Helmholtz equation and plane wave solutions linearity of maxwells equations maxwells equations are linear. Often there is then a cross over from the study of water waves to the study of scattering problems more generally. In the absence of free charge and current densities the maxwell equations are. On solving boundary value problems of modified helmholtz. Spherical harmonics and spherical bessel functions peter young dated. In section 4 it is shown that this equation may be solved iteratively as a neumann series and that the series converges for small enough values of the wave number. The helmholtz equation often arises in the study of physical problems involving partial differential equations pdes in both space and time. Helmholtzs and laplaces equations in spherical polar coordinates. Greens functions for the wave, helmholtz and poisson equations in a twodimensional boundless domain 43plane kc a t t. May 19, 2015 a plane wave virtual element method for the helmholtz problem article pdf available in esaim mathematical modelling and numerical analysis 503 may 2015 with 106 reads how we measure reads. Learning dominant wave directions for plane wave methods. As time evolves, the wavefronts propagate at the wave speed without changing. Simple derivation of electromagnetic waves from maxwells.
Simple derivation of electromagnetic waves from maxwells equations by lynda williams, santa rosa junior college physics department assume that the electric and magnetic fields are constrained to the y and z directions, respectfully, and that they are both functions of only x and t. Plane wave semicontinuous galerkin method for the helmholtz equation anders matheson masters thesis, spring 2015. There are one way wave equations, and the general solution to the two way equation could be done by forming linear combinations of such solutions. Hornbuckle january 27, 2016 1 maxwells equations maxwells equations and the constitutive relations describe the behavior of electromagnetic elds. As is shown in 7, using dominant wave directions in plane wave methods can signi cantly improve e ciency and accuracy for solving the highfrequency helmholtz equation in heterogeneous medium. The angular dependence of the solutions will be described by spherical harmonics. From maxwell to helmholtz maxwells equations and the inhomogeneous wave equation. Consistent with geometrical optics, its magnitude is constrained to be proportional to the refractive index n 2.
The mathematics of pdes and the wave equation michael p. Plane wave discontinuous galerkin pwdg methods are a class of tre. Greens functions for the wave, helmholtz and poisson. It turns out that a propagating sinusoidal wave is a solution to the helmhotz equations which is consistent with our previous understanding of the behavior of electromagnetic radiation and how it propagates as. Figure 1 the contours used to evaluate the integral in eq. We first define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations. Helmholtz s equation as discussed in class, when we solve the di. As a starting point, let us look at the wave equation for the single xcomponent of magnetic field. Chapter maxwells equations and electromagnetic waves. Wave propagation and scattering 12 lectures of 24 part iii. Pdf plane wave discontinuous galerkin methods for the 2d.
There is an interesting parallel between free wave propagation and transmission line propagation. This equation is referred to as helmholtz equation. The solutions of wave equation in cylindrical coordinates the helmholtz equation in cylindrical coordinates is by separation of variables, assume. Since were mostly interested in electromagnetic waves here, and in particular light waves, we have to convert the maxwell equations into a form that easily yields wavelike solutions. The helmholtz equation scattering problems in this document we discuss the. Pdf a plane wave virtual element method for the helmholtz. A plane wave virtual element method for the helmholtz problem article pdf available in esaim mathematical modelling and numerical analysis 503. Plane wave discontinuous galerkin methods for the helmholtz. The interpretation of the unknown ux and the parameters nx. Plane wave discontinuous galerkin methods for the 2d. Physics 116c helmholtzs and laplaces equations in spherical.
This combines the advantages of plane wave methods and phasebased methods. In this paper, we study the approximation of solutions of the homogeneous helmholtz equation. Plane wave discretizations of the helmholtz equation with lagrange multipliers, and a domain decomposition method for resulting systems. Also, if we perform a cylindrical eigenfunction expansion we find that the modes all decay rapidly as distance goes to infinity except for the solutions which satisfy helmholtz s equation. The 3d wave equation plane wave spherical wave mit 2. One example is to consider acoustic radiation with spherical symmetry about a point y fy ig, which without loss of generality can be taken as the origin of coordinates.
We present a raybased finite element method for the highfrequency helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. In this paper we are concerned with plane wave discontinuous galerkin pwdg methods for helmholtz equation and timeharmonic maxwell equations in threedimensional anisotropic media, for which the coefficients of the equations are matrices instead of numbers. Plane wave approximation of homogeneous helmholtz solutions a. The local basis are motivated by the geometric optics ansatz and are composed of polynomials modulated by plane waves prop. Plane wave approximation of homogeneous helmholtz solutions. Learning dominant wave directions for plane wave methods for.
