\magnification=\magstep1

$$
n \quad << \quad N
$$

$$
              \biggl[ {{n \pi}\over{2 (N + 1)}} \biggr] \quad << \quad 1
$$


$$
\omega_n \quad \simeq \quad 2 \left( \sqrt{{k}\over{m}} \right) 
              \biggl[ {{n \pi}\over{2 (N + 1)}} \biggr]
  \quad \simeq \quad 
                      2  \thinspace \omega_0 \thinspace
              \biggl[ {{n \pi}\over{2 (N + 1)}} \biggr] 
$$

$$
\omega_n \quad \simeq \quad n \thinspace \omega_1
$$

$$
{{F}\over{\Delta L}} \quad = \quad {{A}\over{L}} \thinspace Y
$$

$$
{{F}\over{\Delta L}} \quad = \quad {{A}\over{L}} \thinspace Y 
   \quad \quad \quad \quad 
{{F}\over{\Delta x}} \quad = \quad k
$$

$$
\omega_0 \quad = \quad \sqrt{{k}\over{m}} 
$$

$$
\omega_0 \quad = \quad \sqrt{ { A Y / L } \over{ \rho A L } }
$$

$$
f_0 \quad = \quad {{\omega_0}\over{2 \pi}} \quad = \quad 2.4 \times 10^{12} {\rm Hz}
$$

$$
\lambda_0 \quad = \quad {{c}\over{f_0}} \quad = \quad 122 \ \mu{\rm m}
$$

$$
\omega_N \quad = \quad 2 \omega_0 \quad \quad \quad 
      \Rightarrow \quad \quad 
      \lambda_N \quad =  \quad {{1}\over{2}} \ \lambda_0
$$

$$
\omega_0 \quad = \quad {{1}\over{L}} \thinspace  \sqrt{ {{Y}\over{\rho}} }
$$

$$
\omega_N \quad \simeq \quad 3 \times 10^{13} {\rm \ rad/s}
$$

$$
\omega_N \quad \sim \quad 2 \thinspace \sqrt{{k}\over{m_{\rm avg}}} 
$$

$$
F_1 \quad = \quad k(x_2 - x_1) \quad = \quad -k x_1 \ +\  k x_2 
$$

$$
F_2 \quad = \quad -k(x_2 - x_1) \quad = \quad k x_1 \ -\  k x_2 
$$

$$
F_1 \quad = \quad m_C \ {{d^2 x_1}\over{d t^2}}
$$

$$
F_2 \quad = \quad m_O \ {{d^2 x_2}\over{d t^2}}
$$


$$
F_1 \quad = \quad \left( {{3}\over{4}} m_O \right) \ {{d^2 x_1}\over{d t^2}}
$$

$$
F_2 \quad = \quad m_O \ {{d^2 x_2}\over{d t^2}}
$$

$$
  -k x_1 \ +\  k x_2  \quad = \quad \left( {{3}\over{4}} m_O \right) \ 
                        {{d^2 x_1}\over{d t^2}}
$$

$$
  k x_1 \ -\  k x_2   \quad = \quad m_O \ 
                        {{d^2 x_2}\over{d t^2}}
$$

$$
\left( {{k}\over{m_O}} \right) 
  \pmatrix {  -{{4}\over{3}} & {{4}\over{3}} \cr
               \phantom{.} \cr
                1 & -1 \cr } 
  \pmatrix { a  \cr \phantom{.} \cr b  }
     \quad = \quad
  - \omega^2  \pmatrix { a  \cr \phantom{.} \cr b  }
$$

$$
- {{4}\over{3}} {{k}\over{m_o}} a \ + \ 
  {{4}\over{3}} {{k}\over{m_o}} b \quad = \quad
    - \omega^2 a
$$

$$
b \quad = \quad  {  {\left( {{4}\over{3}} {{k}\over{m_o}} \ - \  \omega^2 \right)  a}
    \over{   {{4}\over{3}} {{k}\over{m_o}} } }
$$

$$
{{4}\over{3}} {\left( {{k}\over{m_o}} \right)}^2 \ - \ 
  {{4}\over{3}} {\left( {{k}\over{m_o}} \right)}^2 \ + \ 
       {{k}\over{m_o}} \omega^2 \ + \ 
        {{4}\over{3}}  {{k}\over{m_o}} \omega^2 \ - \  
           \omega^4 \quad = \quad 0
$$

$$
        {{7}\over{3}}  {{k}\over{m_o}} \omega^2 \ - \  
           \omega^4 \quad = \quad 0
$$

$$
\omega^2 \quad = \quad {{7}\over{3}} {{k}\over{m_o}}
$$

$$
k \quad = \quad {{3}\over{7}} \thinspace m_o \thinspace \omega^2 
$$


$$
\eqalign{
F_1 \quad &= \quad k(x_2 - x_1) \quad \qquad \qquad \qquad \quad \ = \quad -k x_1 \ +\  k x_2  \cr
F_2 \quad &= \quad -k(x_2 - x_1) \ + \ k(x_3 - x_2) \quad = 
          \quad k x_1 \ -\  2 k x_2  \ + \ k x_3 \cr
F_3 \quad &= \quad - k(x_3 - x_2) \quad \qquad \qquad \qquad \ = \quad k x_2 \ -\  k x_3  \cr
}
$$

