The homogeneous form of the equation, written in terms of either the electric field e or the magnetic field b, takes the form. Coulombs law and the nomagnetic charge law must still be included with these in order to get a unique maxwell equations in wave form. This form for the solution is the fourier expansion of the spacetime solution, 0 r,t. Apr 30, 2014 quick derivation of the wave equation for electromagnetic waves along with a discussion of the general solution for 1d waves. Both the electric field and the magnetic field are perpendicular to the direction of travel x. Strauss, chapter 4 we now use the separation of variables technique to study the wave equation on a. We first consider the solution of the wave equations in free space, in absence of matter and sources. Simple derivation of electromagnetic waves from maxwells. Electromagnetic wave equation hyperphysics concepts. Linear electromagnetic wave equations in materials article pdf available in photonics and nanostructures fundamentals and applications 26 april 2017 with 401 reads how we measure reads. Electromagnetic boundry conditions, poyntings theorem, timeharmonic em fields. Electromagnetic wave problems 4 solution in detail below. May 14, 2012 quick argument to find solutions of wave equation derivation of general solution of the wave equation. The form of a plane wave solution for the electric field is.
Simple derivation of electromagnetic waves from maxwells equations. We will derive the wave equation from maxwells equations in free space where i and q. It is a threedimensional form of the wave equation. The string has length its left and right hand ends are held. Chapter 6 maxwells equations for electromagnetic waves. E e b b maxwells equations imply that empty space supports the propagation of electromagnetic waves, traveling at a speed 8 00 1.
The solutions to the wave equations, where there is no. Electromagnetic field theory pdf vssut emft pdf vssut. Thus the poynting vector represents the ow of energy in the same way that the current jrepresents the ow of charge. Pdf solution of electromagnetic wave equation in graded interface. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Lei when an electromagnetic wave travels from one medium to another its speed changes either increases or. In the last section, we transform the the generalized wave equation and. Lee shows the electromagnetic wave equation can be derived by using maxwells equation. The 1d wave equation for light waves 22 22 0 ee xt where. What is the wavelength of a microwave that has a frequency of 4. Thus, in cylindrical coordinates the wave equation becomes 2 2 2 2 2 2 2 2 2 2 1 z q c t.
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. Wherever possible, electrodynamic solutions are examined in various limits to illustrate the appropriateness of. Wave equation in 1d part 1 derivation of the 1d wave equation vibrations of an elastic string solution by separation of variables three steps to a solution several worked examples travelling waves more on this in a later lecture dalemberts insightful solution to the 1d wave equation. Matrix transformation and solutions of wave equation of free. The wave equation for a plane electric wave traveling in the x direction in space is. The final solution for a give set of, and can be expressed as. Separation of variables to look for separable solutions to the wave equation in cylindrical coordinates we posit a product solution. Plane wave solutions of the free dirac equation assume solutions of the form. The implication is that electromagnetic waves obey the superposition principle. The simplest solutions to the differential equations 3 and 4 are. The equations above in boxes are maxwells equations in waveequation form. Sinusoidal planewave solutions of the electromagnetic wave. Equating the speed with the coefficients on 3 and 4 we derive the speed of electric and magnetic waves, which is a constant that we symbolize with c.
In this discussion, vectors are denoted by boldfaced underscored lowercase letters, e. In figure 1, the explicit solution for a special case of the di. In the presence of absorption, the wave will diminish in size as it move. For the nonhomogeneous differential equation k2c2 2 is not required and one must make a fourdimensional fourier expansion. The complete family of solutions forms a wide and sometimes rather compli. We deal with both normal and nonnormal angles of incidence.
For the case that and are constant our equation reduces to the standard waveequation which can be found in every electromagnetic book. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. The electromagnetic wave from maxwells equations contd 2 2 t e e w w u u 2 2 2 t e e e o o w w x ph xe 0 using the vector identity becomes, in free space and we are left with the wave equation 0 2 2 2 w w t e e p oh o. Pdf linear electromagnetic wave equations in materials. The homogeneous form of the equation, written in terms of either the electric field e or the magnetic field b, takes the form left abla2 1 over c2 partial2 over partial t2 ight mathbfe 0. The wave equation for e and b in vacuum, each cartesian component of e and b satisfies the threedimensional wave equation 2 2 00 2 2 2 2 22 2 00 2 1 t f f vt t. Consistent with geometrical optics, its magnitude is constrained to be proportional to the refractive index n 2. Static equations and faradays law the two fundamental equations of electrostatics are shown below. The energy and intensity of electromagnetic waves will be. The general solution to the electromagnetic wave equation is a linear superposition of waves of the form.
