\magnification=\magstep1

$$
{\rm impedence\ Z\ } \quad = \quad 
   {{\rm force\ on\ string\ in\ y\ direction\ }
      \over 
    {\rm amplitude\ of\ velocity\ in\ y\ direction}}
$$

$$
Z \quad = \quad {{F_y}\over{v_y}}
$$

$$
F_y \quad = \quad T_y \quad = \quad T \thinspace \sin(\theta) \quad 
          = \quad T \thinspace \Biggl( {{T_y} \over{\sqrt{T_x^2 + T_y^2} }} \Biggr)
$$

$$
\sqrt{T_x^2 + T_y^2} \quad \simeq \quad T_x
$$

$$
F_y \quad \simeq \quad  T \thinspace \biggl( {{T_y}\over{T_x}} \biggr)
   \quad = \quad T \thinspace {\rm ( slope\ at\ P)}
   \quad = \quad T \thinspace 
                {{\partial y}\over{\partial x}}
$$

$$
F_y \quad = \quad - T \thinspace
                {{\partial y}\over{\partial x}}
$$

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

$$
v_y \quad = \quad {{\partial y}\over{\partial t}} \quad = \quad - \omega A_i \thinspace
                    \cos(k_I x - \omega t)
$$


$$
F_y \quad = \quad - T \biggl[ k_I A_i \cos \left( k_I x - \omega t \right) \biggr]
$$

$$
Z_I \quad = \quad {{F_y}\over{v_y}} \quad = \quad
           {{- T k_I A_i \thinspace \cos(k_I x - \omega t) }
                \over
            {- \omega A_i \thinspace \cos(k_I x - \omega t) }}
$$

$$
Z_I \quad = \quad {{T k_I}\over{\omega}} 
$$


$$
Z_I \quad = \quad {{T}\over{v_I}} 
$$

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

$$
Z_I \quad = \quad \sqrt{ T \mu_I}
$$

$$
A_r \quad = \quad \biggl( {{Z_{II} - Z_I}\over{Z_I + Z_{II}} } \biggr) 
                  \thinspace  A_i
$$

$$
A_t \quad = \quad \biggl( {{2 \thinspace Z_I}\over{Z_I + Z_{II}} } \biggr)
                  \thinspace  A_i
$$

$$
Z_{II} \quad = \quad Z_I
$$


$$
Z_I \quad = \quad \sqrt{ T_I \mu_I} \quad = \quad 
                  \sqrt{ T_{II} \mu_{II}} \quad = \quad Z_{II}
$$

$$
F \quad = \quad -b v \qquad \qquad \Rightarrow \quad \quad b \quad = \quad 
          {{F}\over{v}}
$$

$$
b \quad = \quad Z_I
$$

$$
Z \quad = \quad {{V}\over{I}}
$$

                  


\bye
