![]() ![]() The paper develops the results of previous authors’ investigations, ,,. The quasi-steady-state solutions are also considered for the case of non-moving time-harmonic source and time-harmonic boundary condition for temperature. Several particular cases of the considered problem including the heat conduction equation and the wave equation are investigated. Expressions which show the wave fronts and elucidate the Doppler effect are obtained. In this paper, we study the Cattaneo telegraph equation for temperature with moving time-harmonic source on a line and on a half-line domain. When the mean-free-path becomes infinite, the probability ρ ( x, t) satisfies the wave equation when the mean-free-path equals zero, the standard diffusion equation is obtained. For the inertial random walk in which the mean-free-path is finite, the probability ρ ( x, t) that the particle exists at a point ( x, t) obeys the telegraph equation. We would like to mention very interesting unified approach based on random walks proposed by Takayasu. The heat waves as solutions of the telegraph equation are also called the “second sound”,. In 1948 Cattaneo proposed the evolution equation for the heat flux Often, in the previous studies, the quasi-steady-state oscillations were investigated when the solution was represented as a product of a function of the spatial coordinates and time-harmonic term e iωt with the angular frequency ω. The first one consists in adding the harmonic source term, the second one involves the time-harmonic boundary conditions. There are two possibilities to introduce oscillations into the parabolic diffusion equation. An extensive review of literature on this subject can be found in,. To describe such type of physical phenomena the term “oscillatory diffusion” is used, in parallel with the term “diffusion-wave”. This pioneering study had aroused considerable interest of researchers and even was translated into English. ![]() Ångström was the first to investigate the standard parabolic diffusion equation (heat conduction equation) under the time-harmonic (wave) impact. ![]() The term “diffusion-wave” is also used in another context. The speed of light is generally a point of comparison to express that something is fast.With the Caputo time-fractional derivative describes different important physical phenomena in bodies with complex internal structure, interpolates between the diffusion equation when α = 1 and the wave equation when α = 2 and exhibits inherent features of both types of equations,. Placing a coin in contact with both terminals of a 9-volt battery produces electromagnetic waves that can be detected by bringing the antenna of a radio (tuned to a static-producing station) within a few inches of the point of contact. These and many more such devices use electromagnetic waves to transmit data and signals.Īll the above sources of electromagnetic waves use the simple principle of moving charge, which can be easily modeled. Both electric and magnetic fields in an electromagnetic wave will fluctuate in time, one causing the other to change.Įlectromagnetic waves are ubiquitous in nature (i.e., light) and used in modern technology-AM and FM radio, cordless and cellular phones, garage door openers, wireless networks, radar, microwave ovens, etc. This means that an electric field that oscillates as a function of time will produce a magnetic field, and a magnetic field that changes as a function of time will produce an electric field. Once in motion, the electric and magnetic fields created by a charged particle are self-perpetuating-time-dependent changes in one field (electric or magnetic) produce the other. When it accelerates as part of an oscillatory motion, the charged particle creates ripples, or oscillations, in its electric field, and also produces a magnetic field (as predicted by Maxwell’s equations). Now, let us understand what the Doppler effect is with a short description. This charged particle creates an electric field (which can exert a force on other nearby charged particles). Also, we will derive the Doppler effect equation and some illustrating Doppler shift facts. The creation of all electromagnetic waves begins with a charged particle. Notice that the electric and magnetic field waves are in phase. The direction of the electric field is indicated in blue, the magnetic field in red, and the wave propagates in the positive x-direction. These waves oscillate perpendicularly to and in phase with one another.Įlectromagnetic Wave: Electromagnetic waves are a self-propagating transverse wave of oscillating electric and magnetic fields. As it travels through space it behaves like a wave, and has an oscillating electric field component and an oscillating magnetic field. \]Įlectromagnetic radiation, is a form of energy emitted by moving charged particles. ![]()
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