\magnification=\magstep1


$$
x \quad = \quad {{a}\over{2}} \sin(\theta)
$$

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

$$
\sin(\theta) \quad = \quad {{\lambda}\over{a}}
$$

$$
\sin(\theta) \quad \simeq \quad {{y}\over{L}} \qquad \quad \Rightarrow 
             \quad \quad y \quad = \quad 1 \ {{L \lambda}\over{a}}
$$


$$
x \quad = \quad {{a}\over{4}} \sin(\theta)
$$


$$
\sin(\theta) \quad = \quad {{2 \lambda}\over{a}}
$$

$$
\sin(\theta) \quad \simeq \quad {{y}\over{L}} \qquad \quad \Rightarrow 
             \quad \quad y \quad = \quad 2 \ {{L \lambda }\over{a}}
$$

$$
y \quad = \quad M \ {{L \lambda}\over{a}} \quad, \qquad \quad M = 1, 2, 3, ...
$$


$$
x \quad = \quad dz \sin(\theta)
$$

$$
{{x}\over{\lambda}} \quad = \quad {{dz}\over{\lambda}} \thinspace \sin(\theta)
$$

$$
{{2 \pi x}\over{\lambda}} \quad = \quad {{2 \pi dz}\over{\lambda}} 
                                       \thinspace \sin(\theta)
$$

$$
A_m \quad = \quad {{A}\over{{a/dz}}} \quad = \quad A \left( {{dz}\over{a}} \right)
$$

$$
y_1 \quad = \quad A_m \ {{1}\over{r_1}} \sin(k r_1 - \omega t)
$$

$$
y_2 \quad = \quad A_m \ {{1}\over{r_2}} \sin(k r_2 - \omega t)
$$

$$
y_1 \quad = \quad {{A_m}\over{r_{\rm avg}}} \sin(k r_1 - \omega t)
$$

$$
y_2 \quad = \quad {{A_m}\over{r_{\rm avg}}} \sin(k r_2 - \omega t)
$$

$$
{\rm phase\ offset\ } \quad = \quad {{2 \pi \sin(\theta)}\over{\lambda}} \thinspace dz
$$

$$
y_1 \quad = \quad {{A_m}\over{r_{\rm avg}}} \thinspace
       \sin\left( 0 - \ \omega t \right)
$$


$$
y_2 \quad = \quad {{A_m}\over{r_{\rm avg}}} \thinspace
       \sin\left( {{2 \pi \sin(\theta)}\over{\lambda}} z \ - \ \omega t \right)
$$

$$
{\rm sum\ of\ all\ waves\ } \quad = \quad 
      \int_{z=0}^{z=a} \thinspace
         {{A_m}\over{r_{\rm avg}}} \thinspace
         \sin\left( {{2 \pi \sin(\theta)}\over{\lambda}} z \ - \ \omega t \right)
$$

$$
{\rm sum\ of\ all\ waves\ } \quad = \quad 
      \int_{z=0}^{z=a} \thinspace
         \left( {{A \thinspace dz}\over{a }} \right) \  {{1}\over{r_{\rm avg}}} \thinspace
         \sin\left( {{2 \pi \sin(\theta)}\over{\lambda}} z \ - \ \omega t \right) 
$$

$$
{\rm sum\ of\ all\ waves\ } \quad = \quad 
       {{A}\over{r_{\rm avg}}} \thinspace
      \int_{z=0}^{z=a} \thinspace
         {{1}\over{a}}  \thinspace
         \sin\left( {{2 \pi \sin(\theta)}\over{\lambda}} z \ - \ \omega t \right) dz
$$

$$
{\rm sum\ } \quad = \quad 
       {{A}\over{r_{\rm avg}}} \thinspace
      \biggl\{
         - {{1}\over{a}}  {{\lambda}\over{2 \pi \sin(\theta)}}  \thinspace
         \cos\left( {{2 \pi \sin(\theta)}\over{\lambda}} a \ - \ \omega t \right) 
   \ + \   
         {{1}\over{a}}  {{\lambda}\over{2 \pi \sin(\theta)}}  \thinspace
         \cos\left( 0 \ - \ \omega t \right) 
      \biggr\} 
$$

$$
\beta \quad = \quad {{2 \pi \sin(\theta) a}\over{\lambda}}
$$


$$
{\rm sum\ } \quad = \quad 
       {{A}\over{r_{\rm avg}}} \thinspace
      \biggl\{
         - {{1}\over{\beta}}   \thinspace
         \cos\left( \beta \ - \ \omega t \right) 
   \ + \   
         {{1}\over{\beta}}  \thinspace
         \cos\left( 0 \ - \ \omega t \right) 
      \biggr\} 
$$

$$
\cos(P) \ - \ \cos(Q) \quad = \quad - 2 \sin\left( {{P + Q}\over{2}} \right)
                           \sin \left( {{P - Q}\over{2}} \right)
$$

