\magnification=\magstep1

$$
y(x, t) \quad = \quad A \thinspace
                   \sin \left( {{\pi n x}\over{L}} \right) \thinspace
                   \cos( \omega_n t )
$$

$$
y_1 (x, t) \quad = \quad {{A}\over{2}} \thinspace
                   \sin \left( {{\pi n x}\over{L}} \ - \  \omega_n t \right) 
$$

$$
y_2 (x, t) \quad = \quad {{A}\over{2}} \thinspace
                   \sin \left( {{\pi n x}\over{L}} \ + \ \omega_n t \right) 
$$


$$
y_i (x, t) \quad = \quad A \thinspace
                   \sin \left( {k x} \ - \  \omega t \right) 
$$

$$
y_1 (x, t) \quad = \quad B \thinspace
                   \sin \left( {k x} \ - \  \omega t \right) 
$$

$$
y_2 (x, t) \quad = \quad C \thinspace
                   \sin \left( {k x} \ + \  \omega t \right) 
$$

$$
y_3 (x, t) \quad = \quad D \thinspace
                   \cos \left( {k x} \ - \  \omega t \right) 
$$

$$
y_4 (x, t) \quad = \quad E \thinspace
                   \cos \left( {k x} \ + \  \omega t \right) 
$$

$$
y_i (0, t) \ + \ 
    y_2(0, t) \ + \ y_4(0, t) \quad = \quad 0
$$

$$
A \thinspace \sin \left( 0 \ - \  \omega t \right) 
  \ + \ 
            C \thinspace   \sin \left( 0 \ + \  \omega t \right)  
                  \ + \   
            E \thinspace   \cos \left( 0 \ + \  \omega t \right)  
   \quad = \quad 0
$$

$$
{{d y_i (0, t)}\over{dt}}  \ + \ 
    {{d y_2(0, t)}\over{dt}} \ + \ 
    {{d y_4(0, t)}\over{dt}} 
      \quad = \quad 0 
$$

$$
y_r (x, t) \quad = \quad -A \thinspace
                   \sin \left( {k x} \ + \  \omega t \right) 
$$

$$
y_r (x, t) \quad = \quad A \thinspace
                   \sin \left( {k x} \ + \  \omega t \right) 
$$


$$
y(x, t) \quad = \quad 0
$$





\bye
