\magnification=\magstep1

$$
y_1(x, t) \quad = \quad A \sin(kx - \omega t)
$$

$$
y_2(x, t) \quad = \quad A \sin(kx + \omega t)
$$

$$
y_1 + y_2 \quad = \quad A \biggl[ \sin(kx - \omega t) \ +  \ 
                                 \sin(kx + \omega t) \biggr]
$$

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

$$
y_1 + y_2 \quad = \quad
     2 A \thinspace \biggl[ \sin(kx) \thinspace \cos(\omega t) \biggr]
$$


$$
y_1(x, t) \quad = \quad A \sin(k_1 x \ -\  \omega_1 t)
$$

$$
y_2(x, t) \quad = \quad A \sin(k_2 x \ -\  \omega_2 t)
$$

$$
k_1 \quad \simeq \quad k_2 \qquad \quad \Rightarrow 
                  \quad k_2 \quad = \quad k_1 \ + \ \Delta k
$$

$$
\omega_1 \quad \simeq \quad \omega_2 \qquad \quad \Rightarrow 
                  \quad \omega_2 \quad = \quad \omega_1 \ + \ \Delta \omega
$$

$$
v_1 \quad = \quad {{\omega_1}\over{k_1}} \quad \quad = \quad \quad 
                 v_2 \quad = \quad {{\omega_2}\over{k_2}}
$$


$$
y_1 + y_2 \quad = \quad A \sin(k_1 x \ -\  \omega_1 t) \ + \ 
                            A \sin(k_2 x \ -\  \omega_2 t)
$$

$$
\sin(\alpha) \ + \ \sin(\beta) \quad = \quad
  2 \thinspace \sin \biggl( 
            {{\alpha + \beta}\over{2}} \biggr)
                \cos \biggl(
            {{\alpha - \beta}\over{2}} \biggr)
$$

$$
\alpha \quad = \quad k_1 x - \omega_1 t  \quad, \quad \quad 
   \beta \quad = \quad k_2 x - \omega_2 t  
$$

$$
\eqalign{
y_1 + y_2 \quad &= \quad 2 A \thinspace
    \sin{\biggl( {{k_1 x - \omega_1 t + k_2 x - \omega_2 t}\over{2}} \biggr)}
       \thinspace
    \cos{\biggl( {{k_1 x - \omega_1 t - k_2 x + \omega_2 t}\over{2}} \biggr)} \cr
  \phantom{=} \cr
          \quad &= \quad 2 A \thinspace
    \sin{\biggl( ({{k_1 + k_2}\over{2}})x - ({{\omega_1 + \omega_2 }\over{2}}) t  \biggr)}
       \thinspace
    \cos{\biggl( ({{k_1 - k_2}\over{2}}) x - ({{\omega_1 - \omega_2}\over{2}}) t \biggr)} \cr
  \phantom{=} \cr
          \quad &= \quad 2 A \thinspace
    \sin{\biggl( k_{\rm avg}\thinspace x \ -\  \omega_{\rm avg}\thinspace  t  \biggr)}
       \thinspace
    \cos{\biggl( {{\Delta k}\over{2}} \thinspace x \ - \ {{\Delta \omega}\over{2}}  \thinspace t \biggr)} \cr
}
$$

$$
y_1 + y_2    \quad = \quad 2 A \thinspace
    \sin{\biggl( k_{\rm avg}\thinspace x \ -\  \omega_{\rm avg}\thinspace  t  \biggr)}
       \thinspace
    \cos{\biggl( {{\Delta k}\over{2}} \thinspace x \ - \ {{\Delta \omega}\over{2}}  \thinspace t \biggr)} 
$$

$$
v_p \quad = \quad {{\omega_{\rm avg}}\over{k_{\rm avg}}}
$$

$$
v_g \quad = \quad  { {{\Delta \omega / 2}\over{\Delta k / 2}} } \quad = \quad
                   { {\Delta \omega}\over{\Delta k} }  
$$

$$
v_g \quad = \quad  { {d \omega}\over{d k} }  
$$

$$
\omega_1 \quad = \quad \omega \qquad \quad {\rm (no\ change) }
$$

$$
\lambda_1 \quad = \quad {{\lambda}\over{n}} \qquad \qquad \qquad \qquad
$$

$$
\qquad k_1 \quad = \quad {{2 \pi}\over{\lambda_1}} \quad = \quad {{2 \pi n}\over{\lambda}} \quad = \quad n k
$$

$$
v_p \quad = \quad {{\omega_1}\over{k_1}} \quad = \quad {{\omega}\over{n k}} 
                \quad = \quad {{1}\over{n}} \thinspace {{\omega}\over{k}}
$$

$$
v_p \quad = \quad {{c}\over{n}} 
$$

$$
v_p \quad = \quad {{\omega}\over{k}} \quad \quad \Rightarrow 
          \quad \quad \omega \quad = \quad k \thinspace v_p
$$

$$
\omega \quad = \quad {{k c}\over{n}}
$$

$$
{{d \omega}\over{dk}} \quad = \quad {{d}\over{dk}} \left( {{k c}\over{n}} \right) 
$$

$$
v_g \quad = \quad {{d \omega}\over{dk}} \quad = \quad {{d}\over{dk}} \left( {{k c}\over{n}} \right) 
$$


$$
{{d \omega}\over{dk}} \quad = \quad {{c}\over{n}} \ + \ 
                   k c \thinspace {{d}\over{dk}} \left( {{1}\over{n}} \right)
      \quad = \quad 
                                    {{c}\over{n}} \ + \ 
                   k c \thinspace \left( -{{1}\over{n^2}} \thinspace {{dn}\over{dk}} \right)
$$

$$
{{d \omega}\over{dk}} \quad = \quad {{c}\over{n}} \ - \ 
                     {{k c}\over{n^2}} \thinspace {{dn}\over{dk}}
$$

$$
v_g \quad = \quad {{d \omega}\over{dk}} \quad = \quad
                    {{c}\over{n}} \ - \ 
                    {{c}\over{n}} \thinspace \left( {{k}\over{n}} 
                             \thinspace {{dn}\over{dk}} \right) 
   \quad = \quad 
                    {{c}\over{n}} \left( 1 \ - \ {{k}\over{n}} \thinspace
                                          {{dn}\over{dk}} \right)     
$$

$$
v_g \quad = \quad v_p \ \left( 1 \ - \ {{k}\over{n}} {{dn}\over{dk}} \right) 
$$

$$
{{dn}\over{d\lambda}} \quad > \quad 0   \qquad {\rm or\ } \qquad
         {{dn}\over{dk}} \quad < \quad 0
$$

\bye
