9: Plane Waves in Loseless Media
- Page ID
- 3957
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 9.1: Maxwell’s Equations in Differential Phasor Form
- In this section, we derive the phasor form of Maxwell’s Equations from the general time-varying form of these equations. Here we are interested exclusively in the differential (“point”) form of these equations. It is assumed that the reader is comfortable with phasor representation and its benefits.
- 9.2: Wave Equations for Source-Free and Lossless Regions
- The general solution to Maxwell’s Equations includes constants whose values are determined by the applicable electromagnetic boundary conditions. However, this direct approach can be difficult and is often not necessary. In unbounded homogeneous regions that are “source free” (containing no charges or currents), a simpler approach is possible. In this section, we reduce Maxwell’s Equations to wave equations that apply to the electric and magnetic fields in this simpler category of scenarios.
- 9.3: Types of Waves
- Solutions to the electromagnetic wave equations exist in a variety of forms, representing different types of waves. It is useful to identify three particular geometries for unguided waves. Each of these geometries is defined by the shape formed by surfaces of constant phase, which we refer to as phasefronts.
- 9.4: Uniform Plane Waves - Derivation
- A uniform plane wave is one for which both E˜ and H˜ have constant magnitude and phase in a specified plane. Despite being a special case, the solution turns out to be broadly applicable, appearing as a common building block in many practical and theoretical problems in unguided propagation, as well as in more than a few transmission line and waveguide problems.
- 9.5: Uniform Plane Waves - Characteristics
- In this section, we identify some important characteristics of uniform plane waves, including wavelength and phase velocity.
- 9.6: Wave Polarization
- Polarization refers to the orientation of the electric field vector. For waves, the term “polarization” refers specifically to the orientation of this vector with increasing distance along the direction of propagation, or, equivalently, the orientation of this vector with increasing time at a fixed point in space. The relevant concepts are readily demonstrated for uniform plane waves, as shown in this section.
- 9.7: Wave Power in a Lossless Medium
- In many applications involving electromagnetic waves, one is less concerned with the instantaneous values of the electric and magnetic fields than the power associated with the wave. In this section, we address the issue of how much power is conveyed by an electromagnetic wave in a lossless medium. The relevant concepts are readily demonstrated in the context of uniform plane waves, as shown in this section.
Thumbnail: Plan wave. (Public Domain via Wikipedia)