# String vibration/Algebra

[File:Fourier series and transform.gif|thumb|If you don't understand this animated gif, look at the next figure]]

The first four terms of the Fourier series of a square wave, ${\displaystyle \psi (x)={\frac {4}{\pi }}\sin(\pi x)}$ ${\displaystyle +{\frac {4}{3\pi }}\sin(3\pi x)}$ ${\displaystyle +{\frac {4}{5\pi }}\sin(5\pi x)}$ ${\displaystyle +{\frac {4}{7\pi }}\sin(7\pi x)+\ldots }$${\displaystyle =\sum _{{\text{odd }}n=1}^{\infty }{\frac {4}{n\pi }}\sin(n\pi x)}$.

Define ${\displaystyle \kappa _{T}=(1-a)\kappa _{s}\leq \kappa _{s}}$

${\displaystyle m{\ddot {\xi }}=\kappa _{s}\xi ^{\prime \prime }+a\kappa _{s}\eta ^{\prime }\eta ^{\prime \prime }}$, and ${\displaystyle m{\ddot {\eta }}=\kappa _{T}\eta ^{\prime \prime }}$.

Transverse standing wave: ${\displaystyle \omega /k={\sqrt {\kappa _{T}/m}}}$

${\displaystyle \eta =A\sin(kx)\sin(\omega t)}$

${\displaystyle \eta ^{\prime }\eta ^{\prime \prime }=-k^{3}A^{2}\sin(kx)\cos(kx)\sin ^{2}(\omega t)}$

Define ${\displaystyle {\mathcal {L}}=m\partial ^{2}/\partial t^{2}-\kappa _{s}\partial ^{2}/\partial X^{2}}$

Second order differential equation with one variable: https://openstax.org/books/calculus-volume-3/pages/7-2-nonhomogeneous-linear-equations

${\displaystyle {\mathcal {L}}\xi =-k^{3}A^{2}\sin(kx)\cos(kx)\sin ^{2}(\omega t)}$

${\displaystyle \xi =\xi _{p}(X,t)+\xi _{h}(X,t)}$ where ${\displaystyle \xi _{h}}$ is the solution to the homogeneous equation, i.e., solution to ${\displaystyle {\mathcal {L}}\xi _{h}=0}$

Employ two identities:

${\displaystyle \sin(kX)\cos(kX)={\frac {\sin(2kX)}{2}}}$ and ${\displaystyle \sin ^{2}(\omega t)={\frac {1-\cos(2\omega t)}{2}}}$

{\displaystyle {\begin{aligned}{\mathcal {L}}\xi &=-a\kappa _{s}k^{3}A^{2}\sin(2kX)\left({\frac {1-\cos(2\omega t)}{4}}\right)\\&=-{\frac {a\kappa _{s}k^{3}A^{2}}{4}}\sin(2kX)+{\frac {a\kappa _{s}k^{3}A^{2}}{4}}\sin(2kX)\cos(2\omega t)\end{aligned}}}

To find a particular solution, ${\displaystyle \xi _{p},}$ to (?) we first consider two different inhomogeneous equations:

${\displaystyle {\mathcal {L}}\xi _{1}=-{\frac {a\kappa _{s}k^{3}A^{2}}{4}}\sin(2kX)}$

${\displaystyle {\mathcal {L}}\xi _{2}={\frac {a\kappa _{s}k^{3}A^{2}}{4}}\sin(2kX)\cos(2\omega t)}$

## NOW

Recall ${\displaystyle \kappa _{T}=(1-a)\kappa _{s}}$ => ${\displaystyle a={\frac {\kappa _{s}-\kappa _{T}}{\kappa _{s}}}}$

If ${\displaystyle \xi _{1}}$ is proportional to ${\displaystyle \sin(2kX)}$, then ${\displaystyle {\mathcal {L}}\xi _{1}=4k^{2}\kappa _{s}\xi _{1}}$, and: ${\displaystyle \xi _{1}=-{\frac {a\kappa _{s}kA^{2}}{16\kappa _{T}}}\sin(2kX)}$ => ${\displaystyle {\color {red}\xi _{1}=-{\frac {kA^{2}}{16}}\left(1-{\frac {\kappa _{T}}{\kappa _{s}}}\right)\sin(2kX)}}$

