Analysis of Various Propagation Phenomena

The circular ripples caused by the impact of an object on the perfectly flat surface of a body of water illustrate one of the most familiar propagation phenomena. However, while the observation of waves formed on the surface may seem trivial, their physical understanding and mathematical formulation are nonetheless highly sophisticated.

In order to alleviate theoretical difficulties, the article devoted to the analysis of propagation phenomena will approach the subject within a much simpler conceptual framework than the waves cited as an example. In the following, we focus on waves transported along a single dimension of space. For this reason, the two parts of the article will make frequent reference to the theory of transmission lines, as described in the article [D 1 322] .

The first section, devoted entirely to wave propagation on periodic structures, addresses the question of the motion of chains of mechanical or electrical resonators subjected to sustained sinusoidal excitation. Paradoxical as it may seem, the formulation of waves propagated on a periodic, i.e. discontinuous, structure is based on the concepts of travelling wave and retrograde wave derived from the wave equation of continuous media. In the particular case of periodic structures, however, the existence of a cut-off pulsation and a wave dispersion diagram specific to these media will be demonstrated. The developments outlined in this first paragraph will also make extensive use of equivalence criteria, whereby certain mechanical parameters are matched by electrical parameters, or vice versa. This approach will be of great interest in the second section, which focuses on wave propagation in material media such as materials or fluids.

The second paragraph will be devoted to three academic problems involving a link between line theory and the periodic structures analyzed above. It will be shown that a chain of electrical resonators evolves under certain conditions in a similar way to a line. Through the artifice of equivalence, we will succinctly analyze the propagation of acoustic waves in condensed matter and fluids. Two parameters particularly well studied in this section concern the calculation of wave propagation velocities and the extension of impedance concepts to waves transmitted in material media.

To conclude the article, the question of calculating the input impedance of any continuous line or structure terminated by an impedance will be...