Abstract

The problem of scattering from an infinitely long conducting cylinder partially buried in a perfectly conducting ground plane due to an obliquely incident plane wave (a three-dimensional problem) is solved by an exact procedure based on the method of images. The incident field, the reflected field from the ground plane, and the scattered field from the cylinder and its image are expressed in terms of cylindrical vector wave functions. By satisfying the boundary conditions on the surface of the cylinder, a set of two coupled infinite systems of equations for the even-and odd-mode expansion coefficients of the scattered field is obtained. These equations are solved numerically. Related quantities, i.e., the induced current in the cylinder and the scattered power patterns in the far field, are obtained, and their variations with respect to the angle of incidence, the height (depth) above (below) the ground plane, and the electrical radius are studied for the TM case.

© 1989 Optical Society of America

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References

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  1. Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.
  2. M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
    [CrossRef]
  3. V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
    [CrossRef]
  4. V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,”IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
    [CrossRef]
  5. K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,”J. Appl. Opt. 26, 2995–2999 (1987).
    [CrossRef]
  6. C. Flammer, H. Singhaus, “The interaction of electromagnetic pulses with an infinitely long conducting cylinder above a perfectly conducting ground,” EMP Interaction Note 144 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., July1973).
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  8. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

1987 (1)

K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,”J. Appl. Opt. 26, 2995–2999 (1987).
[CrossRef]

1957 (2)

V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,”IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
[CrossRef]

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[CrossRef]

1951 (1)

V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
[CrossRef]

Biot, M. A.

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[CrossRef]

Flammer, C.

C. Flammer, H. Singhaus, “The interaction of electromagnetic pulses with an infinitely long conducting cylinder above a perfectly conducting ground,” EMP Interaction Note 144 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., July1973).

Nahm, K. B.

K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,”J. Appl. Opt. 26, 2995–2999 (1987).
[CrossRef]

Rayleigh,

Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.

Singhaus, H.

C. Flammer, H. Singhaus, “The interaction of electromagnetic pulses with an infinitely long conducting cylinder above a perfectly conducting ground,” EMP Interaction Note 144 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., July1973).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Twersky, V.

V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,”IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
[CrossRef]

V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
[CrossRef]

Wolfe, W. L.

K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,”J. Appl. Opt. 26, 2995–2999 (1987).
[CrossRef]

IRE Trans. Antennas Propag. (1)

V. Twersky, “On scattering and reflection of electromagnetic waves by rough surfaces,”IRE Trans. Antennas Propag. AP-5, 81–90 (1957).
[CrossRef]

J. Appl. Opt. (1)

K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,”J. Appl. Opt. 26, 2995–2999 (1987).
[CrossRef]

J. Appl. Phys. (2)

M. A. Biot, “Some new aspects of the reflection of electromagnetic waves on a rough surface,” J. Appl. Phys. 28, 1455–1463 (1957).
[CrossRef]

V. Twersky, “On the nonspecular reflection of electromagnetic waves,” J. Appl. Phys. 22, 825–835 (1951).
[CrossRef]

Other (4)

Rayleigh, “On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits,” in Scientific Papers (Cambridge U. Press, Cambridge, 1912), Vol. 5, pp. 410–418.

C. Flammer, H. Singhaus, “The interaction of electromagnetic pulses with an infinitely long conducting cylinder above a perfectly conducting ground,” EMP Interaction Note 144 (Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., July1973).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

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Figures (13)

Fig. 1
Fig. 1

Geometry of the problem: A, cylinder residing on the ground plane (h = a); B, cylinder partially buried in the ground plane (h = 0.8a); C, cylinder protruding above the ground plane (h = −0.75a).

Fig. 2
Fig. 2

Geometry of scattering: plane wave and its image incident upon a cylinder and its image.

Fig. 3
Fig. 3

Modulus of normalized induced current as a function of ka for a normally incident plane wave (α = −π/2): —, h = a;– – –, h = 0.75a;— · —, h = 0.5a.

Fig. 4
Fig. 4

Modulus of normalized induced current as a function of ka for an obliquely incident plane wave(α = −π/5): —, h = a;— — —, h = 0.75a;— · —, h = 0 5a.

Fig. 5
Fig. 5

Scattered power (in decibels) as a function of ϕ for an incident plane wave (γ = π/2, α = −π/2) for ka = 1: —, h = a; — — —, h = 0.75a;— · —, h = 0.5a.

Fig. 6
Fig. 6

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for ka = 2: —, h = a; – – –, h = 0.75a;— · —, h = 0.5a;⋯, h = 0.

