Graphs 8

Graphs 1, 2, 3, and 4 show the waveforms for the flute, violin, piano, and oboe. The Fourier Series can be used to explain why each of the instruments have their own unique sound. The flute, violin, piano and oboe have different combinations of frequencies as each waveform is made of an unique combination of sine and cosine waves, and this creates distinct waveforms and allows each instrument to have a unique sound. Recall that the formula of the Fourier Series is f(x)=a_0+∑_(k=1)^∞▒(a_k cos⁡〖πkx/T〗+b_k sin⁡〖πkx/T〗 )
The Fourier Series is the sum of all the waves within a waveform, in other words, it separates all the waves in the waveform into its basic components. For instance, take the waveforms of the flute and oboe. I will use these
From looking at Graphs 1 and 4, the tiny dots along the wave represent each sound that is picked up by Audacity, and they are graphed on a y axis which is a vertical scale from -1 to 1. The range of Graph 1, the flute waveform, is from -0.25 to 0.4, while the range of the oboe waveform is from -0.3 to 0.5. That means that the oboe produces a greater variety of sound at different amplitudes, leading to richer sound as the listener hears the A note being played at multiple frequencies. The graph of the oboe waveform, Graph 4, also has a greater as there are more crests, the highest point of the wave, which also supports the fact that the oboe has a louder and richer sound (Koch). Moreover, since the oboe has more fluctuations than the flute, it will have more component waves, and thus, it will have a higher sum (f(x)). A larger sum will lead to a richer sound as the Fourier coefficients will be larger, and that can explain why the oboe and flute sound different. The Fourier Series allows me to explain why different instruments sound different using their waveforms, but I want to find a way to explain in terms of the amplitudes and frequencies, which can provide a more precise