We deal with both normal and nonnormal angles of incidence. The mathematics of gravitational waves a little over a hundred years ago, albert einstein predicted the existence of gravitational waves as a possible consequence of his theory of general relativity. Planewave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers shouyuan shi, caihua chen, and dennis w. The wave phase is constant along a spherical surface the wavefront. The main reason one would choose to work with the complex exponential form of the plane wave is because complex exponentials are often algebraically easier to work with than the trigonometric sines and cosines. Deriving the fresnel equations 1 introduction the intensity of light re. Cwp646 waveequation migration with dithered plane waves. We have seen that there are solutions of the wave equation which correspond to plane waves, and that any electromagnetic wave can be described as a superposition of many plane waves. Incoming plane wave expikz outgoing spherical waves expikrr fig. The general solution to the em wave equation includes waves that are not squareintegrable, e. Lets start with one that gives a spin up electron in the first two components and plug it into the dirac equation to see what the third and fourth components can be.
A wave is disturbance of a continuous medium that propagates with a fixed shape at constant velocity. Plane waves in lossy media jkz j j z xe j ze z 0 0 0 exponential decay. Kshetrimayum 3192014 polarization is the curve traced out by the tip of the arrow representing the instantaneous electric field. When these vectors coincide in direction, the plane wave is termed homogeneous. But im still a newbie on indices notation and contravariant and covariant objects. Gibbs bivectors are used to give a description of inhomogeneous plane waves in anisotropic homogeneous. Although we will not discuss it, plane waves can be used as a basis for. Expansion of plane waves in spherical harmonics consider a free particle of mass in three dimension. 444 465 may 2012 with 67 reads how we measure reads. This equation determines the properties of most wave phenomena, not only light waves. In the absence of free charge and current densities the maxwell equations are.
Maxwells equations wave equations plane waves boundary conditions a. The sign the product depends on the direction of power. In many realworld situations, the velocity of a wave. Penetration depth, or skin depth what is wavelength, wave velocity in the lossy medium. If and have the same phase but different magnitude we have linear polarization of the field with the polarization vector making an angle with and magnitude. As time evolves, the wavefronts propagate at the wave speed without changing. The polarization of the plane wave describes the relative direction, magnitude, and phase of the electric part of the wave. Towards the schrodinger equation we construct an equation that is valid for matter in the nonrelativistic domain, but also allows for wave like solutions.
The solutions plane wave of maxwells equations are wave like with both e and b satisfying a wave equation. In this issue of notices we focus on the mathematics behind this profound discovery. Sinusoidal planewave solutions of the electromagnetic. The angular dependence of the solutions will be described by spherical harmonics. For the plane waves, one sees that the operators thus the maxwell equations become where. The wave phase is constant along a planar surface the wavefront. Plane electromagnetic waves and wave propagation 7. Cwp646 waveequation migration with dithered plane waves article in geophysical prospecting 603.
Uniform plane waves propagation in free space starting with point form of maxwells equations for time varying fields in free space. This substitution will impose conditions on the constants, and, for the plane wave functions to be solutions of the maxwell equations. In the presence of absorption, the wave will diminish in size as it move. This chapter discusses inhomogeneous plane waves in incompressible elastic materials. 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. Electromagnetic waves travel through empty space with the speed of light c 1 0 0 b b cos kx t e e cos kx t z o y o the plane wave as represented by above is said to be linearly. Thus if we can find a wave equation, we could solve it to find out the fields directly. We shall analyze the source terms in the maxwells equations to simplify maxwells equations in a conductor. Plane wave solutions of the dirac equation stack exchange. The 3d wave equation plane wave spherical wave mit 2. Pdf inhomogeneous plane waves in incompressible elastic. Tem wave the value of the magnetic field is equal to the magnitude of.
For this case the right hand sides of the wave equations are zero. The solutions for nonzero momentum are not as simple. The wave equation in one dimension later, we will derive the wave equation from maxwells equations. A new class of exact solutions of the kleingordon equation. U u one may align propagation of wave k with zdirection, which simplifies the equation. Im trying to understand the plane wave solutions of the dirac equation. In certain special cases, however, it is more convenient to describe the wave field in a different mathematical form. We now look at solving problems involving the laplacian in spherical polar coordinates. This can be written in terms of its real and imaginary parts, representing the real wavenumber vector and the attenuation vector, respectively. Varro s, a new class of exact solutions of the kleingordon equation of a charged particle interacting with an electromagnetic plane wave in a medium.
Wave is unconstrained in plane orthogonal to wave direction, i. The wave equation is an example of a linear differential equation, which means that if. When an electromagnetic wave strikes the surface of a dielectric, both re. In free space the phase velocity is equal to the speed of light,c. Here it is, in its onedimensional form for scalar i. The constant c gives the speed of propagation for the vibrations. We now extend the wave equation to threedimensional space and look at some basic solutions to the 3d wave equation, which are known as plane waves. The timeindependent schrodinger equation for the energy eigenstates in the coordinate representation is given by. If the medium is dispersive different frequencies travel at. This function obeys the wave equation, which is a secondorder partial di erential equation. A plane wave in free space propagates with phase velocity equal to the speed of light.
Planewave expansion method for calculating band structure. Hence, e and h for each travelling wave are related through a simple multiplication operation. Chapter maxwells equations and electromagnetic waves. The conduction current in a conductor is the cause of the di. The implication is that electromagnetic waves obey the superposition principle. For example, a wave on a string is described by the transverse displacement yas a function of xand t. Electric and magnetic field are perpendicular to each other no electric or magnetic field in the direction of propagation.
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