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I have a question. I was looking for the Wave equation (first Eq. of this wikipedia page). I saw for the first time a version of this equation during an Acoustic course, where we obtained it for the sound wave combining the Euler equation, the Continuity equation, the general gas equation.

So, how is a generical wave equation, as the one described in wikipedia, derived? Is there behind a mathematical derivation or is it just a specific form of Differential Eq. that was found the same for some scalars, so we have to take it "as it is"? Thank you in advance

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The wave equation is a "general" differential equation that describes waves in several contexts.

It is given by $$\partial_t ^2 u = v^2 \Delta u$$ and has has general solution (in 1D)

$$u(x, t) = f(x-vt)+g(x+vt)$$

i.e. the sum of a function "moving" to the left with velocity $v$ and one moving to the right. That is, waves that translate: whatever value $f$ has at position $x$ at the beginning it will have it at position $x_2$ such that $x_2=x+vt$ (and same for $g$ with different signs).

You can not "derive" it. What you can do is observe that several phenomena (electromagnetic fields, material waves, etc) are described by an equation having this form i.e. they accept a solution which "moves" like a wave. Roughly what will change between different system is the value of $v$ i.e. the speed of the wave.

The "shape" of the wave instead is given by initial conditions and geometrical/symmetry arguments, usually.

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There is no unique answer to this question.

Domain-specific derivations
In electromagnetism the wave equation arises from the Maxwell equations. In elasticity or hydrodynamics it arises from the correspondinge quations for the media. Note that in these latter cases the wave equation is actually an approximation - more general equations for waves can be derived, which are either non-linear or higher order.

Theory of second order partial differential equations
In general, linear second order partial differential equations can be classified into three types: hyperbolic, parabolic and elliptic. (Note how this classification follows the classification of the conic sections.) The canonical representatives of these types are often referred to wave equation, diffusion equation, and Laplace equation. So wave equation si just one of the general second order PDEs.

Why second order is more important in physics than anything else? One the one hand, unlike the first order, it does not contain inherent asymmetry/direction. On the other hand, higher order equations often result in non-local theories, which are harder to deal with (although sometimes one has to).

See also these threads:
Why do wave equations produce single- or few-valued dispersion relations? Why no continuum of possible $\omega$ for one $|k|$?
Big misconceptions with the fundamentals of “ waves”
Why do we need the Schrödinger equation, if we have wave equation?

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If I understood the question correctly, the generic equation you mentioned in the link is just a mathematical generalization of the wave equation for all dimensions as $x^{2}+y^{2}+z^{2}-c^{2}t^{2}=0$ which can be generalized as $\;\;x^{2}_{1}+....+x_{n}^{2}-c^{2}t^{2}=0$

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If you had to write down a generic Lagrangian for a scalar field that is invariant under rotations and space-time translations it would look something like $$\mathcal L = \frac{1}{2}\left(\frac{\partial \phi}{\partial t}\right)^2 + \frac{1}{2}v^2 \nabla^2\phi + A\phi+B\phi^2 +C\phi^3 + \mu (\nabla^2 \phi)^2 +... $$ The first two terms give us the standard wave equation while the other terms either give mass to the field or cause interactions, so if we want a massless field with no interactions then the lagrangian is pretty much fixed.

Notice how rotational , translational, and temporal symmetry forbids terms like $$\mathbf{a}\cdot\nabla\phi\qquad\rho(\mathbf{x})\nabla^2\phi\qquad\gamma(t)\phi^2.$$ Imposing these symmetries ensures angular momentum, linear momentum, and energy conservation respectively, which you kind of want for a typical physical system.

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