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?
$f_1$ 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 $(x+ct)$.