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<math>~A(y_1)</math>
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<math>~A(n\Delta y_1)</math>
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Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio.  As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored.  Via a related simplification, we will find that the various natural lengths of this problem are can be related via the expression,
+
Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio.  As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored.  Via a related simplification, we also will find that the various natural lengths of this problem are can be related via the expression,
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<math>~y_1\biggr|_{1^\mathrm{st} \mathrm{fringe}}</math>
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<math>~y_1\biggr|_{1^\mathrm{st} \mathrm{fringe}} = \frac{L \Delta y_1}{c}</math>
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<math>~\frac{\lambda Z}{w} \, .</math>
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<math>~\frac{\lambda Z}{2c} \, .</math>
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CAUTION:  Be sure that the "L" in these various expressions is, indeed, always the same quantity.
=See Also=
=See Also=

Revision as of 21:55, 27 December 2017

Contents

CGH: Philosophical Overview

Whitworth's (1981) Isothermal Free-Energy Surface
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Slit Diffraction

Single Aperture

Figure 1
Chapter1Fig1

As has been detailed in an accompanying discussion, we consider, first, the amplitude (and phase) of light that is incident at a location ~y_1 on an image screen that is located a distance ~Z from a slit of width ~w = (Y_1 - Y_2) = 2c. The amplitude is given by the expression,

~A(n\Delta y_1)

~=

~\sum_j
a_j \biggl[ \cos\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) + i   \sin\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) \biggr]
\, ,

where,

~D_j

~=

~
L \biggl[1 - \frac{2y_1 Y_j}{L^2} + \frac{Y_j^2}{L^2} \biggr]^{1 / 2} \, ,

and,

~L

~\equiv

~
[Z^2 + y_1^2  ]^{1 / 2} \, .

Notice that, in the definition of ~D_j, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, ~Y_j/L, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we also will find that the various natural lengths of this problem are can be related via the expression,

~y_1\biggr|_{1^\mathrm{st} \mathrm{fringe}} = \frac{L \Delta y_1}{c}

~\approx

~\frac{\lambda Z}{2c} \, .

CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.

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

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