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Artykuły w czasopismach na temat „One-Way equations”

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Data publikacji: 1 lutego 2025

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1

Bschorr, Oskar, i Hans-Joachim Raida. "Factorized One-Way Wave Equations". Acoustics 3, nr 4 (9.12.2021): 717–22. http://dx.doi.org/10.3390/acoustics3040045.

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The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical finite element (FE) and finite difference (FD) calculations. For seismic applications, a huge number of methods have been developed to reduce these disturbances, but none of these attempts have prevailed to date. However, a priori factorization of the longitudinal wave equation for inhomogeneous media eliminates the above-mentioned ambiguity, and the resulting one-way equations provide the definition of the wave propagation direction by the geometric position of the transmitter and receiver.

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2

Halpern, Laurence, i Lloyd N. Trefethen. "Wide‐angle one‐way wave equations". Journal of the Acoustical Society of America 84, nr 4 (październik 1988): 1397–404. http://dx.doi.org/10.1121/1.396586.

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3

Lee, Myung W., i Sang Y. Suh. "Optimization of one‐way wave equations". GEOPHYSICS 50, nr 10 (październik 1985): 1634–37. http://dx.doi.org/10.1190/1.1441853.

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The theory of wave extrapolation is based on the square‐root equation or one‐way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square‐root equation represents waves propagating in one direction only.

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4

Towne, Aaron, i Tim Colonius. "One-way spatial integration of hyperbolic equations". Journal of Computational Physics 300 (listopad 2015): 844–61. http://dx.doi.org/10.1016/j.jcp.2015.08.015.

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5

Bschorr, Oskar, i Hans-Joachim Raida. "One-Way Wave Equation Derived from Impedance Theorem". Acoustics 2, nr 1 (10.03.2020): 164–71. http://dx.doi.org/10.3390/acoustics2010012.

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The wave equations for longitudinal and transverse waves being used in seismic calculations are based on the classical force/moment balance. Mathematically, these equations are 2nd order partial differential equations (PDE) and contain two solutions with a forward and a backward propagating wave, therefore also called “Two-way wave equation”. In order to solve this inherent ambiguity many auxiliary equations were developed being summarized under “One-way wave equation”. In this article the impedance theorem is interpreted as a wave equation with a unique solution. This 1st order PDE is mathematically more convenient than the 2nd order PDE. Furthermore the 1st order wave equation being valid for three-dimensional wave propagation in an inhomogeneous continuum is derived.

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6

Sheinman, Izhak, i Yeoshua Frostig. "Constitutive Equations of Composite Laminated One-Way Panels". AIAA Journal 38, nr 4 (kwiecień 2000): 735–37. http://dx.doi.org/10.2514/2.1025.

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7

Sheinman, Izhak, i Yeoshua Frostig. "Constitutive equations of composite laminated one-way panels". AIAA Journal 38 (styczeń 2000): 735–37. http://dx.doi.org/10.2514/3.14474.

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8

You, Jiachun, Ru-Shan Wu i Xuewei Liu. "One-way true-amplitude migration using matrix decomposition". GEOPHYSICS 83, nr 5 (1.09.2018): S387—S398. http://dx.doi.org/10.1190/geo2017-0625.1.

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To meet the requirement of true-amplitude migration and address the shortcomings of the classic one-way wave equations on the dynamic imaging, one-way true-amplitude wave equations were developed. Migration methods, based on the Taylor or other series approximation theory, are introduced to solve the one-way true-amplitude wave equations. This leads to the main weakness of one-way true-amplitude migration for imaging the complex or strong velocity — contrast media — the limited imaging angles. To deal with this issue, we apply a matrix decomposition method to accurately calculate the square-root operator and impose the boundary conditions of the one-way true-amplitude wave equations. Our migration method and the conventional one-way true-amplitude Fourier finite-difference (FFD) migration method are used by us to test and compare the imaging performance. The impulse responses in a strong velocity-contrast model prove that our migration method works for larger imaging angles than the one-way true-amplitude FFD method. The amplitude calculations in a strong-lateral velocity variation media with one reflector and in the Marmousi model demonstrate that our migration method provides better amplitude-preserving performance and offers higher structural imaging quality than the one-way true-amplitude FFD method. We also use field data to indicate the imaging enhancement and the feasibility of our method compared with the one-way true-amplitude FFD method. Our one-way true-amplitude migration method using matrix decomposition fully exploits the features of one-way true-amplitude wave equations with less approximation, and it is capable of producing more accurate amplitude estimations and potentially wider imaging angles.

