I'm just curious to know the mathematical relationship between guitar strings and how their frequency changes with the variation of guitar's string length and thickness. Say, I'm vibrating some node (either open or closed) how do I know the frequency of that particular node is generating? Is there any sequence or formula? Please give me a detail if possible.
Thanks :)
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The wave velocity is determined (for a string of constant thickness) by the tension $T$ (force per unit area cross-section) and the density $\rho$ (mass per unit volume) of the string. Purely dimensional arguments show that the velocity is proportional to $\sqrt{T/\rho}$. (The constant of proportionality turns out to be 1.)
The fundamental frequency is then dictated by the wave velocity divided by the string length, but the situation is not quite as simple as that. When a string is plucked most of the energy will go into some combination of higher harmonics (modes in which the string is stationary at one or more points in the middle. Thus in a guitar, the perceived frequency of a plucked string will be twice the fundamental frequency.
Wikipedia gives the frequency of a vibrating string as $f\frac 1{2L}\sqrt{\frac T\mu}$ where $L$ is the length, $T$ is the tension, and $\mu$ is the mass/length ratio. This means long and heavy strings have lower frequency and tight ones have higher frequency. The harmonics are just integer multiples of this fundamental frequency.