Difference between revisions of "User:Tohline/Appendix/CGH/ParallelAperturesConsolidate"

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and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,
and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,
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where,  
where,  
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Note that <math>~L</math> is formally a function of <math>~y_1</math>, but in most of what follows it will be reasonable to assume, <math>~L \approx Z</math>.
Note that <math>~L</math> is formally a function of <math>~y_1</math>, but in most of what follows it will be reasonable to assume, <math>~L \approx Z</math>. Notice, as well, that this last approximate expression for the (complex) amplitude at the image screen may be rewritten in the form,
 
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~A(y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ e^{i 2\pi L/\lambda }  \sum_j  a_j e^{i \phi_j}  \cdot e^{-i \Theta_j }
\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\Theta_j</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \, .</math>
  </td>
</tr>
</table>
 
In a related accompanying derivation titled, [[User:Tohline/Appendix/CGH/ParallelApertures#Analytic_Result|''Analytic Result'']], we made the substitution,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~a_j e^{i \phi_j} </math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~a_0(Y) dY \, ,</math>
  </td>
</tr>
</table>
and changed the summation to an integration, obtaining,
 
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~A(y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ e^{i 2\pi L/\lambda } \int a_0(Y) e^{-i \Theta } dY
\, ,
</math>
  </td>
</tr>
</table>


=See Also=
=See Also=

Revision as of 17:14, 17 March 2020

CGH: Consolidate Expressions Regarding Parallel Apertures

One-dimensional Apertures

From our accompanying discussion of the Utility of FFT Techniques, we start with the most general expression for the amplitude at one point on an image screen, namely,

<math>~A(y_1)</math>

<math>~=</math>

<math>~\sum_j a_j e^{i(2\pi D_j/\lambda + \phi_j)} \, , </math>

and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~\sum_j a_j e^{i[ 2\pi L/\lambda + \phi_j]}\biggl[ \cos\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) - i \sin\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \biggr] \, , </math>

where,

<math>~L</math>

<math>~\equiv</math>

<math>~ Z \biggl[1 + \frac{y_1^2}{Z^2} \biggr]^{1 / 2} \, . </math>

Note that <math>~L</math> is formally a function of <math>~y_1</math>, but in most of what follows it will be reasonable to assume, <math>~L \approx Z</math>. Notice, as well, that this last approximate expression for the (complex) amplitude at the image screen may be rewritten in the form,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~ e^{i 2\pi L/\lambda } \sum_j a_j e^{i \phi_j} \cdot e^{-i \Theta_j } \, , </math>

where,

<math>~\Theta_j</math>

<math>~\equiv</math>

<math>~\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \, .</math>

In a related accompanying derivation titled, Analytic Result, we made the substitution,

<math>~a_j e^{i \phi_j} </math>

<math>~\rightarrow</math>

<math>~a_0(Y) dY \, ,</math>

and changed the summation to an integration, obtaining,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~ e^{i 2\pi L/\lambda } \int a_0(Y) e^{-i \Theta } dY \, , </math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation