\magnification=\magstep1

$$
v_I \quad = \quad \sqrt{ {{T}\over{\mu_I}} } \quad  = \quad {{\omega}\over{k_I}}
$$

$$
k_I \quad = \quad \omega \thinspace \sqrt{ {{\mu_I}\over{T}} }
$$


$$
k_{II} \quad = \quad \omega \thinspace \sqrt{ {{\mu_{II}}\over{T}} }
$$

$$
y_i (x, t) \quad = \quad A_i \thinspace
                   \sin(k_I x - \omega t)
$$

$$
y (x, t) \quad = \quad A_i \thinspace
                   \sin \left(\omega [ t - {{1}\over{v_I}} x]  \right)
$$

$$
y_r (x, t) \quad = \quad A_r \thinspace
                   \sin(k_I x + \omega t)
$$


$$
y_t (x, t) \quad = \quad A_t \thinspace
                   \sin(k_{II} x - \omega t)
$$

$$
y_i  \ + \ y_r \quad = \quad y_t
$$

$$
v_{yi} \ + \ v_{yr} \quad = \quad v_{yt}
$$


$$
A_i \thinspace \sin( 0 - \omega t) \ + \ 
     A_r \thinspace \sin( 0 + \omega t) 
       \quad = \quad 
     A_t \thinspace \sin( 0 - \omega t) 
$$

$$
- A_i \ + \ A_r \quad = \quad - A_t
$$

$$
A_i \quad = \quad  A_r \ + \  A_t
$$

$$
-\omega A_i \thinspace \cos(0 - \omega t) \ + \ 
     \omega A_r \thinspace \cos(0 + \omega t) 
           \quad = \quad
     -\omega A_t \thinspace \cos(0 - \omega t) 
$$

$$
- A_i \ + \ A_r \quad = \quad  - A_t
$$

$$
{{\partial y_i}\over{\partial x}} \ + \ 
   {{\partial y_r}\over{\partial x}} 
      \quad = \quad 
   {{\partial y_t}\over{\partial x}}
$$

$$
k_I A_i \thinspace \cos( 0 - \omega t) \ + \ 
   k_I A_r \thinspace \cos( 0 + \omega t) 
       \quad = \quad 
   k_{II} A_t \thinspace \cos( 0 - \omega t) 
$$

$$
k_I A_i \ + \ k_I A_r \quad = \quad k_{II} A_t
$$

$$
A_t \quad = \quad \biggl( {{2 k_I}\over{k_I + k_{II}}} \biggr) \ A_i
$$

$$
A_r \quad = \quad \biggl( {{k_{II} - k_I}\over{k_I + k_{II}}} \biggr) \ A_i
$$

$$
y_r \quad = \quad A_i \sin(k_I x + \omega t)
$$






\bye
