A WKB based preconditioner for the 1D Helmholtz wave equation

AbstractAsymptotic formulae for the capacitances of two oppositely charged, identical, circular, coaxial discs and two identical, infinite parallel strips separated by dielectric slabs are derived from the dual integral equations approach. The formulation in terms of dual integral equations using transforms gives rise to relatively simple Green's functions in the transformed space and renders the derivation of the asymptotic formulae relatively easy. The solution near the edge of the plates, a two-dimensional problem previously thought not solvable by the Wiener-Hopf technique, is solved indirectly using the method. Solution of the Helmholtz wave equation is first sought, and the solution to Laplace's equation is obtained by letting the wavenumber go to zero. The solution away from the edge is obtained by solving the dual integral equations approximately. The total charge on the plate is obtained by matching the solution near the edge and away from the edge giving the capacitance.

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Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method -- Part III: Fast Algorithms

Ultrasonic Imaging ◽

10.1177/016173468400600109 ◽

1984 ◽

Vol 6 (1) ◽

pp. 103-116

Author(s):

S. A. Johnson ◽

Y. Zhou ◽

M. K. Tracy ◽

M. J. Berggren ◽

F. Stenger

Keyword(s):

Wave Equation ◽

Inverse Scattering ◽

Scattering Problem ◽

Inverse Scattering Problem ◽

Algebraic Equations ◽

The Past ◽

Nonlinear Algebraic Equations ◽

Detector Geometry ◽

Scattering Potential ◽

Helmholtz Wave Equation

Solving the inverse scattering problem for the Helmholtz wave equation without employing the Born or Rytov approximations is a challenging problem, but some slow iterative methods have been proposed. One such method suggested by us is based on solving systems of nonlinear algebraic equations that are derived by applying the method of moments to a sinc basis function expansion of the fields and scattering potential. In the past, we have solved these equations for a 2-D object of n by n pixels in a time proportional to n5. In the present paper, we demonstrate a new method based on FFT convolution and the concept of backprojection which solves these equations in time proportional to n3 • log(n). Several numerical examples are given for images up to 7 by 7 pixels in size. Analogous algorithms to solve the Riccati wave equation in n3 • log(n) time are also suggested, but not verified. A method is suggested for interpolating measurements from one detector geometry to a new perturbed detector geometry whose measurement points fall on a FFT accessible, rectangular grid and thereby render many detector geometrics compatible for use by our fast methods.

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Alternative derivations for the fields inside a waveguide

Journal of Electrical Engineering ◽

10.2478/jee-2021-0018 ◽

2021 ◽

Vol 72 (2) ◽

pp. 129-131

Author(s):

Raghavendra G. Kulkarni

Keyword(s):

Magnetic Field ◽

Wave Equation ◽

Electric Field ◽

Longitudinal Magnetic Field ◽

Transverse Electric ◽

Transverse Magnetic ◽

Longitudinal Electric Field ◽

Microwave Engineering ◽

Introductory Textbooks ◽

Helmholtz Wave Equation

Abstract Generally, the longitudinal magnetic field of the transverse electric (TE) wave inside a waveguide is obtained by solving the corresponding Helmholtz wave equation, which further leads to the derivation of the remaining fields. In this paper, we provide an alternative way to obtain this longitudinal magnetic field by making use of one of the Maxwell’s equations instead of directly relying on the Helmholtz wave equation. The longitudinal electric field of the transverse magnetic (TM) wave inside a waveguide can also be derived in a similar fashion. These derivations, which are different from those found in the introductory textbooks on microwave engineering, make the study of waveguides more interesting.

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