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11 - Harmonic Plane Waves

Published online by Cambridge University Press:  16 October 2026

Jene A. Golovchenko
Affiliation:
Harvard University, Massachusetts
Edlyn V. Levine
Affiliation:
Harvard University, Massachusetts
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Summary

This chapter develops harmonic plane waves as central solutions of the source-free Maxwell equations and connects them to radiation from localized sources. Starting from the homogeneous equations in vacuum, it introduces complex exponential plane-wave solutions, derives their dispersion relation and transversality, and relates the wave vector, electric field, and magnetic field. The far-field radiation of an oscillating electric dipole is then recast in plane-wave form, illustrating how radiation looks locally like a plane wave. The chapter proceeds to construct an idealized current sheet that produces an exact plane wave and evaluates the fields from the Jefimenko integrals. It goes on to analyze polarization by superposing plane waves, identifying linear, circular, and elliptical polarization states. Finally, it defines the intensity of an electromagnetic wave via the time-averaged Poynting vector and works out interference of crossing plane waves, showing how spatial intensity fringes arise from the superposition principle.

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