Gaussian wave packets in inhomogeneous media with curved interfaces

1987 ◽  
Vol 412 (1842) ◽  
pp. 93-123 ◽  
Keyword(s):  
Wave Equation ◽  
Large Scale ◽  
Evolution Equations ◽  
Wave Packets ◽  
Present Theory ◽  
Orthogonal Direction ◽  
Time Harmonic ◽  
Method Of Solution ◽  
Classical Wave Equation ◽  
The Time Domain

Time-dependent particle-like pulses are considered as asymptotic solutions of the classical wave equation. The wave packets are localized in space with gaussian envelopes. The pulse centres propagate along the rays of the wave equation, and the envelope parameters satisfy evolution equations very similar to the ray equations for time-harmonic disturb­ances. However, the present theory contains an extra degree of freedom not found in the time-harmonic theory. Explicit results are presented for media with constant velocity gradients, and interesting new phenomena are identified. For example, a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction, maintains its initial spatial orientation even as the propagation direction rotates. The reflection and transmission of a pulse incident upon an interface are also discussed. The various theoretical results are illustrated by numerical simulations. This method of solution could be very useful for fast forward modelling in large-scale structures. It is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics.

scholarly journals A new iterative solver for the time-harmonic wave equation

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E57-E63 ◽  
Author(s):  
C. D. Riyanti ◽  
Y. A. Erlangga ◽  
R.-E. Plessix ◽  
W. A. Mulder ◽  
C. Vuik ◽  
...  
Keyword(s):  
Wave Equation ◽  
Linear System ◽  
Time Domain ◽  
Multigrid Method ◽  
Harmonic Wave ◽  
Computational Time ◽  
Full Time ◽  
Iterative Solver ◽  
Time Harmonic ◽  
The Time Domain

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

scholarly journals Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation

2015 ◽  
Vol 25 (10) ◽  
pp. 1927-1960 ◽  
Author(s):  
Hoai-Minh Nguyen ◽  
Linh Viet Nguyen
Keyword(s):  
Wave Equation ◽  
Three Dimensions ◽  
Impedance Boundary ◽  
Change Of Variables ◽  
Full Wave ◽  
Time Harmonic ◽  
Time Regime ◽  
Frequency Regime ◽  
The Difference ◽  
The Time Domain

This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the time regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latter is frequency-dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well controlled. This idea is originally due to Nguyen and Vogelius [Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], we revise and extend the work of Haddar, Joly, and Nguyen, [Generalized impedance boundary condition for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci.15 (2005) 1273–1300], where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variable, SIAM J. Math. Anal.44 (2012) 769–807] does not fit directly in our setting. Our proof makes use of a result by Hörmander in [Lp estimates for (pluri-) subharmonic functions, Math. Scand.20 (1967) 65–78]. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.

scholarly journals Exact nonreflecting boundary conditions for exterior wave equation problems

2014 ◽  
Vol 96 (110) ◽  
pp. 103-123 ◽  
Author(s):  
Silvia Falletta ◽  
Giovanni Monegato
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Time Domain ◽  
Region Of Interest ◽  
Boundary Integral ◽  
Implicit Time ◽  
Artificial Boundary ◽  
Equation Problem ◽  
Classical Wave Equation ◽  
The Time Domain

We consider the classical wave equation problem defined on the exterior of a bounded 2D space domain, possibly having far field sources. We consider this problem in the time domain, but also in the frequency domain. For its solution we propose to associate with it a boundary integral equation (BIE) defined on an artificial boundary surrounding the region of interest. This boundary condition is nonreflecting (or transparent) for both outgoing and incoming waves and it does not have to include necessarily the problem datum supports. The problem physical domain can even be a multi-domain, defined by the union of several disjoint domains. These domains can be convex or nonconvex. This transparent boundary condition is imposed pointwise on the chosen artificial boundary; therefore, its (space collocation) discretization can be coupled with a (space) finite difference or finite element method for the associated PDE problem. In the time-domain case, a classical (explicit or implicit) time integrator is also used. We present a consistency result for the BIE discretization and a sample of the intensive numerical testing we have performed.

scholarly journals Application of Multi-Branch Cauer Circuits in the Analysis of Electromagnetic Transducers Used in Wireless Transfer Power Systems

Sensors ◽  
2020 ◽  
Vol 20 (7) ◽  
pp. 2052
Author(s):  
Milena Kurzawa ◽  
Cezary Jędryczka ◽  
Rafał M. Wojciechowski
Keyword(s):  
Power Systems ◽  
Field Model ◽  
Wireless Power Transfer ◽  
Equivalent Model ◽  
Power Transfer ◽  
Wireless Power ◽  
The Finite Element Method ◽  
Harmonic Field ◽  
Time Harmonic ◽  
The Time Domain

In this paper, the feasibility of applying a multi-branch equivalent model employing first- and second-order Cauer circuits for the analysis of electromagnetic transducers used in systems of wireless power transfer is discussed. A method of formulating an equivalent model (EqM) is presented, and an example is shown for a wireless power transfer system (WPTS) consisting of an air transformer with field concentrators. A method is proposed to synthesize the EqM of the considered transducer based on the time-harmonic field model, an optimization algorithm employing the evolution strategy (ES) and the equivalent Cauer circuits. A comparative analysis of the performance of the considered WPTS under high-frequency voltage supply calculated using the proposed EqM and a 3D field model in the time domain using the finite element method (FEM) was carried out. The selected results of the conducted analysis are presented and discussed.

