\magnification=\magstep1

$$
{\rm matrix\ } {\bf A} \quad = \quad \pmatrix{ 2 & 2 \cr 5 & -1 \cr } 
$$

$$
{\rm vector\ } {\bf x} \quad = \quad \pmatrix{ 1 \cr 2 \cr } 
$$

$$
{\bf A x } \quad = \quad 
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ 1 \cr 2 \cr } 
     \quad = \quad 
        \pmatrix{ 2 + 4 \cr 5  -2 \cr }
     \quad = \quad 
        \pmatrix{ 6 \cr 3 \cr }
$$

$$
{\bf x } = \pmatrix{ 1 \cr 2 \cr } \quad {\rm vs\ } \quad 
    {\bf A x } = \pmatrix{ 6 \cr 3 \cr }
$$


$$
{\rm vector\ } {\bf x} \quad = \quad \pmatrix{ 1 \cr 1 \cr } 
$$

$$
{\bf A x } \quad = \quad 
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ 1 \cr 1 \cr } 
     \quad = \quad 
        \pmatrix{ 2 + 2 \cr 5  -1 \cr }
     \quad = \quad 
        \pmatrix{ 4 \cr 4 \cr }
$$

$$
{\bf x } = \pmatrix{ 1 \cr 1 \cr } \quad {\rm vs\ } \quad 
    {\bf A x } = \pmatrix{ 4 \cr 4 \cr }
$$

$$
{\bf A x } \quad = \quad \pmatrix{4 \cr 4 \cr } \quad = \quad
       4 {\bf x }
$$

$$
{\rm vector\ } {\bf x} \quad = \quad \pmatrix{ 2 \cr 2 \cr } 
$$

$$
{\bf A x } \quad = \quad 
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ 2 \cr 2 \cr } 
     \quad = \quad 
        \pmatrix{ 4 + 4 \cr 10  -2 \cr }
     \quad = \quad 
        \pmatrix{ 8 \cr 8 \cr }
$$


$$
{\bf x } = \pmatrix{ 2 \cr 2 \cr } \quad {\rm vs\ } \quad 
    {\bf A x } = \pmatrix{ 8 \cr 8 \cr }
$$

$$
{\bf A x } \quad = \quad \lambda \thinspace {\bf x}
$$


$$
 \pmatrix{ 2 & 2 \cr 5 & -1 \cr } \pmatrix{ x \cr y \cr } 
     \quad = \quad 
        \lambda \thinspace \pmatrix{ x \cr y \cr }
$$

$$
\eqalign{
2 x \ +\  2 y \quad &= \quad \lambda x \cr
5 x \ -\  1 y \quad &= \quad \lambda y \cr
}
$$

$$
2 y \quad = \quad (\lambda - 2) x \quad \quad \Rightarrow  \quad \quad 
      y \quad = \quad {{1}\over{2}} (\lambda - 2) x
$$

$$
5x \ - \ {{1}\over{2}} \lambda x \ + \ x \quad = \quad 
          {{1}\over{2}} \lambda^2 x \ - \ \lambda x
$$

$$
{{1}\over{2}} \lambda^2 \ - \ {{1}\over{2}} \lambda \ - \ 6 \quad = \quad 0
$$

$$
\lambda^2 \ - \ \lambda \ -\  12 \quad = \quad 0
$$

$$
\lambda = 4 \quad , \quad \lambda = -3
$$


$$
\eqalign{
2 x \ +\  2 y \quad &= \quad 4 x \cr
5 x \ -\  1 y \quad &= \quad 4 y \cr
}
$$

$$ 
\eqalign{
2x \quad &= \quad 2y \quad \quad \quad \Rightarrow \quad x \ = \ y \cr
5x \quad &= \quad 5y \quad \quad \quad \Rightarrow \quad x \ = \ y \cr
}
$$

$$
{\rm eigenvector\ corresponding\ to\ } \lambda = 4 \quad \rightarrow \quad 
    \pmatrix{ 1 \cr 1 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ 2 \cr 2 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ a \cr a \cr } 
$$


$$
{\rm eigenvector\ corresponding\ to\ } \lambda = -3 \quad \rightarrow \quad 
    \pmatrix{ -2 \cr 5 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ 4 \cr -10 \cr } 
       \quad {\rm or \ \ } 
    \pmatrix{ -2a \cr 5a \cr } 
$$


