D O C U M E N T 4 3 L I G H T I N D I S P E R S I V E M E D I A 6 5 is then likewise a solution of the equations. A, , , and V are regarded as slowly variable on the curve such that their changes by moving along the curve by are infinitesimally small. The wave length is very small against the length of the curve and this latter, again, is small against the starting point’s distances r to the points on the curve. Calculation of the integral (2) provides a theory for the propagation of light including the Fraunhofer and Fresnel effects in the cylindrical case under consideration here, if is set constant. In the case where depends on s, one obtains nonstationary solutions, i.e., ones whose ray paths depend on time. We are not interested here in the deflection problem but in the optical problem, neglecting deflection. We ask: Which points are illuminated at time t and which are not, specifically neglecting the deflection effects? This question is easily answered from solutions in the form of (2). H depends on the choice of the starting point and the point on the curve and generally varies rapidly as the curve point wanders along the curve then is a rapidly alternating function. That is why only curve loca- tions for which vanishes can contribute substantially to the integral. If such locations exist for the starting point and the point in time under consideration, then the point is “illuminated,” otherwise it is “dark.” We now choose as a curve the part of the x-axis between and and regard the solution only for starting points with a positive y. If we are just inter- ested in the axis of the beam, regarding it as infinitely thin, then it evidently suffices to set the illumination condition for the midpoint of the distance. Thus, for the ray’s path we obtain the condition . (3) Under the geometrical conditions taken into consideration, the wavelength normal evidently has the orientation of the radius vector drawn from the origin of the coor- dinates to the starting point. The case of interest to us is a beam in a dispersive medium that changes its radi- ative direction at a constant angular velocity. We approach this case stepwise by regarding simpler cases. I. Train of waves[5] of constant direction. We specialize (2) according to the con- ditions . Furthermore, we set here, as in the following, to sufficient accuracy[6] ω α λ ω ω ejH ∂H ∂s ------- ξ b –= ξ +b = [p. 20] ξ 0= ∂H ∂ξ ------- ξ 0= 0= ∂ω ∂ξ ------ - 0 = ∂α ∂ξ ------ - 0 =
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