n << N

[\frac{nπ}{2(N + 1)}] << 1

ω_{n} ≃ 2 (\sqrt{\frac{k}{m}}) [\frac{nπ}{2(N + 1)}] ≃ 2 ω_{0} [\frac{nπ}{2(N + 1)}]

ω_{n} ≃ n ω_{1}

\frac{F}{ΔL} = \frac{A}{L} Y

\frac{F}{ΔL} = \frac{A}{L} Y \frac{F}{Δx} = k

ω_{0} = \sqrt{\frac{k}{m}}

ω_{0} = \sqrt{\frac{AY/L}{ρAL}}

f_{0} = \frac{ω_{0}}{2π} = 2.4 × 10^{12}Hz

λ_{0} = \frac{c}{f_{0}} = 122 μm

ω_{N} = 2ω_{0}  ⇒  λ_{N} = \frac{1}{2} λ_{0}

ω_{0} = \frac{1}{L} \sqrt{\frac{Y}{ρ}}

ω_{N} ≃ 3 × 10^{13} rad/s

ω_{N} ∼ 2 \sqrt{\frac{k}{m_{avg}}}

F_{1} = k(x_{2} − x_{1}) = −kx_{1} + kx_{2}

F_{2} = −k(x_{2} − x_{1}) = kx_{1} − kx_{2}

F_{1} = m_{C} \frac{d^{2}x_{1}}{dt^{2}}

F_{2} = m_{O} \frac{d^{2}x_{2}}{dt^{2}}

F_{1} = (\frac{3}{4}m_{O}) \frac{d^{2}x_{1}}{dt^{2}}

F_{2} = m_{O} \frac{d^{2}x_{2}}{dt^{2}}

−kx_{1} + kx_{2} = (\frac{3}{4}m_{O}) \frac{d^{2}x_{1}}{dt^{2}}

kx_{1} − kx_{2} = m_{O} \frac{d^{2}x_{2}}{dt^{2}}

(\frac{k}{m_{O}} ) ( \begin{matrix}−\frac{4}{3} & \frac{4}{3} \\ 1 & −1 \end{matrix} ) ( \begin{matrix}a \\ b \end{matrix} ) = −ω^{2} ( \begin{matrix}a \\ b \end{matrix} )

−\frac{4}{3} \frac{k}{m_{o}} a + \frac{4}{3} \frac{k}{m_{o}} b = −ω^{2}a

b = \frac{(\frac{4}{3} \frac{k}{m_{o}} − ω^{2}) a}{\frac{4}{3} \frac{k}{m_{o}}}

\frac{4}{3}(\frac{k}{m_{o}} )^{2} − \frac{4}{3}(\frac{k}{m_{o}} )^{2} + \frac{k}{m_{o}} ω^{2} + \frac{4}{3} \frac{k}{m_{o}} ω^{2} − ω^{4} = 0

\frac{7}{3} \frac{k}{m_{o}} ω^{2} − ω^{4} = 0

ω^{2} = \frac{7}{3} \frac{k}{m}

k = \frac{3}{7} m_{o} ω^{2}

\begin{aligned}F_{1} & = k(x_{2} − x_{1})  = −kx_{1} + kx_{2} \\ F_{2} & = −k(x_{2} − x_{1}) + k(x_{3} − x_{2}) = kx_{1} − 2kx_{2} + kx_{3} \\ F_{3} & = −k(x_{3} − x_{2})  = kx_{2} − kx_{3} \end{aligned}

m_{o} \frac{d^{2}x_{1}}{dt^{2}} = −kx_{1} + kx_{2} + 0x_{3}

m_{c} \frac{d^{2}x_{2}}{dt^{2}} = kx_{1} − 2kx_{2} + kx_{3}

m_{o} \frac{d^{2}x_{3}}{dt^{2}} = 0x_{1} + kx_{2} − kx_{3}

( \begin{matrix}−\frac{k}{m_{o}} & \frac{k}{m_{o}} & 0 \\ \frac{k}{m_{c}} & −2\frac{k}{m_{c}} & \frac{k}{m_{c}} \\ 0 & \frac{k}{m_{o}} & −\frac{k}{m_{o}} \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \\ x_{3} \end{matrix} ) = ( \frac{d^{2}x_{1}}{dt^{2}} ; \frac{d^{2}x_{2}}{dt^{2}} ; \frac{d^{2}x_{3}}{dt^{2}} )

m_{c} = \frac{3}{4} m_{o}

( \begin{matrix}−\frac{k}{m_{o}} & \frac{k}{m_{o}} & 0 \\ \frac{4}{3} \frac{k}{m_{o}} & −\frac{8}{3} \frac{k}{m_{o}} & \frac{4}{3} \frac{k}{m_{o}} \\ 0 & \frac{k}{m_{o}} & −\frac{k}{m_{o}} \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \\ x_{3} \end{matrix} ) = ( \frac{d^{2}x_{1}}{dt^{2}} ; \frac{d^{2}x_{2}}{dt^{2}} ; \frac{d^{2}x_{3}}{dt^{2}} )

(\frac{k}{m_{o}} ) ( \begin{matrix}−1 & 1 & 0 \\ \frac{4}{3} & −\frac{8}{3} & \frac{4}{3} \\ 0 & 1 & −1 \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \\ x_{3} \end{matrix} ) = ( \frac{d^{2}x_{1}}{dt^{2}} ; \frac{d^{2}x_{2}}{dt^{2}} ; \frac{d^{2}x_{3}}{dt^{2}} )

(\frac{k}{m_{o}} ) ( \begin{matrix}−1 & 1 & 0 \\ \frac{4}{3} & −\frac{8}{3} & \frac{4}{3} \\ 0 & 1 & −1 \end{matrix} ) ( \begin{matrix}a \\ b \\ c \end{matrix} ) = −ω^{2} ( \begin{matrix}a \\ b \\ c \end{matrix} )

ω_{1}^{2} = 0 \frac{k}{m_{o}} = 0

ω_{2}^{2} = \frac{k}{m_{o}}

ω_{3}^{2} = (\frac{11}{3}) \frac{k}{m_{o}}

ω_{2} = \sqrt{\frac{k}{m_{o}}}  ⇒  s_{2} = ( \begin{matrix}−1 \\ 0 \\ 1 \end{matrix} ) ω_{3} = \sqrt{\frac{11k}{3m_{o}}}  ⇒  s_{3} = ( \begin{matrix}1 \\ −8/3 \\ 1 \end{matrix} )

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