# Is a sinusoidal electromagnetic wave just a special case of electromagnetic radiation?

The only solution of the wave equation that we are taught in elementary level Optics describes only sinusoidal oscillations. Is this a mathematical tool for us to describe electromagnetic radiation or is there something fundamental that dictates an electromagnetic wave of sinusoidal form to be produced? If the first is the case, what other types of electromagnetic waves are there?

• The basic reason that sinusoidal waves are concentrated at the early level because any periodic motion can be decomposed to an infinite series of sinusoidal waves related to the principle time period and properly chosen amplitudes - this is the Fourier's Theorem; so indeed there are periodic waves which are not sinusoidal.
– user36790
Dec 9, 2016 at 18:33
• Thank you for your comment. So is any periodic wave equally valid and much more complex functions are equally probable; or is the majority of electromagnetic waves we observe sinusoidal? Dec 9, 2016 at 18:43

The term wave equation normally denotes a very specific object, which is the partial differential equation $$\frac{\partial^2}{\partial t^2}f = c^2\nabla^2 f, \tag 1$$ where $\nabla^2$ is the Laplacian operator, usually on 3D, and $f$ can be e.g. the scalar potential, or some component of a vector field. In many situations one can simplify this to waves that propagate along a single dimension, in which case it reads $$\frac{\partial^2}{\partial t^2}f = c^2\frac{\partial^2}{\partial x^2} f, \tag 2$$ and it is still called the wave equation in that form. In its essence, the wave equation is an equation of motion: you supply the configuration of the wave $f(x,0)$ at some initial time $t=0$, and the wave equation will tell you how that initial configuration will evolve.

It is hard to know from your question, but you've probably only been exposed to some particular solutions of $(2)$, likely of the form $$f(x,t) = A\cos(kx-\omega t), \tag 3$$ or related constructions. This is not a 'wave equation', it is only one particular solution of $(2)$. We often concentrate on this specific solutions in introductory courses, but this is only to help build intuition, because they are easy to analyse.

In fact, there is a huge infinity of different solutions of the wave equation. In one dimension, for example, for any two arbitrary functions $F$ and $G$ of a single variable you can build a solution to the wave equation, of the form $$f(x,t) = F(x-c t)+ G(x+ct). \tag 4$$ In two and three dimensions the solution space is even bigger.

On the other hand, part of the reason that introductory courses focus so much on sinusoidal solutions is that they are indeed very useful solutions for describing physical phenomena, both by themselves or by building other solutions as superpositions of sinusoidal oscillations. Some phenomena (like AC power propagating down a high-voltage line, or a highly coherent laser beam) are very well described by a sinusoidal oscillations; other phenomena (like, say, sunlight) a bit less so. Whether the 'majority' of solutions are sinusoidal or not, however, depends on whether you spend your time looking at lasers or AC power line, or whether you look at other phenomena with different behaviour.

• I have replaced the part 'the wave equation' with ' the only solution of the wave equation' to be more precise. Dec 9, 2016 at 19:08
• The sinusoidal solutions are the eigenvectors of the energy operator, so are representative of individual photons. Dec 10, 2016 at 1:45
• @Whit3rd No, that is incorrect. You can have a single-photon state that is not an energy eigenstate (and therefore not stationary), simply by superposing single-photon states at different frequencies. The sinusoidal solutions are also not eigenvectors of the energy operator, they're the translationally symmetric eigenfunctions of the Helmholtz operator on coordinate space (not on Hilbert space) and they're therefore completely decoupled from the hamiltonian - you can have e.g. a single photon with well-defined energy in a cylindrical Bessel resonance. Dec 10, 2016 at 1:54
• (And, in any case, discussion of QM is out of the scope of the question as posed.) Dec 10, 2016 at 1:54
• Monochromatic radiation IS an interesting special case, regardless of the reason. Dec 10, 2016 at 3:01