Three-dimensional waves
We want now to turn to the subject of three-dimensional waves. We have already seen that the vector $\FLPE$ satisfies the wave equation. It is also easy to arrive at the same conclusion by arguing directly from Maxwell’s equations. Suppose we start with the equation \begin{equation*} \FLPcurl{\FLPE}=-\ddp{\FLPB}{t} \end{equation*}  and take the curl of both sides: \begin{equation} \label{Eq:II:20:26} \FLPcurl{(\FLPcurl{\FLPE})}=-\ddp{}{t}(\FLPcurl{\FLPB}). \end{equation}  You will remember that the curl of the curl of any vector can be written as the sum of two terms, one involving the divergence and the other the Laplacian, \begin{equation*} \FLPcurl{(\FLPcurl{\FLPE})}=\FLPgrad{(\FLPdiv{\FLPE})}-\nabla^2\FLPE. \end{equation*}  In free space, however, the divergence of $\FLPE$ is zero, so only the Laplacian term remains. Also, from the fourth of Maxwell’s equations in free space [Eq. (20.12)] the time derivative of $c^2\,\FLPcurl{\FLPB}$ is the second derivative of $\FLPE$ with respect to $t$: \begin{equation*} c^2\,\ddp{}{t}(\FLPcurl{\FLPB})=\frac{\partial^2\FLPE}{\partial t^2}. \end{equation*}  Equation (20.26) then becomes \begin{equation*} \nabla^2\FLPE=\frac{1}{c^2}\,\frac{\partial^2\FLPE}{\partial t^2}, \end{equation*}  which is the three-dimensional wave equation. Written out in all its glory, this equation is, of course, \begin{equation} \label{Eq:II:20:27} \frac{\partial^2\FLPE}{\partial x^2}+ \frac{\partial^2\FLPE}{\partial y^2}+ \frac{\partial^2\FLPE}{\partial z^2}- \frac{1}{c^2}\,\frac{\partial^2\FLPE}{\partial t^2}=\FLPzero. \end{equation}  How shall we find the general wave solution? The answer is that all the solutions of the three-dimensional wave equation can be represented as a superposition of the one-dimensional solutions we have already found. We obtained the equation for waves which move in the $x$-direction by supposing that the field did not depend on $y$ and $z$. Obviously, there are other solutions in which the fields do not depend on $x$ and $z$, representing waves going in the $y$-direction. Then there are solutions which do not depend on $x$ and $y$, representing waves travelling in the $z$-direction. Or in general, since we have written our equations in vector form, the three-dimensional wave equation can have solutions which are plane waves moving in any direction at all. Again, since the equations are linear, we may have simultaneously as many plane waves as we wish, travelling in as many different directions. Thus the most general solution of the three-dimensional wave equation is a superposition of all sorts of plane waves moving in all sorts of directions. Try to imagine what the electric and magnetic fields look like at present in the space in this lecture room. First of all, there is a steady magnetic field; it comes from the currents in the interior of the earth—that is, the earth’s steady magnetic field. Then there are some irregular, nearly static electric fields produced perhaps by electric charges generated by friction as various people move about in their chairs and rub their coat sleeves against the chair arms. Then there are other magnetic fields produced by oscillating currents in the electrical wiring—fields which vary at a frequency of $60$ cycles per second, in synchronism with the generator at Boulder Dam. But more interesting are the electric and magnetic fields varying at much higher frequencies. For instance, as light travels from window to floor and wall to wall, there are little wiggles of the electric and magnetic fields moving along at $186{,}000$ miles per second. Then there are also infrared waves travelling from the warm foreheads to the cold blackboard. And we have forgotten the ultraviolet light, the x-rays, and the radiowaves travelling through the room. Flying across the room are electromagnetic waves which carry music of a jazz band. There are waves modulated by a series of impulses representing pictures of events going on in other parts of the world, or of imaginary aspirins dissolving in imaginary stomachs. To demonstrate the reality of these waves it is only necessary to turn on electronic equipment that converts these waves into pictures and sounds. If we go into further detail to analyze even the smallest wiggles, there are tiny electromagnetic waves that have come into the room from enormous distances. There are now tiny oscillations of the electric field, whose crests are separated by a distance of one foot, that have come from millions of miles away, transmitted to the earth from the Mariner II space craft which has just passed Venus. Its signals carry summaries of information it has picked up about the planets (information obtained from electromagnetic waves that travelled from the planet to the space craft). There are very tiny wiggles of the electric and magnetic fields that are waves which originated billions of light years away—from galaxies in the remotest corners of the universe. That this is true has been found by “filling the room with wires”—by building antennas as large as this room. Such radiowaves have been detected from places in space beyond the range of the greatest optical telescopes. Even they, the optical telescopes, are simply gatherers of electromagnetic waves. What we call the stars are only inferences, inferences drawn from the only physical reality we have yet gotten from them—from a careful study of the unendingly complex undulations of the electric and magnetic fields reaching us on earth. There is, of course, more: the fields produced by lightning miles away, the fields of the charged cosmic ray particles as they zip through the room, and more, and more. What a complicated thing is the electric field in the space around you! Yet it always satisfies the three-dimensional wave equation.