Pdf the wave equation and general plane wave solutions in. Contrary to the previous case, the wave presents constant amplitude and phase on a plane orthogonal to k. 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. For this case the right hand sides of the wave equations are zero. Note that when we plug in the other two points into this equation, they satisfy the. Mechanical waves those waves which require a material medium for their propagation, are called mechanical waves, e. Multidimensional waves plane waves as we saw in the previous chapter, a sinusoidal wave of amplitude, wavenumber, and angular frequency, propagating in the positive direction, can be represented in terms of a. 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. In physics, a plane wave is a special case of wave or field. The constant c gives the speed of propagation for the vibrations. Hirschberg eindhoven university of technology 28 nov 2019 this is an extended and revised edition of iwde 9206. In the lightcone gauge, the action is quadratic in the zi 2, and hence the theory is, at least in principle, exactly solvable. In practice, plane wave does not exist because practical wave sources are always finite in extent. This magnitude varies as we move in z and in t but is always uniform in the plane.
Weak vti type anisotropy causes an apparent shift away from the plane wave response toward smaller depth. Thus, in addition to a direction k the wave is moving, there is a direction e0 describing which way the electric. Then the variable in the exponent must be replaced by, the projection of in the direction. General plane wave solution to the wave equation for.
As expec ted, the larger the curvature radius of spherical waves, the closer the avaresponse to the plane wave comparison. We can think of a wave front as a line along the crest of the wave. Lee shows the electromagnetic wave equation can be derived by using maxwells equation. The most popular form in algebra is the slopeintercept form. For propagation in a cube face the characteristic equation factors into a linear tcrm and a quadratic term. The exciting realization is that the speed of the em wave matches with the speed of light. This equation determines the properties of most wave phenomena, not only light waves. Electromagnetic fields in media, uniaxial media, quarterwave plate. In this case, the solutions can be hard to determine. In this paper, we study the approximation of solutions of the homogeneous helmholtz equation. In many realworld situations, the velocity of a wave.
The 3d wave equation, plane waves, fields, and several 3d differential operators. Hence, e and h for each travelling wave are related through a simple multiplication operation. Electromagnetic plane wave of frequency and wave vector suppose an electromagnetic plane wave with direction of propagation to be constructed, where is a unit vector. There are really two wave equations, one for e and one for b we could also derive a wave equation for the magnetic field, using a very similar approach. This equation implies that the wave number of the electron, which is also a quantum number characterizing its state, increases linearly with time under the.
The sign the product depends on the direction of power. Since there is nothing special about the direction, it follows that if is reinterpreted as a unit vector pointing in an arbitrary direction then can be reinterpreted as the general equation of a plane. Lecture 1 course syllabus overview lecture 2 harmonic oscillations complex numbers lecture 3 two coupled oscillators normal modes lecture 4 normal mode coordinates initial value problem homework 1 lecture 5 linear chain normal modes lecture 6 traveling waves, standing waves, and the dispersion relation. 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. In free space the phase velocity is equal to the speed of light,c. For example, a wave on a string is described by the transverse displacement yas a function of xand t. Basic equations of lines and planes equation of a line.
The screen readerfriendly pdf version of this book is tagged structurally and includes. The mathematics of pdes and the wave equation mathtube. This latter solution represents a wave travelling in the z direction. Pdf in electrodynamics courses and textbooks, the properties of plane electromagnetic waves in both conducting and nonconducting media. This equation implies that the magnetic field in a plane wave is completely. There is a pure shear wave polarized normal to the cuhe. The plane containing both the normal to the surface and the direction of. Plane waves a uniform plane wave is a particular solution of maxwells equations with e assuming the same direction, same magnitude, and same phase in infinite planes perpendicular to the direction of propagation similarly for h. The derivative matrix defines the direction of planewave propagation. Three dimensional geometry equations of planes in three. A plane wave is a solution of the propagation equation that propagates in an direction orthogonal to a plane, with normal k, and its electric field is erte0ej.
Plane wave approximation of homogeneous helmholtz solutions a. Pdf this work presents the analytical solution of vector wave equation in fractional space. From inspection of fresnels equations, we can see that the relative. Fortunately, this is not the case for electromagnetic waves. As before, the plane is normal to, and its distance of. Boundary conditions, kr, phase matching, nonuniform plane wave, snells law. One dimensional wave equation the wave equation is a partial differential equation that any arbitrary wave form will satisfy it 22 222 the wave equation is a partial differential equation that any arbitrary wave form will satisfy it. The twophoton wave function exists in six spatial dimensions, and its equation of motion can be called the twophoton maxwells equations. Waves, the wave equation, and phase velocity what is a wave. Here it is, in its onedimensional form for scalar i. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. In the last lecture, we showed that maxwells equations admit plane wave solutions.
The wave equation in one dimension later, we will derive the wave equation from maxwells equations. Our deduction of the wave equation for sound has given us a formula which connects the wave speed with the rate of change of pressure with the density at the normal pressure. Magnetic field in a uniform electromagnetic plane wave x. Plane waves, spherical waves and angledependent pwave. The large arrow is a vector called the wave vector, which defines 1 the direction of wave propagation by its orientation perpendicular to the wave fronts, and 2 the wavenumber by its length. For a gaussian beam, the wave is a superposition of plane waves in various directions, resulting a delay in the zdirection of the composite waves. For the dielectric materials used to guide high frequency electromagnetic waves which we will mainly consider here the conductivity is typically zero, while the permeability is that of free space. We will now exploit this to perform fourier analysis on the.
Chapter 2 uniform plane wave equation and power balance. A plane wave in free space propagates with phase velocity equal to the speed of light. Solutions of the maxwell equations and photon wave functions. Chapter maxwells equations and electromagnetic waves. The angularmomentum operator and the corresponding eigenfunctions are given and shown to be complete in sec.
This function obeys the wave equation, which is a secondorder partial di erential equation. In many realworld situations, the velocity of a wave depends on its amplitude, so v vf. Chapter 1 transport properties university of virginia. Thus an electromagnetic plane wave with direction of. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. Wavesplane waves wikibooks, open books for an open world. Reflection and transmission of light from multilayer films. An important topic of high school algebra is the equation of a line. The boundary conditions also require that the angular frequencies of all the waves be the same, the complex wave vectors of each plane wave lie in a common plane the plane of incidence and that.
This means an equation in x and y whose solution set is a line in the x,y plane. Reflection and refraction of a plane wave at oblique incidence let us consider a plane wave that obliquely incidents at the boundary of two media that are characterized by their permittivity and permeability see figure 1. Reflection and refraction of a plane wave at oblique incidence. A wave is a vibratory disturbance in a medium which carries energy from one point to another point without any actual movement of the medium. Plane wave approximation of homogeneous helmholtz solutions. Most of you have seen the derivation of the 1d wave equation from newtons and. The latex source files also include alternative text for all images and figures. We call e 0 the polarization vector for the electric. Propagation of a uniform plane wave in an arbitrary direction. We now want to consider a general plane wave that has arbitrary linear polarization, and travels in any direction. To conclude, the usual classical maxwell field is the quantum wave function for a single photon. Plane waves form a basis of all possible solutions to the wave equation. Compared to conventional planewave migration, the cost of planewave migration in tilted coordinates is a little higher because of the data and.