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9

Bérczes, Attila, J. Ködmön i Attila Pethő. "A one-way function based on norm form equations". Periodica Mathematica Hungarica 49, nr 1 (2004): 1–13. http://dx.doi.org/10.1023/b:mahu.0000040535.45427.38.

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10

Chen, Jing-bo, i Shu-yuan Du. "Multisymplectic Structures and Discretizations for One-way Wave Equations". Letters in Mathematical Physics 79, nr 2 (15.11.2006): 213–20. http://dx.doi.org/10.1007/s11005-006-0119-x.

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11

Пачев, У. М., А. Х. Кодзоков, А. Г. Езаова, А. А. Токбаева i З. Х. Гучаева. "On One Way to Solve Linear Equations Over a Euclidean Ring". Вестник КРАУНЦ. Физико-математические науки 46, nr 1 (8.03.2024): 9–21. http://dx.doi.org/10.26117/2079-6641-2024-46-1-9-21.

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Линейным уравнениям, т.е. уравнениям первой степени, а также системам из таких уравнений уделяется большое внимание как в алгебре, так в теории чисел. Наибольший интерес представляет случай таких уравнений с целыми коэффициентами и при этом их нужно решать в целых числах. Такие уравнения с указанными условиями называют линейными диофантовыми уравнениями. Еще Эйлер рассматривал способы решения линейных диофантовых уравнений с двумя неизвестными, причем один из этих способов был основан на применении алгоритма Евклида. Другой способ решения таких уравнений, основанный на цепных дробях, применялся также Лагранжем. Более удобным и перспективным оказался способ Эйлера, чем способ цепных дробей. В настоящей работе рассматривается один новый способ решения линейных уравнений над евклидовым кольцом, основанный на сравнениях по подходящим модулям. Известный ранее матричный метод решения таких уравнений с увеличением числа неизвестных является довольно громоздким в виду того, что он связан с нахождением обратных к унимодулярным целочисленным матрицам. Существенным в нашем способе решения линейных уравнений над евклидовым кольцом является использование алгоритма Евклида и линейного представления НОД элементов в евклидовом кольце. Доказанная в работе теорема применяется к нахождению решения линейного уравнения с тремя неизвестными над кольцом целых гауссовых чисел, являющимся, как известно, евклидовым кольцом. В заключении приводятся замечания о возможных путях дальнейшего развития изложенного исследования. Linear equations, i.e. Equations of the first degree, as well as systems of such equations, receive much attention both in algebra and in number theory. Of greatest interest is the case of such equations with integer coefficients, and in this case they need to be solved in integers. Such equations with the specified conditions are called linear Diophantine equations. Euler also considered ways to solve linear Diophantine equations with two unknowns, and one of these methods was based on the use of the Euclid algorithm. Another method for solving such equations, based on continued fractions, was also used by Lagrange. Euler’s method turned out to be more convenient and promising than the method of continued fractions. In this paper, we consider one new method for solving linear equations over a Euclidean ring, based on comparisons over suitable moduli. The previously known matrix method for solving such equations with an increasing number of unknowns is quite cumbersome due to the fact that it is associated with finding the inverses of unimodular integer matrices. Essential in our method of solving linear equations over a Euclidean ring is the use of the Euclidean algorithm and the linear GCD representation of elements in the Euclidean ring. The theorem proved in the work is applied to finding a solution to a linear equation in three unknowns over a ring of Gaussian integers, which, as is known, is a Euclidean ring. In conclusion, comments are made on possible ways of further development of the presented research.

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12

Wapenaar, C. P. A. "Representation of seismic sources in the one‐way wave equations". GEOPHYSICS 55, nr 6 (czerwiec 1990): 786–90. http://dx.doi.org/10.1190/1.1442892.

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One‐way extrapolation of downgoing and upgoing acoustic waves plays an essential role in the current practice of seismic migration (Berkhout, 1985; Stolt and Benson, 1986; Claerbout, 1985; Gardner, 1985). Generally, one‐way wave equations are derived for the source‐free situation. Sources are then represented as boundary conditions for the one‐way extrapolation problem. This approach is valid provided the source representation is done with utmost care. For instance, it is not correct to represent a monopole source by a spatial delta function and to use this as input data for a standard one‐way extrapolation scheme. This yields an erroneous directivity pattern as illustrated below.