If ${\displaystyle \xi _{2}}$ is proportional to ${\displaystyle \sin(2kX)\cos(2\omega t)}$, then ${\displaystyle {\mathcal {L}}\xi _{2}=\left(-4m\omega ^{2}+4k^{2}\kappa _{s}\right)\xi _{2}}$ and:${\displaystyle \xi _{2}={\frac {a\kappa _{s}}{16}}{\frac {k^{3}A^{2}}{k^{2}\kappa _{s}-m\omega ^{2}}}\sin(2kX)\cos(2\omega t)}$ =${\displaystyle \xi _{2}={\frac {a\kappa _{s}kA^{2}}{16}}{\frac {1}{\kappa _{s}-m\omega ^{2}/k^{2}}}\sin(2kX)\cos(2\omega t)}$ =>${\displaystyle \xi _{2}={\frac {a\kappa _{s}kA^{2}}{16}}{\frac {1}{\kappa _{s}-\kappa _{T}}}\sin(2kX)\cos(2\omega t)}$ =>${\displaystyle \xi _{2}={\frac {kA^{2}}{16}}\sin(2kX)\cos(2\omega t)}$. Now use ${\displaystyle \cos(2\omega t)=1-2\sin ^{2}\omega t)}$.

${\displaystyle {\color {red}\xi _{1}={\frac {kA^{2}}{16}}\left(1-2\sin ^{2}\omega t\right)\sin(2kX)}}$

By the linearity of the operator ${\displaystyle {\mathcal {L}},}$ we see that a particular solution to (?) is the sum of ${\displaystyle \xi _{p}=\xi _{1}+\xi _{2}:}$

${\displaystyle \xi _{1}={\frac {kA^{2}}{8}}\left(-\sin ^{2}(\omega t)+{\frac {\kappa _{T}}{2\kappa _{s}}}\right)\sin(2kX)}$

In these units the speed of a ${\displaystyle \left\{{\text{transverse, longitudinal}}\right\}}$ wave is ${\displaystyle \left\{c_{T}={\sqrt {\kappa _{T}/m}},\,c_{s}={\sqrt {\kappa _{s}/m}}\right\}}$. This permits us to write an expression that does not depend on the choice of units.[1] Relating the wavenumber of the lowest order mode to string length by ${\displaystyle kL_{0}=\pi }$:

${\displaystyle \xi _{1}={\frac {\pi A^{2}}{8L_{0}}}\left(-\sin ^{2}(\omega t)+{\frac {c_{T}^{2}}{2c_{s}^{2}}}\right)\sin(2kX)}$

${\displaystyle {\color {red}xx}$}$ [itex]$

## Other identities

${\displaystyle \sin(2\theta )=2\sin \theta \cos \theta }$   *    ${\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }$   *    ${\displaystyle \sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2}}}$   *    ${\displaystyle \cos ^{2}\theta ={\frac {1+\cos(2\theta )}{2}}}$   *    ${\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$   *    ${\displaystyle \cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$   *    ${\displaystyle 2\sin \theta \cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi )}}$   *    ${\displaystyle 2\cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi )}}$   *

After modifying an equation from Wikipedia:

{\displaystyle {\begin{aligned}\sin \alpha \cos \beta &={\frac {e^{i\alpha }-e^{-i\alpha }}{2i}}\cdot {\frac {e^{i\beta }+e^{-i\beta }}{2}}\\&={\frac {1}{2}}\cdot {\frac {e^{i(\alpha +\beta )}+e^{i(\alpha -\beta )}-e^{i(-\alpha +\beta )}-e^{i(-\alpha -\beta )}}{2i}}\\&={\frac {1}{2}}{\bigg (}\underbrace {\frac {e^{i(\alpha +\beta )}-e^{-i(\alpha +\beta )}}{2i}} _{\sin(\alpha +\beta )}+\underbrace {\frac {e^{i(\alpha -\beta )}-e^{-i(\alpha -\beta )}}{2i}} _{\sin(\alpha -\beta )}{\bigg )}.\end{aligned}}}

1. See David R Rowland 2011 Eur. J. Phys. 32 1475, equation 11