Fig. 7
Fig. 7

Scattered power (in decibels) as a function of ϕ for a normally incident plane wave (γ = π/2, α = −π/2) for ka = 2: —, h = a, – – –, h = 0.75a;— · —, h = 0.5a.

Fig. 8
Fig. 8

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for ka = 2: —, h = a; – – –, h = 0.75a;— · —, h = 0.5a;⋯, h = 0.

Fig. 9
Fig. 9

Scattered power (in decibels) as a function of ϕ for a normally incident plane wave (γ = π/2,α = −π/2) for ka = 4:—, h = a;– – –,h = 0.75a;— · —, h = 0.5a.

Fig. 10
Fig. 10

Scattered power (in decibels) as a function of ϕ for an obliquely incident plane wave (γ = π/2, α = −π/5) for ka = 4:—, h = a;– – –, h = 0.75a; — · —, h= 0.5a;⋯, h = 0.

Fig. 11
Fig. 11

Variation of the scattered power (in decibels) as a function of the angles ϕ and γ for a normally incident plane wave (α = −π/2) for ka = 2 and h =a: ⋯, γ =90°,— · —, γ = 60°; —, γ= 45°,– – –, γ = 30°.

Fig. 12
Fig. 12

Scattered power (in decibels) as a function of ϕ for a normally incident plane wave (α = −π/2) for a partially protruding cylinder: −, h = −0.5a;– – –, h = −0.625a; — · —, h = −0.625a;— · —, h = −0.75a.

Fig. 13
Fig. 13

Geometry of translated coordiante system for the cylindrical wave addition theorem.

Equations (73)

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cos Θ i = cos θ cos γ + sin θ sin γ cos ( ϕ - α ) ,
k · r = k r cos Θ i = k z cos γ + k x sin γ cos α + k y sin γ sin α = k z cos γ + k ρ sin γ cos ( ϕ - α ) = k z z + k x x + k y y .
exp ( i k · r ) × a ^ z = i k sin γ { ( sin α ) a ^ x - ( cos α ) a ^ y } × exp ( i k r cos Θ i ) ,
H inc = ( 0 μ 0 ) 1 / 2 E 0 [ a ^ x ( sin α ) - a ^ y ( cos α ) ] exp ( i k r cos Θ i ) .
H inc = ( 0 μ 0 ) 1 / 2 E 0 i k sin γ × exp ( i k r cos Θ i ) a ^ z .
exp ( i k r cos Θ i ) = exp [ i k ρ sin γ cos ( ϕ - α ) ] exp ( i k z cos γ ) ,
exp ( i k · ρ ) = n = - i n J n ( k ρ sin γ ) exp [ i n ( α - ϕ ) ] ,
exp ( i k r cos Θ i ) = m = 0 ( 2 - δ o m ) i m J m ( k ρ sin γ ) × cos [ m ( ϕ - α ) ] exp ( i k z cos γ ) .
M e o m ( i ) ( r , γ ) = × ψ e o m ( i ) ( r , γ ) a ^ z = ψ e o m ( i ) ( r , γ ) × a ^ z ,
N e o m ( i ) ( r , γ ) = 1 k × M e o m ( i ) ( r , γ ) ,
ψ e o m ( i ) ( r , γ ) = Z m ( i ) ( k ρ sin γ ) sin cos m ϕ exp ( i k z cos γ ) .
Z m ( 1 ) = J m , Z m ( 3 ) = H m ( 1 ) , Z m ( 2 ) = Y m , Z m ( 4 ) = H m ( 2 ) .
M e o m ( i ) ( r , γ ) = [ m ρ Z m ( i ) ( k ρ sin γ ) cos sin m ϕ a ^ ρ - ρ × { Z m ( i ) ( k ρ sin γ ) } sin cos m ϕ a ^ ϕ ] exp ( i k z cos γ ) ,
N e o m ( i ) ( r , γ ) = [ i cos γ ρ { Z m ( i ) ( k ρ sin γ ) } sin cos m ϕ a ^ ρ i m cos γ ρ ] × Z m ( i ) ( k ρ sin γ ) cos sin m ϕ a ^ ϕ + k sin 2 γ Z m ( i ) × ( k ρ sin γ ) sin cos m ϕ a ^ z ] exp ( i k z cos γ ) .
× × A - k 2 A = 0.
H inc = ( 0 μ 0 ) 1 / 2 E 0 i k sin γ m = 0 ( 2 - δ o m ) i m × [ cos m α M e m ( 1 ) ( r , γ ) + sin m α M o m ( 1 ) ( r , γ ) ] .
E inc = i k ( μ 0 0 ) 1 / 2 × H inc
= E 0 k [ - k z cos α a ^ x - k z sin α a ^ y + ( k x cos α + k y sin α ) a ^ z ] exp ( i k r cos Θ i )
= E 0 k sin γ m = 0 ( 2 - δ o m ) i m [ cos m α N e m ( 1 ) ( r , γ ) + sin m α N o m ( 1 ) ( r , γ ) ] .
H ref = ( 0 μ 0 ) 1 / 2 E 0 ( sin α a ^ x + cos α a ^ y ) exp ( i k r cos Θ r ) = ( 0 μ 0 ) 1 / 2 E 0 i k sin γ m = 0 ( 2 - δ o m ) i m [ - cos m α M e m ( 1 ) ( r , γ ) + sin m α M o m ( 1 ) ( r , γ ) ] ,
E 0 = E 0 exp ( - i 2 k h sin γ sin α )
k r cos Θ r = k x x - k y y + k z z = k ρ sin γ cos ( ϕ + α ) + k z cos γ .
E ref = E 0 k [ k z cos α a ^ x - k z sin α a ^ y - ( k x cos α + k y sin α ) a ^ z ] × exp ( i k r cos Θ γ ) = E 0 k sin γ m = 0 ( 2 - δ o m ) i m [ - cos m α N e m ( 1 ) ( r , γ ) + sin m α N o m ( 1 ) ( r , γ ) ] .
E scat = E 0 k sin γ m = 0 ( 2 - δ o m ) i m [ C e m cos m α N e m ( 3 ) ( r , γ ) + C o m sin m α N o m ( 3 ) ( r , γ ) ] ,
E scat = E 0 k sin γ m = 0 ( 2 - δ o m ) i m [ - C e m cos m α N e m ( 3 ) ( r , γ ) + C o m sin m α N o m ( 3 ) ( r , γ ) ] ,
H scat = 1 i k sin γ ( 0 μ 0 ) 1 / 2 E 0 m = 0 ( 2 - δ o m ) i m × [ C e m cos m α M e m ( 3 ) ( r , γ ) + C o m sin m α M o m ( 3 ) ( r , γ ) ] ,
H scat = 1 i k sin γ ( 0 μ 0 ) 1 / 2 E 0 m = 0 ( 2 - δ o m ) i m × [ - C e m cos m α M e m ( 3 ) ( r , γ ) + C o m sin m α M o m ( 3 ) ( r , γ ) ] .
C e o m = C e o m ,
