scholarly journals Multiple Scattering of P1 Waves by Arbitrarily Arranged Cavities in Saturated Soils

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Miaomiao Sun ◽  
Huajian Fang ◽  
Xiaokang Zheng ◽  
Ru Zhang ◽  
Shimin Zhang ◽  
...  
Keyword(s):  
Wave Function ◽  
Multiple Scattering ◽  
Expansion Method ◽  
Wave Theory ◽  
Saturated Soil ◽  
Displacement Amplitude ◽  
Function Expansion ◽  
Wave Function Expansion Method ◽  
Hexagonal Arrangement ◽  
Complex Coefficients

Based on Biot’s saturated soil wave theory, using wave function expansion method, theoretical solutions of multiple scattering of plain P1 waves are achieved by rows of cavities as barrier with arbitrarily arranged cavities in saturated soil. Undetermined complex coefficients after wave function expansion are obtained by cavities-soil stress and displacement free boundary conditions. Numerical examples are used to investigate variation of dimensionless displacement amplitude at the back and force of cavities barrier under P1 wave incident, and it is also discussed that the main parameters influenced isolation effect such as scattering orders, separation of cavities, distances between cavity rows, numbers of cavities, and arrangement of barriers. The results clearly demonstrate optimum design proposals with rows of cavities: with the multiple scattering order increases, the displacement amplitude tends to converge and the deviation caused by subsequent scattering cannot be neglected; it will obtain higher calculation accuracy when the order of scattering is truncated at m=4; it is considered to select 2.5≤sp/as≤3.0 and 2.5≤h/as≤3.5, while designing cavity spacing and row-distance, respectively. The isolation properties of elastic waves with rectangular arrangement (counterpoint) are weaker than that with hexagonal arrangement (counterchanged) when the row-distance of barrier is uniform.

scholarly journals Surface Motion of a Half-Space Containing an Elliptical-Arc Canyon under Incident SH Waves

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1884
Author(s):  
Hui Qi ◽  
Fuqing Chu ◽  
Jing Guo ◽  
Runjie Yang
Keyword(s):  
Wave Function ◽  
Half Space ◽  
Wave Scattering ◽  
Scattering Problem ◽  
Expansion Method ◽  
Numerical Results ◽  
Mathieu Function ◽  
Axis Ratio ◽  
Function Expansion ◽  
Wave Function Expansion Method

The existence of local terrain has a great influence on the scattering and diffraction of seismic waves. The wave function expansion method is a commonly used method for studying terrain effects, because it can reveal the physical process of wave scattering and verify the accuracy of numerical methods. An exact, analytical solution of two-dimensional scattering of plane SH (shear-horizontal) waves by an elliptical-arc canyon on the surface of the elastic half-space is proposed by using the wave function expansion method. The problem of transforming wave functions in multi-ellipse coordinate systems was solved by using the extra-domain Mathieu function addition theorem, and the steady-state solution of the SH wave scattering problem of elliptical-arc depression terrain was reduced to the solution of simple infinite algebra equations. The numerical results of the solution are obtained by truncating the infinite equation. The accuracy of the proposed solution is verified by comparing the results obtained when the elliptical arc-shaped depression is degraded into a semi-ellipsoidal depression or even a semi-circular depression with previous results. Complicated effects of the canyon depth-to-span ratio, elliptical axis ratio, and incident angle on ground motion are shown by the numerical results for typical cases.

scholarly journals Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 246
Author(s):  
Hui Qi ◽  
Fuqing Chu ◽  
Yang Zhang ◽  
Guohui Wu ◽  
Jing Guo
Keyword(s):  
Wave Function ◽  
Stress Concentration ◽  
Wave Scattering ◽  
Dynamic Stress ◽  
Scattering Problem ◽  
Soil Layer ◽  
Expansion Method ◽  
Dynamic Stress Concentration ◽  
Sh Wave ◽  
Wave Function Expansion Method

Wave diffusion in the composite soil layer with the lined tunnel structure is often encountered in the field of seismic engineering. The wave function expansion method is an effective method for solving the wave diffusion problem. In this paper, the wave function expansion method is used to present a semi-analytical solution to the shear horizontal (SH) wave scattering problem of a circular lined tunnel under the covering soil layer. Considering the existence of the covering soil layer, the great arc assumption (that is, the curved boundary instead of the straight-line boundary) is used to construct the wavefield in the composite soil layer. Based on the wave field and boundary conditions, an infinite linear equation system is established by adding the application of complex variable functions. The finite term is intercepted and solved, and the accuracy of the solution is analyzed. Although truncation is inevitable, due to the Bessel function has better convergence, a smaller truncation coefficient can achieve mechanical accuracy. Based on numerical examples, the influence of SH wave incident frequency, soil parameters, and lining thickness on the dynamic stress concentration factor of lining is analyzed. Compared with the SH wave scattering problem by lining in a single medium half-space, due to the existence of the cover layer and the influence of its stiffness, the dynamic stress of the lining can be increased or inhibited. In addition, the lining thickness has obvious different effects on the dynamic stress concentration coefficient of the inner and outer walls of different materials.

