China
Africa
Explore chapters and articles related to this topic
Photoelasticity
Published in Rajpal S. Sirohi, Optical Methods of Measurement, 2018
where δy and δx are the phases of the waves. These waves satisfy the wave equation. Owing to the superposition principle, the sum of the waves will also satisfy the wave equation. In general, a wave will have both x and y components and can be written as () E(z;t)=iEx(z;t)+jEy(z;t)
Electromagnetic Waves in Homogeneous Media without Absorption
Published in Vladimir V. Mitin, Dmitry I. Sementsov, An Introduction to Applied Electromagnetics and Optics, 2016
Vladimir V. Mitin, Dmitry I. Sementsov
Mathematically, the principle of superposition is a consequence of the linearity of the wave equation that describes the propagation of light waves in vacuum. Indeed, if the fields E1, E2, E3,… are solutions of the wave equation, their sum E = E1 + E2 + … is also a solution. You can verify this by substituting into the wave equation the solutions in the form of a sum of waves (e.g., plane waves): () E=∑iEi=∑iAi0cos(ωit−ki⋅r), where ki are the wave vectors and ki = ωi/c are their magnitudes (wave numbers) in the sum. In this case, the wave equation is a sum of terms describing the individual waves.
Analysis in Electromagnetic Product Design
Published in S. Ratnajeevan H. Hoole, Yovahn Yesuraiyan R. Hoole, Finite Elements-based Optimization, 2019
S. Ratnajeevan H. Hoole, Yovahn Yesuraiyan R. Hoole
The best-known differential methods are the finite-difference and finite element methods. The finite-difference method is best suited to homogeneous problems and therefore is very popular in solving the wave equation which usually involves modeling in free homogeneous space. It is not so popular in the many problems where inhomogeneous materials are present such as in a stuffed waveguide. While we will focus on the finite element method because of its wide applications, flexibility, and suitability to the most efficient matrix-solution methods, here we will briefly mention the finite-difference method.
Domain truncation methods for the wave equation in a homogenization limit
Published in Applicable Analysis, 2022
Mathias Schäffner, Ben Schweizer, Yohanes Tjandrawidjaja
Let us turn to the numerical treatment of the wave equation. The wave equation can be discretized with finite differences or finite elements, the resulting finite dimensional problems can be solved numerically. The various aspects of stability and convergence for these schemes are well understood, see [10, 11] and the references therein.
Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
S.M. Dsouza, T. Khajah, X Antoine, S.P.A. Bordas, S. Natarajan
Spectral finite element method is one of the most commonly used methods for wave propagation [9,10]. The distribution of the domain nodes is such that oscillations that occur because of the Runge phenomenon can be reduced [11]. This method requires fewerdegrees of freedom (DOFs) per wavelength compared to the conventional Finite Element Method (FEM) [12]. Using vector and parallel computing algorithms [13], it easily reduces the storage requirements and the computational complexity of this method. The Chebyshev polynomials [14] are typically employed to minimize the dispersion error [15]. Higher accuracy is obtained compared to the low-order -FEM (quadratic) [16,17]. The major drawback of SEM is its difficulties to deal with complex geometries. The finite difference method (FDM) based on the Taylor series expansion is a classical way to solve the wave equation. The terms are truncated to arbitrary number and the dominant power of the truncated terms determines the accuracy. The Cartesian grids are usually necessary to obtain the solution unless generalized finite differences (GFD) are used. The GFD is a meshless technique used on the domain, eliminating the mesh generation and the numerical quadrature. This results in requiring a high number of degrees of freedom to obtain higher accuracy. In the case of curved complex geometries which is a major characteristic and difficulty of scattering problems, this method cannot produce accurate results due to the well-known staircase effect [18]. The semi analytical finite element (SAFE) [19,20] is another way of combining the advantages of numerical and analytical methods. This technique which uses Fourier Transforms to recover time-domain analysis was used to study Lamb wave propagation, whose behaviour is independent of the direction of propagation [21]. For a given accuracy, the computational cost for SAFE is less than that of FEM. The disadvantage here is that the wave propagation over complex geometrical features cannot be handled. When the geometries are complex, it is needed to make some of the approximations and assumptions which affect the accuracy. In these cases, the FEM becomes more useful.
A hybrid computational method for local fractional dissipative and damped wave equations in fractal media
Published in Waves in Random and Complex Media, 2022
Ved Prakash Dubey, Jagdev Singh, Ahmed M. Alshehri, Sarvesh Dubey, Devendra Kumar
A wave equation is a linear PDE of second order which provides the general characteristics of mechanical or light waves that arise in classical physics. The wave equation informs about the propagation of oscillations at a constant speed in some quantity. The problem related to a vibration of a string was initially studied by d’ Alembert [22]. A classical 1D wave equation was discovered by d’ Alembert [22] and was described as , where denotes the displacement of the vibration of the string in a vertical direction or along the direction of motion and denotes the speed of propagation of the wave. It is notable that the string can be fixed at both ends or just at one end or an infinite string. In each case, different boundary conditions will be obtained for the wave equation. The equation for the damped wave is given by , where symbolizes the coefficient of friction arises from displacement of string. The wave equation describes how the displacement of a string changes as a function of time and position. A damped wave is that which possesses successive cycles of progressively diminishing amplitudes. It is notable that the amplitude decreases exponentially with distance due to energy loss. On the other hand, dissipation is used to mark out the course of actions in which energy is wasted. A dissipative wave is that which loses its amplitude. The dissipative wave equation indicates the loss of wave energy and consequent decline in wave height due to turbulence, wave breaking and viscous effects, and in case of shallow water waves by virtue of bottom friction. The damped wave equation appears in various disciplines of engineering and science. The normal supposition is that the damping mechanics is proportional to the velocity of propagation. The quantitative magnitude of damping is frequency-dependent and its behavior mostly follows a power law. This phenomenon usually occurs in various kinds of applications, for instance, viscoelasticity, acoustics, seismic wave propagation, and structural vibration.