E z inc ( a , ϕ , z ) + E z ref ( a , ϕ , z ) + E z scat ( a , ϕ , z ) + [ a ^ z · E scat ( r , γ ) ] ρ = a = 0.
E 0 m = 0 ( 2 - δ o m ) i m [ cos m α N e m z ( 1 ) ( a , ϕ , z ; γ ) + sin m α N o m z ( 1 ) ( a , ϕ , z ; γ ) ] + E o m = 0 ( 2 - δ o m ) i m × [ - cos m α N e m z ( 1 ) ( a , ϕ , z ; γ ) + sin m α N o m z ( 1 ) ( a , ϕ , z ; γ ) ] + E 0 m = 0 ( 2 - δ o m ) i m [ C e m cos m α N emz ( 3 ) ( a , ϕ , z ; γ ) + C o m sin m α N o m z ( 3 ) ( a , ϕ , z ; γ ) ] + E 0 m = 0 ( 2 - δ o m ) i m × [ - C e m cos m α n , μ , j ( - 1 ) n A n ( e , m j , μ ) N j , n + μ , z ( 1 ) ( a , ϕ , z ; γ ) + C o m sin m α n , μ , j ( - 1 ) n A n ( o , m j , μ ) N j , n + μ , z ( 1 ) ( a , ϕ , z ; γ ) ] = 0 ,
m = 0 d m n = 0 ( - 1 ) n A n N n + m
m = 0 d m n = 0 ( - 1 ) n A n N n - m ,
m = 0 d m n = 0 ( - 1 ) n A n N n + m = m = 0 m = 0 ( - 1 ) m = m A m - m d m N m .
m = 0 d m n = 0 ( - 1 ) n A n N n - m = m = 0 m = 0 ( - 1 ) m + m A m + m d m N m + m = 0 ( 1 - δ o m ) m = 0 ( - 1 ) m - m A m - m d m N - m ,
C e m = - [ 1 - exp ( - i 2 λ h sin α ) ] J m ( λ a ) H m ( 1 ) ( λ a ) + 1 2 ( 1 + δ o m ) cos m α × J m ( λ a ) H m ( 1 ) ( λ a ) { m = 0 m ( 2 - δ o m ) i m - m A m - m ( e , m e , m ) × cos m α C e m + m = 0 ( 2 - δ o m ) i m - m A m + m ( e , m e , - m ) × cos m α C e m + ( - 1 ) m ( 1 - δ o m ) 2 m = m 1 i m - m A m - m × ( e , m e , - m ) cos m α C e m - 2 m = 1 m i m - m A m - m ( o , m e , m ) × sin m α C o m - 2 m = 1 i m - m A m + m ( o , m e , - m ) sin m α C o m - ( - 1 ) m ( 1 - δ o m ) 2 m = m 1 i m - m A m - m ( o , m e , - m ) × sin m α C o m }             ( m 0 )
C o m = - [ 1 + exp ( - i 2 λ h sin α ) ] J m ( λ a ) H m ( 1 ) ( λ a ) - 1 2 sin m α J m ( λ a ) H m ( 1 ) ( λ a ) { 2 m = 1 m i m - m A m - m ( o , m o , m ) sin m α C o m + 2 m = 1 i m - m × A m + m ( o , m o , - m ) sin m α C o m + ( - 1 ) m + 1 ( 1 - δ o m ) 2 × m = m i m - m A m - m ( o , m o , - m ) sin m α C o m - m = 0 m ( 2 - δ o m ) i m - m A m - m ( e , m o , m ) cos m α C e m - m = 0 ( 2 - δ o m ) i m - m A m + m ( e , m o , - m ) cos m α C e m - ( - ) m + 1 ( 1 - δ o m ) 2 m = m i m - m A m - m ( e , m o , - m ) × cos m α C e m }             ( m 1 ) , with λ = k sin γ .
C e m = 0 , C o m = - 2 J m ( λ a ) H m ( 1 ) ( λ a ) .
E z scat = - E o sin γ m = 0 ( 2 - δ o m ) i m 2 J m ( a ) H m ( 1 ) ( λ a ) sin m α × sin m ϕ H m ( 1 ) ( k ρ sin γ ) = - E 0 sin γ m = - i m J m ( λ a ) H m ( 1 ) ( λ a ) { exp [ i m ( α - ϕ ) ] - exp [ i m ( α + ϕ ) ] } H m ( 1 ) ( k ρ sin γ ) .
E z scat ~ E 0 ( 2 π k ρ sin γ ) 1 / 2 exp [ i k z cos γ + i k ρ sin γ - i π / 4 ] × m = 0 ( 2 - δ o m ) { C e m cos m α cos m ϕ + C o m sin m α sin m ϕ } ,
E ρ scat ~ - E 0 cos γ k sin γ ( 2 π k ρ sin γ ) 1 / 2 exp [ i k z cos γ + i k ρ sin γ - i π / 4 ] m = 0 ( 2 - δ o m ) { C e m cos m α cos m ϕ + C o m sin m α sin m ϕ } ,
E ϕ scat ~ - i E 0 cos γ k ρ sin γ ( 2 π k ρ ) 1 / 2 exp [ i k z cos γ + i k ρ sin γ - i π / 4 ] m = 0 ( 2 - δ o m ) m × { C e m cos m α sin m ϕ - C o m sin m α cos m ϕ } .
