In this paper, I present three studies on high-frequency seismic attenuation and propose that the physical mechanism results from the interference of elastic microscopic multiple scattering waves. First, I propose a new theory on wave propagation in a two-phase medium which is based on the concept that the basic unit for wave propagation is a nano- mass point. As a result of the elasticity variations of pore fluid and rock framework, micro multiple scattering waves would emerge at the wavelength of the seismic waves passing through the two-phase medium and their interference and overlap would generate high- frequency seismic attenuation. Second, I present a study of the frequency response of seismic transmitted waves by modeling thin-layers with thicknesses no larger than pore diameters. Results indicate that high-frequency seismic waves attenuate slightly in a near-surface water zone but decay significantly in a near-surface gas zone. Third, I analyze the seismic attenuation characteristics in near-surface water and gas zones using dual-well shots in the Songliao Basin, and demonstrate that the high-frequency seismic waves attenuate slightly in water zones but in gas zones the 160-1600 Hz propagating waves decay significantly. The seismic attenuation characteristics from field observations coincide with the modeling results. Conclusions drawn from these studies theoretically support seismic attenuation recovery.