Maxwell's Equations and Waves

We have full described the whole of electromagnetism with Maxwell’s equations,

$$\nabla \cdot \mathbf{E} = \dfrac{\rho}{\varepsilon_0} \qquad \nabla \cdot \mathbf{B} = 0$$ $$\nabla \times \mathbf{E} = -\dfrac{d\mathbf{B}}{dt} \qquad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \varepsilon_0 \mu_0 \dfrac{d\mathbf{E}}{dt}$$

Consider a region of space where there are no chargesor currents. Show that Maxwell’s equations give rise to a pair of uncoupled wave equations. At what speed do these waves travel? How do you know?

Solutions to these wave equations are the subject of our future work, but consider solutions that are monochromatic (one frequency), one dimensional traveling wave solutions (i.e., take $z$ to be the direction of propagation).

What can you say about these waves (e.g., are they logitudinal, transverse, neither?)? What is the relationship between the electric and magnetic field for these kinds of solutions? Remember that these solutions must still satisfy the Maxwell’s equations!