About the evident description of distribution of beams and "wavy geometrical trajectories" in long thin pipes
firstname.lastname@example.org; email@example.com pp. (accepted)
In work  it was shown that process of distribution of light in lasers can be described in the form of the branching system of beams and sequences like Fibonacci's ranks. Also in work  was possibility of the description of such system of beams by means of wavy trajectories ("waves") where "waves" consist of a set of direct pieces (links) is noted.
In works [2, 3] distribution of the branching system of paraxial (Gaussian) bunches on the basis of consideration of binomial distribution of the 2nd type  was visually shown. In works [2, 3] the assumption was accepted, process of branching of beams happens in open space without obstacles or walls.
In the presented work similar process of branching of beams in long and thin pipes with walls is considered. Geometrically (it is approximate in a form) it is possible to carry a deep one-dimensional potential hole, the laser resonator in which light passes between two mirrors infinite number of times, etc. to such tubes. Usually, for the description of such models probabilistic approach is used.
In the presented work geometrical properties of wavy trajectories in thin pipes are considered. The new description of trajectories of beams by means of "a package of wavy trajectories" ("a wave package") is offered. It is in number shown that in ours "a wave package", to increase in number of passes of rays of light between resonator mirrors (that is equivalent to increase in length of a thin pipe) there is a redistribution of energy from long "waves" - to shorter up to establishment of stationary distribution. Distinction of forms of the beams which are bending around angular distribution and distribution of beams on the section of pipes is shown. The parallel between geometrical and probabilistic approaches for our model is drawn.
1. A. V. Yurkin. Quasi-resonator a new interpretation of scattering in lasers // Quantum Electron., v. 24, p. 359, 1994.
2. A. V. Yurkin. System of rays in lasers and a new feasibility of light coherence control // Optics Communications, v.114, p.393, 1995.
3. A. V. Yurkin. New binomial and new view on light theory. (Lambert Academic Publishing, 2013). ISBN 978-3-659-38404-2.
4. N. J. A. Sloane, S Plouffe. The Encyclopedia of Integer Sequences. New York: Academic Press, 1995. http://oeis.org/Seis.html. Sequences A053632.