Wave Propagation Part 9 ppt

Wave Propagation 272 The analysis of these dispersion equations show that under the following conditions [33] 00 1,1 nn nn θδ θδ − ≥≈ − ≥≈   AA (1.27) we become to the region of weak interaction between the signal wave and the modulation wave where the characteristic indexes n μ  and n μ are real and then the Mathie-Hill equations have the stable solutions. With help of obtained solutions of dispersion equations () 2 2 00 () , nn nn μ θμθ ==   (1.28) and the expressions for the coefficients ( ) ( ) 10 10 11 00 , 41 41 nn nn nn nn CC CC θθ θ θ ±± ⋅⋅ == ±±     , (1.29) where 0 n C  and 0 n C are defined from the conditions of normalizing, we obtained the analytical expressions for the z H and z E of TE and TM fields in the waveguide in the region of weak interaction. They have a form [33] () () () 00 0 1 0 0 01 1 ,, n iPz t ikk z ut nn zn k nk HxyCVee ω μ ∞ − − ==− =Ψ ⋅ ⋅ ∑∑     (1.30) () () () 00 0 1 0 0 01 1 ,, n iPz t ikk z ut nn zn k nk ExyCVee ω ε ∞ − − ==− =Ψ ⋅ ⋅ ∑∑ (1.31) where 0 0 , 22 k n n n k k n m C Vk C μ ⎛⎞ Δ =⋅ + − ⎜⎟ ⎜⎟ ⎝⎠     0 0 22 k n n n k k n Cm Vk C ε ⎛⎞ Δ =⋅ + − ⎜⎟ ⎜⎟ ⎝⎠ , (1.32) 0 0 0 2 n n m ku μ μω Δ= +   A , 0 0 0 2 n n m ku ε μω Δ= + A , (1.33) () () 22 2 10 0 22 22 00 24 nn n n m kb kb μ θχλχ β ± ⎡⎤ =+− ⎢⎥ ⎣⎦     A , (1.34) () () 22 2 10 0 22 22 00 24 nn n n m kb kb ε θχλχ β ± ⎡⎤ =+− ⎢⎥ ⎣⎦ A , (1.35) () () () () 22 22 00 22 000000 22 , nn nn PP cc ωω ε μλ εμλ =−=−  . (1.37) As is seen from the expressions (1.30) and (1.31) TE and TM fields in the waveguide with modulated filling are represented as the set of space-time harmonics with different On the Electrodinamics of Space-Time Periodic Mediums in a Waveguide of Arbitrary Cross Section 273 amplitudes. At that time the amplitude of the zero (fundamental) harmonic are independent of small modulation indexes, while the amplitudes of the plus and minus first harmonics (side harmonics) are proportional to the small modulation indexes in the first degree. At the realization the following condition [31], [33] 0 1, n n θδ −≤   (1.38) where 22 2 , 42 n n ε ε ε ηβ δ β − =   A 2 2 0 4 1, n n kb ε λ η =+   2 , m b ε ε ε ε β =A (1.39) () 0 2 2 022 00 2 222 0 4 , n n u C kub ε ε ε ω ελ βω θ ⎛⎞ ⎜⎟ −− ⎜⎟ ⎝⎠ =   2 0 ,1 u b c ε εε β εβ ==− (1.40) (here are shown the results for the TE field when 1 μ = ) the strong (resonance) interaction between the signal wave and the modulation wave takes place, when occurs the considerable energy exchange between them. The analytical expression for the frequency of strong interaction is found in the form 0, 0 0 0, 0 , ss ω ωωω ω − Δ≤ ≤ +Δ () 0 0, 2 sn uk ε ε ω ηβ β =+  (1.41) and is shown that the width of strong interaction is small and proportional to the modulation index in the first degree [31], [33] () ( ) 22 0 0 1 . 82 nn n ku εε ε ε ηβ η β ω η β +− Δ=   A  (1.42) In the region of strong interaction the dispersion equation (1.25) has complex solutions in the following form () 2 2 1 , 1, 2 n n n i ε ε θ δ μ − =±    ,1 22 . n n ε θ δ =   (1.43) (1.43) allow to receive the analytical expressions for the amplitudes of different harmonics in the form [31], [33] ( ) ( ) ,1 ,1 2 3 1, . 16 nn nn VV εε ε εε ε ηβη β β − ++ ≈≈     A (1.44) The analysis of these expressions shows that in the case of forward modulation, when the directions of propagation of the signal wave and the modulation wave coincide, the Wave Propagation 274 amplitude of minus first harmonic doesn’t depend from the modulation index, while the amplitude of the plus first harmonic is proportional to the modulation index in the first degree. In other words in the region of strong interaction besides the fundamental harmonic the substantial role plays the minus first harmonic reflected from the periodic structure of the filling on the frequency () 0 1, 0, 0 , 2 ss n ku ku ε ε ωω ηβ β − =−= −  n ε η β >  . (1.45) In the backward modulation case, when the directions of propagation of the signal wave and the modulation wave don’t coincide, the minus first and plus first harmonics change their roles. The results received above admit the visual physical explanation of the effect of strong interaction between the signal wave and the modulation wave. Below the physical explanation we show by example of TE field in the case of forward modulation. The zero harmonic in the modulated filling of the waveguide is incident on the density maxima of the filling at the angle ,0 n ε ϕ  and is reflected from them at the angle ,1 n ε ϕ  (Fig.2). These angles are defined from the following correlations [33] 0 2 2 0 ,0 2 00 cos , n n c c ε ω ϕ ελ ωε =−   ( ) 0 1 0 2 , 2 , 1cos2 cos . 12cos n n n ε εε ε εεε β ϕβ ϕ ββϕ +− = +−    (1.46) At that time the incident and reflection angles are different because of the moving of the modulation wave of the filling and the frequencies of incident and reflected waves satisfy to the following correlation [33] 10 1,0, sin sin . nn εε ωϕωϕ ⋅=⋅   (1.47) Fig. 2. The physical explanation of the effect of strong interaction. On the Electrodinamics of Space-Time Periodic Mediums in a Waveguide of Arbitrary Cross Section 275 If now we apply the first-order Wolf-Bragg condition, when the waves reflected from high- density points of the interference pattern are amplified, we obtain the following equation ( ) () 0 00 , 2 0 2cos 1. 1 n kc εε ε εω ϕ β β ⋅− = −  (1.48) It is not difficult to note, taking into account (1.46), that the solution of the equation (1.48) precisely coincides with the expression of the frequency of strong interaction (see (1.41)), received above. 3. Propagation of electromagnetic waves in a waveguide with a periodically modulated anisotropic insert Consider a waveguide of arbitrary cross section with an anisotropic nonmagnetic ( ) 1 μ = modulated insert (modulated uniaxial crystal) the permittivity tensor of which has the form () 1 1 2 (,) 0 0 0(,)0, 00 , zt zt zt ε εε ε ⎛⎞ ⎜⎟ = ⎜⎟ ⎜⎟ ⎝⎠  (2.1) where components 1 (,)zt ε and 2 (,)zt ε are modulated by the pumping wave in space and time according to the harmonic law 0 1110 ( , ) [1 cos ( )],zt m k z ut εε =+ − 0 2220 (,) [1 cos ( )].zt m k z ut εε =+ − (2.2) Here, 0 1 ε and 0 2 ε are the permittivities in the absence of a modulating wave; m, and m 2 are the modulation indices; and k 0 and и are, respectively, the wavenumber and velocity of the modulating wave. Consider the propagation of a signal electromagnetic wave at frequency 