Whispering Gallery Orbits in the Bunimovich Stadium C. A. Kruelle 3 , A. Kittel b , J. P e i n k e b , and R. R i c h t e t a b c
Max-Planck-Institut für Metallforschung, Heisenbergstraße 1, D-70569 Stuttgart Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth Universität Magdeburg, Institut für Experimentelle Physik, Abteilung Nichtlineare Phänomene, D-39016 Magdeburg
Z. Naturforsch. 52 a, 581-584 (1997); received June 10, 1997 A new visualization method is found for investigating the complex ray dynamics of whispering gallery orbits inside the Bunimovich stadium. Regarding the whispering gallery as mirror cabinet, a fractal system of virtual images of the mirror walls is found which reflects the sensitive dependence of the dynamic behavior on the initial conditions. Keywords: Chaotic systems; fractals; optics; whispering gallery; Ray tracing.
1. I n t r o d u c t i o n In 1904 Lord Rayleigh noticed in the d o m e of St. P a u l ' s Cathedral in L o n d o n that sound waves are p r o p a g a t e d particularly easily along the inner surface of a c o n c a v e wall [1, 2]. If a person w h o is situated at position B in Fig. 1 whispers along the wall, ano t h e r p e r s o n at position F can hear his voice clearly.
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S i n c e the g u i d a n c e of w a v e s along the inner surf a c e of a circular b o u n d a r y w a s first discovered inside such a w h i s p e r i n g gallery, the c o r r e s p o n d i n g ray traj e c t o r i e s w e r e n a m e d " w h i s p e r i n g gallery orbits".
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Fig. 1. Lord Rayleigh's experimental setup for the investigation of the whispering gallery. B: bird-call, F: flickering flame, W: wooden bar, (from [2]). R a y l e i g h noticed that this is not due to a f o c u s i n g e f f e c t as in, for e x a m p l e , a parabolic mirror. Here, the s o u n d waves are " g u i d e d " by multiple reflections
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along the semicircle of the wall. T h e effect is enh a n c e d if the s p e a k e r w h i s p e r s , since the wavelength of a high-pitched voice is shorter and therefore the main b e a m of the radiation pattern "irradiated" f r o m the finite aperture of the m o u t h b e c o m e s narrower. Rayleigh investigated this p h e n o m e n o n in his laboratory, using a bird-call B (A = 2 c m ) as source and a flickering flame F as d e t e c t o r inside a twelve feet long strip of zinc. His interpretation was supported by the fact that a direct barrier b e t w e e n the bird-call and the flame did not destroy the flicker, while a small w o o d e n bar at position W did.
All trajectories inside the circle propagate, dep e n d i n g on the initial conditions, along periodic or q u a s i p e r i o d i c orbits, respectively. T h e initial angular m o m e n t u m of any ray is c o n s e r v e d . In contrast, for e x a m p l e , the B u n i m o v i c h s t a d i u m (two semicircles separated by a square) is a s y s t e m w h o s e trajectories can be classified into a discrete set of periodic orbits of m e a s u r e zero separated by an i n n u m e r a b l e n u m b e r of chaotic orbits [ 3 , 4 ] , Fig. 2a s h o w s a typical chaotic trajectory. H o w e v e r , in the s t a d i u m exists a system of whispering gallery orbits as well. T h e first three s p e c i m e n s of an infinite, discrete set of periodic whispering gallery orbits inside the s t a d i u m can be seen in Fig. 2b. T h e u n d e r l y i n g construction rule is easily f o u n d : T h e
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C. A. Kruelle et al. • Whispering Gallery Orbits in the Bunimovich Stadium
Fig. 2. (a) Chaotic trajectory in the Bunimovich stadium, (b) 3 periodic whispering gallery orbits, (c) nonperiodic whispering gallery orbit. regular 4 n - p o l y g o n s (square, o c t a g o n , and so on) are cut in half and connected by t w o straight lines parallel to the flanks of the stadium. In addition to these periodic orbits, an i n n u m e r a b l e n u m b e r of trajectories exists that are not periodic but guided along the curves of the s t a d i u m as well (see Figure 2c). For the whisperer r e m a i n s the p r o b l e m : H o w can he find the initial direction for s e n d i n g a message to the listener on the o t h e r side of the stadium? 2. M i r r o r C a b i n e t M a p To solve this p r o b l e m the recently i n t r o d u c e d "mirror cabinet m a p " [5, 6] can be applied. We c o n s i d e r a whisperer w h o is situated at an e m i s s i o n point ( x , 0) very close to the b o u n d a r y and sends out his m e s sage into a c h o s e n direction ^p. T h e stadium is seen as a mirror cabinet. A c c o r d i n g to reflection laws, the initial sound ray will be multiply reflected at the mirror cabinet walls. Using a ray tracing t e c h n i q u e w e determine the path lengths b e t w e e n t w o c o n s e c u t i v e
Fig. 3. Section of the system of virtual image walls as seen by an observer inside a mirror cabinet with a Bunimovich stadium shape who is situated at position (1.98,0). The pronounced channels correspond to the whispering gallery orbits inside the stadium. reflections. T h e obtained sequence of path lengths is plotted along a straight line departing f r o m the whisperer into the initial direction
axis, respectively. They correspond to trajectories inside the stadium that are injected into the left curve with a grazing angle and will slide in the following along the inner surface of the curve b e f o r e being reflected into the stadium again (see Figure 2c). A m e s s a g e sent by the whisperer into one of these 14 directions will reach the listener on the other end of the stadium via a whispering gallery orbit.
583 C. A. Kruelle et al. • Whispering Gallery Orbits in the Bunimovich Stadium f r o m the wall, it would be desirable to have a tool for m a p p i n g this d e p e n d e n c e of the trajectory d y n a m i c s on the starting position as well. d7 d6 d5
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d N ^ " ^ Fig. 4. Principle of the whispering gallery map, detailed description in the text.
T h e n u m b e r of such suitable directions d e p e n d s heavily on the distance of the whisperer to the wall. To b e m o r e specific w e have to e x a m i n e the series of periodic whispering gallery orbits (Fig. 2b) m o r e carefully. Since they are constructed by inscribing r e g u l a r 4 n - p o l y g o n s into the semicircles of the stad i u m , it can easily be shown that the intersection of the n t h periodic orbit with the x - a x i s is at the position xn = 1 + c o s ( 7 r / 4 n ) . A p p l y i n g the mirror cabinet m a p f o r d i f f e r e n t starting positions (a;, 0) on the x - a x i s , o n e c a n observe the following: • F o r x
