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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.