More about the deflection formula applied to guitars
Although the formula that relates length, pitch and tension for a vibrating string is widely found on the Internet, the equation for string displacement isn't.
Like the Simple Harmonic Motion equation for a vibrating string, it only applies for excursions that are modest in amplitude.
The diagram below shows a string plucked at its centre...
For small d, the tension T is not altered significantly by the excursion of the string, meaning that the restoring force for each half string is T sin(theta) where sin(theta) = d/(L/2)
Summing the two half strings we get the formula above - 2dT/L + 2dT/L = 4dT/L
Plucking near the bridge...
As any guitarist knows, the string gets progressively harder to deform the nearer we get to the bridge. That's because the restoring force grows when the two portions of string are unequal in length.
For example ...
- Plucked 1/4 of the way along, we get 4dT/L + 4dT/3L = 5.33dT/L
- Plucked 1/10 of the way along, we get 10dT/L + 10dT/9L = 11.1dT/L
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