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

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


$$
F_1 \quad = \quad - 2 k x_1 \ + \ k x_2  
$$

$$
F_2 \quad = \quad k x_1 \ - \ 2 k x_2 \ + \ k x_3 
$$

$$
F_3 \quad = \quad            k x_2 \ - \ 2 k x_3 
$$


$$
\left( {{k}\over{m}} \right)  
  \pmatrix { -2 & 1 & 0 \cr 1 & -2 & 1 \cr 0 & 1 & -2 \cr }
    \pmatrix { a x_1 \cr b x_2 \cr c x_3 } \quad = \quad
      - \omega^2 
    \pmatrix { a x_1 \cr b x_2 \cr c x_3 } 
$$



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




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

$$
 {{k}\over{m}} x_1 \ - \ 2 {{k}\over{m}} x_2 \quad = \quad {{d^2 x_2}\over{dt^2}} \ 
$$

$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { x_1 \cr x_2 \cr } 
  \quad = \quad 
%    \pmatrix { {d^2 x_1} / {dt^2} \cr
%               {d^2 x_2} / {dt^2} \cr }
    \pmatrix { {{d^2 x_1}\over{dt^2}} \cr
                 \phantom{x} \cr
               {{d^2 x_2}\over{dt^2}} \cr }
$$

$$
s \quad = \quad \pmatrix { a x_1 \cr b x_2 }
$$

$$
{{d^2 s}\over{dt^2}} \quad = \quad  - \omega^2 s
$$

$$
{{d^2 x_1}\over{dt^2}} \quad = \quad  - \omega^2 x_1 
$$

$$
{{d^2 x_2}\over{dt^2}} \quad = \quad  - \omega^2 x_2 
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a x_1 \cr b x_2 \cr } 
  \quad = \quad 
%    \pmatrix { {d^2 x_1} / {dt^2} \cr
%               {d^2 x_2} / {dt^2} \cr }
    - \omega^2 \pmatrix { a x_1  \cr
                          b x_2  \cr }
$$


$$
{\bf A} {\bf s} \quad = \quad \lambda \thinspace {\bf s}
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad = \quad 
    - \omega^2 \pmatrix { a   \cr
                          b  \cr }
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad 
    + \quad \omega^2 \pmatrix { a   \cr
                          b  \cr }
  \quad = \quad 0
$$

$$
- 2 ({{k}\over{m}}) a \ + \ 1 ({{k}\over{m}}) b \ + \ \omega^2 a \quad = \quad 0
$$

$$
  1 ({{k}\over{m}}) a \ - \ 2 ({{k}\over{m}}) b \ + \ \omega^2 b \quad = \quad 0
$$

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

$$
1 ({{k}\over{m}}) a  \ + \
    \left( -2 {{k}\over{m}} + \omega^2 \right) b \quad = \quad 0
$$

$$
1 ({{k}\over{m}}) a  \ + \
    \left( -2 {{k}\over{m}} + \omega^2 \right) 
      \left(  {{\omega^2 - 2{{k}\over{m}} } \over{- {{k}\over{m}} } } \right) a
 \quad = \quad 0
$$

$$
-1 \left({{k}\over{m}}\right)^2 \ + \ 
   4 \left({{k}\over{m}}\right)^2 \ - \ 
    4 \left({{k}\over{m}}\right) \omega^2 \ + \
   \omega^4 \quad = \quad 0
$$

$$
\lambda \quad = \quad \omega^2
$$

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


$$
\lambda \quad = \quad \omega^2 \quad = \quad 
     1 {{k}\over{m}} \quad {\rm or \ } \quad 3 {{k}\over{m}} 
$$

$$
\eqalign{
\omega_1 \quad &= \quad \sqrt{ {{k}\over{m}} } \cr
\omega_2 \quad &= \quad \sqrt{ {{3k}\over{m}} } \cr
}
$$

