I have never seen a formal definition of a wave-like function. By wave-like function, I mean, intuitively, something similar to the sine function. The closest definition I have gotten is "finite linear combination of sine functions". But this still leaves a lot of stuff out. Is there a text where there is a good definition of wave-like functions?
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What a good question! Unfortunately I think there is no answer. Somebody said “physics is full of good theorems. Unfortunately, there are no definitions”. This is one of those missing definitions. – Giuseppe Negro Feb 24 '21 at 00:27
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2I would generally consider a "wave" to be a solution to the wave-equation, $(\partial_t - c^2\Delta)u= 0$. The general solution to this equation is $u(x,t) = F(x-ct)+G(x+ct)$ which is the sum of two "travelling waves" with $F(x-ct)$ moving "rightward", and $G(x+ct)$ moving "leftward" (though waves could exist in any dimension of course, taking $x\in\mathbb R^n$). – Nico Feb 24 '21 at 03:06
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@Nico: however, another common use of the word is quantum-mechanical, and there a “wave” or “wave-function” is a solution to the Schrödinger equation, instead. Still, I broadly agree with your suggestion; in general, a wave is a solution to some “oscillatory” partial differential equation. – Giuseppe Negro Feb 24 '21 at 11:37
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The closest I have found for the definition of a “wave” in physics is “a disturbance that propagates through a medium, or through a field of some sort”. (The “field” refers to a vector field, or a more general tensor field). This is, of course, not a precise definition but it is the best I can find, at this level of generality.
I have forgotten where I read this definition, some physics textbook.
Giuseppe Negro
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