Waves in Infinite Media

Abstract

A wave is a disturbance, usually periodic, that travels with finite velocity through a medium. Sound waves, water waves, and electromagnetic waves are some examples. All wave motions have two important characteristics in common: First, energy is propagated to distant points and, second, the disturbance travels through the medium without giving the medium as a whole any permanent displacement. Each successive particle of the medium performs a motion similar to its predecessor’s but later in time, and returns to its origin. Whatever the nature of the medium that transmits the waves, be it air, a stretched string, a liquid, or an electrical cable, these two properties enable us to relate all wave motions together. Indeed, many types of waves are governed by a second-. order linear partial differential equation

$${\nabla ^2}\Psi = {1 \over {{c^2}}}{{{\partial ^2}\Psi } \over {\partial {t^2}}},$$

where Ψ(r, t) represents the disturbance traveling with the velocity c. Equation (2.1) is known as the wave equation.

The enormous and the minute are interchangeable manifestations of the eternal.

(William Blake)