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13

Hakimov, Abdusalom, Baxiyor Hayitovich Ungarov i Maftuna Abdinazarova. "The roots of some algebraic equations one way to determine". ACADEMICIA: An International Multidisciplinary Research Journal 11, nr 9 (2021): 255–59. http://dx.doi.org/10.5958/2249-7137.2021.01904.2.

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14

Towne, Aaron, i Tim Colonius. "Efficient jet noise models using the one-way Euler equations". Journal of the Acoustical Society of America 136, nr 4 (październik 2014): 2081. http://dx.doi.org/10.1121/1.4899470.

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15

Wetton, Brian T. R., i Gary H. Brooke. "One‐way wave equations for seismoacoustic propagation in elastic waveguides". Journal of the Acoustical Society of America 87, nr 2 (luty 1990): 624–32. http://dx.doi.org/10.1121/1.398931.

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16

Dong, Liangguo, Zhongyi Fan, Hongzhi Wang, Benxin Chi i Yuzhu Liu. "Correlation-based reflection waveform inversion by one-way wave equations". Geophysical Prospecting 66, nr 8 (26.07.2018): 1503–20. http://dx.doi.org/10.1111/1365-2478.12668.

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17

Porter, Michael B., Carlo M. Ferla i Finn B. Jensen. "The problem of energy conservation in one‐way wave equations". Journal of the Acoustical Society of America 86, S1 (listopad 1989): S54. http://dx.doi.org/10.1121/1.2027554.

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18

Kamel, A. H. "Time‐domain behavior of wide‐angle one‐way wave equations". GEOPHYSICS 56, nr 3 (marzec 1991): 382–84. http://dx.doi.org/10.1190/1.1443054.

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The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

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19

Zhu, Min, i Aaron Towne. "Recursive one-way Navier-Stokes equations with PSE-like cost". Journal of Computational Physics 473 (styczeń 2023): 111744. http://dx.doi.org/10.1016/j.jcp.2022.111744.

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20

Vivas, Flor A., i Reynam C. Pestana. "True-amplitude one-way wave equation migration in the mixed domain". GEOPHYSICS 75, nr 5 (wrzesień 2010): S199—S209. http://dx.doi.org/10.1190/1.3478574.

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One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

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21

Bernstein, Alexander, Richard Rand i Robert Meller. "The Dynamics of One Way Coupling in a System of Nonlinear Mathieu Equations". Open Mechanical Engineering Journal 12, nr 1 (30.04.2018): 108–23. http://dx.doi.org/10.2174/1874155x01812010108.

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Background: This paper extends earlier research on the dynamics of two coupled Mathieu equations by introducing nonlinear terms and focusing on the effect of one-way coupling. The studied system of n equations models the motion of a train of n particle bunches in a circular particle accelerator. Objective: The goal is to determine (a) the system parameters which correspond to bounded motion, and (b) the resulting amplitudes of vibration for parameters in (a). Method: We start the investigation by examining two coupled equations and then generalize the results to any number of coupled equations. We use a perturbation method to obtain a slow flow and calculate its nontrivial fixed points to determine steady state oscillations. Results: The perturbation method reveals the existence of an upper bound on the amplitude of steady state oscillations. Conclusion: The model predicts how many bunches may be included in a train before instability occurs.

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22

Sun, Yan Jun, Chang Ming Liu, Hai Yu Li i Zhe Yuan. "One-Way Function Construction Based on the MQ Problem and Logic Function". Applied Mechanics and Materials 220-223 (listopad 2012): 2360–63. http://dx.doi.org/10.4028/www.scientific.net/amm.220-223.2360.

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Multivariate quadratic based public-key cryptography called MQ problem which based on calculation of a secure cryptography of multivariate equations and MQ cryptography security is based on the difficulty of the solution of multivariate equations. But computer and mathematician scientists put a lot of effort and a long time to research MQ cryptography and they have proved that MQ cryptography is NP complete problem. Therefore, before the P problem Equal to the NP problem we do not figure out selected multivariate equations by random in polynomial time. So we can use this feature to construct the relative safety method of the public key encryption. A new type of public-key cryptosystem has been brought up in this paper that one-way shell core function which has such advantages as more security and flexibility, and provides a more inclusive public-key cryptosystem.

