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
Best Answer
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.