\vskip 1in

$$
m_o {{d^2 x_1}\over{dt^2}} \quad = \quad -k x_1 \ +\  k x_2  \ + \ 0 x_3
$$ 

$$
m_c {{d^2 x_2}\over{dt^2}} \quad = \quad k x_1 \ -\  2 k x_2  \ + \ k x_3 
$$

$$
m_o {{d^2 x_3}\over{dt^2}} \quad = \quad 0 x_1 \ + \ k x_2 \ -\  k x_3  
$$

$$
\pmatrix { -{{k}\over{m_o}} & {{k}\over{m_o}} & 0 \cr
                \phantom{.} \cr
           {{k}\over{m_c}} & -2{{k}\over{m_c}} & {{k}\over{m_c}} \cr
                \phantom{.} \cr
           0 & {{k}\over{m_o}} & -{{k}\over{m_o}} \cr }
  \pmatrix { x_1 \cr \phantom{.} \cr x_2 \cr \phantom{.} \cr x_3 } 
    \quad = \quad 
  \pmatrix { {{d^2 x_1}\over{dt^2}}  \cr 
                   \phantom{.} \cr 
             {{d^2 x_2}\over{dt^2}}  \cr 
                   \phantom{.} \cr 
             {{d^2 x_3}\over{dt^2}}  \cr   }
$$

$$
m_c \quad = \quad {{3}\over{4}} \ m_o
$$
 

$$
\pmatrix { -{{k}\over{m_o}} & {{k}\over{m_o}} & 0 \cr
                \phantom{.} \cr
           {{4}\over{3}} {{k}\over{m_o}} & -{{8}\over{3}} {{k}\over{m_o}} & 
                                 {{4}\over{3}} {{k}\over{m_o}} \cr
                \phantom{.} \cr
           0 & {{k}\over{m_o}} & -{{k}\over{m_o}} \cr }
  \pmatrix { x_1 \cr \phantom{.} \cr x_2 \cr \phantom{.} \cr x_3 } 
    \quad = \quad 
  \pmatrix { {{d^2 x_1}\over{dt^2}}  \cr 
                   \phantom{.} \cr 
             {{d^2 x_2}\over{dt^2}}  \cr 
                   \phantom{.} \cr 
             {{d^2 x_3}\over{dt^2}}  \cr   }
$$


$$
\left( {{k}\over{m_o}} \right) 
   \pmatrix { -1  & 1 & 0 \cr
                \phantom{.} \cr
           {{4}\over{3}}  & -{{8}\over{3}}  & {{4}\over{3}} \cr
                \phantom{.} \cr
           0 & 1  & -1 \cr }
  \pmatrix { x_1 \cr \phantom{.} \cr x_2 \cr \phantom{.} \cr x_3 } 
    \quad = \quad 
  \pmatrix { {{d^2 x_1}\over{dt^2}}  \cr 
                   \phantom{.} \cr 
             {{d^2 x_2}\over{dt^2}}  \cr 
                   \phantom{.} \cr 
             {{d^2 x_3}\over{dt^2}}  \cr   }
$$


$$
\left( {{k}\over{m_o}} \right) 
   \pmatrix { -1  & 1 & 0 \cr
                \phantom{.} \cr
           {{4}\over{3}}  & -{{8}\over{3}}  & {{4}\over{3}} \cr
                \phantom{.} \cr
           0 & 1  & -1 \cr }
  \pmatrix { a \cr \phantom{.} \cr b \cr \phantom{.} \cr c } 
    \quad = \quad 
        -\omega^2 \ 
  \pmatrix { a  \cr 
                   \phantom{.} \cr 
             b  \cr 
                   \phantom{.} \cr 
             c  \cr   }
$$

$$
\qquad \quad \quad \omega_1^2 \quad = \quad 0 \ {{k}\over{m_o}}  \quad = \quad 0
$$

$$
\omega_2^2 \quad = \quad  {{k}\over{m_o}} 
$$

$$
\qquad \quad \omega_3^2 \quad = \quad  \left( {{11}\over{3}} \right) \ {{k}\over{m_o}} 
$$

$$
\omega_2 \quad = \quad \sqrt{{k}\over{m_o}} \qquad \quad 
       \Rightarrow 
        \quad \quad 
           s_2 \quad = \quad \pmatrix{ -1 \cr 0 \cr 1 }
$$


$$
\omega_3 \quad = \quad \sqrt{{11 k}\over{3 m_o}} \qquad \quad 
       \Rightarrow 
        \quad \quad 
           s_3 \quad = \quad \pmatrix{ 1 \cr -8/3 \cr 1 }
$$







\bye