Maxwalls equations in differential and integral forms. The electromagnetic wave equation is a secondorder partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. There are some attempts to consider propagation of the electromagnetic waves in the context of the general equation given in 2. Ex,t is the electric field is the magnetic permeability is the dielectric permittivity this is a linear, secondorder, homogeneous differential equation. Both equations 3 and 4 have the form of the general wave equation for a wave \, xt traveling in the x direction with speed v. For this case the right hand sides of the wave equations are zero. The only possible solution of the above is where, and are constants of, and. Electromagnetic field theory a problemsolving approach. The solutions of wave equation in cylindrical coordinates the helmholtz equation in cylindrical coordinates is by separation of variables, assume. Pdf the electromagnetic wave equation in graded interface for exponential permeability and exponential permittivity profile was solved using. This is the threedimensional wave equation in vector form. The wave equation in cylindrical coordinates overview and. Wave equation maxwells equations contain the wave equation for electromagnetic waves. Sinusoidal planewave solutions are particular solutions to the electromagnetic wave equation the general solution of the electromagnetic wave equation in homogeneous, linear, timeindependent media can be written as a linear superposition of planewaves of different frequencies and polarizations.
For example, plane electromagnetic waves in free space may be labelled using the three components of the wave vector that describes the direction and wavelength of the wave. He also shows the progressing em waves can be reflected by a perfect conductor. The general solution to the electromagnetic wave equation is a linear superposition of waves of the form and for virtually any wellbehaved function g of dimensionless argument. The wave equation for electromagnetic waves arises from maxwells equations. If the medium is dispersive different frequencies travel at. 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. Solution of the wave equation by separation of variables. Electromagnetic waves example problems what is the frequency green light that has a wavelength of 5. The most general solution has two unknown constants, which. The electromagnetic wave equation is a secondorder partial differential equation that describes the propagation of electromagnetic wave s through a medium or in a vacuum. In the absence of free charge and current densities the maxwell equations are. Matrix transformation and solutions of wave equation of. In particular, it can be used to study the wave equation in higher.
Quick derivation of the wave equation for electromagnetic waves along with a discussion of the general solution for 1d waves. The symbol c represents the speed of light or other electromagnetic waves. First we need to consider what would be the best equation to use. This equation is mathematically the same as the harmonic oscillator equation and has solutions where is a constant vector.
Plane wave propagation in lossless and lossy dielectric medium and conductiong. To be consistent with maxwells equations, these solutions must be related by. Substitute amperes law for a charge and currentfree region. The electromagnetic wave from maxwells equations contd 2 2 t e e w w u u 2 2 2 t e e e o o w w x ph xe 0 using the vector identity becomes, in free space and. As mentioned above, this technique is much more versatile. Obviously, the equation involving energy change, planks constant, and frequency is the best way to go. The strategy is to take a plane wave solution to 18. The exciting realization is that the speed of the em wave matches with the speed of light. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. A wave is disturbance of a continuous medium that propagates with a fixed shape at constant velocity. Solution of the wave equation by separation of variables the problem let ux,t denote the vertical displacement of a string from the x axis at position x and time t.
Formally this corresponds the upper solution corresponds to a particle with momentum p and energy e while the lower solution has. The solutions of wave equation in cylindrical coordinates. Chapter maxwells equations and electromagnetic waves. Take the point source as an example, the spectral decomposition or the planewave expansion of the.
Maxwells equations have given us the further information that electromagnetic waves have field components only at right angles to the direction of the wave propagation. E total 0 coulombs law in differential form coulombs law is the statement that electric charges create diverging electric fields. To see what wave equations the electromagnetic elds. Imrana ashraf zahid quaidiazam university, islamabad. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. Next, we need to figure out what we are solving for.
1282 1366 540 156 893 1350 620 1590 1190 700 1392 725 1317 144 1597 1372 710 817 1312 441 588 1179 1000 1425 817 1438 896 1354 1442