New PDF release: Nondiscrete Induction and Iterative Processes

By Potra F.A., Ptak V.

ISBN-10: 0273086278

ISBN-13: 9780273086277

ISBN-10: 1919671331

ISBN-13: 9781919671338

ISBN-10: 2219779599

ISBN-13: 9782219779595

ISBN-10: 5219721011

ISBN-13: 9785219721014

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Sample text

Keller and Robert M. 176) to give z1, and zf on the interface. 174) can be used to calculate them off the interface along the reflected and transmitted rays. We shall not carry out this calculation here. The function z 1 has been calculated in [ 18] in this way. We shall calculate [Jz~! ay at y = 0 because this quantity will be needed shortly for another purpose. -(x- htana) = - . - - - - . l- 2) tan 3 a J . 184) sin {3, Differentiation of the first equation with respect to y yields 1 Hence, when y (3 = 0, = 7f /2, = oa ay a cos (3 --cos(3- a - - - .

184) sin {3, Differentiation of the first equation with respect to y yields 1 Hence, when y (3 = 0, = 7f /2, = oa ay a cos (3 --cos(3- a - - - . 185) imply the following relations: . l, 01 . 186) ' y = 0. The computation is greatly = 0 when y = 0. l)1/2 (x- htanao)-3/2. 1 0. If 11 < 1 total reflection occurs for rays incident at angles a greater than the critical angle a 0 = sin - I 11· The transmitted rays, corresponding to angles of incidence a in the interval 0 ::; a ::; a 0 , cover the entire lower half-space, and the critically transmitted ray, for which (3 = rr /2, lies in the interface y = 0.

But u e is not required for the calculation of z ~, so z ~ can be found as before. 179) in the region of total reflection. Form > 0, ue is needed to determine z::-n, but we will not determine ue here. We only remark that it decays exponentially with distance from the interface. The wave ud is completely determined by ut and the boundary conditions. To calculate u d we write rv e iksd ""' L_. m=O 2: z;,(ik)-m. 190) Joseph B. Keller and Robert M. 191) . (x, 0) = az;,(x, 0), x > h tana 0 ; asd d - z +l ay m az!

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Nondiscrete Induction and Iterative Processes by Potra F.A., Ptak V.


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