0 ω in this waveguide under the assumption that the modulation indices are small ( ) 1212 1, 1, .mmmm<< << ≈ Note that, when the condition 1 0.8 β ≤ is satisfied, where 0 11 /uc βε = , not only the modulation indices, but also parameter () 2 111 1 /1lm β β =− are small ( ) 1 1.l << As in my earlier works (see, e.g., [23], [31], [34-37]), transverse electric (ТЕ) and transverse magnetic (TM) waves in the waveguide will be described through the longitudinal components of the magnetic ( z H ) and electric ( z E ) field. Then, bearing in mind that ( ) ( ) ( ) 112 ,, ,, , xx yy zz D ztE D ztE D ztE εεε === and В = H and using the Maxwell equations, we obtain equations for H z (x, y, z, f) and E,(x, y, z, 0; namely, for the TE wave 2 1 22 1 0 zz z HH H tt zc ε ⊥ ∂∂ ∂ ⎡⎤ Δ +− = ⎢⎥ ∂∂ ∂ ⎣⎦ , (2.3) Wave Propagation 276 for the TM wave 2 2 2 22 1 1 0 zz z EE E zz ct ε ε ε ⊥ ⎛⎞ ∂∂ ∂ Δ +−= ⎜⎟ ∂∂ ∂ ⎝⎠   , (2.4) where ⊥ Δ is the two-dimensional Laplacian and 2zz EE ε =  . It is easy to check in this case that the transverse components of the ТЕ and TM fields can be expressed in terms of (1.6) as: for TE wave ( ) () 2 0 , ,, n nn n Hzt Hxy z τ λ ∞ − = ∂ =∇Ψ ∂ ∑ G   (2.5) ( ) () 2 0 0 , 1 , n nn n Hzt Ezxy ct τ λ ∞ − = ∂ ⎡ ⎤ =∇Ψ ⎣ ⎦ ∂ ∑ G   G , (2.6) for TM wave ( ) () 2 2 0 0 , 1 , n nn n Ezt Hzx y ct τ ε λ ∞ − = ∂⎡ ⎤ ⎣⎦ ⎡ ⎤ =− ∇Ψ ⎣ ⎦ ∂ ∑ G G , (2.7) ( ) () 2 2 1 0 , 1 , n nn n Ezt Ex y z τ ε λ ε ∞ − = ∂⎡ ⎤ ⎣⎦ =∇Ψ ∂ ∑ G . (2.8) Let us introduce the new variables , zut ξ = − () 20 11 1 0 1 1/ zd uu ξ ξ η β εξ ε =− − ∫ (2.9) into equations (2.3) and (2.4) and seek for solutions to the above equations in the form () () () 00 0 , n iPu znzn n He H xy ωη ξ ∞ − = =Ψ ∑   , (2.10) () () () 00 0 , n iPu znzn n Ee E xy ωη ξ ∞ − = =Ψ ∑  . (2.11) Taking into account that functions ( ) , n x y ψ  and ( ) , n x y ψ satisfy the Helmholtz equations (1.4) and (1.5), we get ordinary second-order differential equations in variable ξ to find () nz H ξ and ( ) nz E ξ , 2 2 1 1 0 2 1 1 1 0 1 10 1 nz n nz dH d H dd χ ε β ε ξξ ε β ε ⎡⎤ ⎛⎞ − += ⎢⎥ ⎜⎟ ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ −  , (2.12) On the Electrodinamics of Space-Time Periodic Mediums in a Waveguide of Arbitrary Cross Section 277 2 2 1 21 0 2 1 1 1 1 0 1 1 10 1 nz n nz dE d E dd χ ε εβ ε ξε ξ ε β ε ⎡⎤ ⎛⎞ − += ⎢⎥ ⎜⎟ ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ −   . (2.13) Here, () 2 00 222 1 11 20 1 1 n nn up c ω ε χελβ ε − ⎛⎞ =−− ⎜⎟ ⎜⎟ ⎝⎠    , () 2 00 222 1 11 20 1 1 n nn up c ω ε χελβ ε − ⎛⎞ =−− ⎜⎟ ⎜⎟ ⎝⎠ , (2.14) () 2 2 02 0 01 2 n n p c ω ε λ =−   , () 2 002 2 02 0 211 011 0022 212 n n p b c ω εεβ ελ εεβ − =− − , 2 11 1b β =− , 0 2 2 u c ε β = . (2.15) In terms of the new variables 01 2 1 0 1 0 1 , 2 1 kb d s ξ ξ ε β ε = − ∫ , 01 1 0 2 1 1 0 1 0 1 2 1 kb d s ξ ε ξ ε ε β ε = − ∫ (2.16) Equations (2.12) and (2.13) take the form of the Mathieu-Hill equations 22 222 01 4 0 nz n nz dH H ds k b χ + =   , () 2 20 2 1 222 0112 4 0 n nz nz dE E ds k b χε εε + =   . (2.17) Note that the frequency domain described by the conditions 1 0 1 22 n n n θ θδ −>>≅    , 1 0 1 22 n n n θ θδ −>>≅ , (2.18) where 2 2 01 00 22 01 4 nn p u kb ωβ θ ⎛⎞ =− ⎜⎟ ⎜⎟ ⎝⎠   , 2 2 01 00 22 01 4 nn p u kb ωβ θ ⎛⎞ =− ⎜⎟ ⎜⎟ ⎝⎠ , (2.19) () 2 22 00 01 11 01 22 2 01 2 4 n nn n up kb kb u ω θθ θ − ⎡ ⎤ − ⎢ ⎥ == + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦    A , 02 10 11 2 1 220 012 2 2 n nn n mm kb ελ θ θθ ε − == − , (2.20) is the domain of weak interaction between the signal wave and the wave that modulates the insert. Solving (2.17) by the method developed in [23], [31], [34-37] and discarding the terms proportional to the modulation indices in the first power, we obtain the following expressions for the ТЕ and TM field in the frequency domain defined by formulas (2.18): [38] Wave Propagation 278 for the TE wave () () () 00 0 1 0 01 , n iPz t ikk z ut nn zn k nk HxyecVe ω ∞ − − ==− =Ψ ∑∑     , (2.21) where 0 1 0 0 2 k n n k k n c Vk uk c ω ⎛⎞ =+ ⎜⎟ ⎜⎟ ⎝⎠   A  , ( ) 10 1 0 , 41 nn n n c c θ θ ± = ±     (2.22) for the TM wave () () () 00 0 1 0 01 ,, n iPz t ikk z ut nn zn k nk ExyecVe ω ∞ − − ==− =Ψ ∑∑ (2.23) where () 0 112 0 0 1 22 k n n k k n c Vk mm uk c ω ⎡ ⎤ =+−+ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ A , ( ) 10 1 0 41 nn n n c c θ θ ± = ± . (2.24) Note that quantities 0 n c  and 0 n c in (2.21) and (2.23) are found from the normalization condition. As follows from (2.21) and (2.23), when an electromagnetic wave propagates in a waveguide with an insert harmonically modulated in space and time, the ТЕ and TM fields represent a superposition of space-time harmonics of different amplitudes. In the domain of weak interaction between the signal and modulation waves, the amplitudes of harmonics +1 and -1 prove to be small (they are linearly related to the modulation indices) compared with the amplitude of the fundamental harmonic (which is independent of modulation indices). It is known [21] that, when 0 n θ  and 0 n θ tend to unity, i.e., when the conditions 0 1, n n θ δ −≤   0 1 n n θ δ − ≤ (2.25) are satisfied, the signal wave and the wave that modulates the insert strongly interact (the first-order Bragg condition for waves reflected from a high-density area is met) and vigorously exchange energy. Condition (2.25) can be recast (for the TM field) as 0, 0 0 0, 0 , ss ω ωωω ω − Δ≤ ≤ +Δ (2.26) where 0,s ω given by () 2 0 22 0 1 0, 1 20 1 10 2 4 ,1 , , 2 n snn nn uk bk λ ε ω βη η λ λ β ε =+=+ = (2.27) is the frequency near which the strong interaction takes place, and On the Electrodinamics of Space-Time Periodic Mediums in a Waveguide of Arbitrary Cross Section 279 ( ) 01 01 1 1 82 n n n ku βη ω θ βη + Δ= (2.28) is the width of the domain of strong interaction. Calculations show that 1 ~1 n V − , 112 ~, n Vmm (2.29) in frequency domain (2.26). From relationships (2.29), it follows that the amplitude of reflected harmonic -1 is independent of modulation indices in the domain where the signal wave and the wave that modulates the anisotropic insert strongly interact. In other words, not only the zeroth harmonic of the signal, but also reflected harmonic -1 of frequency ()() 0 1, 1 1 1 , 2 snn uk ω ηβ η β β − =−> (2.30) plays a significant role in this domain. Note in conclusion that the results obtained here turn into those reported in [37] in the limit 1 0m → ; in the limit 0u → , one arrives at results for a waveguide with an inhomogeneous but stationary anisotropic insert. 