From this the corresponding fundamental solutions for the. Also, if we perform a cylindrical eigenfunction expansion we find that the modes all decay rapidly as distance goes to infinity except for the solutions which satisfy helmholtzs equation. The helmholtz equation, which represents a timeindependent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. An equally important solution to the wave equation which we will encounter many times in this course in the spherical wave, which is solved by considering the helmholtz equation written in spherical coordinates.
The ortiz formulation is a kind of halfway house, in which the depth is allowed to vary in the governing partial differential equation, but a set of plane waves is still used in the solution space. Wave functions waveguides and cavities scattering separation of variables the special functions vector potentials the spherical bessel equation each function has the same properties as the corresponding cylindrical function. Assume the modulation is a slowly varying function of z slowly here mean slow compared to the wavelength a variation of a can be written as so. Helmholtzs equation as discussed in class, when we solve the di. Comparisons of three kinds of plane wave methods for the helmholtz equation and timeharmonic maxwell equations with complex wave numbers. Elementary waves in free space the electromagnetic plane wave. Sep, 2019 in this paper we are concerned with plane wave discontinuous galerkin pwdg methods for helmholtz equation and timeharmonic maxwell equations in threedimensional anisotropic media, for which the coefficients of the equations are matrices instead of numbers. Wave equations we shall use throughout the linearised wave equation, and restrict ourselves to timeharmonic waves only, i. Gander 1 introduction we consider in this paper the iterative solution of linear systems of equations arising from the discretization of the inde. The above wave finite elements are strictly only effective for the helmholtz equation, or other equations with a constant wave speed. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Ts perspve paves e way for marryg plane wave aroxation with many of e various dg ods devoped for sond orr. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. The method requires a xed number of grid points per wavelength to represent the wave eld.
Uniform plane wave solution to maxwells equations brian k. Recently, the use of special local test functions other than polynomials in discontin. The paraxial helmholtz equation start with helmholtz equation consider the wave which is a plane wave propagating along z transversely modulated by the complex amplitude a. Hirschberg eindhoven university of technology 28 nov 2019 this is an extended and revised edition of iwde 9206. Barnett december 28, 2006 abstract i gather together known results on fundamental solutions to the wave equation in free space, and greens functions in tori, boxes, and other domains.
Hornbuckle january 27, 2016 1 maxwells equations maxwells equations and the constitutive relations describe the. The method requires a fixed number of grid points per wavelength to represent the wave field. Greens functions for the wave equation dartmouth college. In this thesis we concentrate on the plane wave dg pwdg method which uses plane wave basis functions, see 9, 10, 12. Plane wave discontinuous galerkin methods for the 2d helmholtz equation. Jul 12, 2011 in this paper, we study the approximation of solutions of the homogeneous helmholtz equation. An equally important solution to the wave equation which we will encounter many times in this course in the spherical wave, which. It is the purpose of this paper to examine the issue of solving modified helmholtz equations by plane wave functions. The wave equation handbook of optical systems wiley. They include the socalled ultra weak variational formulation from o. The solutions of wave equation in cylindrical coordinates. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. The only possible solution of the above is where, and are constants of, and.
In 9 it was shown that the ultra weak variational formulation uwvf for helmholtz equation, 3, is a special case of pwdg. Indeed, as pointed out in, modified helmholtz equations naturally arise in solving the diffusion equations with initial and boundary value problems by difference methods. A survey of trefftz methods for the helmholtz equation. Elementary waves in free space the electromagnetic plane. Wave equation in homogeneous media and the scalar wave equation.
To accomplish this, we will derive the helmholtz wave equation from the maxwell equations. Pdf plane wave discretizations of the helmholtz equation. We combine approximation estimates for homogeneous helmholtz solutions by generalized harmonic polynomials, obtained from vekuas theory, with estimates for the approximation of generalized harmonic. Plane wave semicontinuous galerkin method for the helmholtz. In this case, strictly speaking, the plane wave basis is no longer valid, since the plane waves are not solutions to the inhomogeneous helmholtz equation.
Dg od oyg local trial spas anned by a w plane waves, as poind out in. The wave phase is constant along a planar surface the wavefront. This will result in a linearly polarized plane wave travelling. This will result in a linearly polarized plane wave travelling in the x direction at the speed of light c. Optimization of plane wave directions in plane wave. The dispersion relation of the harmonic wave solution. In particular, we examine questions about existence and. Plane electromagnetic waves and wave propagation 7.
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