# Solved >   12.Equate the approximations in Steps 10 and:1681587 …

12.Equate the approximations in Steps 10 and 11, and let the height of the rectangle ?z approach zero to show

that

13.Assume that E and B and their first and second partial derivatives are continuous. Differentiate the results of Steps 9 and 12 in two ways to show that both E and B satisfy the wave equation:

where c2 = εμ is the speed of propagation (in a vacuum c = 3 × 108 m/s is the speed of light).

14.Show that wave functions of the form

E(z, t) = A sin (kz − ωt) B(z, t) = A sin (kz − ωt)

satisfy the wave equations in Step 13, where k is the wave number, ω is the frequency of the wave, and

k 2 = c2ω 2(the guided project Wave Propagation).

15.Conclude that electromagnetic waves, as described in Step 13, consist of coupled E and Bwaves that are orthogonal to each other and which travel in a direction orthogonal to both waves; they are called plane transverse waves (Figure 6).

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