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

From VistrailsWiki
Jump to navigation Jump to search
Line 61: Line 61:
</tr>
</tr>
</table>
</table>
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>.


=See Also=
=See Also=

Revision as of 16:45, 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>.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
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