The molecular orbital theory of chemical valency IX. The interaction of paired electrons in chemical bonds

1951 ◽  
Vol 210 (1101) ◽  
pp. 190-206 ◽  
Keyword(s):  
Wave Function ◽  
Molecular Orbital ◽  
Wave Theory ◽  
Molecular Orbital Theory ◽  
Orbital Theory ◽  
Coupled Equations ◽  
Function Expansion ◽  
Orbital Function ◽  
Group A ◽  
Approximate Wave

In previous parts of this series the molecular orbital theory has been developed in a way which, while dealing satisfactorily with the interaction of electrons in different orbitals, is not adequate to describe the wave theory of two paired electrons in the same orbital. This paper is an attempt to improve this part of the theory. Most of the work is restricted to the problem of two electrons in the bond of a homonuclear diatomic molecule. The method is based on expansions of the wave function and the interelectronic repulsion term of the Hamiltonian over the irreducible representations of the symmetry group. A series of coupled equations are obtained for the terms of the wave-function expansion. These exact equations form a useful background against which to examine some approximate wave functions, including the simple molecular orbital function and the electron-pair function. Some approximate calculations on the hydrogen molecule indicate that inclusion of higher terms in the wave-function expansion substantially reduces the calculated electron repulsion energy.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

scholarly journals Influence of Bearing on Pier Failure Considering the Separation Condition under Near-Fault Earthquake

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 692
Author(s):  
Wenjun An ◽  
Guquan Song
Keyword(s):  
Expansion Method ◽  
Transient Wave ◽  
Earthquake Excitation ◽  
Near Fault ◽  
Continuous Beam Bridge ◽  
Wave Function Expansion Method ◽  
Earthquake Action ◽  
Stiffness And Damping ◽  
Beam Bridge ◽  
Bending Failure

To study the influence of the near-fault vertical earthquake, the beam-spring-damper-pier model is used to simulate the double-span continuous beam bridge. The transient wave function expansion method and the indirect mode function method are used to calculate the seismic response of the bridge. The theoretical solutions of the contact force and displacement response of the bridge under vertical earthquake excitation near-fault are derived. By using piers with three different heights, the influence of vertical separation on pier-bending failure is analyzed reasonably. The results show that under the near-fault earthquake action, the split has a certain influence on the pier failure. Moreover, the stiffness and damping of the bearing have an influence on the pier failure, and the change of the maximum pier height has different effects. Therefore, for bridges of different sizes, it is of great significance to select the appropriate stiffness and damping bearings to reduce pier failure.

Average wave function method for multiple scattering

1986 ◽  
Vol 84 (3) ◽  
pp. 1373-1378 ◽  
Author(s):  
Harjinder Singh ◽  
Dalcio K. Dacol ◽  
Herschel Rabitz
Keyword(s):  
Wave Function ◽  
Multiple Scattering ◽  
Function Method ◽  
Average Wave

scholarly journals Unified double- and single-sided homogeneous Green’s function representations

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160162 ◽  
Author(s):  
Kees Wapenaar ◽  
Joost van der Neut ◽  
Evert Slob
Keyword(s):  
Green’S Function ◽  
Multiple Scattering ◽  
Green's Function ◽  
Time Reversal ◽  
Wave Theory ◽  
Boundary Integral ◽  
Open Boundary ◽  
Quantum Mechanical ◽  
Holographic Imaging ◽  
Scattering Time

In wave theory, the homogeneous Green’s function consists of the impulse response to a point source, minus its time-reversal. It can be represented by a closed boundary integral. In many practical situations, the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only. The inherent approximations are acceptable as long as the effects of multiple scattering are negligible. However, in case of strongly inhomogeneous media, the effects of multiple scattering can be severe. We derive double- and single-sided homogeneous Green’s function representations. The single-sided representation applies to situations where the medium can be accessed from one side only. It correctly handles multiple scattering. It employs a focusing function instead of the backward propagating Green’s function in the classical (double-sided) representation. When reflection measurements are available at the accessible boundary of the medium, the focusing function can be retrieved from these measurements. Throughout the paper, we use a unified notation which applies to acoustic, quantum-mechanical, electromagnetic and elastodynamic waves. We foresee many interesting applications of the unified single-sided homogeneous Green’s function representation in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green’s function retrieval.

The Wave Theory of the Electron

1928 ◽  
Vol 24 (4) ◽  
pp. 501-505 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Commutative Algebra ◽  
Wave Theory ◽  
Wave Functions ◽  
Linear Differential Equations ◽  
Wave Mechanics ◽  
First Order ◽  
Linear Differential

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

Analytical Solutions of Two Space-Time Fractional Nonlinear Models Using Jacobi Elliptic Function Expansion Method

2021 ◽  
pp. 173-188
Author(s):  
Zillur Rahman ◽  
M. Zulfikar Ali ◽  
Harun-Or-Roshid ◽  
Mohammad Safi Ullah
Keyword(s):  
Elliptic Function ◽  
Nonlinear Models ◽  
Applied Mathematics ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Space Time ◽  
Jacobi Elliptic Function ◽  
Function Expansion ◽  
Partial Models ◽  
Jacobi Elliptic Function Expansion

In this manuscript, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM) models have been investigated which are frequently arises in nonlinear optics, solid states, fluid mechanics and shallow water. Jacobi elliptic function expansion integral technique has been used to build more innovative exact solutions of the s-tfEW and s-tfWBBM nonlinear partial models. In this research, fractional beta-derivatives are applied to convert the partial models to ordinary models. Several types of solutions have been derived for the models and performed some new solitary wave phenomena. The derived solutions have been presented in the form of Jacobi elliptic functions initially. Persevering different conditions on a parameter, we have achieved hyperbolic and trigonometric functions solutions from the Jacobi elliptic function solutions. Besides the scientific derivation of the analytical findings, the results have been illustrated graphically for clear identification of the dynamical properties. It is noticeable that the integral scheme is simplest, conventional and convenient in handling many nonlinear models arising in applied mathematics and the applied physics to derive diverse structural precise solutions.

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