Standing Electromagnetic Waves

7 Key excerpts on "Standing Electromagnetic Waves"

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  Electromagnetic Fields of Wireless Communications: Biological and Health Effects

  • Dimitris J. Panagopoulos(Author)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  33 Defning Wireless Communication Electromagnetic Fields When two waves of the same polarization have opposite phases at another location, in other words, when their phase difference is: ˜ ˛ (2n ˝ 1 ) ° [1.14], then the result of their superposition is destructive interference, i.e., a wave of the same polarization but with diminished intensity (or even zero when the two amplitudes are equal). The electrical components of two such waves (plane harmonic waves of the same polarization and frequency) reaching a certain location after having traveled different distances, r 1 , and r 2 , from their two coherent sources are given by the equations: E 1 ˜ E o1 sin ˛ ˙t k w r 1 ˝ ° [1.15] E 2 ˜ E o2 sin ˙t k w 2 ˛ ° r ˝ [1.16] ˜ Again, the amplitude, E o , of the resultant electric feld, E , (electric component of the resultant elec- tromagnetic wave) is: E ˜ E 2 ° E 2 ° 2E E cos˛ [1.17] o o1 o2 o1 o2 where the phase difference among the two vectors is: ˜ ˝ 2° ˆ r r 1 ˙ 2 ˇ depending, in this case, only ˛ upon the difference in the distances traveled by the two waves. At any location where: φ = 2nπ, Eq. 1.17 gives: 2 2 E ˜ E ° E ° 2E E ˛ ˜ ˝ [1.18] o o1 o2 o1 o2 E o1 ° E o2 At these locations, we have constructive interference. At any location where: φ = (2n+1)π, Eq. 1.17 gives: 2 2 ˙ [1.19] E o ˜ E o1 ° E o2 ˛ 2E E o1 o2 ˝ ˜ E o1 ˛ E o2 At these locations, we have destructive interference. The intensity of the resultant wave at any location is: ˜ ˜ ˜ J J ˜ 1 ° J 2 [1.20] The amplitude of the resultant wave intensity will be, correspondingly: 2 J o ˜ c˙˙ o ˛ E o1 ° E o2 ˝ [1.21] 2 J o ˜ c˙˙ o ˛ E o1 ° E o2 ˝ [1.22] (at the locations of constructive interference and at the locations of destructive interference, respectively). Thus, at the locations of constructive interference, the electric feld vectors of the two waves/ felds are parallel and in the same direction, and both the resultant feld and the resultant wave intensity are maximum (Eqs. 1.18 and 1.21).

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  The Physics and Mathematics of Electromagnetic Wave Propagation in Cellular Wireless Communication

  • Tapan K. Sarkar, Magdalena Salazar Palma, Mohammad Najib Abdallah(Authors)
  • 2018(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Hence, it is a wave that oscillates in time but has a stationary spatial dependence. In that case one may encounter locations of zero field strength for all times but for that to occur one has to operate in an environment which has multiple reflections. Reflections from buildings, trees and the like which are located in an open environment, it is most probable that a standing wave does not occur in such circumstances as seen in the Chapter 2 as one is operating in the near field of the transmitting antenna where there are no pattern nulls and the rays are not defined. Another point to be made here is the following: when one turns on a switch in the power line how does the energy travel on the wires? The contribution of Maxwell which is often missed in this context is that the energy does not travel through the electrons in the wires but through the E and B fields which reside outside the wire and they essentially travel at the speed of light in the media in which they are located. The electrons in the conducting wire travel typically at a velocity given by v = I /( n A Q ), where v is the velocity, I is the magnitude of the current flow, n is the number of atoms in a cubic meter of the conductor, A is the cross section area of the wire and Q is the charge of the electron. So for a current flow of 1 A in a copper wire of radius 1 mm, n = 8.5 ×10 28 m 3 and the charge on the electron being –1.6×10 –19 Coulombs, the velocity of the flow of electrons becomes v = [8.5 × 10 28 × ( π × 10 −6 ) × (−1.6 × 10 −19 )] −1 = − 0.000023 m/sec. When a DC voltage is applied, the electron velocity will increase in proportion to the strength of the electric field. AC voltages cause no net move-ment of the electrons as they oscillate back and forth in response to the 1 The Mystery of Wave Propagation and Radiation from an Antenna 16 alternating electric field (over a distance of a few micrometers).

