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.
Let $u_1(x, t) = f_0(x -\sqrt{b} t)$ and $u_2(x, t) = f_0(x + \sqrt{b} t)$, we can verify that $u_1(x,t)$ and $u_2(x,t)$ both satisfy the wave equation. The general solution is $u(x, t) = a u_1(x, t) + b u_2(x,t)$. The solution represents the wave front (at the beach, facing ocean, and let the time stop, the wave in front of you is the shape of wave) traveling along the time direction.