Potential Energy U = 1 , 2 , 3 , . . . n {\displaystyle U=1,2,3,...n\;}
U ≅ 1 2 k ∑ n ( U n − U n − 1 ) {\displaystyle U\cong {\frac {1}{2}}k\sum _{n}(U_{n}-U_{n-1})\;}
⇒ m U ¨ n = − k [ 2 U n − U n − 1 − U n + 1 ] = − m ω U ( t ) {\displaystyle \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 ω t u m {\displaystyle u_{m}(t)=e^{i\omega t}u_{m}\;}
⇒ − m ω 2 u → ¨ n = − k M u → {\displaystyle \Rightarrow -m\omega ^{2}{\ddot {\vec {u}}}_{n}=-k\mathbf {M} {\vec {u}}\;}
Band Matrix
M = | 2 1 0 0 1 2 1 0 0 1 2 1 0 0 1 2 | {\displaystyle \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 i k ( n a ) − ω t = e i k R n {\displaystyle u_{m}(t)=e^{ik(na)-\omega t}=e^{ikR_{n}}\;}
where
u m → u n e i k ( n a ) {\displaystyle u_{m}\rightarrow u_{n}e^{ik(na)}\;}
R n {\displaystyle R_{n}\;} = distance on some coordinate system
⇒ − m ω 2 u → = − k [ 2 − e i k α − e − i k α ] {\displaystyle \Rightarrow -m\omega ^{2}{\vec {u}}=-k[2-e^{ik\alpha }-e^{-ik\alpha }]\;}
Derive and get:
⇒ ω ( k ) = 2 k m | s i n ( k a ) | {\displaystyle \Rightarrow \omega (k)=2{\sqrt {\frac {k}{m}}}|sin(ka)|}
u ( R n ) ≡ e i k R n = c o s ( k a ) {\displaystyle u(R_{n})\equiv e^{ikR_{n}}=cos(ka)\;}
⇒ − m ω 2 e i k n a = − k [ 2 e i k ( n a ) − e i k ( n + 1 ) a − e i k ( n − 1 ) a ] {\displaystyle \Rightarrow -m\omega ^{2}e^{ikna}=-k[2e^{ik(na)}-e^{ik(n+1)a}-e^{ik(n-1)a}]\;}
⇒ m ω 2 = k [ 2 − ( e i k a + e − i k a ) ] {\displaystyle \Rightarrow m\omega ^{2}=k[2-(e^{ika}+e^{-ika})]\;}
⇒ ω 2 = 2 k m [ 1 − c o s ( k a ) ] {\displaystyle \Rightarrow \omega ^{2}={\frac {2k}{m}}[1-cos(ka)]\;}
⇒ ω 2 = Δ k m 1 2 [ 1 − c o s ( k a ) ] {\displaystyle \Rightarrow \omega ^{2}={\frac {\Delta k}{m}}{\frac {1}{2}}[1-cos(ka)]\;}
⇒ ω 2 = Δ k m s i n 2 ( k a 2 ) {\displaystyle \Rightarrow \omega ^{2}={\frac {\Delta k}{m}}sin^{2}({\frac {ka}{2}})\;}
⇒ ω ( k ) = 2 k m | s i n ( k a 2 ) | {\displaystyle \Rightarrow \omega (k)=2{\sqrt {\frac {k}{m}}}|sin({\frac {ka}{2}})|\;}
![{\displaystyle \Rightarrow m{\ddot {U}}_{n}=-k[2U_{n}-U_{n-1}-U_{n+1}]=-m\omega U(t)\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/461b812664679855fe2267f3b2cb29ec477ba6b6)
= distance on some coordinate system
![{\displaystyle \Rightarrow -m\omega ^{2}{\vec {u}}=-k[2-e^{ik\alpha }-e^{-ik\alpha }]\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6a218fe9a00344679e78f96eb71bc19aa04fd9)
![{\displaystyle \Rightarrow -m\omega ^{2}e^{ikna}=-k[2e^{ik(na)}-e^{ik(n+1)a}-e^{ik(n-1)a}]\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf21eaf0cd6fff39dbaadf7d6c171483dc3d65fb)
![{\displaystyle \Rightarrow m\omega ^{2}=k[2-(e^{ika}+e^{-ika})]\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c7aad9c07cedc555f91814e4bb4eaa24a1810b)
![{\displaystyle \Rightarrow \omega ^{2}={\frac {2k}{m}}[1-cos(ka)]\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd3a86e311f275e9c6e2dc3778c673e96de10fa)
![{\displaystyle \Rightarrow \omega ^{2}={\frac {\Delta k}{m}}{\frac {1}{2}}[1-cos(ka)]\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89704ae3a53c5b0d68e89b57e0881bd41abd54fd)