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23

Godin, Oleg A. "Three‐dimensional, energy‐conserving, and reciprocal one‐way acoustic wave equations". Journal of the Acoustical Society of America 102, nr 5 (listopad 1997): 3149. http://dx.doi.org/10.1121/1.420705.

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Godin, Oleg A. "On reciprocity and energy conservation for one‐way acoustic wave equations". Journal of the Acoustical Society of America 100, nr 4 (październik 1996): 2835. http://dx.doi.org/10.1121/1.416693.

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25

Trefethen, Lloyd N., i Laurence Halpern. "Well-posedness of one-way wave equations and absorbing boundary conditions". Mathematics of Computation 47, nr 176 (1986): 421. http://dx.doi.org/10.1090/s0025-5718-1986-0856695-2.

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26

Fishman, Louis. "Derivation of one‐way wave equations for vector wave propagation problems". Journal of the Acoustical Society of America 89, nr 4B (kwiecień 1991): 1895. http://dx.doi.org/10.1121/1.2029420.

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27

Charara, Marwan, i Albert Tarantola. "Boundary conditions and the source term for one‐way acoustic depth extrapolation". GEOPHYSICS 61, nr 1 (styczeń 1996): 244–52. http://dx.doi.org/10.1190/1.1443945.

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The one‐way acoustic wave equations can be derived, using eigenvalue decomposition of the two‐way wave equation in the Fourier domain, in such a manner that the source term and the free‐surface boundary condition are explicitly introduced. The proposed form of the one‐way wave equations is well adapted to seismic reflection modeling because it allows a pressure field recorded at the surface to be extrapolated directly in depth. A numerical example illustrates the appropriate implementation of the source term and the free‐surface boundary conditions. A comparison with a two‐way modeling shows a good agreement of the computed wave fields.

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28

Berjamin, Harold. "On the accuracy of one-way approximate models for nonlinear waves in soft solids". Journal of the Acoustical Society of America 153, nr 3 (marzec 2023): 1924–32. http://dx.doi.org/10.1121/10.0017681.

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Simple strain-rate viscoelasticity models of isotropic soft solid are introduced. The constitutive equations account for finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation. A nonlinear viscous wave equation for the shear strain is obtained exactly and corresponding one-way Burgers-type equations are derived by making standard approximations. Analysis of the travelling wave solutions shows that these partial differential equations produce distinct solutions, and deviations are exacerbated when wave amplitudes are not arbitrarily small. In the elastic limit, the one-way approximate wave equation can be linked to simple wave theory and shock wave theory, thus, allowing direct error measurements.

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29

Yusupov, A. K., Kh M. Muselemov i R. I. Vishtalov. "Calculation models of beams with one-way connections". Herald of Dagestan State Technical University. Technical Sciences 51, nr 4 (20.01.2025): 236–48. https://doi.org/10.21822/2073-6185-2024-51-4-236-248.

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Objective. The article considers the operational features of beam structures with one-way ties. Method. By studying the operational features of beam structure supports, analytical and graphical dependencies between beam displacements and their support reactions are derived. The properties of generalized functions are used to describe the nonlinearities that occur during structural deformations. Differential equations for transverse bending of beams are presented, taking into account nonlinearities and methods for their solution. Result. Design and calculation schemes have been developed that take into account the operation of beam structure supports, allowing us to obtain precise values of deflections and internal forces of sections on this basis, which ensures the necessary reliability of the structure. The results are given in the form of analytical expressions, graphs, and tables. Conclusion. Calculation models of beams with one-way connections can find wide application in the design of beam structures with supports with one-way connections.

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30

Angus, D. A. "True amplitude corrections for a narrow-angle one-way elastic wave equation". GEOPHYSICS 72, nr 2 (marzec 2007): T19—T26. http://dx.doi.org/10.1190/1.2430694.

