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Integral solutions of Maxwell’s equations
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
In the far zone E˜, H˜, r^ are mutually orthogonal. Because of this, and because the fields vary as e−jkr/r, the electromagnetic field in the far zone takes the form of a spherical TEM wave, which is consistent with the Sommerfeld radiation condition.
Integral solutions of Maxwell’s equations
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
In the far zone E~,H~,r^ $ \tilde{\mathbf{E}},\tilde{\mathbf{H}},\hat{\mathbf{r}} $ are mutually orthogonal. Because of this, and because the fields vary as e-jkr/r $ e^{-jkr}/r $ , the electromagnetic field in the far zone takes the form of a spherical TEM wave, which is consistent with the Sommerfeld radiation condition.
Cloaking and channelling of elastic waves in structured solids
Published in A. B. Movchan, N. V. Movchan, I. S. Jones, D. J. Colquitt, Mathematical Modelling of Waves in Multi-Scale Structured Media, 2017
A. B. Movchan, N. V. Movchan, I. S. Jones, D. J. Colquitt
Let v(x) be a continuous piecewise smooth solution of the Helmholtz equation in R2 $ {\mathbb{R}}^{2} $ satisfying the Sommerfeld radiation condition at infinity. Integrating the difference u(x)Lv(x)-v $ u(x){\mathcal{L}}v(x) - v $ over a disc Dr $ {\mathcal{D}}_{r} $ of radius r containing Ω-(1) $ \iOmega_{ - }^{(1)} $ yields
An inverse problem of a simultaneous reconstruction of the dielectric constant and conductivity from experimental backscattering data
Published in Inverse Problems in Science and Engineering, 2021
Vo Anh Khoa, Grant W. Bidney, Michael V. Klibanov, Loc H. Nguyen, Lam H. Nguyen, Anders J. Sullivan, Vasily N. Astratov
Suppose that we only consider non-magnetic targets. Then their relative permeability has to be unity. With being the vacuum permittivity and vacuum permeability, respectively, and , (1) becomes In (2), the fraction is essentially close to the so-called characteristic impedance of free space , which is approximately 377 (). Denote as the dielectric constant. We therefore rewrite the Helmholtz equation (2) and impose the Sommerfeld radiation condition:
Regularization in ultrasound tomography using projection-based regularized total least squares
Published in Inverse Problems in Science and Engineering, 2020
Mohamed Almekkawy, Anita Carević, Ahmed Abdou, Jiayu He, Geunseop Lee, Jesse Barlow
To model the total pressure field , , of the system composed of homogeneous acoustic medium and a scattering object, we use the Helmholtz equation [12]: and where is the Laplacian operator and is the wave number. The incident and the scattered field at position are denoted with and , respectively. The Helmholtz equation with Sommerfeld radiation condition can be written in the Lippmann–Schwinger integral form [46] as: where ℜ denotes the bounded spatial domain of the whole ROI, and are the position vectors of two points in the ROI.
Bayesian approach for limited-aperture inverse acoustic scattering with total variation prior
Published in Applicable Analysis, 2022
Xiao-Mei Yang, Zhi-Liang Deng, Ailin Qian
Let be a bounded, simply connected domain with boundary . Define . Consider a time-harmonic plane wave propagating in the exterior of Ω. The velocity potential is of this form where d is the propagating direction, is the frequency, is the wavenumber and c is the speed of sound. Hereafter, the term is factored out and only the space dependent part of all waves is considered. Let Ω be an impenetrable sound-soft obstacle. Clearly, the obstacle will ‘scatter’ the wave so that we may write the total field as , where is the scattered wave. The classical physical problem is to explore the scattering phenomenon, i.e. find . As we know, the scattered field is governed by an exterior boundary value problem for Helmholtz equation where Equation (2b) is the sound-soft boundary condition and (2c) is the Sommerfeld radiation condition.