scholarly journals EN The wave model of secondary flows and coherent structures in pipes

2020 ◽  
Vol 55 (5-6) ◽  
pp. 273-281
Author(s):  
S. Surkov
Keyword(s):  
Wave Equation ◽  
Stream Function ◽  
Coherent Structures ◽  
Large Scale ◽  
Experimental Studies ◽  
Viscous Friction ◽  
Orthogonal Decomposition ◽  
Secondary Flows ◽  
Friction Forces ◽  
Special Cases

In this article, a theoretical analysis of the flows arising in the cross sections of fluid and gas flows is performed. Such flows are subdivided into secondary flows and coherent structures. From experimental studies it is known that both types of flows are long-lived large-scale movements (LSM) stretched along the flow. The relative stability of the vortices is traditionally explained by the fact that the viscous friction forces that inhibit the rotation are compensated by the intensification of the swirl when moving slowly rotating peripheral layers to the center of the vortex due to longitudinal tension. An analysis of this mechanism made it possible to develop a relatively simple model of vortex structures in which the viscous friction forces and axial expansion are considered to be infinitesimal. Under these assumptions, one can use the equations of motion of an ideal fluid in the variables “stream function - vorticity”. It is shown that under certain assumptions these equations take the form of a wave equation, and the boundary conditions are the condition that the stream function on the solid walls of the flow equals zero. The obtained solutions of the wave equation describe the following special cases: Goertler’s vortices between rotating cylinders, secondary flows in a pipe with a square cross section, swirling flow in a round pipe, paired vortex after bend of the pipe. The physical sense of more complex solutions of the wave equation has become clear relatively recently. Very similar structures were found in experimental studies using orthogonal decomposition (POD) of a turbulent pulsations field. This may mean that the eigenfunctions in the POD correspond to coherent structures that really arise in the flow. The results obtained confirm the hypothesis that secondary flows and coherent structures have a common nature. The solutions obtained in this paper can be used in processing the experiment as eigenfunctions for the orthogonal decomposition method. In addition, they can be used in direct numerical simulation (DNS) of turbulent flows

Spatiotemporal light control with active metasurfaces

Science ◽  
2019 ◽  
Vol 364 (6441) ◽  
pp. eaat3100 ◽  
Author(s):  
Amr M. Shaltout ◽  
Vladimir M. Shalaev ◽  
Mark L. Brongersma
Keyword(s):  
Large Scale ◽  
Negative Refraction ◽  
New Physics ◽  
New Materials ◽  
Power Efficient ◽  
Wavefront Shaping ◽  
Control Light ◽  
The Time Domain ◽  
External Stimuli ◽  
Optical Behavior

Optical metasurfaces have provided us with extraordinary ways to control light by spatially structuring materials. The space-time duality in Maxwell’s equations suggests that additional structuring of metasurfaces in the time domain can even further expand their impact on the field of optics. Advances toward this goal critically rely on the development of new materials and nanostructures that exhibit very large and fast changes in their optical properties in response to external stimuli. New physics is also emerging as ultrafast tuning of metasurfaces is becoming possible, including wavelength shifts that emulate the Doppler effect, Lorentz nonreciprocity, time-reversed optical behavior, and negative refraction. The large-scale manufacturing of dynamic flat optics has the potential to revolutionize many emerging technologies that require active wavefront shaping with lightweight, compact, and power-efficient components.

Consistent nonlinear deterministic and stochastic wave evolution equations from deep water to the breaking region

2019 ◽  
Vol 877 ◽  
pp. 373-404
Author(s):  
T. Vrecica ◽  
Y. Toledo
Keyword(s):  
Wave Field ◽  
Large Scale ◽  
Evolution Equations ◽  
Deterministic Model ◽  
Two Dimensional ◽  
Spectral Evolution ◽  
Wave Interactions ◽  
Nonlinear Part ◽  
Mild Slope Equation ◽  
Wave Models

Modelling the evolution of the wave field in coastal waters is a complicated task, partly due to triad nonlinear wave interactions, which are one of the dominant mechanisms in this area. Stochastic formulations already implemented into large-scale operational wave models, whilst very efficient, are one-dimensional in nature and fail to account for the majority of the physical properties of the wave field evolution. This paper presents new two-dimensional (2-D) formulations for the triad interactions source term. A quasi-two-dimensional deterministic mild slope equation is improved by including dissipation and first-order spatial derivatives in the nonlinear part of equation, significantly enhancing the accuracy in the breaking zone. The newly defined deterministic model is used to derive an updated stochastic model consistent from deep waters to the breaking region. It is localized following the approach derived in Vrecica & Toledo (J. Fluid Mech., vol. 794, 2016, pp. 310–342), to which several improvements are also presented. The model is compared to measurements of breaking and non-breaking spectral evolution, showing good agreement in both cases. Finally, the model is used to analyse several interesting 2-D properties of the shoaling wave field including the evolution of directionally spread seas.

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