We deal with both normal and nonnormal angles of incidence. Application and implication of using complex vectors and complex transformations in solutions of maxwells equations is investigated. 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. This includes electromagnetic waves radio waves, visible light, xrays, etc. In the absence of free charge and current densities the maxwell equations are. If we define, and then use, a number that satisfies the equation that is, or then we can always find the n roots of a polynomial of degree n. If the medium is dispersive different frequencies travel at. T is the velocity of the wave, which can be determined from electrostatics and magnetostatics.
Representing plane waves in three dimensional space we may always depend on it that algebra, which cannot be translated into good english and sound common sense, is bad algebra. Plane waves and refractive index chapter 2 31 w p l 2 c vac. The irreducible representations of the lorentz group are given by, as a gen. Pesky inverse tangent whenever doing a phasor or geometrical conversion, the inverse tangent formula in equation 1. Complex vectors are used in complex plane waves and help to represent this type of waves geometrically. In accordance with maxwell equations, the complex amplitude of the object wave u o and the reference wave u r at the hologram plane can be represented by. Representation of waves via complex numbers in mathematics, the symbol is conventionally used to represent the squareroot of minus one. A real number, say, can take any value in a continuum of values lying between and. The most general solution has two unknown constants, which. This substitution will impose conditions on the constants, and, for the plane wave functions to be solutions of the maxwell equations. On the other hand, an imaginary number takes the general. There is a very nice set of complex solutions to the wave equation, known as complex travelling waves, which take the form fx. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines.
Multiplication and division of complex numbers is easier. They can be complex numbers, as in a complex exponential plane wave. Now, consider what happens if we take the real part of eq. E xois complex in general so it has a magnitude je xojand a phase ej since the.
Let us examine this set of examples in further detail. Complex waves complex numbers are extremely useful for describing the propagation of waves. The homogeneous form of the equation, written in terms of either the electric field e or the magnetic field b, takes the form. Ex,t is the electric field is the magnetic permeability is the dielectric permittivity this is a linear, secondorder, homogeneous differential equation. The sc hr o ding er w av e equati on so far, w e ha ve m ad e a lot of progr ess con cerni ng th e prop erties of, an d inte rpretation of th e w ave fu nction, bu t as yet w e h ave h ad very little to sa y ab out ho w the w ave fu nction ma y b e deriv ed in a general situ ation, th at is to say, w e d o not h ave on han d a ow ave. The dependence of r on various parameters is best displayed in the complex plane r complex plane as shown in. A wave is disturbance of a continuous medium that propagates with a fixed shape at constant velocity. This is because it comes from the expansion of the plane wave part only. 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. Here, we write down the complex amplitudes for the eb and fields. In the older theory often described as the classical theory the behaviour of. On the complex conjugate roots of the rayleigh equation. Uniform plane waves have uniform constant properties in a plane perpendicular to their direction of propagation. In the presence of absorption, the wave will diminish in size as it move.
H wm2 44 equation 44 is known as the poyting vector. The wave equations, written for the complex amplitudes monochromatic waves, are referred to as the helmholtz equations. It couples into a plane shear wave that propagates in the medium. Thus this em wave approaches the boundary from the left and is at normal incidence to the boundary. We will now exploit this to perform fourier analysis on the. Complex amplitude an overview sciencedirect topics. Polarization 1 polarization vectors in the last lecture, we showed that maxwells equations admit plane wave solutions e e 0e i kxt, b b 0e i kxt 1 here, e 0 and b0 are called the polarization vectors for the electric and magnetic. Negative energies with no lower bounds is incompatible with stability. A plane electromagnetic wave is therefore a transverse wave. It is a threedimensional form of the wave equation. The complex constant ais called the complex amplitude of the wave. Here, is a real number representing the magnitude of, and represents the angle of in the complex plane. What this equation says is that the wave converging on the scatterer e. Plane waves representation of waves via complex functions in mathematics, the symbol is conventionally used to represent the squareroot of minus one.
Imrana ashraf zahid quaidiazam university, islamabad pakistan. Mimicking the dalembert solution for the wave equation, we anticipate that the solutions to the laplace equation 2. Electron diffraction in mathematics, the symbol is conventionally used to represent the squareroot of minus one. For the plane waves, one sees that the operators thus the maxwell equations become where. Lecture 14 time harmonic fields cornell university. Complex analysis proves a useful tool for solving a wide variety of problems in engineering science the analysis of ac electrical circuits, the solution of linear differential equations with constant coef. We now give a brief derivation of sufficient conditions. Ece 303 fall 2005 farhan rana cornell university waves in a dielectric medium dispersion relation j k r e r n eo e rr rr. Which in many cases means taking the real part but not always. For neatness, we sometimes write the wave equation by putting everything on. There is more information contained in maxwells equations than there is in the wave equation.
