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. This is because it comes from the expansion of the plane wave part only. Of course the wave function is not inherently complex. Boundary conditions poyntings theorem and conservation laws complex function formalism time average of sinusoidal products wave equation 2. Plane electromagnetic waves and wave propagation 7. Chapter maxwells equations and electromagnetic waves. Ex,t is the electric field is the magnetic permeability is the dielectric permittivity this is a linear, secondorder, homogeneous differential equation. What this equation says is that the wave converging on the scatterer e.
We will now exploit this to perform fourier analysis on the. A real number, say, can take any value in a continuum of values lying between and. It is shown that they are also useful in representing inhomogeneous plane waves in plasma, singlenegative and. A plane wave with a fixed direction of the electric field vector e0 is termed lin. Describing real circling motion in a complex way weve seen that any complex number can be written in the form zre i. This substitution will impose conditions on the constants, and, for the plane wave functions to be solutions of the maxwell equations. Thus this em wave approaches the boundary from the left and is at normal incidence to the boundary.
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. Negative energies with no lower bounds is incompatible with stability. 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. Representing plane waves in three dimensional space. The leaky surface wave arises from the complex conjugate roots of the rayleigh equation. Here, we write down the complex amplitudes for the eb and fields. For the plane waves, one sees that the operators thus the maxwell equations become where. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. The wave equations, written for the complex amplitudes monochromatic waves, are referred to as the helmholtz equations. The dirac equation can be thought of in terms of a square root of the kleingordon equation. For neatness, we sometimes write the wave equation by putting everything on. Imrana ashraf zahid quaidiazam university, islamabad pakistan. 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.
Lecture 14 time harmonic fields cornell university. 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. 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. Er rr for the efield of a plane wave we had timeharmonic fields and vector phasors ece 303. Repeating this sort of thing using one of the the curl eqns say, faradays law one gets. Region of lw stability in the b, a plane for a single complex ginzburglandau equation. If f 1 x,t and f 2 x,t are solutions to the wave equation, then. The 1d wave equation for light waves 22 22 0 ee xt where. They can be complex numbers, as in a complex exponential plane wave.
The equation has no solution for in the set of real numbers. T is the velocity of the wave, which can be determined from electrostatics and magnetostatics. In optics, we are fortunate that two highly developed and accurate theories are available. There is a very nice set of complex solutions to the wave equation, known as complex travelling waves, which take the form fx. The wave equation in cylindrical coordinates overview and. It is a threedimensional form of the wave equation. Stability of plane wave solutions in complex ginzburglandau equation with delayed feedback d. 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.
Consistent with geometrical optics, its magnitude is constrained to be proportional to the refractive index n 2. 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. Thus any cartesian component of e or b obeys a classical wave equation of the form. Which in many cases means taking the real part but not always. It is therefore very important for physicists to have a good understanding of the complex description of wave. The wave equation and the speed of light chapter 1 physics 208, electrooptics. For an inhomogeneous medium, these equations have the form. 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. Plane waves and refractive index chapter 2 31 w p l 2 c vac. And is a complex wave vector the length of this vector is inversely proportional to the wavelength of the wave. For plane waves from the maxwells equations, the following. We deal with both normal and nonnormal angles of incidence.
The electromagnetic wave equation is a secondorder partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. If the wave function were real, performing a fourier transform in time will lead to pairs of positivenegative energy eigenstates. Let us examine this set of examples in further detail. Here, k is the wave number, which is related to the wavelength of the wave by the equation 2 k. H wm2 44 equation 44 is known as the poyting vector. Gurevich abstract we perform bifurcation analysis of plane wave solutions in onedimensionalcomplexcubic. E xois complex in general so it has a magnitude je xojand a phase ej since the. Uniform plane waves have uniform constant properties in a plane perpendicular to their direction of propagation. Application and implication of using complex vectors and complex transformations in solutions of maxwells equations is investigated. Electron diffraction in mathematics, the symbol is conventionally used to represent the squareroot of minus one. The wave equation in cylindrical coordinates overview and motivation. Plane waves representation of waves via complex functions in mathematics, the symbol is conventionally used to represent the squareroot of minus one. It means that light beams can pass through each other without altering each other. The 3d wave equation, plane waves, fields, and several 3d differential operators.
