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Deep earth imaging
Published in Rajib Biswas, Recent Developments in Using Seismic Waves as a Probe for Subsurface Investigations, 2023
where dl is differential path length, x is the position vector, and v is velocity. In order to evaluate this integral, the path taken by the seismic ray should be known. However, we face difficulty performing this integration because the wave's travel path is not known as it depends on the velocity structure. The equation that governs the geometry of ray paths can be derived from the eikonal equation by considering how a slight change in time dt affects a point x on a wavefront (see Aki and Richards, 2002). The resultant ray equation:ddl1vxdxdl=∇1vx
Fundamentals of Optical Propagation
Published in Yasuo Kokubun, Lightwave Engineering, 2018
Next, h is expressed in terms of e using Equation (1.149), and then substituted in Equation (1.150). Using the vector formula, () ∇S×(∇S×e)=(∇. e)∇S−(∇S. ∇S)e and the orthogonality e and ∇S, then () |∇S|2=n2 is obtained. This equation is called an Eikonal equation and represents the propagation of a wave front in a medium with a refractive index that is not uniform and has a spatial distribution.
Acoustic Wave Propagation
Published in Indrakshi Dey, Propagation Modeling for Wireless Communications, 2022
Acoustic waves are the choicest mode of communication in underwater and underground scenarios. Acoustic waves are longitudinal waves that travel through the medium by adiabatic compression and relaxation of the collection of particles in the medium. Acoustic waves are characterized by acoustic pressure, particle velocity, particle displacement and acoustic intensity. Rays can be used to represent acoustic waves emitted from a source while propagating in different directions. Sums of the contributions from these rays is used to calculate the acoustic pressure field at the receiver. the acoustic pressure field existing between the source and the receiver are referred to as eigenrays. The eigenrays follow straight lines if the medium is homogeneous. The rays are refracted on the way if the medium is heterogeneous and the velocity of the acoustic waves varies with distance. The flow of acoustic waves is guided by Eikonal equation. The Eikonal equation is a first-order non-linear partial differential equations (PDE) that can be solved by different techniques. The most convenient of these is to use the method of characteristics. In this case, a family of curves (rays) are introduced, which are perpendicular to the level curves (wave-fronts). This family of rays defines a new coordinate system, and it turns out that in ray coordinates the Eikonal equation reduces to linear ordinary differential equation. However, modeling the interaction within the underwater or underground environment is extremely challenging. Therefore, different techniques are used in different scenarios. This chapter elaborates on different techniques that are used for modeling acoustic wave propagation and the impact of the environment on their movement.
Modeling and simulation of urban air pollution from the dispersion of vehicle exhaust: A continuum modeling approach
Published in International Journal of Sustainable Transportation, 2019
Liangze Yang, Tingting Li, S. C. Wong, Chi-Wang Shu, Mengping Zhang
The eikonal equation is a special type of steady-state Hamilton-Jacobi equations. Solutions to Hamilton-Jacobi equations are usually continuous but not differentiable everywhere and are usually not unique. The viscosity solution Crandall and Lions (1983) is the physically relevant solution and the numerical solution for approximating the viscosity solutions of the Hamilton–Jacobi equations follows the lines similar to those for solving conservation laws. Thus, the WENO schemes for conservation laws are extended to those for Hamilton–Jacobi equations in (Jiang & Peng, 2000; Shu, 2007). For the steady-state equation, if we use the time-dependent WENO scheme of (Jiang & Peng, 2000; Shu, 2007), we would need to introduce a pseudo-time and then march to a steady state. The fast sweeping method (Zhang et al., 2006) is much faster in terms of computational time than pseudo-time marching. We use the fast sweeping method to solve the eikonal equation (8) here. The first and third order fast sweeping methods can be found in (Zhao, 2004; Zhang et al., 2006; Xiong et al., 2011).