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Weather Radar Polarimetry Wave Scattering by a Single Particle This article was downloaded by: 10.3.98.104 On: 05 Oct 2021 Access details: subscription number Publisher: CRC Press Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: 5 Howick Place, London SW1P 1WG, UK Weather Radar Polarimetry Guifu Zhang Wave Scattering by a Single Particle Publication details https://www.routledgehandbooks.com/doi/10.1201/9781315374666-4 Guifu Zhang Published online on: 01 Aug 2016 How to cite :- Guifu Zhang. 01 Aug 2016, Wave Scattering by a Single Particle from: Weather Radar Polarimetry CRC Press Accessed on: 05 Oct 2021 https://www.routledgehandbooks.com/doi/10.1201/9781315374666-4 PLEASE SCROLL DOWN FOR DOCUMENT Full terms and conditions of use: https://www.routledgehandbooks.com/legal-notices/terms This Document PDF may be used for research, teaching and private study purposes. Any substantial or systematic reproductions, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Wave Scattering by 3 a Single Particle Weather radars measure electromagnetic (EM) wave scattering by the hydromete- ors that are in clouds and precipitation. To understand polarimetric radar signatures as they relate to the shape, orientation, and composition of these hydrometeors for specific weather conditions, it is important to understand EM wave scattering by various types of hydrometeors. This chapter deals with wave scattering by a single particle, focusing on the sphere and the spheroid shapes on which many hydromete- ors can be modeled. This chapter describes the basic physics behind and mathemati- cal representation of EM waves and wave scattering. The approaches for scattering calculations, including the Rayleigh approximation, Mie theory, and T-matrix method, are provided. The basic scattering characteristics of hydrometeors are also described. 3.1 WAVE AND ELECTROMAGNETIC WAVE A wave is the propagation of a vibration or oscillation and is an efficient way of transporting energy and information. Waves are omnipresent in our daily lives. We hear each other through sound, which are acoustic waves, and see our environment by light, which are EM waves at extremely high frequencies. An acoustic wave is a longitudinal wave, which vibrates air molecules along the direction of the wave propagation. As such, only a one-dimensional variable of the displacement from its reference is needed to describe wave characteristics. A longitudinal wave is also called a scalar wave. An EM wave, however, is a transverse wave, which vibrates electrons (if present) in a direction perpendicular to the direction of propagation. Since a two-dimensional vector is needed to define the vibration direction, a trans- verse wave is also called a vector wave. 3.1.1 VIBRATION AND WAVE To mathematically describe a wave, we must start with its vibration. A vibration is a back-and-forth movement that is caused by a force, f = −κz, in the opposite direction of the displacement z (κ is the restoring coefficient). Examples of vibration include a spring and a pendulum, as shown in Figure 3.1. Based on Newton’s second law, the equation for the displacement is written as dz2 fm==−κz, (3.1) dt2 51 Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 52 Weather Radar Polarimetry m z m FIGURE 3.1 Conceptual illustration of a spring vibration. The restoring force is opposite to the displacement. z t FIGURE 3.2 Sketch of a vibration with instantaneous displacement z as a function of time. where m is the mass. We can rewrite Equation 3.1 to yield dz2 +ω2z = 0, (3.2) dt2 where ω2 = κ/m. Solving Equation 3.2 yields z(t) = A cos (ωt + ϕ0), (3.3) where A is the amplitude, which is defined as the maximal displacement of vibration from its equilibrium position, ω = 2πf is the angular frequency ( f = 1/T is the fre- quency, with T as the period), and ϕ0 in Equation 3.3 is the initial phase at t = 0. The phase is the argument of the cosine function, which is zero in Figure 3.2. As a wave is the propagation of a vibration, its mathematical expression can be obtained by changing the time t to t − x/v in Equation 3.3, yielding z(x,t) = Acos[ω(t − x/v) + ϕ0], (3.4) where v is the phase velocity of the wave, and the wavelength is λ = vT (as shown from peak to peak in Figure 3.3). Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Wave Scattering by a Single Particle 53 z at (t=t1) x z at (t=t2) x z at (t=t3) x z at (t=t4) x FIGURE 3.3 Sketch of wave propagation: asterisk (*) propagates in the x-direction. 3.1.2 PHASOR REPRESENTATION OF TIME-HARMONIC WAVES Although a time-harmonic wave can be simply represented by a sinusoidal function like Equation 3.4, it can be more conveniently represented by a complex quantity. For example, Equation 3.4 can be rewritten as jt(/ω+φ−0 ωωxv) jt zx(,tA)c=ωos tx+φ()0 −ω /RvA = eR ez = e( xe), (3.5) where zx()= Ae jx()φ−0 ω /v is a complex quantity, which is equivalent to the real quan- tity z(x,t) in representing the wave, except for the omission of [ejωt] in writing. The complex quantity z(x) is called a phasor because it is a complex quantity with phase information, but omits the time dependence term. The advantage of introducing the phasor in the complex representation is the convenience it allows in mathematical manipulation. For example, a derivative with respect to time t is simply a multiplica- ∂zx(,t) tion of jω : ↔ωjz(,xt). ∂t 3.1.3 EM WAVE As mentioned earlier, an EM wave is a transverse wave that needs a vector in the form of an electric field Er(),t to represent it. An EM wave propagates in space or a medium such that a changing electric field causes a magnetic field. This mag- netic field is also changing, which in turn yields a varying electric field, and so on. Using the analogy of the phasor for a scalar wave, the phasor representation of the Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 54 Weather Radar Polarimetry jtω electric field is Er(),RtE= e ()re . In the case of the time-harmonic wave, the Maxwell equations (Equation 2.14) for the phasors of the electric and magnetic fields become ∇×Ej=− ωµH (3.6) ∇×Hj=ωεE (3.7) ∇=i E 0 (3.8) ∇=i H 0. (3.9) Taking the curl of Equation 3.6 and using the vector identity of ∇×∇×EE=∇()∇• −∇2E, and using Equation 3.7, we have ∇+22EkE = 0 (3.10) with the wave number 2π kk=ω µε =ω µε00 ε=r 0mm= . (3.11) λ0 In the uniform medium where the permittivity and permeability are constant, a plane wave is the solution of Equation 3.10, specifically: −•jk r Ee= ˆEe0 (3.12) and E Hk=׈ eˆ 0 e−•jk r , (3.13) η µ where η= is the intrinsic impedance of the medium, eˆ is the unit vector for ε polarization, kk= kˆ is the wave vector, and kˆ is the unit vector of wave propagation (Figure 3.4). In the case of a spherical wave, the electric field can be expressed by e− jkr Ee= ˆE0 (3.14) r and it is illustrated as in Figure 3.5, where the equal phase constitutes spherical surfaces. Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 Wave Scattering by a Single Particle 55 z O y k x E FIGURE 3.4 A sketch of a uniform plane wave propagation, where phases of the wave are equal on a plane that is perpendicular to the wave vector. k E FIGURE 3.5 A sketch of a spherical wave propagation, where equal phase surfaces are perpendicular to the wave vector. 3.1.4 WAVE POLARIZATION AND REPRESENTATION An EM wave is a transverse wave, and the electric field E varies sinusoidally with time (vibration) at a given location. The wave polarization is a description of the vibration direction, which is the locus of the tip of the E vector as time progresses. Figure 3.6 shows examples of the polarization of an electric wave field from a coherent source or sources. If the locus of the E vector is a straight line, as shown in Figure 3.6a, the wave is said to be linearly polarized (or in linear polarization); if the loci form a circle as in Figure 3.6b, it is circularly polarized (circular polarization); and if the locus is an ellipse as in Figure 3.6c, it is elliptically polarized (elliptical polarization). If the locus is random, however, the wave is unpolarized. Sunlight or Downloaded By: 10.3.98.104 At: 20:46 05 Oct 2021; For: 9781315374666, chapter3, 10.1201/9781315374666-4 56 Weather Radar Polarimetry z z (0, A) Az E Ez Ez E y y Ey Ay Ey (A, 0) (a) (b) z (0, Az) E Ez y Ey (Ay, 0) (c) FIGURE 3.6 Polarization of an electric wave field.
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