If you remember, when in textbooks* they derive the wave equation by considering a small element of string and applying Newton's 2nd law on it, they make the assumption that the angles the tension makes at the two ends of the element with the horizontal is smallish, such that sin ~ tan. Without this assumption, the wave equation is NOT a good approximation of the equation of motion. The motion of the rope is only well described by the wave equ. when the angles are small, i.e. when the shape of the wave is flat-looking. How can we ally the fact that this analysis shows that the motion of a wave on a rope/string is described by the wave equation only for small angles, but that in reality, a wave that looks like this is observable and its motion is a solution of the wave equation [y(x,t)=g(x-vt)] even though the angles are obviously not small (they're near 90° at some points). *See Griffiths E&M pp.365 for exemple.