\magnification=\magstep1

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

$$
L_{\rm tot} \quad = \quad N L
$$

$$
n \quad = \quad 1 \quad \qquad \lambda_1 \quad = \quad {{2 L_{\rm tot}}\over{1}}
$$

$$
n \quad = \quad 2 \quad \qquad \lambda_2 \quad = \quad {{2 L_{\rm tot}}\over{2}}
$$

$$
n \quad = \quad 3 \quad \qquad \lambda_3 \quad = \quad {{2 L_{\rm tot}}\over{3}}
$$

$$
n \quad = \quad n \quad \qquad \lambda_n \quad = \quad {{2 L_{\rm tot}}\over{n}}
$$

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

$$
N >> 1
$$

$$
n << N 
$$

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

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

$$
v \quad = \quad \lambda f \quad = \quad \lambda {{\omega}\over{2 \pi}}
$$

$$
v_n \quad = \quad \ \left( {{2 L_{\rm tot}}\over{n}} \right)
                  \ {{1}\over{2 \pi}} \
                  \sqrt{ {{4 T}\over{m L}} } 
                 \ \left( {{\pi}\over{2 (N + 1)}} \right) 
                 \ n
$$

$$
n \quad \sim \quad {{1}\over{2}} \thinspace N
$$

$$
 \sin {\left( {{n \pi} \over{2 (N+1)}} \right) } \quad \neq \quad
             {{n \pi} \over{2 (N+1)}} 
$$
         
$$
\omega_p \quad \simeq \quad 56 \times 10^3 \sqrt{ n_e} {\rm \ \ \ rad/s}
$$

$$
v_{pl} \quad = \quad c \ \sqrt{ 1 - {{\omega_p^2}\over{\omega^2}} }
$$
                     
$$
v_{pl} \quad = \quad c \ \sqrt{ 1 - {{9.4 \times 10^7}\over{\omega^2}} }
$$

$$
t_0 \quad = \quad {{D}\over{c}} \quad = \quad 6500 {\rm \ years\ } 
       \quad \simeq \quad 2 \times 10^{11} {\rm sec} 
$$

$$
t_2 - t_1 \quad = \quad \Delta t \quad = \quad 
       {{D}\over{c}} \ \left( \phantom{1.76 x 10^{-13} } - \phantom{1.03 x 10^{-13}} \right) 
$$

$$
\Delta t \quad = \quad {{D}\over{c}} \ \left( \phantom{0.73 x 10^{-13}} \right)
$$
                     
$$
D \quad = \quad {{c \Delta t}\over{(\phantom{0.73 x 10^{-13}} }) } 
$$




\bye

