Talk:Physics/A/String vibration

$\lambda \to 0$

Partial derivatives

• Subscripts denote partial derivatives evaluated at $x=y=0$. Formulas obtained from https://www.symbolab.com/solver/
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• I think this is now right. But am troubled by terms with ?:

$f={\frac {1+x}{\sqrt {(1+x)^{2}+y^{2}}}}$

$f_{x}=0$

$f_{y}=0$

$f_{xx}=0$

$f_{xy}=0$

$f_{yy}=-1$

---

$g={\frac {1}{\sqrt {(1+x)^{2}+y^{2}}}}$

$g_{x}=-1$

$g_{y}=0$

$g_{xx}=-2$?

$g_{xy}=0$

$g_{yy}=-1$

Substitute

• $f={\frac {1+x}{\sqrt {(1+x)^{2}+y^{2}}}}=1-{\tfrac {1}{2}}y^{2}+{\mathcal {O}}z^{3}$
• $f={\frac {1+\xi }{\sqrt {(1+\xi )^{2}+\eta ^{2}}}}=1-{\tfrac {1}{2}}\eta ^{2}+{\mathcal {O}}z^{3}$
• $g={\frac {1}{\sqrt {(1+x)^{2}+y^{2}}}}=1-x-x^{2}?-{\tfrac {1}{2}}y^{2}+{\mathcal {O}}z^{3}$
• $g={\frac {1}{\sqrt {(1+\xi )^{2}+\eta ^{2}}}}=1-\xi -\xi ^{2}?-{\tfrac {1}{2}}\eta ^{2}+{\mathcal {O}}z^{3}$

not TRASH

$m{\frac {d^{2}{\vec {r}}_{j}}{dt^{2}}}=-\kappa _{s}\left(\ell _{j,j-1}-a\right)\left[{\frac {{\vec {r}}_{j}-{\vec {r}}_{j-1}}{\ell _{j,j-1}}}\right]+\kappa _{s}\left(\ell _{j,j+1}-a\right)\left[{\frac {{\vec {r}}_{j+1}-{\vec {r}}_{j}}{\ell _{j+1,j}}}\right]$

$m{\frac {d^{2}{\vec {r}}_{j}}{dt^{2}}}=-\kappa _{s}\left(\ell _{j,j-1}-a\right)\left[{\frac {{\vec {r}}_{j}-{\vec {r}}_{j-1}}{\ell _{j,j-1}}}\right]+\kappa _{s}\left(\ell _{j,j+1}-a\right)\left[{\frac {{\vec {r}}_{j+1}-{\vec {r}}_{j}}{\ell _{j+1,j}}}\right]$

$\ell _{j,j-1}=\left|{\vec {r}}_{j}-{\vec {r}}_{j-1}\right|={\sqrt {(x_{j}-{\vec {x}}_{j\pm 1})^{2}+(y_{j}-{\vec {y}}_{j-1})^{2}}}$

${\vec {\ell }}_{j,j-1}={\vec {r}}_{j}-{\vec {r}}_{j-1}$

FOOTNOTE${\vec {\gamma }}_{j}=\xi _{j}{\hat {x}}\,+\,\eta _{j}{\hat {y}}\,+\,\zeta _{j}{\hat {z}}$

${\vec {\gamma }}_{j}=\xi _{j}{\hat {x}}\,+\,\eta _{j}{\hat {y}}$

2

${\vec {F}}_{76}={\vec {F}}_{\text{on 7}}^{\,{\text{ by 6}}}$

${\vec {\ell }}_{76}=\ell {\hat {x}}+{\vec {\gamma }}_{7}-{\vec {\gamma }}_{6}$

$\ell =1$

${\vec {\ell }}_{76}={\hat {x}}+{\vec {\gamma }}_{7}-{\vec {\gamma }}_{6}=(1+\Delta \xi _{76}){\hat {x}}+(\Delta \eta _{76}){\hat {y}}$

Recall that for small $\epsilon$,

${\sqrt {1+\epsilon }}=1+{\tfrac {1}{2}}\epsilon -{\tfrac {1}{8}}\epsilon ^{2}+{\tfrac {1}{16}}\epsilon ^{3}-{\tfrac {5}{128}}\epsilon ^{4}+{\tfrac {7}{256}}\epsilon ^{5}-\ldots ,,$

$\ell _{76}={\sqrt {1+2\Delta \xi _{76}+(\Delta \xi _{76})^{2}+(\Delta \eta _{76})^{2}}}=1+\Delta \xi _{76}+{\tfrac {1}{2}}(\Delta \eta _{76})^{2}+{\mathcal {O}}(\Delta \xi )^{2}+{\mathcal {O}}(\Delta \eta )^{4}+...$

Templates

Aligned wikitable

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Equation

We create a four-term expression using (4) and (5). Then we move the terms involving energy density and the Poynting vector to the LHS to obtain:

$f=ma$

(10)

Key equation

$e=mc^{2}$

(11)

References (Rowland)

First mention of Rowland

First non-Rowland reference

Second mention of roland

2nw non-Rowland reference

Second mention of roland