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  Optics for Materials Scientists

  • Myeongkyu Lee(Author)
  • 2019(Publication Date)
  • Apple Academic Press

    (Publisher)

  CHAPTER 1

  Electromagnetic Waves

  1.1 MATHEMATICS OF WAVE MOTION

  1.1.1 ONE-DIMENSIONAL WAVE EQUATION

  A wave can be described as a disturbance that travels through a medium from one position to another position. In physics, waves are time-varying oscillations of a physical quantity around fixed locations. Wave motion transfers energy from one point to another. A wave can be transverse or longitudinal. Transverse waves such as electromagnetic waves occur when a disturbance generates oscillations that are perpendicular to the propagation of energy transfer. Longitudinal waves (e.g., sound waves) occur when the oscillations are parallel to the propagation direction. Waves are described by a wave equation that sets out how the disturbance proceeds over time. Consider a transverse pulsed wave traveling in the positive x-direction with a constant speed v, as shown in Figure 1.1 . Since the disturbance is a function of both position and time, it can be written as

  ψ = f ( x , t ) ⋅

  (1.1)

  We here deal with a wave whose shape does not change as it propagates through space. The shape of the disturbance at any time, say t = 0, can be obtained by holding time constant at that value.

  ψ ( x , t ) | t = 0 = f ( x , 0 ) = f ( x ) ⋅

  (1.2)

  Figure 1.1 shows the wave profiles at t = 0 and t = t, that is, f (x, 0) and f (x, t). They represent the shapes of the disturbance taken at the beginning and end of a time interval t. We now introduce a new coordinate x’ that is defined as

  x ′ = x − v t .

  (1.3)

  In this new coordinate system, the disturbance ψ = f (x´) is no longer a function of time and looks the same at any value of t as it did at t = 0 in the original stationary coordinate system. It follows from Figure 1.1

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  General Physics Electromagnetism Optics

  • Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
  • 2022(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Electromagnetic Waves 14.1 Introduction A wave, or vibration, represents the propagation of a perturbation in a material. Exam- ples of waves are the vibrations of a stretched string, due to induced elastic deformation, the circular waves in a liquid, due to the energy transferred to it by a perturbation in a point, sound waves, due to local pressure variations in a gas. Hence, a wave is generated by an external action which transfers energy to a medium in which the wave propagates: no matter displacement ever takes place, but only an oscillation of the matter which returns to its original position once the perturbation has passed by. A possible solution of Maxwell equations implies the existence of electromagnetic waves which appear to be how time dependent electromagnetic phenomena, i.e. time de- pendent electric and magnetic fields, propagate. The general properties of waves are easier to understand if the propagating oscillation is generated by mechanical effects. In order to find the general properties of waves, the easiest case, that is, transverse waves along a stretched string, is discussed first: the case for elec- tromagnetic waves will be analysed later. 14.2 Elastic Waves along a Stretched String Consider a vibration produced in a stretched elastic string of mass m by a small defor- mation obtained by exerting a force in the direction normal to the string. Chosen an x axis directed as the string and a y axis so that the deformation is contained in plane xy, it is clear that the deformation must be described by a function y = y x, t ( ) : in fact in a given position x on the string, its deformation is given by its coordinate y, which changes in time as the perturbation passes by; likewise, at a given time, the y deformation is different in different positions along the string. A short portion Δ of the string is deformed by a small quantity Δy.

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  Mechanical and Electromagnetic Vibrations and Waves

  • Tamer Bécherrawy(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  A large number of fibers may be assembled to form an optical fiber cable, which may be bent with a radius as small as 1 cm, since light propagates separately in each fiber. This type of cable may be used to transmit images, thus allowing us to observe normally inaccessible places, such as internal organs of the body (endoscopy). It may be used to transmit an intense laser beam to be used in medical surgery and in industry. Because of the high frequency of light waves, fibers and lasers may be associated with telecommunications for the transmission of an enormous quantity of information as optical signals for distances of up to thousands of kilometers.

  9.12 Summary

  A standing wave may be established in a bounded medium of propagation. Its amplitude vanishes at the nodes and it is at a maximum at the antinodes. The region between two adjacent nodes form a wave zone. All the points of the same wave zone oscillate in phase and points of adjacent wave zones oscillate in opposite phase.

  The one-dimensional wave equation has solutions of the form u st = f (x ) g (t ). The wave equation is then equivalent to two independent equations for g and f . The harmonic solution, which satisfies the boundary conditions, is a standing wave u = A cos (ωx /v = ϕ) cos(ωt + α) The two energy densities are not equal. The energy flux vanishes at the nodes and antinodes; thus, there is no transfer of physical quantities from one half-zone to the other. The intensity of the wave is zero at all points.