4. Interaction of electromagnetic waves with space-time periodic anisotropic magneto-dielectric filling of a waveguide Let the axis of a regular waveguide of an arbitrary cross section coincides with the OZ axis of a Certain Cartesian coordinate frame. Assume that the waveguide is filled with a periodically modulated anisotropic magneto- dielectric filling whose tensor permittivity and permeability are specified by the formulas () 1 1 2 00 00 00 ,zt ε εε ε ⎛⎞ ⎜⎟ = ⎜⎟ ⎜⎟ ⎝⎠  , () 1 1 2 00 00 00 ,zt μ μμ μ ⎛⎞ ⎜⎟ = ⎜⎟ ⎜⎟ ⎝⎠  . (3.1) In (3.1) 11 ,const const ε μ == and the ( ) 2 ,zt ε and ( ) 2 ,zt μ components are harmonic functions in space and time: ( ) ( ) 0 22 00 ,1coszt m kz k ut ε εε ⎡ ⎤ =+ − ⎣ ⎦ , (3.2) ( ) ( ) 0 22 00 ,1coszt m kz k ut μ μμ ⎡ ⎤ =+ − ⎣ ⎦ , (3.3) where 1m ε << and 1m μ < < are small modulation indexes, 0 2 const ε = and 0 2 const μ = are, respectively, the permittivity and permeability of the filling in the absence of a modulation wave. Let a signal wave unit amplitude with frequency 0 ω propagates in such a waveguide in a positive Direction of an axis OZ. After some algebra, the wave equations for the longitudinal components ( ) ,,, z Hxyztand ( ) ,,, z Exyztof TE and TM fields can be obtained from Maxwell equations Wave Propagation 280 D curlH t ∂ = ∂ G G , , B curlE t ∂ =− ∂ G G 0divD = G , 0divB = G , (3.4) 0 DE ε ε = GG  , 0 BH μ μ = G G  , ( ) 9 0 1/4 9 10 /Fm επ =⋅⋅ , 7 0 410 /Hm μπ − =⋅ (3.5) with allowance for the equalities 01xx DE ε ε = , 01 yy DE ε ε = , ( ) 02 , zz DztE εε = , (3.6) 01xx BH μμ = , 01 yy BH μ μ = , ( ) 02 , zz BztH μμ = . (3.7) We arrive at the following equations: for TE waves ( ) () 22 2 0012 22 1 , , zz z zt HH Hzt zt μ εμεμ μ ⊥ ∂∂ Δ+ = ∂∂   , (3.8) for the TM waves ( ) () 22 2 0012 22 1 , , zz z zt EE Ezt zt ε εμμε ε ⊥ ∂∂ Δ+ = ∂ ∂   , (3.9) where ( ) 2 ,, zz HztH μ =  ( ) 2 , zz EztE ε =  . (3.10) With the use of Maxwell equations (3.4) and (3.5), the transverse components of ТЕ and TM fields can be represented in terms of () () 0 ,, zn n HHztxy ∞ = =Ψ ∑  , () () 0 ,, zn n EEztxy ∞ = =Ψ ∑ (3.11) as follows: for TE waves ( ) ( ) () 2 2 1 0 ,, 1 , n nn n zt H zt Hx y z τ μ λ μ ∞ − = ∂⎡ ⎤ ⎣⎦ =∇Ψ ∂ ∑ G   , (3.12) ( ) ( ) () 2 2 00 0 ,, , n nn n zt H zt Ezx y t τ μ μλ ∞ − = ∂⎡ ⎤ ⎣⎦ ⎡ ⎤ =∇Ψ ⎣ ⎦ ∂ ∑ G   G , (3.13) for TM waves ( ) ( ) () 2 2 00 0 ,, , n nn n zt E zt Hzx y t τ ε ελ ∞ − = ∂⎡ ⎤ ⎣⎦ ⎡ ⎤ =− ∇Ψ ⎣ ⎦ ∂ ∑ K G , (3.14) [...]... Press, MRI-17, 299 ( 196 7) [8] Barsukov K A., Gevorkyan E A., Zvonnikov N A Radiotekhnika i Elektronika, 20, No 5, 90 8 ( 197 5) [9] Tamir T., Wang H C., Oliner A A IEEE, Transactions on Microwave Theory and Techniques, MTT-12, 324 ( 196 4) [10] Averkov S I., Boldin V P Izvestiya Vuzov Seriya Radiofizika, 23, No 9, 1060 ( 198 0) [11] Askne J TIIER, 59, No 9, 244 ( 196 8) [12] Rao TIIER, 65, No 9, 244 ( 196 8) [13] Yeh... 