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...