Consider playing multiple frequencies at the same time; the different waves don’t match exactly(obviously), so the resultant graph of of the combined waves has a seemingly complex pattern: in some areas the amplitudes of the waves combine constructively (negative or positive), sometimes destructively.
The general strategy is to treat signals with a certain frequency different from other signals.
‘Winding up’ an intensity vs. time graph on a two-dimensional plane in a circle around the origin creates a visual akin to a wire wound about a physical point. Tracking the x-coordinate of the centre of mass of that ‘wire’ will show that it shifts most significantly at the value for the original wave’s frequency.
One example of where this may be useful is sound processing; taking the Fourier transform of an audio helps dissect its components, and this can be useful in processes including noise isolation and amplification of certain elements.
Rather than calculate the average over time with the term outside the integral, we can instead scale up the Fourier transform(within the integral) by an amount of time.