\magnification=\magstep1

$$
y_1(r_1, t) \quad = \quad A \left( {{1}\over{r_1}} \right) 
                      \sin(k r_1 - \omega t)
$$

$$
y_2(r_2, t) \quad = \quad A \left( {{1}\over{r_2}} \right) 
                      \sin(k r_2 - \omega t)
$$

$$
y_{\rm sum} \quad = \quad A \thinspace
                      \biggl\{ {{1}\over{r_1}} \sin(k r_1 - \omega t) \ + \ 
                               {{1}\over{r_2}} \sin(k r_2 - \omega t) \biggr\}
$$


$$
y_{\rm sum} \quad \simeq \quad {{A}\over{r_{\rm avg}}}  \thinspace
                      \biggl\{ \sin(k r_1 - \omega t) \ + \ 
                               \sin(k r_2 - \omega t) \biggr\}
$$

$$
\sin(\alpha) \ + \ \sin(\beta) \quad = \quad 
         2 \thinspace \sin\left( {{\alpha + \beta}\over{2}} \right)
                \thinspace
                      \cos\left( {{\alpha - \beta}\over{2}} \right)
$$
 
$$
y_{\rm sum} \quad \simeq \quad {{2 A}\over{r_{\rm avg}}}  \thinspace
                      \biggl\{ \sin\left( {{k r_1 - \omega t + k r_2 - \omega t}\over{2}} \right)  
               \thinspace
                               \cos\left( {{k r_1 - \omega t - k r_2 + \omega t}\over{2}} \right) \biggr\}
$$


 
$$
y_{\rm sum} \quad \simeq \quad {{2 A}\over{r_{\rm avg}}}  \thinspace
                      \biggl\{ \sin\left( {{k [r_1 + r_2]}\over{2}} - \omega t \right)  
               \thinspace
                               \cos\left( {{k [r_1 - r_2]}\over{2}} \right) \biggr\}
$$


$$
   \cos\left( {{k [r_1 - r_2]}\over{2}} \right)  \quad = \quad 0
$$

$$
    {{k [r_1 - r_2]}\over{2}}  \quad = \quad  (2 N - 1) \thinspace {{\pi}\over{2}} \quad , \quad \quad N = 1, 2, 3, ...
$$

$$
k \quad = \quad {{2 \pi}\over{\lambda}}
$$

$$
{\rm sum\ =\ zero\ when} \qquad \quad |r_1 \ - \ r_2| \quad = \quad {{(2 N - 1)}\over{2}} \thinspace \lambda
$$


$$
   \cos\left( {{k [r_1 - r_2]}\over{2}} \right)  \quad = \quad 1
$$

$$
{\rm sum\ =\ max\ when} \qquad \quad |r_1 \ - \ r_2| \quad = \quad  N \thinspace \lambda
$$


$$
x \quad = \quad d \thinspace \sin(\theta)
$$

$$
{\rm zeroes\ } \qquad \quad 
      x \quad = \quad {{2 N - 1}\over{2}} \thinspace \lambda 
                \quad = \quad 
                d \thinspace \sin(\theta)
$$

$$
{\rm if\ \ \ } d  <<  L \ , \qquad \sin(\theta) \quad = \quad 
               {{y}\over{r_{\rm avg}}} \quad \simeq \quad {{y}\over{L}}
$$

$$
{\rm zeroes\ } \qquad \quad {{ y \thinspace d }\over{L}} \quad = \quad {{2N - 1}\over{2}} \thinspace \lambda
$$


\bye