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Wavefield extrapolators using one-way wave equations are computationally efficient methods for accurate traveltime modeling in laterally heterogeneous media, and have been used extensively in many seismic forward modeling and migration problems. However, most leading-order, one-way wave equations do not simulate waveform amplitudes accurately and this is primarily because energy flux is not accounted for correctly. I review the derivation of a leading-order, narrow-angle, one-way elastic wave equation for 3D media. I derive correction terms that enable energy-flux normalization and introduce a new higher-order, narrow-angle, one-way elastic wave extrapolator. By implementing these correction terms, the new true amplitude wave extrapolator allows accurate amplitude estimates in the presence of strong gradients. I present numerical examples for 1D velocity transition models to show that (1) the leading-order, narrow-angle propagator accurately models traveltimes, but overestimates transmitted- or primary-wave amplitudes and (2) the new amplitude corrected narrow-angle propagator accurately models both the traveltimes and amplitudes of all forward-traveling waves.

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Khoma, H., Viktor Chornyi i Svitlana Khoma-Mohylska. "About one way of construction of t-periodic solutions to hyperbolic type equations". Visnyk of Zaporizhzhya National University. Physical and Mathematical Sciences, nr 1 (2018): 153–60. http://dx.doi.org/10.26661/2413-6549-2018-1-15.

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Zhang, Yu, Guanquan Zhang i Norman Bleistein. "True amplitude wave equation migration arising from true amplitude one-way wave equations". Inverse Problems 19, nr 5 (5.09.2003): 1113–38. http://dx.doi.org/10.1088/0266-5611/19/5/307.

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Cheng, Yingda, Ching-Shan Chou, Fengyan Li i Yulong Xing. "$L^2$ stable discontinuous Galerkin methods for one-dimensional two-way wave equations". Mathematics of Computation 86, nr 303 (3.03.2016): 121–55. http://dx.doi.org/10.1090/mcom/3090.

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McCoy, John J., i L. Neil Frazer. "Pseudodifferential operators, operator orderings, marching algorithms and path integrals for one-way equations". Wave Motion 9, nr 5 (wrzesień 1987): 413–27. http://dx.doi.org/10.1016/0165-2125(87)90030-8.

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35

ZHU, JIANXIN, i YA YAN LU. "VALIDITY OF ONE-WAY MODELS IN THE WEAK RANGE DEPENDENCE LIMIT". Journal of Computational Acoustics 12, nr 01 (marzec 2004): 55–66. http://dx.doi.org/10.1142/s0218396x0400216x.

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Numerical solutions of the Helmholtz equation and the one-way Helmholtz equation are compared in the weak range dependence limit, where the overall range distance is increased while the range dependence is weakened. It is observed that the difference between the solutions of these two equations persists in this limit. The one-way Helmholtz equation involves a square root operator and it can be further approximated by various one-way models used in underwater acoustics. An operator marching method based on the Dirichlet-to-Neumann map and a local orthogonal transform is used to solve the Helmholtz equation.

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Zhang, Yu, Guanquan Zhang i Norman Bleistein. "Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration". GEOPHYSICS 70, nr 4 (lipiec 2005): E1—E10. http://dx.doi.org/10.1190/1.1988182.

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One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By “true-amplitude” one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in the standard wave-equation migration imaging condition. The boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation. When the full wave-form solutions are replaced by their ray-theoretic approximations, the imaging formula reduces to the common-shot Kirchhoff inversion formula. In this sense, the migration is true amplitude as well. On the other hand, this new method retains all of the fidelity features of wave equation migration. Computer output using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data must be collected from a single common-shot experiment. Multiexperiment data, such as common-offset data, cannot be used with this method as presently formulated.

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Javanmardi, Nasrin, i Parviz Ghadimi. "Hydroelastic analysis of surface-piercing propeller through one-way and two-way coupling approaches". Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment 233, nr 3 (3.08.2018): 844–56. http://dx.doi.org/10.1177/1475090218791617.

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Semi-submerged operation of surface-piercing propellers can cause severe structural stress and deflection due to the high-rate variation of the hydrodynamic loads on each blade in one cycle of revolution. Therefore, a proper hydroelastic analysis regarding these propellers seems imperative. Accordingly, in this article, hydrodynamic performance of a surface-piercing propeller is studied by considering the structural flexibility. The one-way and two-way coupled unsteady Reynolds-averaged Navier–Stokes equations (finite volume method used for the fluid analysis) and direct finite element method (used for the structural analysis) are utilized to analyze an 821-b surface-piercing propeller model. The obtained ventilation patterns and hydrodynamic loads on a single blade of this propeller are compared with the published experimental data. ANSYS multiphysics solvers are used to conduct the targeted simulations. The results are presented for different velocity ratios in the range of [0.8, 1.2] and Froude number Fr = 6.0. The blade maximum deformation and von Mises stress are presented in one cycle of revolution. Based on the obtained results, one may conclude that both the adopted approaches predict the propeller transient hydrodynamic characteristics with good accuracy, compared with the experimental data. However, the two-way approach displays lower maximum structural displacement and stress than one-way simulation. This is attributed to the fact that the two-way approach takes into consideration the added mass and damping effects of the surrounding fluid around the blade.

