Equations for Waves

Abstract

Many of the earlier sections of this book may be summarised by saying that they dealt with superposition properties of waves of various kinds. As we saw in section 1.2, if it is assumed that the equation of motion of the wave is linear, the amplitude of different waves may be added. This linear superposition has many interesting consequences, such as the interference pattern produced in the Young’s slits experiment (figure 1.7), the distinction between phase and group velocity discussed in section 2.5, the various diffraction effects of chapter 3, and so on. However, there are some problems which cannot be tackled without explicit knowledge of the equation of motion itself. In particular, we always need the equation of motion when an expression for the intensity of the wave is required. The fact that the intensity is proportional to the square of the amplitude has been used at various points, but we did not know the constant of proportionality, which clearly depends on the properties of the medium through which the wave is travelling. Similarly, we need an equation of motion to deal with interface problems of various kinds. For example, in the microwave tunnelling experiment described in section 2.7, although it is clear on general grounds that there is a tunnel wave transmitted across the air gap, we cannot find its magnitude without knowing the equation of motion of the microwaves, and also the boundary conditions which apply at the interface between the dielectric and the air.