# Why is the wave equation so pervasive?

The homogenous wave equation can be expressed in covariant form as

$$\Box^2 \varphi = 0$$

where $\Box^2$ is the D'Alembert operator and $\varphi$ is some physical field.

The acoustic wave equation takes this form.

Classical electromagnetism is described by the inhomogenous wave equation

$$\Box^2 A^\mu = J^\mu$$

where $A^\mu$ is the electromagnetic four-potential and $J^{\mu}$ is the electromagnetic four-current.

Relativistic heat conduction is described by the relativistic Fourier equation

$$( \Box^2 - \alpha^{-1} \partial_t ) \theta = 0$$

where $\theta$ is the temperature field and $\alpha$ is the thermal diffusivity.

The evolution of a quantum scalar field is described by the Klein-Gordon equation

$$(\Box^2 + \mu^2) \psi = 0$$

where $\mu$ is the mass and $\psi$ is the wave function of the field.

Why are the wave equation and its variants so ubiquitous in physics? My feeling is that it has something to do with the Lagrangians of these physical systems, and the solutions to the corresponding Euler-Lagrange equations. It might also have something to do with the fact that hyperbolic partial differential equations, unlike elliptic and parabolic ones, have a finite propagation speed.

Are these intuitions correct? Is there a deeper underlying reason for this pervasiveness?

EDIT: Something just occurred to me. Could the ubiquity of the wave equation have something to do with the fact that the real and imaginary parts of an analytic function are harmonic functions? Does this suggest that the fields that are described by the wave equation are merely the real and imaginary components of a more fundamental, complex field that is analytic?

EDIT 2: This question might be relevant: Why are differential equations for fields in physics of order two?

• Related: physics.stackexchange.com/q/159021/2451 , physics.stackexchange.com/q/78500/2451 and links therein. Aug 22, 2015 at 4:02
• In a very real sense nature only evolves hyperbolic equations; elliptic and parabolic equations are what we get when we make various approximations that change the question from "how does this evolve?" to "what does this evolve into?" or "approximately how does this evolve?" Even then, though, the wave equation isn't the only hyperbolic equation out there. Conservation laws (e.g. $\nabla_\mu T^{\mu\nu} = 0$) naturally beget first-order hyperbolic systems.
– user10851
Aug 22, 2015 at 7:03

This is an answer by an experimentalist who had been fitting data with mathematical models since 1968.

When fitting data one goes to the simplest mathematical models. When the data display variations in time and space the Fourier expansion is extremely useful because it gives the frequencies and amplitudes that will fit a periodic data set. One gets as solutions sines and cosines and the purest differential equations are the wave differential equations.

At a very simplified level, wave equations are ubiquitous similar to the ubiquity of the harmonic oscillator potential: the first term in even potentials is the harmonic oscillator potential. String theory for example is using that , and now one is going into M theory and maybe higher "terms/functions" on the idea of periodicity on dimensions.

So it seems to me it is the KISS principle (keep it simple stupid :) ) at work. After all physics theories are "invented" to fit observables and predict new observation, and simplicity is a rule of thumb in physics.

• The key part here is the fact that near equilibrium most physical systems are harmonic. That's really all there is to it. Aug 22, 2015 at 3:16
• But some of these wave equations appear to be fundamental and not merely approximations, like the Klein-Gordon equation or covariant potential equation for classical electromagnetism for example. Wouldn't this suggest that there is something deeper going on here? Nov 4, 2015 at 20:32
• Klein Gordon Equation can be viewed as a gaussian fixed point of a RG flow. Nov 4, 2015 at 21:35
• I think the OP is not asking why people keep trying versions of the wave equation, I think it's more why it keeps working. KISS applies to human behavior, not necessarily the behavior the physical systems observed (or not) by the humans.
– uhoh
Sep 20, 2016 at 6:04
• @uhoh Fitting curves on data is human behavior. If the model is predictive, then it is validated and it is assumed that it holds even if noone observes . Further than that one is into metaphysics, not physics. Do you exist when I am not looking at your comment? Sep 20, 2016 at 6:44