Talk:Solution to Set 5

Class Notes for Problem 1

Potential Energy $U=1,2,3,...n\;$

$U\cong {\frac {1}{2}}k\sum _{n}(U_{n}-U_{n-1})\;$

$\Rightarrow m{\ddot {U}}_{n}=-k[2U_{n}-U_{n-1}-U_{n+1}]=-m\omega U(t)\;$

Eigenvectors of Modes A and B (defined arbitrarily)

$u_{m}(t)=e^{i\omega t}u_{m}\;$

$\Rightarrow -m\omega ^{2}{\ddot {\vec {u}}}_{n}=-k\mathbf {M} {\vec {u}}\;$

Band Matrix

$\mathbf {M} ={\begin{vmatrix}2&1&0&0\\1&2&1&0\\0&1&2&1\\0&0&1&2\end{vmatrix}}$

Running waves through a solid

$u_{m}(t)=e^{ik(na)-\omega t}=e^{ikR_{n}}\;$

where

$u_{m}\rightarrow u_{n}e^{ik(na)}\;$

$R_{n}\;$ = distance on some coordinate system

$\Rightarrow -m\omega ^{2}{\vec {u}}=-k[2-e^{ik\alpha }-e^{-ik\alpha }]\;$

Derive and get:

$\Rightarrow \omega (k)=2{\sqrt {\frac {k}{m}}}|sin(ka)|$

Class Notes for Problem 1

$u_{m}(t)=e^{ik(na)-\omega t}=e^{ikR_{n}}\;$

$u(R_{n})\equiv e^{ikR_{n}}=cos(ka)\;$

$\Rightarrow -m\omega ^{2}e^{ikna}=-k[2e^{ik(na)}-e^{ik(n+1)a}-e^{ik(n-1)a}]\;$

$\Rightarrow m\omega ^{2}=k[2-(e^{ika}+e^{-ika})]\;$

$\Rightarrow \omega ^{2}={\frac {2k}{m}}[1-cos(ka)]\;$

$\Rightarrow \omega ^{2}={\frac {\Delta k}{m}}{\frac {1}{2}}[1-cos(ka)]\;$

$\Rightarrow \omega ^{2}={\frac {\Delta k}{m}}sin^{2}({\frac {ka}{2}})\;$

$\Rightarrow \omega (k)=2{\sqrt {\frac {k}{m}}}|sin({\frac {ka}{2}})|\;$