  The boundary conditions can be satisfied only in discrete modes of the form u n =f n (r) g n (t), where

  n = (n

  1 , n 2 , etc.) and the n i are integer numbers. The functions u n are orthogonal and complete: any function u (r, t ), that verifies the same boundary conditions, is a linear superposition of the modes: u (r, t ) = ∑n a n u n (r, t ).

  If a string is fixed at both ends, the condition u (0, t ) = u (L , t ) = 0 is verified if the string is formed by an integer number n of wave zones of length λ/2; thus, L = n λ/2 and u n = A n sin(ωn x /v ) cos(ωn t + αn ), where n = 1, 2 …, ωn = n πv/L and .The total energy of the string in the mode A string that is fixed at one end and free at the other is formed by (n − ½) wave zones of length λ/2; thus, with n

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  Principles of Engineering Physics 1

  • Md Nazoor Khan, Simanchala Panigrahi(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  At all the points, P 1 , P 2 , P 3 , P 4 , P 5 , P 6 , P 7 , P 8 , P 9 , the particles of the medium have zero amplitude. See Fig. 1.17(e). 56 Principles of Engineering Physics 1 1.10.2 Characteristics of stationary waves The followings are the characteristics of stationary waves. i. A stationary wave is produced when two identical progressive waves (i.e., waves having the same wavelength, the same time period, the same frequency, and the same speed) travelling in the same line but in opposite directions are superimposed. ii. The points at which particles are not oscillating are called nodes. iii. The points at which particles are oscillating with maximum displacements are called anti nodes iv. The distance between any two consecutive nodes is half the wavelength. v. The distance between any two consecutive anti nodes is half the wavelength vi. The distance between any two consecutive nodes and anti nodes is one-fourth of the wavelength. vii. Except at nodes, each particle of the medium executes simple harmonic oscillation about its mean position with a time period equal to the time period of the wave motion. viii. Amplitudes of all the particles of the medium are not the same. It is zero at the node and maximum at the anti node. 1.10.3 Differences between progressive and stationary waves Progressive waves Stationary waves 1 A progressive wave is produced due to the oscillation of the particles of the medium. A stationary wave is produced when two identical progressive waves travelling in the same line but opposite directions are superimposed. 2 The waves move with a velocity depending upon the properties of the medium. The waves remain stationary and do not move. 3 Each particle of the medium executes periodic motion about their mean position with the same amplitude. Except the node, all the particles of the medium execute SHO with varying amplitude. 4 There is a continuous change of phase from particle to particle.

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  Electromagnetism

  Maxwell Equations, Wave Propagation and Emission

  • Tamer Becherrawy(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  (±) /ω. In the general case of a progressive wave, a standing wave or a superposition of both types of waves, we write

  [10.82]

  In the case of a purely progressive wave, the Poynting vector is S = vU EM,v e . The momentum of the wave being directed toward the electron, the orbital angular momentum carried by the wave is equal to zero and s EM,v (±) is effectively the density of intrinsic angular momentum (or spin ) of the wave.

  In the case of a left-handed circularly polarized wave

  E(−) ,

  s v (−) points in the direction of propagation (positive helicity), while in the case of the right-handed circularly polarized wave

  E(+) ,

  s v (+) points in the opposite direction of the direction of propagation (negative helicity).

  Generally, an electromagnetic wave has orbital angular momentum , which adds to its intrinsic angular momentum; this increases the effect of angular momentum transfer. The total angular momentum density may be written as

  [10.83]

  If a body receives an electromagnetic wave, the transfer of angular momentum may set the body in rotation. This effect was verified experimentally by Berth in 1936 by observing the rotation of a plate of quartz if it intercepts a circularly polarized wave. Actually, the effect may be easily observed by using a laser beam.

  10.6. A simple model of dispersion

  The speed of propagation is independent of the signal profile if the magnetic permeability μ and the electric permittivity ε are constant characteristics of the medium. The reality is more complex: if an electromagnetic wave is incident on a dielectric, a fraction of it penetrates as a primary wave . It acts on the charged particles in the medium. These particles (especially the electrons) emit secondary electromagnetic waves. The superposition of the primary wave and secondary waves is the transmitted wave, while the reflected wave is a superposition of secondary waves emitted back, toward the incidence medium. We first note that on the microscopic scale, the electrons, atoms, and molecules are in permanent thermal agitation, the fields undergo large fluctuations and they become infinite at the positions of the particles. These are the microscopic fields . On the contrary, the Maxwell's equations are written in terms of the macroscopic

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