# DERIVATION OF THE WAVE EQUATION

DERIVATION OF THE WAVE EQUATION

Here is an alternative derivation of Eq. (33-9) for the speed of electromagnetic waves. It is more mathematical than our other treatment, but it includes a derivation of the wave equation for electromagnetic waves. This part of the section can be omitted without loss of continuity in the chapter.
During our discussion of mechanical waves in Section 19-4, we showed that a function y(x, t) that represents the displacement of any point in a mechanical wave traveling along the x-axis must satisfy a differential equation, Eq. (19-12):

This equation is called the wave equation, and u is the speed of propagation of the wave. To derive the corresponding equation for an electromagnetic wave, we again consider a plane wave. That is, we assume that at each instant, E, and B, are uniform over any plane perpendicular to the x-axis, the direction of propagation. But now we let E, and B, vary continuously as we go along the x-axis; then each is a function of x and t. We consider the values of E, and B, on two planes perpendicular to the x-axis, one at x and one at x + ~x .

Following the same procedure as previously, we apply Faraday’s law to a rectangle lying parallel to the xy-plane, as in Fig. 33-6. This figure is similar to Fig. 33-4. Let the left end gh of the rectangle be at position z, and let the right end ef be at position (x + ~x). At time t, the values of E, on these two sides are E,(x, t) and E,(x + ~x, t), respectively. When we apply Faraday’s law to this rectangle, we find that instead of
~E. di = -Eaas before, we have To find the magnetic flux IlIB through this rectangle, we assume that ~x is small enough that B, is nearly uniform over the rectangle. In that case, IlIB = Bz(x, t)A ;:

Bz(x, t)a ~x, and