# Why is the wave equation applicable to the EM wave?

I was first exposed to the wave equation in which it was derived from a model of 1-D string. But afterwards the derived wave equation somehow was universal and thus applicable to EM waves. But because wave equations was found out of physical relationship of materials involving mass and tension, I don't see how it is naturally applicable to EM waves which don't have qualities of materials. So, if wave equation is universal, it seems that it must be speaking of geometric nature. So is there a more general derivation of wave equation that justifies its generality?

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

The study of electricity and magnetism took some century. During that time "laws" were found that were dependent on the experimental observations. Then came Maxwell's equations, which used those laws as axioms to develop his theory, and that theory came up with the wave equation for light. So the simple answer is : "the wave equation for light depends on experimental observations" on electricity and magnetism, and is continually validated with new observations and experiments.

Now your:

But because wave equations was found out of physical relationship of materials involving mass and tension, I don't see how it is naturally applicable to EM waves which don't have qualities of materials.

Was a question for the first researchers too, who wanted a medium for those waves to propagate, the luminiferous aether. That brought the light differential equations in line with your thinking. The Michelson Morley experiment showed that light waves propagated in vacuum.

So, if wave equation is universal, it seems that it must be speaking of geometric nature.So is there a more general derivation of wave equation that justifies its generality?

It is a second derivative differential equation. If we accept that geometry was the start of mathematics all differential equations describe a geometry. How and in what spaces depends on the use that is made for the various models.

They all come from Maxwell equations. Solutions of Maxwell equations provide oscillations of both electric fields and magnetic fields (not propagation). The questioner stated in the first sentence that one dimensional string problem gives wave differential equation, for which solution need not give a propagating wave but only an oscillation. Since Maxwell equations also provide wave equations, em waves also satisfy wave equations, if waves are produced. So, answer is ready. However, I ask the questioner to assume the followings (for which a basic theory may be found in a book entitled “Planets and electromagnetic waves”), for more understandings. The followings are based on mathematical interpretations for solutions of Maxwell equations. (1) If current exceeds some limit (depending on source of production), then magnetic field waves are produced. These waves are called in general radio waves. (2) If voltage exceeds some limit (depending on source of production), then electric field waves are produced. These waves are called in general light waves. (3) If current exceeds the corresponding limit and the voltage also exceeds the corresponding limit then both magnetic field waves and electric field waves exist separately; not in combined form (that is, not as everyone gives a figure for em waves with two component waves). A peculiar situation for (3) is the Hertz’s experiment. It may also happen that both types of waves may not be produced, and only oscillations may happen when the limits are not reached.

The answer to your title question is that it follows from Maxwell's equations.

The answer to the question why the wave equation is universal is that it follows from special relativity. The wave equation is the wave form of the special relativistic energy momentum relation $$E^2=m^2+p^2$$ (or $$E^2=m^2c^4+p^2c^2$$ if you don't like setting $$c = 1$$). Inspired by this and the massless wave equation following from Maxwell's equations, De Broglie proposed to make the replacement $$E \rightarrow \frac{\hbar}{i} \frac{\partial}{\partial_t}$$ and $$\vec p \rightarrow \frac{\hbar}{i} \vec \nabla ~.$$ This gave birth to wave mechanics and the wave equation, to which any non-interaction free particle wave function is a solution.