The Fourier Series and The Superposition Principle
It was Joseph Fourier (1768 - 1830) who showed that any periodic wave can be represented by a sum of simple sine waves. This sum is called the Fourier Series. The Fourier Series is based on the physics of real waves, whether electromagnetic waves, sound waves, or whatever the medium they are in. The Fourier Series follows from the Superposition Principle.
The Superposition Principle Does Not Always Hold
The Superposition Principle does not always hold, and the Fourier Series only holds when the Superposition Principle does. The Superposition Principle does not hold when the waves are so big that they "break" the medium and it does not bounce back. This is akin to breaking a spring in an oscillating spring-mass system because the oscillations are too big for the spring to handle. In the air, an explosion is an example of something that can break the linear effect so that sound waves are no longer sinusoidal.