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I have a question regarding the $1$D wave equation: $$ \cfrac{\partial^2y}{\partial x^2} = \cfrac{1}{c^2} \cfrac{\partial^2y}{\partial t^2}$$

I have seen in several physics books that its complete solution is: $$ y(x,t)=f_1(x-ct)+f_2(x+ct) $$

I do not understand what $f_1$ and $f_2$ exactly are

i) Since, given some boundary conditions, the solution is an infinite sum of sine and cosine functions, shouldn't each one of those functions ($f_1$ and $f_2$) be an infinite sum of $g_n(x \pm ct)$ functions, for instance, accompanied by arbitrary $c_n$ constants ?

ii) I understand that $f_1$ and $f_2$ satisfy the PDE, but how do we know there should not be another $f_3( ... )$ term with another argument in the $y(x,t)$ expression?

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    $\begingroup$ i) Given the representation of $y$ above, the assumption is that you are solving the PDE for $x \in \mathbb{R}$. Otherwise yes, if the set is compact then some particular boundary boundary conditions imply an infinite set of discrete modes and in turn an infinite set of eigenfunctions and constants ii) You can 'factor' the operator to get $$\left( \partial_{t} - c \partial_{x} \right)\left( \partial_{t} + c \partial_{x} \right) y = L_{1} L_{2} y = 0$$ The solution is then a sum of the solutions to $L_{1}y = 0$ and $L_{2}y = 0$. $\endgroup$ – mattos Mar 29 '20 at 2:13
  • $\begingroup$ @mattos Thank you! So, $f_1$ and $f_2$ could each represent an infinite sum of functions in some cases (boundary conditions mentioned) right? $\endgroup$ – ggrin Mar 29 '20 at 2:31
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$f_1$ and $f_2$ are general expressions. The point of the representation is that the two argument function of $(x,t)$ is the sum of two one argument functions of $(x-ct)$ and $(x+ct)$.

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