An equally important solution to the wave equation which we will encounter many times in this course in the spherical wave, which. This will result in a linearly polarized plane wave travelling in the x direction at the speed of light c. From maxwell to helmholtz maxwells equations and the inhomogeneous wave equation. We will derive the wave equation from maxwells equations in free. In this case, strictly speaking, the plane wave basis is no longer valid, since the plane waves are not solutions to the inhomogeneous helmholtz equation. Learning dominant wave directions for plane wave methods. Pdf plane wave discontinuous galerkin methods for the 2d. Greens functions for the wave, helmholtz and poisson equations in a twodimensional boundless domain 43plane kc a t t. Helmholtz s equation as discussed in class, when we solve the di. Optimization of plane wave directions in plane wave. We combine approximation estimates for homogeneous helmholtz solutions by generalized harmonic polynomials, obtained from vekuas theory, with estimates for the approximation of generalized harmonic. Simple derivation of electromagnetic waves from maxwells. Plane wave discontinuous galerkin methods for the 2d. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract.
This will result in a linearly polarized plane wave travelling. The wave phase is constant along a planar surface the wavefront. The helmholtz equation often arises in the study of physical problems involving partial differential equations pdes in both space and time. Jul 12, 2011 in this paper, we study the approximation of solutions of the homogeneous helmholtz equation. Wave equation in homogeneous media and the scalar wave equation. As time evolves, the wavefronts propagate at the wave speed without changing. Physics 116c helmholtzs and laplaces equations in spherical. Chapter maxwells equations and electromagnetic waves. A plane wave virtual element method for the helmholtz problem article pdf available in esaim mathematical modelling and numerical analysis 503. Pdf plane wave discretizations of the helmholtz equation. 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. 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. 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. The only possible solution of the above is where, and are constants of, and.
A survey of trefftz methods for the helmholtz equation. It is the purpose of this paper to examine the issue of solving modified helmholtz equations by plane wave functions. 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 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. Plane wave discontinuous galerkin methods for the 2d helmholtz equation. 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. Ts perspve paves e way for marryg plane wave aroxation with many of e various dg ods devoped for sond orr.
Greens functions for the wave equation dartmouth college. Wave equations we shall use throughout the linearised wave equation, and restrict ourselves to timeharmonic waves only, i. 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. Plane wave discretizations of the helmholtz equation with lagrange multipliers, and a domain decomposition method for resulting systems. Comparisons of three kinds of plane wave methods for the helmholtz equation and timeharmonic maxwell equations with complex wave numbers. Helmholtz equation and plane wave solutions linearity of maxwells equations maxwells equations are linear. It arises in fields like acoustics, electromagnetics, and fluid dynamics. The interpretation of the unknown ux and the parameters nx.
Elementary waves in free space the electromagnetic plane wave. As a starting point, let us look at the wave equation for the single xcomponent of magnetic field. Helmholtzs equation as discussed in class, when we solve the di. 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. Helmholtzs and laplaces equations in spherical polar coordinates. 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. The solutions of wave equation in cylindrical coordinates. The wave equation handbook of optical systems wiley. Hirschberg eindhoven university of technology 28 nov 2019 this is an extended and revised edition of iwde 9206. This equation is referred to as helmholtz equation.
Plane wave approximation of homogeneous helmholtz solutions a. 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 behavior of electromagnetic elds. Pdf a plane wave virtual element method for the helmholtz. Plane wave approximation of homogeneous helmholtz solutions. The mathematics of pdes and the wave equation michael p. There is an interesting parallel between free wave propagation and transmission line propagation. Ortiz assumes that in each element the plane wave functions still form a valid solution space, since the wave speed does not vary greatly over any element. 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. 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.
The solutions of wave equation in cylindrical coordinates the helmholtz equation in cylindrical coordinates is by separation of variables, assume. Consistent with geometrical optics, its magnitude is constrained to be proportional to the refractive index n 2. On solving boundary value problems of modified helmholtz. 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. Plane electromagnetic waves and wave propagation 7. This combines the advantages of plane wave methods and phasebased methods. Plane wave discontinuous galerkin methods for the helmholtz. Gander 1 introduction we consider in this paper the iterative solution of linear systems of equations arising from the discretization of the inde. Elementary waves in free space the electromagnetic plane.
In the absence of free charge and current densities the maxwell equations are. The 3d wave equation plane wave spherical wave mit 2. 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. We first define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations. The helmholtz equation scattering problems in this document we discuss the. 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. From this the corresponding fundamental solutions for the. To accomplish this, we will derive the helmholtz wave equation from the maxwell equations. Plane wave semicontinuous galerkin method for the helmholtz.
The angular dependence of the solutions will be described by spherical harmonics. Spherical harmonics and spherical bessel functions peter young dated. The dispersion relation of the harmonic wave solution. 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. 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 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. Figure 1 the contours used to evaluate the integral in eq.
Learning dominant wave directions for plane wave methods for. Dg od oyg local trial spas anned by a w plane waves, as poind out in. The local basis are motivated by the geometric optics ansatz and are composed of polynomials modulated by plane waves prop. Recently, the use of special local test functions other than polynomials in discontin. The above wave finite elements are strictly only effective for the helmholtz equation, or other equations with a constant wave speed. Uniform plane wave solution to maxwells equations brian k. The method requires a xed number of grid points per wavelength to represent the wave eld. In particular, we examine questions about existence and. Greens functions for the wave, helmholtz and poisson. 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. In this paper, we study the approximation of solutions of the homogeneous helmholtz equation. In this thesis we concentrate on the plane wave dg pwdg method which uses plane wave basis functions, see 9, 10, 12.
Often there is then a cross over from the study of water waves to the study of scattering problems more generally. They include the socalled ultra weak variational formulation from o. Indeed, as pointed out in, modified helmholtz equations naturally arise in solving the diffusion equations with initial and boundary value problems by difference methods. 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. 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. In 9 it was shown that the ultra weak variational formulation uwvf for helmholtz equation, 3, is a special case of pwdg. Wave propagation and scattering 12 lectures of 24 part iii. Plane wave discontinuous galerkin pwdg methods are a class of tre.
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