Consider the one dimensional wave equation: $$\frac{\partial^2 f(x, t)}{\partial t^2} - c^{2}\frac{\partial^2 f(x, t)}{\partial x^2} = 0. $$
I understand that one may find "wavy" solutions to this equation. But, $f(x, t) = x$ is a solution and it's just a simple linear equation. I'm working through a physics text, and whenever we arrive at a function which satisfies the wave equation, we always write the solution as $A\sin (\omega t - kx)$. I understand that this is a solution to the wave equation, but without some deep theorem stating that "any function which solves the wave equation can be represented as this sine function" I do not feel it is just to assume the function has this form. For the linear example, I don't believe it can be represented by a sine function.