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?
