Radiowave Propagation: Physics and Applications

3

PLANE WAVES

3.1 INTRODUCTION

Maxwell's equations (equations (2.1)–(2.4)) are very elegant and concise, but elegance should not be confused with simplicity. There is nothing simple about four coupled vector partial differential equations! Nor should we expect simplicity since these equations are to describe all propagation modes in all possible media. The complexity of the mathematics reflects the complexity of the physical phenomena it serves to describe.

Since the general case is too complex to solve analytically, one is forced to make simplifying assumptions to find some simpler analytical solutions. Plane waves turn out to be the simplest solutions of Maxwell's equations [1–4]. Despite their analytical simplicity, plane waves find physical applications in a wide range of scenarios. More complex solutions are generally required to describe electromagnetic fields in the vicinity of sources, or close to material discontinuities and/or inhomogeneities in the propagation medium. Far from such regions, plane waves are in general a very good description for the local electromagnetic field behavior. Furthermore, in more complex cases, the total solution can often be represented as the superposition of a set of plane waves with varying amplitudes and propagation directions, in a manner analogous to a Fourier (or spectral) decomposition. Therefore, plane wave solutions are worthy of special attention.

3.2 D'ALEMBERT'S SOLUTION

To simplify the derivations below, we consider a charge-free ...