$$
{\rm sum\ } \quad = \quad 
       {{A}\over{r_{\rm avg}}} \thinspace
      \biggl\{
         {{2}\over{\beta}}   \thinspace
         \sin\left( {{\beta \ - \ 2 \omega t}\over{2}} \right) 
         \thinspace
         \sin\left( {{\beta}\over{2}} \right) 
      \biggr\} 
$$

$$
{\rm intensity\ } \quad = \quad {\rm amplitude}^2 \quad = \quad 
       {{A^2}\over{r^2_{\rm avg}}} \thinspace
      \biggl\{
         {\left[ \sin\left( {{\beta \ - \ 2 \omega t}\over{2}} \right) \right]}^2
         \thinspace
        {\left( { \sin( {{\beta}\over{2}} )}\over{ {{\beta}\over{2}} } \right)}^2
      \biggr\} 
$$

$$
{\rm time-averaged\ intensity\ } \quad = \quad 
       {{A^2}\over{r^2_{\rm avg}}} \thinspace
        {\left( { \sin( {{\beta}\over{2}} )}\over{ {{\beta}\over{2}} } \right)}^2
$$



$$
\sin(\theta) \quad \simeq \quad {{y}\over{L}}
$$

$$
\beta \quad = \quad {{2 \pi a}\over{\lambda}} \thinspace {{y}\over{L}}
$$




$$
y_{\rm sum} \quad = \quad 
    A \ {{1}\over{r_1}} \sin(k r_1 - \omega t)
  \ + \  
    A \ {{1}\over{r_2}} \sin(k r_2 - \omega t)
  \ + \  
    A \ {{1}\over{r_3}} \sin(k r_3 - \omega t)
$$

$$
{\rm neighbor\ phase\ offset\ } \phi \quad = \quad
          {{2 \pi d \sin(\theta)}\over{\lambda}}
   \quad \quad \quad ({\rm radians})
$$

$$
y_{\rm sum} \quad = \quad \left( {{A}\over{r_{\rm avg}}} \right) \ 
                \biggl[
                        \sin(0 - \omega t)
  \ + \  
                        \sin(\phi - \omega t)
  \ + \  
                        \sin(2 \phi - \omega t)
                \biggr]
$$


$$
y_{\rm sum} \quad = \quad \left( {{A}\over{r_{\rm avg}}} \right) \ 
                        2 \sin\left( {{2 \phi - 2 \omega t}\over{2}} \right)
                        \cos\left( {{2 \phi}\over{2}} \right)
  \ + \ 
              \left( {{A}\over{r_{\rm avg}}} \right) \thinspace 
                        \sin(\phi - \omega t)
$$


$$
y_{\rm sum} \quad = \quad \left( {{A}\over{r_{\rm avg}}} \right) \ 
                        \biggl[ 2 \cos(\phi) + 1 \biggr]
     \ 
                        \sin(\phi - \omega t)
$$

$$
{\rm intensity \ = \ (amplitude)^2} \quad = \quad 
                          {\left( {{A}\over{r_{\rm avg}}} \right)}^2 \ 
                        {\biggl[ 2 \cos(\phi) + 1 \biggr]}^2
                        {\sin(\phi - \omega t)}^2
$$

$$
{\rm time-averaged\ intensity} \quad = \quad 
                          {{1}\over{2}} \thinspace {\left( {{A}\over{r_{\rm avg}}} \right)}^2 \ 
                        {\biggl[ 2 \cos(\phi) + 1 \biggr]}^2
$$


$$
{\rm sum\ of\ N\ slits\ } \quad \propto \quad A \thinspace
               \left[ {{\sin({N \phi/2)}}\over{\sin(\phi/2)}} \right]
$$

$$
{\rm intensity\ of\ N\ slits} \quad \propto \quad A^2 \thinspace
               {\left[ {{\sin({N \phi/2)}}\over{\sin(\phi/2)}} \right]}^2
$$

$$
d \sin(\theta) \quad = \quad \lambda
$$

$$
d \sin(\theta) \quad = \quad \lambda \ + \ \epsilon
$$

$$
10 \thinspace \biggl[ d \sin(\theta) \biggr] \quad = \quad 
              10 \lambda  \ + \ 10 \epsilon
$$

$$
y_{\rm sum} \quad = \quad \left( {{A}\over{r_{\rm avg}}} \right) \ 
                \biggl[
                        \sin(0 - \omega t)
  \ + \  
                        \sin(\phi - \omega t)
                \biggr]
$$

$$
y_{\rm sum} \quad = \quad \left( {{A}\over{r_{\rm avg}}} \right) \ 
                \biggl[
                        \sin(0 - \omega t)
  \ + \  
                        \sin(\phi - \omega t)
  \ + \  
                        \sin(2 \phi - \omega t)
  \ + \  
                        \sin(3 \phi - \omega t)
                \biggr]
$$

$$
y_{\rm sum} \quad = \quad A \thinspace 
           \biggl[ {{\sin(N \phi/2)}\over{\sin(\phi/2)}} \biggr]
$$



\bye