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Sun, Weijia, i Li-Yun Fu. "Compensation for transmission losses based on one-way propagators in the mixed domain". GEOPHYSICS 77, nr 3 (1.05.2012): S65—S72. http://dx.doi.org/10.1190/geo2011-0460.1.

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Most true-amplitude migration algorithms based on one-way wave equations involve corrections of geometric spreading and seismic [Formula: see text] attenuation. However, few papers discuss the compensation of transmission losses (CTL) based on one-way wave equations. Here, we present a method to compensate for transmission losses using one-way wave propagators for a 2D case. The scheme is derived from the Lippmann-Schwinger integral equation. The CTL scheme is composed of a transmission term and a phase-shift term. The transmission term compensates amplitudes while the wave propagates through subsurfaces. The transmission term is a function of the vertical wavenumbers of two adjacent heterogeneous screens. The phase-shift term is a Fourier finite-difference (FFD) propagator implemented in a mixed domain via Fourier transform. The transmission term can be flexibly incorporated into the conventional phase-shift migration algorithm, i.e., FFD, at every depth step. We analyze the effects of frequency, lateral velocity contrast, and vertical velocity ratio on the accuracy of the presented formulae. Numerical examples from a flat model and a fault model with lateral velocity variations are presented to demonstrate the ability of the proposed scheme for compensation of transmission losses.

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Naghadeh, Diako Hariri, i Mohamad Ali Riahi. "One-way wave-equation migration in log-polar coordinates". GEOPHYSICS 78, nr 2 (1.03.2013): S59—S67. http://dx.doi.org/10.1190/geo2012-0229.1.

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We obtained acoustic wave and wavefield extrapolation equations in log-polar coordinates (LPCs) and tried to enhance the imaging. To achieve this goal, it was necessary to decrease the angle between the wavefield extrapolation axis and wave propagation direction in the one-way wave-equation migration (WEM). If we were unable to carry it out, more reflection wave energy would be lost in the migration process. It was concluded that the wavefield extrapolation operator in LPCs at low frequencies has a large wavelike region, and at high frequencies, it can mute the evanescent energy. In these coordinate systems, an extrapolation operator can readily lend itself to high-order finite-difference schemes; therefore, even with the use of inexpensive operators, WEM in LPCs can clearly image varied (horizontal and vertical) events in complex geologic structures using wide-angle and turning waves. In these coordinates, we did not encounter any problems with reflections from opposing dips. Dispersion played important roles not only as a filter operator but also as a gain function. Prestack and poststack migration results were obtained with extrapolation methods in different coordinate systems, and it was concluded that migration in LPCs can image steeply dipping events in a much better way when compared with other methods.

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Amazonas, Daniela, Rafael Aleixo, Gabriela Melo, Jörg Schleicher, Amélia Novais i Jessé C. Costa. "Including lateral velocity variations into true-amplitude common-shot wave-equation migration". GEOPHYSICS 75, nr 5 (wrzesień 2010): S175—S186. http://dx.doi.org/10.1190/1.3481469.

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In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

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Pelykh, V., i Y. Taistra. "On the null one-way solution to Maxwell equations in the Kerr space-time". Mathematical Modeling and Computing 5, nr 2 (1.12.2018): 201–6. http://dx.doi.org/10.23939/mmc2018.02.201.

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Dolgaya, A. A., H. R. Zainulabidova, A. S. Mozzhukhin, Sh Sh Nazarova, A. A. Nazarov, G. V. Sorokina i A. M. Uzdin. "About one way to integrate the equations of seismic vibrations of seismically isolated structures". Herald of Dagestan State Technical University. Technical Sciences 51, nr 2 (27.07.2024): 190–96. http://dx.doi.org/10.21822/2073-6185-2024-51-2-190-196.

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Objective. The purpose of the study is to develop a new method of analyzing seismic vibrations of seismically isolated structures.Method. The structure is modeled by a system with one degree of freedom with a nonlinear stiffness characteristic. In the absence of damping an analytical solution of oscillation equations on the phase plane is obtained. In this case the connection between speeds and displacements is accurate, and the law of uniformly variable motion is used to calculate displacements.Result. To take into account the attenuation, the law of change in mechanical energy at each integration interval is used.Conclusion. An example of calculating a seismically isolated system with a sliding belt and a spring with a linearly varying characteristic of stiffness is considered. The influence of dry friction and viscous damping on the displacement and acceleration of the system is analyzed.

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Zhong, Jize, i Zili Xu. "An energy method for flutter analysis of wing using one-way fluid structure coupling". Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 231, nr 14 (14.09.2016): 2560–69. http://dx.doi.org/10.1177/0954410016667146.

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In this paper, an energy method for flutter analysis of wing using one-way fluid structure coupling was developed. To consider the effect of wing vibration, Reynolds-averaged Navier–Stokes equations based on the arbitrary Lagrangian Eulerian coordinates were employed to model the flow. The flow mesh was updated using a fast dynamic mesh technology proposed by our research group. The pressure was calculated by solving the Reynolds-averaged Navier–Stokes equations through the SIMPLE algorithm with the updated flow mesh. The aerodynamic force for the wing was computed using the pressure on the wing surface. Then the aerodynamic damping of the wing vibration was computed. Finally, the flutter stability for the wing was decided according to whether the aerodynamic damping was positive or not. Considering the first four modes, the aerodynamic damping for wing 445.6 was calculated using the present method. The results show that the aerodynamic damping of the first mode is lower than the aerodynamic damping of higher order modes. The aerodynamic damping increases with the increase of the mode order. The flutter boundary for wing 445.6 was computed using the aerodynamic damping of the first mode in this paper. The calculated flutter boundary is consistent well with the experimental data.

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Austin, Joe Dan, i H. J. Vollrath. "Representing, Solving, and Using Algebraic Equations". Mathematics Teacher 82, nr 8 (listopad 1989): 608–12. http://dx.doi.org/10.5951/mt.82.8.0608.

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Students of beginning algebra are quickly expected to solve linear equations. The solution procedures are generally abstract, involving the manipulation of numbers and algebraic symbols. Many students, even after completing a year of algebra, do not understand variables, equations, and solving equations (cf. Carpenter et al. [1982]). One way to help students learn to solve equations is to use physical objects, diagrams, and then symbols to represent equations. (Bruner [1964, 1967] calls such representations enactive (concrete), iconic (pictorial), and symbolic.) Although solving equations symbolically is essential, many students can benefit from working with physical problems that can also be symbolized mathematically. This article describes one way for students to learn to solve certain linear equations using pan balances, diagrams, and then symbols.

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Gustavson, Richard, i Sarah Rosen. "A Reduction Algorithm for Volterra Integral Equations". PUMP Journal of Undergraduate Research 6 (30.05.2023): 172–91. http://dx.doi.org/10.46787/pump.v6i0.3631.

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An integral equation is a way to encapsulate the relationships between a function and its integrals. We develop a systematic way of describing Volterra integral equations – specifically an algorithm that reduces any separable Volterra integral equation into an equivalent one in operator-linear form, i.e., one that only contains iterated integrals. This serves to standardize the presentation of such integral equations so as to only consider those containing iterated integrals. We use the algebraic object of the integral operator, the twisted Rota-Baxter identity, and vertex-edge decorated rooted trees to construct our algorithm.

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Bleistein, Norman, Yu Zhang i Guanquan Zhang. "Analysis of the neighborhood of a smooth caustic for true-amplitude one-way wave equations". Wave Motion 43, nr 4 (kwiecień 2006): 323–38. http://dx.doi.org/10.1016/j.wavemoti.2006.01.003.

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Zhu, Rui, Da-duo Chen i Shi-wei Wu. "Unsteady Flow and Vibration Analysis of the Horizontal-Axis Wind Turbine Blade under the Fluid-Structure Interaction". Shock and Vibration 2019 (31.03.2019): 1–12. http://dx.doi.org/10.1155/2019/3050694.

