Mechanical and Electromagnetic Vibrations and Waves

Chapter 9

Standing Waves and Guided Waves

In an open space, a wave propagates very often in all directions; this causes the decrease of its intensity with the travelled distance r. A waveguide conserves the intensity of the wave by confining it within a structure, which limits the medium of propagation in the transverse directions. Guided waves propagate in well-determined modes. Each mode is characterized by a cut-off frequency, a phase velocity and a group velocity, which depend on the frequency of the wave and on the geometry of the waveguide. The propagation properties in the infinite medium are recovered if the transverse dimensions of the waveguide are much larger than the wavelength. The waves on the surface of the water in a canal are an example of guided waves (section 5.10). If all the dimensions of the medium are limited, it can support only standing or stationary waves in normal modes of discrete frequencies (also called normal frequencies). In each mode, the propagation medium is a juxtaposition of wave zones with no transfer of physical quantities from one zone to the other. More generally, the guided or standing wave may be a superposition of modes.

In this chapter we study one-dimensional standing waves and we generalize to two-dimensional and three-dimensional waves. Afterwards, we analyze guided waves. We may analyze their propagation by studying the successive reflections on the guide walls. However, a more practical and general method consists of directly finding ...