Wave mode computing method using the step-split Padé parabolic equation

Models based on a parabolic equation(PE)can accurately predict sound propagation problems in range-dependent ocean waveguides.Consequently,this method has developed rapidly in recent years.Compared with normal mode theory,PE focuses on numerical calculation,which is difficult to use in the mode domain analysis of sound propagation,such as the calculation of mode phase velocity and group velocity.To broaden the capability of PE models in analyzing the underwater sound field,a wave mode calculation method based on PE is proposed in this study.Step-split Pade PE recursive matrix equations are combined to obtain a propagation matrix.Then,the eigenvalue decomposition technique is applied to the matrix to extract sound mode eigenvalues and eigenfunctions.Numerical experiments on some typical waveguides are performed to test the accuracy and flexibility of the new method.Discussions on different orders of Padéapproximant demonstrate angle limitations in PE and the missing root problem is also discussed to prove the advantage of the new method.The PE mode method can be expanded in the future to solve smooth wave modes in ocean waveguides,including fluctuating boundaries and sound speed profiles.

Spatial-temporal characterization of the San Andreas Fault by fault-zone trapped waves at seismic experiment site,Parkfield,California

In this article,we review our previous research for spatial and temporal characterizations of the San Andreas Fault(SAF)at Parkfield,using the fault-zone trapped wave(FZTW)since the middle 1980s.Parkfield,California has been taken as a scientific seismic experimental site in the USA since the 1970s,and the SAF is the target fault to investigate earthquake physics and forecasting.More than ten types of field experiments(including seismic,geophysical,geochemical,geodetic and so on)have been carried out at this experimental site since then.In the fall of 2003,a pair of scientific wells were drilled at the San Andreas Fault Observatory at Depth(SAFOD)site;the main-hole(MH)passed a~200-m-wide low-velocity zone(LVZ)with highly fractured rocks of the SAF at a depth of~3.2 km below the wellhead on the ground level(Hickman et al.,2005;Zoback,2007;Lockner et al.,2011).Borehole seismographs were installed in the SAFOD MH in 2004,which were located within the LVZ of the fault at~3-km depth to probe the internal structure and physical properties of the SAF.On September 282004,a M6 earthquake occurred~15 km southeast of the town of Parkfield.The data recorded in the field experiments before and after the 2004 M6 earthquake provided a unique opportunity to monitor the co-mainshock damage and post-seismic heal of the SAF associated with this strong earthquake.This retrospective review of the results from a sequence of our previous experiments at the Parkfield SAF,California,will be valuable for other researchers who are carrying out seismic experiments at the active faults to develop the community seismic wave velocity models,the fault models and the earthquake forecasting models in global seismogenic regions.

Guided Wave Propagation in Multilayered Two-dimensional Quasicrystal Plates with Imperfect Interfaces

An analytical solution of the guided wave propagation in a multilayered twodimensional decagonal quasicrystal plate with imperfect interfaces is derived.According to the elastodynamic equations of quasicrystals(QCs),the wave propagating problem in the plate is converted into a linear control system by employing the state-vector approach,from which the general solutions of the extended displacements and stresses can be obtained,These solutions along the thickness direction are utilized to derive the propagator matrix which connects the physical variables on the lower and upper interfaces of each layer.The special spring model,which describes the discontinuity of the physical quantities across the interface,is introduced into the propagator relationship of the multilayered structure.The total propagator matrix can be used to propagate the solutions in each interface and each layer about the multilayered plate.In addition,the traction-free boundary condition on the top and bottom surfaces of the laminate is considered to obtain the dispersion equation of wave propagation,Finally,typical numerical examples are presented to illustrate the marked influences of stacking sequence and interface coeficients on the dispersion curves and displacement mode shapes of the QC laminates.

Bending Analysis of Functionally Graded One-Dimensional Hexagonal Piezoelectric Quasicrystal Multilayered Simply Supported Nanoplates Based on Nonlocal Strain Gradient Theory

In this study,the nonlocal strain gradient theory is adopted to investigate the static bending deformation of a functionally graded(FG)multilayered nanoplate made of onedimensional hexagonal piezoelectric quasicrystal(PQC)materials subjected to mechanical and electrical surface loadings.The FG materials are assumed to be exponential distribution along the thickness direction.Exact closed-form solutions of an FG PQC nanoplate including nonlocality and strain gradient micro-size dependency are derived by utilizing the pseudo-Stroh formalism.The propagator matrix method is further used to solve the multilayered case by assuming that the layer interfaces are perfectly contacted.Numerical examples for two FG sandwich nanoplates made of piezoelectric crystals and PQC are provided to show the influences of nonlocal parameter,strain gradient parameter,exponential factor,length-to-width ratio,loading form,and stacking sequence on the static deformation of two FG sandwich nanoplates,which play an important role in designing new smart composite structures in engineering.