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E011 , and to a position (b) – phase distribution of an interference of waves of the same types The diffracted wave in spherical waveguides of spherical diffraction antenna array according to (5), (6) can be written as 298 Wave Propagation а) b) Fig 9 Distribution of amplitude (a) and phase (b) of fundamental mode E00 of spherical waveguide ⎛ Im⎜ arctg ⎝ ⎛ Re⎜ ⎝ 11 ∑U 0i i =1 ⎞ ⎟ ⎠' 11 ⎞ U 0i ⎟ i =1... interpretation the phenomena’s of waves diffraction 3 Spherical hybrid reflector antennas 3.1 Surfaces electromagnetic waves For the first time surface wave properties were investigated by Rayleigh and were named “whispering gallery” waves, which propagate on the concave surface of circumferential gallery (Rayleigh, 194 5) It was determined that these waves propagate in thin layer with equal wave length This layer... Sovetskoe Radio ( 197 1) [27] Yariv A Kvantovaya Elektronika I Nelineynaya Optika Perevod s Angliyskovo, M.: Sovetskoe Radio ( 197 3) [28] Volnovodnaya Optoelektronika Pod Redaktsiey T Tamir Perevod s Angliyskovo, M.: Mir ( 197 4) [ 29] Markuze D Opticheskie Volnovodi Perevod s Angliyskovo, M.: Mir ( 197 4) [30] Yariv A., Yukh P Opticheskie Volni v Kristallakh Perevod s Angliyskovo, M.: Mir ( 198 7) [31] Barsukov... Radiotekhnika i Elektronika, 28, No 2, 237 ( 198 3) [32] Mak-Lakhlan N V Teoriya i Prilozheniya Funktsiy Mathe Perevod s zAngliyskovo, M.: Fizmatgiz ( 196 3) [33] Gevorkyan E A Uspekhi Sovremennoy Radioelektroniki, No 1, 3 (2006) [34] Barsukov K A., Gevorkyan E A Radiotekhnika i Elektronika, 31, 1733 ( 198 6) [35] Barsukov K A., Gevorkyan E A Radiotekhnika i Elektronika, 39, 1170 ( 199 4) [36] Gevorkyan E A Proceedings... M., Volf E Osnovi Optiki Perevod s Angliyskovo, M.: Nauka, 197 3 [3] Cassedy E S., Oliner A A TIIER, 51, No 10, 1330 ( 196 3) [4] Barsukov K A., Bolotovskiy B M Izvestiya Vuzov Seriya Radiofizika, 7, No 2, 291 ( 196 4) [5] Barsukov K A., Bolotovskiy B M Izvestiya Vuzov Seriya Radiofizika, 8, No 4, 760 ( 196 5) [6] Cassedy E S TIIER, 55, No 7, 37 ( 196 7) [7] Peng S T., Cassedy E S Proceedings of the Symposium... (Miller & Talanov, 195 6) It was showed that their energy is concentrated at the layer with approximate width a − (ν − ν 1/3 ) k The same conclusion was made after viewing the diffraction of waves on the bent metallic list that illuminated by waveguide source and in bent waveguides (Shevchenko, 197 1) A lot of letters were aimed at investigating the properties of surface electromagnetic waves (EMW) on the... − ∑ ⎪ 2 2 ( kr )1/2 l =1 ⎪ ⎭ where Al , C1 , C 2 are the constant coefficients The analysis of the equations (1) shows that eigenwaves propagate into hemisphere with angles ±θ Each wave propagates with its constant of propagation γ m( n ) along virtual curves 290 Wave Propagation with radiuses rm( n ) = γ m( n ) / k with maintaining vectors of polarization about component Er and Eϕ (fig.2) In the . Radiofizika, 23, No 9, 1060 ( 198 0). [11] Askne J. TIIER, 59, No 9, 244 ( 196 8). [12] Rao. TIIER, 65, No 9, 244 ( 196 8). [13] Yeh C., Cassey K. F., Kaprielian Z. A. IEEE, Transactions on Microwave Theory. Techniques, MTT-13, 297 ( 196 5). Wave Propagation 284 [14] Elachi Ch. TIIER, 64, No 12, 22 ( 197 6). [15] Karpov S. Yu., Stolyarov S. N. Uspekhi Fizicheskikh Nauk, 163, No 1, 63 ( 199 3). [16] Elachi. shows that eigenwaves propagate into hemisphere with angles θ ± . Each wave propagates with its constant of propagation ()mn γ along virtual curves Wave Propagation 290 with radiuses