\magnification=\magstep1

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

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

$$
-\left( {{k}\over{m}} + {{k_c}\over{m}} \right)  x_1 \ + \ 
        {{k_c}\over{m}} x_2 
  \quad = \quad
       {{d^2 x_1}\over{dt^2}}
$$

$$
 {{k_c}\over{m}} x_1 \ - \ 
\left( {{k}\over{m}} + {{k_c}\over{m}} \right)  x_2
  \quad = \quad
       {{d^2 x_2}\over{dt^2}}
$$

$$
\pmatrix { -\left( {{k}\over{m}} + {{k_c}\over{m}} \right)  & 
                 {{k_c}\over{m}}   \cr
            \phantom{.} \cr
                 {{k_c}\over{m}}   &
           -\left( {{k}\over{m}} + {{k_c}\over{m}} \right) \cr } 
   \quad
   \pmatrix { x_1 \cr   \phantom{.}  \cr  x_2 } 
   \quad = \quad
   \pmatrix { {{d^2 x_1}\over{dt^2}} \cr \phantom{.} \cr {{d^2 x_2}\over{dt^2}} }
$$
   
$$
s \quad = \quad a x_1 \ + \ b x_2
$$
           
$$
\pmatrix { -\left( {{k}\over{m}} + {{k_c}\over{m}} \right)  & 
                 {{k_c}\over{m}}   \cr
            \phantom{.} \cr
                 {{k_c}\over{m}}   &
           -\left( {{k}\over{m}} + {{k_c}\over{m}} \right) \cr } 
   \quad
   \pmatrix { a \cr   \phantom{.}  \cr  b } 
   \quad = \quad
     - \omega^2 
   \pmatrix { a  \cr \phantom{.} \cr b }
$$

$$
\gamma \quad \equiv \quad \left( {{k}\over{m}} \ + \ {{k_c}\over{m}} \right) 
$$

$$
\Delta \quad \equiv \quad  {{k_c}\over{m}} 
$$
   
$$
\pmatrix { -\gamma  & \Delta   \cr 
                \phantom{.}  \cr
            \Delta   & -\gamma \cr } 
   \quad
   \pmatrix { a \cr   \phantom{.}  \cr  b } 
   \quad = \quad
     - \omega^2 
   \pmatrix { a  \cr \phantom{.} \cr b }
$$

$$
- \gamma a \ + \ \Delta b \ + \ \omega^2 a \quad = \quad 0
$$

$$
 \Delta a \ - \ \gamma b \ + \ \omega^2 b \quad = \quad 0
$$

$$
b \quad = \quad {{(\gamma - \omega^2)}\over{\Delta}} a
$$

$$
\Delta \thinspace a \ - \ \gamma \left( {{\gamma - \omega^2}\over{\Delta}} \right) a
    \ + \ \omega^2 \left( {{\gamma - \omega^2}\over{\Delta}} \right) a 
   \quad = \quad 0
$$

$$
( \Delta^2 - \gamma^2 ) \ + \ 2 \gamma \omega^2 \ - \ \omega^4 
    \quad = \quad 0
$$

$$
\lambda \quad \equiv \quad \omega^2 
$$

$$
( \Delta^2 - \gamma^2 ) \ + \ 2 \gamma \lambda \ - \ \lambda^2
    \quad = \quad 0
$$

$$
\lambda \quad = \quad \gamma \ \pm \ \Delta  \quad = \quad
        \left( {{k}\over{m}} + {{k_c}\over{m}} \right) \ \pm \ {{k_c}\over{m}}
$$

$$
\lambda \quad = \quad {{k}\over{m}} \ + \ {{2 k_c}\over{m}} 
     \quad \quad {\rm or \ } \quad \quad 
                      \lambda \quad = \quad {{k}\over{m}} 
$$

$$
\omega_1 \quad = \quad \sqrt {{{k}\over{m}} }
$$

$$
\omega_2 \quad = \quad \sqrt {{{k}\over{m}} \ + \ {{2 k_c}\over{m}} }
$$




$$
- \gamma a \ + \ \Delta b \ + \ 
    \left( {{k}\over{m}} + {{2 k_c}\over{m}} \right)  a \quad = \quad 0
$$


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

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

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

$$
b \quad = \quad a
$$


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


$$
s_1(t) \quad = \quad A_1 \cos(\omega_1 t + \phi_1 )
$$
$$
s_2(t) \quad = \quad A_2 \cos(\omega_2 t + \phi_2 )
$$


$$
s_1(t) \quad = \quad A_1 \cos(\omega_1 t + \phi_1 ) \quad = \quad
           x_1(t) \ + \ x_2(t)
$$
$$
s_2(t) \quad = \quad A_2 \cos(\omega_2 t + \phi_2 ) \quad = \quad
           x_2(t) \ - \ x_1(t)
$$

$$
a = a
$$

$$
x_1(t) \quad = \quad {{1}\over{2}} \biggl( 
           A_1 \cos(\omega_1 t + \phi_1 ) \ - \ 
           A_2 \cos(\omega_2 t + \phi_2 )   \biggr)
$$

$$
x_2(t) \quad = \quad {{1}\over{2}} \biggl( 
           A_1 \cos(\omega_1 t + \phi_1 ) \ + \ 
           A_2 \cos(\omega_2 t + \phi_2 )   \biggr)
$$

$$
x_1(t) \quad = \quad {{L}\over{2}} \biggl[ \cos(\omega_1 t) \ - \ 
                      \cos(\omega_2 t) \biggr] 
$$

$$
x_2(t) \quad = \quad {{L}\over{2}} \biggl[ \cos(\omega_1 t) \ + \ 
                      \cos(\omega_2 t) \biggr] 
$$


\bye
