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?
Also: Why don't differential equations of physics go beyond the second order?
 A: 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.
