Traveling waves making up another kind of wave

In lecture we learned that the general solution of the wave equation can be written as $y(x,t) = f(x - ct) + g(x + ct)$ (for waves realized on a vibrating string).

Find an explicit expression for the solution $y(x,t)$ for the particular choice $f(u) = g(u) = A\cos(ku)$ of the functions $f$ and $g$ ($A$ is a constant). What kind of wave is this?

Hint: Rewrite $A\cos(ku)$ using the complex representation.