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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- J. F. Carlson and A. E. Heins, The reflection of an electromagnetic plane wave by an infinite set of plates. I, Quart. Appl. Math. 4 (1947), 313–329. MR 19523, DOI https://doi.org/10.1090/S0033-569X-1947-19523-6
- Albert E. Heins and J. F. Carlson, The reflection of an electromagnetic plane wave by an infinite set of plates. II, Quart. Appl. Math. 5 (1947), 82–88. MR 20929, DOI https://doi.org/10.1090/S0033-569X-1947-20929-7
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.
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 \[ 0 < \frac {1}{{1 + \sin \theta }} < \frac {{dk}}{{2\pi }} < \left \{ {\frac {1}{2}or\frac {2}{{1 + \sin \theta }}or\frac {1}{{1 - \sin \theta }}} \right \}\]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.
See CH I, p. 321 for some remarks regarding the present use of the term regular.
See CH I, sec. 4.
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.
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.
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.
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 \[ 0 < \frac {1}{{1 + \sin \theta }} < \frac {{dk}}{{2\pi }} < \left \{ {\frac {1}{2}or\frac {2}{{1 + \sin \theta }}or\frac {1}{{1 - \sin \theta }}} \right \}\]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.
See CH I, p. 321 for some remarks regarding the present use of the term regular.
See CH I, sec. 4.
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© Copyright 1950
American Mathematical Society