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A 1.5 MW horizontal-axis wind turbine blade and fluid field model are established to study the difference in the unsteady flow field and structural vibration of the wind turbine blade under one- and two-way fluid-structure interactions. The governing equations in ﬂuid field and the motion equations in structural were developed, and the corresponding equations were discretized with the Galerkin method. Based on ANSYS CFX fluid dynamics and mechanical structural dynamics calculation software, the effects of couplings on the aerodynamic and vibration characteristics of the blade are compared and analyzed in detail. Results show that pressure distributions at different sections of the blade are concentrated near the leading edge, and the leeward side of two-way coupling is slightly higher than that of one-way coupling. Deformation along the blade span shows a nonlinear change under the coupling effect. The degree of amplitude attenuation in two-way coupling is significantly greater than that in one-way coupling because of the existence of aerodynamic damping. However, the final amplitude is still higher than the one-way coupling. The Mises stress fluctuation in the windward and leeward sides is more obvious than one-way coupling, and the discrepancy must not be ignored.

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Namazian, Z., A. F. Najafi i S. M. Mousavian. "Numerical Simulation of Particle-Gas Flow Through a Fixed Pipe, Using One-Way and Two-Way Coupling Methods". Journal of Mechanics 33, nr 2 (15.07.2016): 205–12. http://dx.doi.org/10.1017/jmech.2016.53.

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AbstractA numerical simulation of the particle-gas flow in a vertical turbulent pipe flow was conducted. The main objective of the present article is to investigate the effects of dispersed phase (particles) on continuous phase (gas). In so doing, two general forms of Eulerian-Lagrangian approaches namely, one-way (when the fluid flow is not affected by the presence of the particles) and two-way (when the particles exert a feedback force on the fluid) couplings were used to describe the equations of motion of the two-phase flow. Gas-phase velocities which are within the order of magnitude as that of particles, volume fraction, and particle Stokes number were calculated and the results were subsequently compared with the available experimental data. The simulated results show that when the particles are added, the fluid velocity is attenuated. With an increase in particle volume fraction, particle mass loading and Stokes number, velocity attenuation also increases. Moreover, the results indicate that an increase in particle Stokes number reduces the special limited particle volume fraction, according to which one-way coupling method yields plausible results. The results have also indicated that the significance of particle fluid interaction is not merely a function of volume fraction and particle Stokes number.

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Wu, Bangyu, Ru-Shan Wu i Jinghuai Gao. "Preliminary Investigation of Wavefield Depth Extrapolation by Two-Way Wave Equations". International Journal of Geophysics 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/968090.

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Most of the wavefield downward continuation migration approaches are relying on one-way wave equations, which move the seismic energy always in one direction along depth. The one-way downward continuation migrations only use the primaries for imaging and do not treat secondary reflections recorded on the surface correctly. In this paper, we investigate wavefield depth extrapolators based on the full acoustic wave equations, which can propagate wave components to opposite directions. Several two-way wavefield downward continuation propagators are numerically tested in this study. Recursively implementing of the depth extrapolator makes it necessary and important to eliminate the unstable wave modes, that is, evanescent waves. For the laterally varying velocity media, distinction between the propagating and evanescent wave mode is less clear. We demonstrate that the spatially localized two-way beamlet propagator is an effective way to remove the evanescent waves while maintain the propagating mode in laterally inhomogeneous media.

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Golikov, Pavel, i Alexey Stovas. "Traveltime parameters in tilted transversely isotropic media". GEOPHYSICS 77, nr 6 (1.11.2012): C43—C55. http://dx.doi.org/10.1190/geo2011-0457.1.

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Traveltime parameters define the coefficients of the Taylor series for traveltime or traveltime squared as a function of offset. These parameters provide an efficient tool for analyzing the effect of the medium parameters for short- and long-offset reflection moveouts. We derive the exact equations for one-way and two-way traveltime parameters in a homogeneous transversely isotropic medium with the tilted symmetry axis (TTI). It is shown that most of the one-way traveltime parameters in TTI differ from the two-way traveltime parameters, and we observe strong dependence of all traveltime parameters on tilt. The equations for traveltime parameters are extended to a vertically heterogeneous TTI medium, and weak-anisotropy and weak-anellipticity approximations are considered. We also apply the exact and approximate equations to invert the traveltime parameters into the model parameters for different acquisition setups. Using the traveltime parameters in a weak-anisotropy approximation, our tests show that an analytical inversion is not applicable, whereas the numerical inversion with exact equations yields a good accuracy for strongly anisotropic models.

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