chap7 - Plane Waves and Wave Propagation Augustin Jean...

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Unformatted text preview: Plane Waves and Wave Propagation Augustin Jean Fresnel (1788 - 1827) November 9, 2001 Contents 1 Plane Waves in Uniform Linear Isotropic Nonconducting Media 2 1.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Conditions Imposed by Maxwell’s Equations . . . . . . . . . . . . . . 4 2 Polarization 6 3 Boundary Conditions; Waves at an Interface 9 3.1 Kinematic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Conditions from Maxwell’s Equations . . . . . . . . . . . . . . . . . . 12 3.2.1 Polarization of E Perpendicular to the Plane . . . . . . . . . 15 3.2.2 Polarization of E Parallel to the Plane . . . . . . . . . . . . . 16 3.3 Parallel Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Reflection and Transmission Coefficients 19 5 Examples 21 5.1 Polarization by Reflection . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . 23 1 6 Models of Dielectric Functions 26 6.1 Dielectric Response of Free Electrons . . . . . . . . . . . . . . . . . . 30 7 A Model for the Ionosphere 31 8 Waves in a Dissipative Medium 35 8.1 Reflection of a Wave Normally Incident on a Conductor . . . . . . . . 39 9 Superposition of Waves; Pulses and Packets 41 9.1 A Pulse in the Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . 46 10 Causality and the Dielectric Function 47 11 Arrival of a Signal in a Dispersive Medium 53 A Waves in a Conductor 57 2 In this chapter we start by considering plane waves in infinite or semi-infinite me- dia. We shall look at their properties in both insulating and conducting materials and shall give some thought to the possible properties of materials of different kinds. We will also look at the reflection and refraction of waves at planar boundaries between different materials, a topic which forms the basis for much of physical optics. If time allows, we shall also look at some of the more abstract aspects of wave propagation having to do with causality and signal propagation. 1 Plane Waves in Uniform Linear Isotropic Non- conducting Media 1.1 The Wave Equation One of the most important predictions of the Maxwell equations is the existence of electromagnetic waves which can transport energy. The simplest solutions are plane waves in infinite media, and we shall explore these now. Consider a material in which B = μ H D = ² E J = ρ = 0 . (1) Then the Maxwell equations read ∇ · E = 0 ∇ · B = 0 ∇ × E =- 1 c ∂ B ∂t ∇ × B = μ² c ∂ E ∂t . (2) Now we do several simple manipulations that will become second nature. First take the curl of one of the curl equations, e.g., Faraday’s law, to find ∇ × ( ∇ × E ) = ∇ ( ∇ · E )- ∇ 2 E =- 1 c ∂ ∂t ( ∇ × B ) =- μ² c 2 ∂ 2 E ∂t 2 , (3) where the generalized Amp` ere’s law was employed in the last step. Because the divergence of E is zero, this equation may be written as ˆ ∇ 2- μ² c 2 ∂ 2 ∂t 2 !...
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This note was uploaded on 01/08/2012 for the course PHYSICS 707 taught by Professor Electrodynamics during the Fall '11 term at LSU.

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chap7 - Plane Waves and Wave Propagation Augustin Jean...

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