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There is a slight confusion in this question. In quantum field theory, the Dirac equation and the Schrödinger equation have very different roles. The Dirac equation is an equation for the field, which is not a particle. The time evolution of a particle, ie, a quantum state, is always given by the Schrödinger equation.
The equation $\frac{\partial^2 f}{\partial t^2} = c^2\nabla^2 f$, despite being called "the wave equation," is not the only equation that does this. If you plug the wave solution into the Schroedinger equation for constant potential, using $\xi = x - vt$
You can invent an obfuscated definition of a "wave" under which the Schrodinger equation is a "wave equation", but it would still be conceptually different from the wave equation $\partial^2\psi/\partial x^2=\partial^2\psi/\partial t^2$.
Some derivations of the Schrodinger equation start from wave-particle duality for light and argue that matter should also exhibit this phenomenon. In some notes by Fermi , it was derived by comparing the Fermat least time principle $\delta \int n \;ds = 0 $ and Maupertuis least action principle $\delta \int 2T(t) \; dt = 0 $.
However, Schrödinger derived the equation from previous knowledge. Schrödinger thought his equation from Hamilton-Jacobi formalism. If you take the classical limit in that equation you'll find the Hamilton-Jacobi equation. You can also read the original Schrödinger papers in English introducing his wave mechanics formalism.
If you want a relativistic theory, you would want to find a wave equation that reproduced the relativistic relationship \begin{equation} E^2 = m^2 c^4 + p^2 c^2 \end{equation} The Klein-Gordon equation is an example of such a wave equation (and indeed Schrodinger tried it first). But, there are problems interpreting the solutions of the Klein ...
1) Both: it is apparently a heat equation in imaginary time and it is a wave equation because its solutions are waves. 2) Nonstationary Schrodinger equation (let us assume free particle) $$ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2\nabla^2}{2m}\psi $$ is essentially complex: it can never be satisfied by a real function, only by a ...
$\begingroup$ @ACuriousMind Might be stupid question but further on there is an example. Example: Imagine a system in which there are just two linearly independent states: $$|1\rangle =\begin{pmatrix}1\\0\end{pmatrix}~~~\text{and}~~~|2\rangle=\begin{pmatrix}0\\1\end{pmatrix}$$ The most general state is a normalized linear combination: $$\ \mathcal{S} = a|1\rangle + b|2\rangle= \begin{pmatrix}a ...
There are plenty of wave equations that are not 'the' wave equation. (Say, the KdV equation, or the paraxial wave equation for the propagation of light in a graded-index optical fiber (a.k.a. the Schrödinger equation).) The notation is somewhat unfortunate (so perhaps 'the' wave equation should have a better name), but it is what it is ...
Within a year, wave mechanics was born and shown to be physically equivalent to matrix mechanics, mostly by Dirac and partly by Schrödinger. Wave mechanics instantly became popular, perhaps more popular than matrix mechanics, due to its mathematical similarity to classical field theory which is why it looked and still looks simpler to many.