All of the things you’ve learned about the vibrating string – noninterfering superpositions of normal modes whose frequency is independent of amplitude – relate to solutions of the ideal wave equation. But deriving this equation from our model requires making the small-angle approximation to trigonometric functions, and this is only valid in the limit . Now it’s time to do something the pen-and-paper crowd can’t do (without a great deal of pain): get away from that limit.