16.2.2 The Displacement

A traveling wave causes the particles of the medium to be displaced from their equilibrium positions. Because one of our goals is to develop a mathematical representation to describe all types of waves, we’ll use the generic symbol D to stand for the displacement of a wave of any type. But what do we mean by a “particle” in the medium? And what about electromagnetic waves, for which there is no medium?

A photograph of spectators making a wave at a stadium. — You’ve probably seen or participated in “the wave” at a sporting event. The wave moves around the stadium, but the people (the medium) simply undergo small displacements from their equilibrium positions.

For a string, where the atoms stay fixed relative to each other, you can think of either the atoms themselves or very small segments of the string as being the particles of the medium. D is then the perpendicular displacement $Δ y$ of a point on the string. For a sound wave, D is the longitudinal displacement $Δ x$ of a small volume of fluid. For any other mechanical wave, D is the appropriate displacement. Even electromagnetic waves can be described within the same mathematical representation if D is interpreted as a yet-undefined electromagnetic field strength, a “displacement” in a more abstract sense as an electromagnetic wave passes through a region of space.

Because the displacement of a particle in the medium depends both on where the particle is (position x) and on when you observe it (time t), D must be a function of the two variables x and t. That is,

D (x, t) = the displacement at time t of a particle at position x

The values of both variables—where and when—must be specified before you can evaluate the displacement D.