$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad 
    + \quad 1 ({{k}\over{m}}) \pmatrix { a   \cr
                          b  \cr }
  \quad = \quad 0
$$

$$
- 2 ({{k}\over{m}}) a \ + \ 1 ({{k}\over{m}}) b \ + \ 1 ({{k}\over{m}}) a \quad = \quad 0
$$

$$
 1 ({{k}\over{m}}) a \ - \ 2 ({{k}\over{m}}) b \ + \ 1 ({{k}\over{m}}) b \quad = \quad 0
$$

$$
a \quad = \quad b
$$

$$
{\rm eigenvalue\ } \omega_1 \ = \ \sqrt{{k}\over{m}} \  \quad 
   {\rm corresponds\ to\ eigenvector\ } \quad 
   \pmatrix { 1 \cr 1 \cr }
$$

$$
s_1 \quad = \quad 
   \pmatrix { 1 x_1 \cr 1 x_2 \cr } 
      \quad \quad {\rm or } \quad \quad 
s_1 \quad = \quad x_1 \ + \ x_2
$$


$$
\pmatrix{ -2 {{k}\over{m}} &  {{k}\over{m}} \cr 
             {{k}\over{m}} & -2 {{k}\over{m}} \cr }  
   \pmatrix { a \cr b \cr } 
  \quad 
    + \quad 3 ({{k}\over{m}}) \pmatrix { a   \cr
                          b  \cr }
  \quad = \quad 0
$$

$$
- 2 ({{k}\over{m}}) a \ + \ 1 ({{k}\over{m}}) b \ + \ 3 ({{k}\over{m}}) a \quad = \quad 0
$$

$$
 1 ({{k}\over{m}}) a \ - \ 2 ({{k}\over{m}}) b \ + \ 3 ({{k}\over{m}}) b \quad = \quad 0
$$



$$
a \quad = \quad -b
$$

$$
{\rm eigenvalue\ } \omega_2 \ = \ \sqrt{{3k}\over{m}} \  \quad 
   {\rm corresponds\ to\ eigenvector\ } \quad 
   \pmatrix { 1 \cr -1 \cr }
$$


$$
s_2 \quad = \quad 
   \pmatrix { 1 x_1 \cr -1 x_2 \cr } 
      \quad \quad {\rm or } \quad \quad 
s_2 \ = \ x_1 \ - \ x_2 
      \quad {\rm or\ } \quad 
s_2 \ = \ x_2 \ - \ x_1 
$$

$$
-\omega_1^2 \quad = \quad (\sqrt{2} - 2) {{k}\over{m}} \quad \quad 
               \Rightarrow \quad \quad \omega_1 \quad = \quad \sqrt{ (2 - \sqrt{2} )  {{k}\over{m}} }
            \quad = \quad 0.765 \thinspace \sqrt{{k}\over{m}} 
$$

$$
-\omega_2^2 \quad = \quad -2 {{k}\over{m}} \quad \quad 
               \Rightarrow \quad \quad \omega_2 \quad = \quad \sqrt{ 2 {{k}\over{m}} } 
            \quad = \quad 1.414 \thinspace \sqrt{{k}\over{m}} 
$$

$$
-\omega_3^2 \quad = \quad (-\sqrt{2} - 2) {{k}\over{m}} \quad \quad 
               \Rightarrow \quad \quad \omega_3 \quad = \quad \sqrt{ (2 + \sqrt{2} )  {{k}\over{m}} }
            \quad = \quad 1.848 \thinspace \sqrt{{k}\over{m}} 
$$

$$
s_1 \quad = \quad \pmatrix{ 1 \cr \sqrt{2} \cr 1 }  \quad = \quad 
          x_1 \ + \  \sqrt{2} x_2 \ + \  x_3
$$

$$
s_2 \quad = \quad \pmatrix{ -1 \cr 0 \cr 1 }  \quad = \quad 
          -x_1 \ + \  0 x_2 \ + \  x_3
$$

$$
s_3 \quad = \quad \pmatrix{ 1 \cr -\sqrt{2} \cr 1 }  \quad = \quad 
          x_1 \ - \  \sqrt{2} x_2 \ + \  x_3
$$

