Guitar strings and fret positions ed

String motion ed

Bending energy ed

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Elongation energy ed

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Action ed

inspired by https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory

\[ S = \int dt \int_0^L dx \left[ \frac{\mu}{2} (\dot y)^2 - \frac{E \, I}{2} (y'')^2 - \frac{F_0}{2} (y')^2 \right] = \iint \mathcal{L} \]

with \(E\) the Young modulus (\(\approx 100 GPa\)), \(\mu = \rho \, A = \rho \, \pi \, r^2\) the mass per unit length, \(I = \frac{\pi}{2} r^4\) the second moment of area, \(F_0 \approx 80N\) the initial tension/force on the string.

Differential equation ed

Euler-Lagrange\[ 0 = \frac{\partial \mathcal L}{\partial y} - \frac{\partial}{\partial t}\frac{\partial \mathcal L}{\partial \dot y} - \frac{\partial}{\partial x}\frac{\partial \mathcal L}{\partial y'} + \frac{\partial^2}{\partial x^2}\frac{\partial \mathcal L}{\partial y''} \]leads to\[ 0 = - \mu \, \ddot y + F_0 \, y'' - E \, I \, y'''' \]

We're looking for eigen-frequencies, using the Ansatz\[ y(x,t) = \hat y(x) \, \cos \omega t \]Now, the \(t\)-dependency splits off, leaving\[ 0 = \mu \, \omega^2 \, \hat y + F_0 \, \hat y'' - E\,I\,\hat y'''' \]Linear dgl., bla bla bla, insert \(e^{\beta x}\), solve for \(\beta^2\) gives\[ \beta^2 = \frac{F_0 \pm \sqrt{{F_0}^2 + 4 \mu\,\omega^2\,E\,I}}{2 E\,I} \]with \({\beta_+}^2 \gt 0 \Rightarrow \beta_+ \in \mathbb R\) and \({\beta_-}^2 \lt 0 \Rightarrow \beta_- \in i \mathbb R\). The 4 basic solutions correspond to\[ \cosh(\beta_+ x), \quad \sinh(\beta_+ x), \quad \cos(|\beta_-| x), \quad \sin(|\beta_-| x) \]

With the boundary conditions \(\hat y(0) = \hat y(L) = 0\), we'll pick the \(\sin\).

For the fundamental mode, we get \(\sin(|\beta_-| L) = 0 \Rightarrow |\beta_-| L = \pi\).