\magnification=\magstep1


$$
\eqalign{
s_1 \quad &= \quad x_1 \ + \ x_2 \cr
s_2 \quad &= \quad x_2 \ - \ x_1 \cr
}
$$

$$
\eqalign{
s_1(t) \quad &= \quad A_1 \cos( \omega_1 t + \phi_1 )  \cr
s_2(t) \quad &= \quad A_2 \cos( \omega_2 t + \phi_2 )  \cr
}
$$


$$
\eqalign{
s_1(t) \quad &= \quad L \cos( \omega_1 t )  \cr
s_2(t) \quad &= \quad L \cos( \omega_2 t )  \cr
}
$$

$$
s_1 \ - \ s_2 \quad = \quad (x_1 + x_2) \ - \ (x_2 - x_1) \quad = \quad 2 x_1
$$

$$
x_1 \quad = \quad {{1}\over{2}} (s_1 - s_2) 
$$

$$
x_2 \quad = \quad {{1}\over{2}} (s_1 + s_2) 
$$

$$
\eqalign{
x_1(t) \quad &= \quad {{1}\over{2}} L \left[ \cos (\omega_1 t) \ - \ 
                        \cos( \omega_2 t ) \right] \cr
      \quad &= \quad 
               {{1}\over{2}} L \left[ \cos \left( \sqrt{{k}\over{m}} t \right) \ - \ 
                        \cos \left( \sqrt{{3k}\over{m}} t \right) \right] \cr
}
$$

$$
\eqalign{
x_2(t) \quad &= \quad {{1}\over{2}} L \left[ \cos (\omega_1 t) \ + \ 
                        \cos( \omega_2 t ) \right] \cr
      \quad &= \quad 
               {{1}\over{2}} L \left[ \cos \left( \sqrt{{k}\over{m}} t \right) \ + \ 
                        \cos \left( \sqrt{{3k}\over{m}} t \right) \right] \cr
}
$$



\bye