$$
s_1(t) \quad = \quad A_1 \cos(\omega_1 t + \phi_1)
$$

$$
s_2(t) \quad = \quad A_2 \cos(\omega_2 t + \phi_2)
$$

$$
s_3(t) \quad = \quad A_3 \cos(\omega_3 t + \phi_3)
$$


$$
x_1 \quad = \quad  {{1}\over{4}} s_1 \ - \ 
                   {{1}\over{2}} s_2 \ + \ 
                   {{1}\over{4}} s_3 
$$

$$
s_1 \quad = \quad 
   \pmatrix { 1  \cr 1  \cr } 
$$

$$
s_2 \quad = \quad 
   \pmatrix { 1  \cr -1  \cr } 
$$

$$
s_1 \quad = \quad 
   \pmatrix { 1  \cr 1  \cr } 
$$

$$
s_2 \quad = \quad 
   \pmatrix { 1  \cr -1  \cr } 
$$



$$
\left( {{k}\over{m}} \right)  
  \pmatrix { -2 & 1 & 0 & 0 \cr 1 & -2 & 1 & 0 \cr 0 & 1 & -2 & 1 \cr 0 & 0 & 1 & -2 }
    \pmatrix { a x_1 \cr b x_2 \cr c x_3 \cr d x_4 } \quad = \quad
      - \omega^2 
    \pmatrix { a x_1 \cr b x_2 \cr c x_3 \cr d x_4} 
$$



$$
\omega_1^2 \quad = \quad 0.38197 {{k}\over{m}} \quad \quad 
               \Rightarrow \quad \quad \omega_1 \quad = \quad 0.618 \sqrt{ {{k}\over{m}} }
$$

$$
\omega_2^2 \quad = \quad 3.61803 {{k}\over{m}} \quad \quad 
               \Rightarrow \quad \quad \omega_2 \quad = \quad 1.902 \sqrt{  {{k}\over{m}} }
$$

$$
\omega_3^2 \quad = \quad 2.61803  {{k}\over{m}} \quad \quad 
               \Rightarrow \quad \quad \omega_3 \quad = \quad 1.618 \sqrt{  {{k}\over{m}} }
$$

$$
\omega_4^2 \quad = \quad 1.38197  {{k}\over{m}} \quad \quad 
               \Rightarrow \quad \quad \omega_4 \quad = \quad 1.176 \sqrt{  {{k}\over{m}} }
$$

%%  I think the below are wrong.  Better values follow
%%  MWR 2/29/2024 
%%
%% $$
%% s_1 \quad = \quad 
%%    \pmatrix { 0.37175 \cr -0.37175 \cr  0.60150 \cr  0.60150  }
%% $$

%% $$
%% s_2 \quad = \quad
%%    \pmatrix { 0.60150  \cr  0.60150  \cr -0.37175  \cr  0.37175  }
%% $$

%% $$
%% s_3 \quad = \quad
%%    \pmatrix { 0.60150 \cr -0.60150 \cr -0.37175 \cr -0.37175  }
%% $$

%% $$
%% s_4 \quad = \quad
%%    \pmatrix { 0.37175 \cr  0.37175 \cr  0.60150 \cr -0.60150  }
%% $$

$$
s_1 \quad = \quad 
   \pmatrix { 1       \cr 1.61803  \cr  1.61803  \cr  1        }
$$

$$
s_2 \quad = \quad
   \pmatrix { -1       \cr  -0.61803 \cr 0.61803  \cr  1        }
$$

$$
s_3 \quad = \quad
   \pmatrix { 1       \cr -0.61803 \cr -0.61803 \cr 1         }
$$

$$
s_4 \quad = \quad
   \pmatrix { -1      \cr  1.61803 \cr -1.61803 \cr  1        }
$$


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

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

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


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


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

$$
\omega_1 \quad \simeq  \quad 2  \thinspace \omega_0 \thinspace
              \biggl[ {{\pi}\over{2 (N + 1)}} \biggr] 
$$

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

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

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

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

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

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

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

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


\bye
