Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

The reflection of an electromagnetic plane wave by an infinite set of plates. III


Author: Albert E. Heins
Journal: Quart. Appl. Math. 8 (1950), 281-291
MSC: Primary 78.0X
DOI: https://doi.org/10.1090/qam/38239
MathSciNet review: 38239
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  • [1] J. F. Carlson and A. E. Heins The reflection of an electromagnetic plane wave by an infinite set of plates, I, this Quarterly, 4, 313-329 (1947). Hereafter we shall refer to this paper as CH I. MR 0019523
  • [2] A. E. Heins and J. F. Carlson The reflection of an electromagnetic plane wave by an infinite set of plates, II, this Quarterly, 5, 82-88 (1947). Hereafter we shall refer to this paper as CH II. MR 0020929
  • [3] If the propagation normal of the incident wave falls to the left of ON, angle $ \beta $ is a positive acute angle, while if it falls to the right of ON, the angle $ \beta $ is a negative acute angle. The case $ \beta = 0$ requires separate treatment.
  • [4] At this point we realize that to get more than two reflected waves, the inequalities in sec. 2 have to be modified. As we carry on this modification we find that an indefinite number of reflected waves cannot enter. For example, if $ \alpha = \pi /2$, then the inequality for two reflected waves reads
    $\displaystyle 0 < \frac{1}{{1 + \sin \theta }} < \frac{{dk}}{{2\pi }} < \left\{... ...rac{1}{2}or\frac{2}{{1 + \sin \theta }}or\frac{1}{{1 - \sin \theta }}} \right\}$
    for $ 0 < \theta < \pi /2$. This is impossible since $ sin\theta < 1$. Hence there is only one reflected wave in this case. The formulation we gave in CH I makes no assumptions as to the form of $ \psi \left( {y,z} \right)$ to the left of the parallel plates. The convergence study in sec 2, of the present paper gives us conditions for one, two, etc., reflected waves.
  • [5] See CH I, p. 321 for some remarks regarding the present use of the term regular.
  • [6] See CH I, sec. 4.

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DOI: https://doi.org/10.1090/qam/38239
Article copyright: © Copyright 1950 American Mathematical Society

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