The wave equation in $3$ dimensions is simply:
$$\nabla^2\psi = \dfrac{1}{v^2} \dfrac{\partial^2}{\partial t^2}\psi,$$
and the intuition behind this is that at each point of space with coordinates $(x,y,z)$ we have some quantity oscillating there. If it's a sound wave what is oscillating are molecules, if it's an electromagnetic wave what is oscillating are electromagnetic fields and so on. The important thing is: $\psi$ represents the association $t\mapsto \psi_t$ where $\psi_t$ represents a quantity on each point of space and this association from $t$ into $\psi_t$ is to be thought of "oscillatory".
Basic physics texts take the oscillation of points on a string, derive from Newton's second law that the wave equation in $1$-dimension is obeyed and then say: "because of that we have good reasons already to call wave something that satisfies this equation".
The problem is that I'm not yet convinced. Is there any other way to "derive" the wave equation without referring to the particular case of waves on strings? That is, starting from the fact that we want that association $t\to\psi_t$ as I've said, is there a way to conclude that the $\psi$ function should satisfy that equation?
I've tried to reason with the harmonic oscillator. So at each point $(x_0,y_0,z_0)$ the function $t\mapsto \psi(t,x_0,y_0,z_0)$ should satisfy the harmonic oscillator equation, that is:
$$\dfrac{\partial^2}{\partial t^2}\psi + \omega^2 \psi =0,$$
But I think that's not the way, since I can't see a way to put the Laplacian in there. So how can we do it?
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