The wave equation in cylindrical coordinates overview and. The sc hr o ding er w av e equati on macquarie university. And is a complex wave vector the length of this vector is inversely proportional to the wavelength of the wave. So, complex wave functions are needed for stability. Consistent with geometrical optics, its magnitude is constrained to be proportional to the refractive index n 2. There is a simple set of complex traveling wave solutions to this equation. Let us see what additional constraints are thereby imposed. Er rr for the efield of a plane wave we had timeharmonic fields and vector phasors ece 303 fall 2007 farhan rana cornell university. A plane wave with a fixed direction of the electric field vector e0 is termed lin. The wave equation and the speed of light chapter 1 physics 208, electrooptics. Simple derivation of electromagnetic waves from maxwells. The equation has no solution for in the set of real numbers. The 3d wave equation for the electric field and its solution whose solution is where is a constant, complex vector. Since e, h, and k form a righthanded system, the power ows in the direction of wave propagation.
Repeating this sort of thing using one of the the curl eqns say, faradays law one gets. Very often the propagation constant k is taken as a vec tor k. The leaky surface wave is an inhomogeneous wave that propagates along the surface with a phase velocity larger than the shear wave but smaller than the pressure wave. If f 1 x,t and f 2 x,t are solutions to the wave equation, then. Application of complex vectors and complex transformations in. It also means that waves can constructively or destructively interfere. If the wave function were real, performing a fourier transform in time will lead to pairs of positivenegative energy eigenstates. The rate of energy transfer by an electromagnetic wave is described by the poynting vector, s, defined as the rate at which energy passes through a unit surface area perpendicular to the direction of wave propagation wm2. Boundary conditions poyntings theorem and conservation laws complex function formalism time average of sinusoidal products wave equation 2. Compare it to the case of the plane wave without scattering eq. The 3d wave equation, plane waves, fields, and several 3d differential operators. The electromagnetic wave equation is a secondorder partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. In physics, a plane wave is a special case of wave or field.
While cartesian coordinates are attractive because of their simplicity, there are many problems whose symmetry makes it easier to use a different system of coordinates. The complex form of the plane wave just introduced can be simplified by using a complex valued amplitude in. 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. Stability of plane wave solutions in complex ginzburglandau equation with delayed feedback d. Complex numbers are extremely useful for describing the propagation of waves. In addition to satisfying the wave equation, the complex. The quantity, which is a timeindependent complex vector, is a vector phasor for the plane wave er rr in the book, the vector phasor has an additional underline and written as. Gurevich abstract we perform bifurcation analysis of plane wave solutions in onedimensionalcomplexcubic. Representing plane waves in three dimensional space.
The leaky surface wave arises from the complex conjugate roots of the rayleigh equation. The wave is polarized in the x direction by convention, we use the direction of e to specify the polarization of an. So the answer is that, you need to solve your equation for a complex function which is simpler and at the end of your calculations remember that your function mus be real. For an inhomogeneous medium, these equations have the form. Now, a real number, say, can take any value in a continuum of different values lying between and. The wave represented above is conventionally termed a threedimensional plane wave. In addition to satisfying the wave equation, the complex fields must be solutions of the maxwell. Region of lw stability in the b, a plane for a single complex ginzburglandau equation. Er rr for the efield of a plane wave we had timeharmonic fields and vector phasors ece 303.
When the values of f \displaystyle f are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector n \displaystyle \vec n, and a transverse wave if they are always orthogonal perpendicular to it. This will result in a linearly polarized plane wave travelling. The dirac equation can be thought of in terms of a square root of the kleingordon equation. For this case the right hand sides of the wave equations are zero. Stability of plane wave solutions in complex ginzburglandau. Thus any cartesian component of e or b obeys a classical wave equation of the form. In the onedimensional scalar case, that is ex,t, eq. For plane waves from the maxwells equations, the following. Here, k is the wave number, which is related to the wavelength of the wave by the equation 2 k. It arises in fields like acoustics, electromagnetics, and fluid dynamics.
Describing real circling motion in a complex way weve seen that any complex number can be written in the form zre i. It is therefore very important for physicists to have a good understanding of the complex description of wave. The 1d wave equation for light waves 22 22 0 ee xt where. It is threedimensional because its wavefunction, depends on all three cartesian coordinates. Moreover, it is a plane wave because the wave maxima are located at. 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. Chapter maxwells equations and electromagnetic waves.
It is shown that they are also useful in representing inhomogeneous plane waves in plasma, singlenegative and. In optics, we are fortunate that two highly developed and accurate theories are available. It means that light beams can pass through each other without altering each other. Plane electromagnetic waves and wave propagation 7. The wave equation is the simplest example of a hyperbolic differential equation. We first consider the solution of the wave equations in free space, in absence of.
The 3d wave equation and plane waves before we introduce the 3d wave equation, lets think a bit about the 1d wave equation, 2 2 2 2 2 x q c t. For the uniform plane wave described by 19 the plane of uniformity is the xyplane. The plane wave represents a solution for the propagation equation that can be easily handled and is largely used to simulate the response of objects or to interpret complex phenomena linked to the interaction between wave and matter, mainly because of its local relationship with the more general spherical solution. Of course the wave function is not inherently complex. Complex roots of a characteris tic polynomial are associated with an oscillatory sinusoidal natural response. This might be a huge obstacle on the way to an intuitive interpretation of the fundamental laws of nature.
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