Sound waves don't add up linearly. However, it is a good enough idealization for many uses.
Fourier analysis is also approachable from the discrete setting of finite vectors instead of functions, where the fourier analysis is just an orthogonal (orthonomal when sanely defined) linear function, i.e. it acts by matrix multiplication and is represented as that matrix.
This appropriately extended to the continous setting leads to the fourier transform on functions, and also gives intuition why the fourier transform uses integrals.
Comments
Sound waves don't add up linearly. However, it is a good enough idealization for many uses.
Fourier analysis is also approachable from the discrete setting of finite vectors instead of functions, where the fourier analysis is just an orthogonal (orthonomal when sanely defined) linear function, i.e. it acts by matrix multiplication and is represented as that matrix.
This appropriately extended to the continous setting leads to the fourier transform on functions, and also gives intuition why the fourier transform uses integrals.