Thus the wave train cannot penetrate much below the interface. We now give a brief derivation of sufficient conditions. Since e, h, and k form a righthanded system, the power ows in the direction of wave propagation. On the complex conjugate roots of the rayleigh equation. 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. In the onedimensional scalar case, that is ex,t, eq. For this case the right hand sides of the wave equations are zero. 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. Mimicking the dalembert solution for the wave equation, we anticipate that the solutions to the laplace equation 2. The homogeneous form of the equation, written in terms of either the electric field e or the magnetic field b, takes the form.
Complex numbers are extremely useful for describing the propagation of waves. 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. In addition to satisfying the wave equation, the complex. In the absence of free charge and current densities the maxwell equations are. The wave represented above is conventionally termed a threedimensional plane wave. Compare it to the case of the plane wave without scattering eq. Waves, the wave equation, and phase velocity what is a wave. It couples into a plane shear wave that propagates in the medium. Simple derivation of electromagnetic waves from maxwells. The irreducible representations of the lorentz group are given by, as a gen. This might be a huge obstacle on the way to an intuitive interpretation of the fundamental laws of nature. It also means that waves can constructively or destructively interfere.
Now, a real number, say, can take any value in a continuum of different values lying between and. A wave is disturbance of a continuous medium that propagates with a fixed shape at constant velocity. 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. A plane electromagnetic wave is therefore a transverse wave. In the presence of absorption, the wave will diminish in size as it move. 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. 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. The 3d wave equation for the electric field and its solution whose solution is where is a constant, complex vector. Complex waves complex numbers are extremely useful for describing the propagation of waves. Now, consider what happens if we take the real part of eq. It arises in fields like acoustics, electromagnetics, and fluid dynamics.
Stability of plane wave solutions in complex ginzburglandau. Representation of waves via complex numbers in mathematics, the symbol is conventionally used to represent the squareroot of minus one. There is a simple set of complex traveling wave solutions to this equation. Complex vectors are used in complex plane waves and help to represent this type of waves geometrically. The complex form of the plane wave just introduced can be simplified by using a complex valued amplitude in.
Application of complex vectors and complex transformations in. In addition to satisfying the wave equation, the complex fields must be solutions of the maxwell. The wave is polarized in the x direction by convention, we use the direction of e to specify the polarization of an. The most general solution has two unknown constants, which. 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. For the uniform plane wave described by 19 the plane of uniformity is the xyplane. Multiplication and division of complex numbers is easier. 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 complex constant ais called the complex amplitude of the wave. 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. If the medium is dispersive different frequencies travel at. Complex amplitude an overview sciencedirect topics. In physics, a plane wave is a special case of wave or field. The dependence of r on various parameters is best displayed in the complex plane r complex plane as shown in. We first consider the solution of the wave equations in free space, in absence of. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines. So, complex wave functions are needed for stability. In the older theory often described as the classical theory the behaviour of. Pesky inverse tangent whenever doing a phasor or geometrical conversion, the inverse tangent formula in equation 1. Er rr for the efield of a plane wave we had timeharmonic fields and vector phasors ece 303 fall 2007 farhan rana cornell university. Very often the propagation constant k is taken as a vec tor k. Here, is a real number representing the magnitude of, and represents the angle of in the complex plane. There is more information contained in maxwells equations than there is in the wave equation.
Complex roots of a characteris tic polynomial are associated with an oscillatory sinusoidal natural response. Let us see what additional constraints are thereby imposed. 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. The wave equation is the simplest example of a hyperbolic differential equation. The sc hr o ding er w av e equati on macquarie university. This will result in a linearly polarized plane wave travelling. 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. This includes electromagnetic waves radio waves, visible light, xrays, etc. On the other hand, an imaginary number takes the general.
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