The amplitudes and time relationship of harmonic number

 When a musical note is played, apart from the fundamental frequency, there are overtones above are generated. On a classical string, the harmonic number and time are related. 

1.1

A as the amplitude of the frequency. k is the harmonic number. t is time dominant.

This also implies that a frequency takes time to vibrate, ringing a note. As time pass, the sound fades out. For any fundamental or overtone,

1.2
For the intensity of the fundamental, it provides about 50% of the power of a note. The next overtone provides about half of the previous etc. resembling as an exponential decay.

For a complex soundwave that we hear daily, a tone for an instrument consists of a fundamental wave and multiple overtones. For k is any individual waves, a note can be described as,
1.3

Using cosine function 
The cosine function can also be used for expressing a wave by adding the variable e to shift the phase to achieve the same waveform as what the sine function did.
1.4

Fourier Series
Since a wave can be expressed in sine or cosine terms, the Fourier series brings these two terms together and can express any periodic wave. w=2*pi/t, n = any positive integer
1.5a
1.5b
dt is the integral of the time domain, the area between the function (the waveform) and time axis.

Fourier Transform
An alternative and better way to express a waveform are using the imaginary number i = sqrt(-1). It can deal with a waveform which is not period or continuous.
1.6

When an imaginary number is used, a complex number is formed. There are a real part and an imaginary part. With Euler's identity, 1.6 can be expressed in sine and cosine functions, back to where we started at the beginning of this article. We can substitute .
1.7





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