\magnification=\magstep1

$$
y(x, t) \quad = \quad f(x, t)
$$

$$
\sqrt{ L^2 + (y_{p+1} - y_p)^2 } \quad \sim \quad L
$$

$$
\sin\left( {{y_{p+1} - y_p}\over{L}} \right) \quad \sim  \quad {{y_{p+1} - y_p}\over{L}} 
$$

$$
\sin( \theta_{p+1})  \quad \sim  \quad {{y_{p+1} - y_p}\over{L}} 
$$

$$
F_y \quad = \quad T \left( {{y_{p+1} - y_p}\over{L}}  \right) -
                  T \left( {{y_{p} - y_{p-1}}\over{L}} \right)
  \quad = \quad
                 {{T}\over{L}} ( y_{p+1} - 2 y_{p} + y_{p-1} )
$$


$$
{{d^2 y_p}\over{dt^2}} \quad = \quad 
                 {{T}\over{m L}} ( y_{p+1} - 2 y_{p} + y_{p-1} )
$$

$$
y_p(t) \quad = \quad A_p \cos(\omega t)
$$

$$
-\omega^2 A_p \cos(\omega t) \quad = \quad 
                 {{T}\over{m L}} \biggl[ A_{p+1} \cos(\omega t)  
                \ - \ 2 A_{p} \cos(\omega t)  \ +\  A_{p-1} \cos(\omega t) \biggr] 
$$

$$
-\omega^2 A_p \quad = \quad 
                 {{T}\over{m L}} \biggl[ A_{p+1} 
                \ - \ 2 A_{p}  \ +\  A_{p-1}  \biggr] 
$$

$$
\left( {{2 T}\over{m L}} -\omega^2 \right)  A_p \quad = \quad 
                 {{T}\over{m L}} \biggl[ A_{p+1} 
                \ +\  A_{p-1}  \biggr] 
$$

$$
{{A_{p+1} + A_{p-1}}\over{A_p}} \quad = \quad
      \biggr[ 2 \ - \ {{m L}\over{T}} \omega^2 \biggr]
$$

$$
A_p \quad = \quad C \thinspace \sin( p \alpha )
$$

$$
A_1 \quad = \quad C \thinspace \sin( \alpha )
$$


$$
A_2 \quad = \quad C \thinspace \sin( 2 \alpha )
$$

$$
A_3 \quad = \quad C \thinspace \sin( 3 \alpha )
$$

$$
{{ A_{p+1} \ +\  A_{p-1} } \over{A_p}}  \quad = \quad
   {{ C \sin\left[ (p-1) \alpha \right] \ + \ 
   C \sin\left[ (p+1) \alpha \right] } \over
     { C \sin\left( p \alpha \right) } }
$$

$$
\sin \beta \ + \ \sin \gamma \quad = \quad 
          2 \sin \biggl [ {{1}\over{2}} (\beta + \gamma) \biggr] 
            \cos \biggl [ {{1}\over{2}} (\beta - \gamma) \biggr] 
$$


$$
C \sin\left[ (p-1) \alpha \right] \ + \ C \sin\left[ (p+1) \alpha \right] 
       \quad = \quad
     2 C \sin(p \alpha) \cos(\alpha)
$$


$$
{{ A_{p+1} \ +\  A_{p-1} } \over{A_p}}  \quad = \quad
   {{ 2 C \sin( p \alpha ) \cos(\alpha) }
        \over
     { C \sin\left( p \alpha \right) } }  
       \quad = \quad
          2 \cos(\alpha)
$$

$$
A_0 \quad = \quad 0 \qquad \Rightarrow \quad C \sin(0 \alpha) \quad = \quad 0
$$

$$
A_{N+1} \quad = \quad 0 \qquad \Rightarrow \quad C \sin\biggl((N+1) \alpha \biggr) \quad = \quad 0
$$

$$
(N+1) \alpha \quad = \quad \pi, \ 2 \pi, \ 3 \pi, \ ..., \ n \pi
$$

$$
\alpha \quad = \quad {{n \pi}\over{ (N+1) }}
$$

$$
A_p \quad = \quad C \ \sin{ \biggl( {{p n \pi}\over{N + 1}} \biggr) }
$$

$$
y_p(t) \quad = \quad C \ \sin{ \biggl( {{p n \pi}\over{N + 1}} \biggr) }
                  \ \cos( \omega t )
$$

$$
2 \cos \biggl[ {{n \pi}\over{N+1}} \biggr] \quad = \quad
      \left( 2 - {{m L}\over{T}} \omega^2 \right)
$$

$$
\left( {{m L}\over{T}} \right) \ \omega^2 \quad = \quad
        2 \ -\  2 \cos \biggl[ {{n \pi}\over{N+1}} \biggr] 
$$

$$
\omega^2 \quad = \quad \biggl( {{2 T}\over{m L}} \biggr) 
          \biggl[ 1 \ - \ \cos{ \left( {{n \pi}\over{N + 1}} \right) } \biggr]
$$

$$
2 \sin^2 (\beta) \quad = \quad 1 \ - \ \cos(2 \beta)
$$

$$
\beta \quad = \quad {{n \pi}\over{2(N + 1)}}
$$

$$
\omega_n^2 \quad = \quad \biggl( {{2 T}\over{m L}} \biggr) \ 
           2 \sin^2{ \biggl( {{n \pi}\over{2(N + 1)}} \biggr) }
$$

$$
\omega_n \quad = \quad \sqrt{ {{4 T}\over{m L}}  } \ 
            \sin{ \biggl( {{n \pi}\over{2(N + 1)}} \biggr) }
$$


$$
v \quad = \quad \sqrt{{ {T}\over{\mu}} }
                \quad = \quad 
                \sqrt{{ {T}\over{M/L}} }
$$






\bye
