\magnification=\magstep1


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

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

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

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


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


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

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

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


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

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

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

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


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

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

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

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

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

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

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


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

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

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

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

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

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


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

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

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

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

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



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

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

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



$$
{\rm bright\ spots\ ?? \ }\qquad x \quad = \quad 1 \thinspace \lambda
    \quad = \quad {{a}\over{2}} \sin(\theta)
    \qquad \qquad {\rm warning!\ not\  correct!}
$$


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

$$
{\rm for\ } x << 1, \quad \quad \sin(x) \quad \simeq \quad
        x \quad - \quad {{x^3}\over{3!}} \quad + \quad {{x^5}\over{5!}} \quad
        - \quad ...
$$

$$
{\rm if\ } (\beta/2) \rightarrow 0, \quad \quad {{\sin(\beta/2)}\over{\beta/2}} 
        \quad \rightarrow \quad 
         {{(\beta/2) - {{{(\beta/2)}^3}\over{3!}} + {{{(\beta/2)}^5}\over{5!}} - ...}\over{\beta/2}}
        \quad \rightarrow \quad 
         {{\beta/2}\over{\beta/2}} \quad = \quad 1
$$

$$
{\rm time-averaged\ intensity\ } \quad = \quad 
       {{1}\over{2}} \thinspace {{A^2}\over{r^2_{\rm avg}}} 
$$

$$
\sin(\beta/2) \quad = \quad 0
$$

$$
{{\beta}\over{2}} \quad = \quad 0, 
        \thinspace  \pi, \thinspace 2 \pi, \thinspace 3 \pi, 
        \thinspace ..., \thinspace N \pi
   \qquad \Rightarrow \qquad 
   \beta \quad = \quad 2 \pi,  \thinspace 4 \pi, \thinspace 6 \pi, \thinspace ...
$$

$$
{{2 \pi \thinspace \sin(\theta) \thinspace a}\over{\lambda}} 
   \quad = \quad 2 \pi, \thinspace 4 \pi, \thinspace 6 \pi, \thinspace ...
  \qquad \Rightarrow \qquad 
   \sin(\theta) \quad = \quad {{\lambda}\over{a}}, \thinspace 
                              {{2 \lambda}\over{a}}, \thinspace 
                              {{3 \lambda}\over{a}}, \thinspace ...
$$

$$
{\rm dark\ spots:\ } \qquad \sin(\theta) \quad = \quad
                              {{\lambda}\over{a}}, \ 
                              2 \thinspace {{\lambda}\over{a}}, \ 
                              3 \thinspace {{\lambda}\over{a}}, \ 
                              ...
$$

$$
{\rm bright\ spots:\ } \qquad \sin(\theta) \quad {\buildrel ?\over{=}} \quad
                              1.5 \thinspace {{\lambda}\over{a}}, \ 
                              2.5 \thinspace {{\lambda}\over{a}}, \ 
                              3.5 \thinspace {{\lambda}\over{a}}, \ 
                              ...
  \qquad \qquad {\rm warning!\ not\ correct!}
$$


\bye
