Mechanical and Electromagnetic Vibrations and Waves

Chapter 4

Propagation in Infinite Media

A wave is a disturbance, which propagates in a medium carrying energy and other physical quantities. The wave is characterized by its speed, direction of propagation and profile. If it is a simple harmonic its profile is specified by its amplitude, angular frequency and phase. Certain properties are common to waves and free particles: rectilinear propagation in homogeneous media, reflection, transfer of energy, etc. However, some properties are specifically characteristic of waves: space and time extension, interference, diffraction, etc.

In this chapter we study the propagation of waves in infinite media. We write the wave equation in one, two and three dimensions. We consider the harmonic waves and we introduce the Fourier analysis of waves. Finally, we discuss the energy density and its propagation.

In order to simplify the notation, we designate the partial derivatives by and , etc. A summation over the components will be written Σ_i, where i = x, y and z. For instance, Σ_i ∂_iu e_i means ∂_xu e_x + ∂_yu e_y + ∂_zu e_z.

4.1. Propagation of one-dimensional waves

Everyday experience shows that a disturbance, which is produced in a medium, propagates from one point to another. The propagation of mechanical waves, such as elastic waves and sound waves, ...