I = a 0 2 π H ϕ ( a , ϕ , z ) d ϕ ,
H ϕ ( a , ϕ , z ) = H ϕ inc ( a , ϕ , z ) + H ϕ ref ( a , ϕ , z ) + H ϕ scat ( a , ϕ , z ) + [ a ^ ϕ · H scat ( r ) ] ρ = a = 0 / μ 0 E 0 i k sin γ m = 0 ( 2 - δ o m ) i m [ cos m α M e m ϕ ( 1 ) ( a , ϕ , z ; γ ) + sin m α M o m ϕ ( 1 ) ( a , ϕ , z ; γ ] + 0 / μ 0 E 0 i k sin γ m = 0 ( 2 - δ o m ) × i m [ - M e m ϕ ( 1 ) ( a , ϕ , z ; γ ) cos m α + sin m α M o m ϕ ( 1 ) ( a , ϕ , z ; γ ] + 0 / μ 0 E 0 i k sin γ m = 0 ( 2 - δ o m ) i m [ cos m α C e m M e m ϕ ( 3 ) ( a , ϕ , z ; γ ) + sin m α C o m M o m ϕ ( 3 ) ( a , ϕ , z ; γ ] + 0 / μ 0 E 0 i k sin γ m = 0 ( 2 - δ o m ) i m × [ - cos m α C e m n , μ , j ( - 1 ) n A n ( e , m j , μ ) M j , n + μ , ϕ ( 1 ) ( a , ϕ , z ; γ ) + sin m α C o m n , μ , j ( - 1 ) n A n ( o , m j , μ ) M j , n + μ , ϕ ( 1 ) ( a , ϕ , z ; γ ) ] ,
I ˜ ( μ 0 0 ) 1 / 2 I 2 π a E 0 = { - i [ 1 - exp { - i 2 k h sin γ sin α } ] × J 1 ( k a sin γ ) - i H 1 ( 1 ) ( k a sin γ ) C e o + i 1 2 H 0 ( 1 ) ( 2 k h sin γ ) × J 1 ( k a sin γ ) C e o + i 1 2 m = 0 , 2 , 4 , ( 2 - δ o m ) 2 ( - 1 ) m H m ( 1 ) × ( 2 k h sin γ ) J 1 ( k a sin γ ) cos m α C e m - 2 m = 1 , 3 , 5 , ( - 1 ) m × H m ( 1 ) ( 2 k h sin γ ) J 1 ( k a sin γ ) sin m α C o m } exp ( i k z cos γ ) .
ψ e o m ( i ) ( r ) = ψ e o m ( i ) ( ρ ) exp ( i k z cos γ ) ,
ψ e o m ( i ) ( r ) = ψ e o m ( i ) ( ρ ) exp ( i k z cos γ ) ,
H m ( 1 ) ( λ ρ ) = 1 π C 1 exp [ i λ ρ cos α + i m α - i m π / 2 ] d α = 1 π i m C exp [ i λ ρ cos ( α - ϕ ) + i m ( α - ϕ ) ] d α ,
H m ( 1 ) ( λ ρ ) e i m ϕ = 1 π i m C exp [ i λ ρ cos ( α - ϕ ) + i m α ] d α .
H - m ( 1 ) ( z ) = ( - 1 ) m H m ( 1 ) ( z ) ,
H m ( 1 ) ( λ ρ ) exp ( - i m ϕ ) = 1 π i m C exp [ i λ ρ cos ( α - ϕ ) - i m α ] d α .
exp ( i k · ρ ) = n = - i n J n ( k ρ sin γ ) exp [ i n ( α - ϕ ) ] .
H m ( 1 ) ( k ρ sin γ ) exp ( - i m ϕ ) = 1 π i m C exp ( i k · ρ - i m α ) d α = 1 π i m C exp [ i k · ( ρ 0 + ρ ) - i m α ] d α = 1 π i m C exp ( i k · ρ 0 ) d α n = - i n J n ( k ρ sin γ ) × exp [ i ( n - m ) α - i n ϕ ] .
H m ( 1 ) ( k ρ sin γ ) exp ( - i m ϕ ) = n = - J n ( k ρ sin γ ) H m - n ( 1 ) ( k ρ 0 sin γ ) × exp [ i ( n - m ) ϕ 0 ] exp ( - i n ϕ ) = n = - ( - 1 ) n H n ( 1 ) ( k ρ 0 sin γ ) J n + m ( k ρ sin γ ) × exp [ - i ( n + m ) ϕ ] exp ( + i n ϕ 0 ) .
H m ( 1 ) ( k ρ sin γ ) exp ( i m ϕ ) = n = - ( - 1 ) n H n ( 1 ) ( k ρ 0 sin γ ) × J n + m ( k ρ sin γ ) exp [ i ( n + m ) ϕ ] exp ( - i n ϕ 0 ) .
H m ( 1 ) ( k ρ sin γ ) sin cos m ϕ = 1 2 n = 0 ( 2 - δ o n ) ( - 1 ) n H n ( 1 ) ( k ρ 0 sin γ ) × [ J n + m ( k ρ sin γ ) { cos n ϕ 0 sin cos ( n + m ) ϕ ± sin n ϕ 0 cos sin ( n + m ) ϕ } + ( - 1 ) m J n - m ( k ρ sin γ ) { ± cos n ϕ 0 sin cos ( n - m ) ϕ + sin n ϕ 0 cos sin ( n - m ) ϕ } ] .
H m ( 1 ) ( k ρ sin γ ) sin cos m ϕ = 1 2 n = 0 ( 2 - δ o n ) ( - 1 ) n H n ( 1 ) ( k ρ sin γ ) × [ J n + m ( k ρ 0 sin γ ) { cos n ϕ 0 sin cos ( n + m ) ϕ ± sin n ϕ 0 cos sin ( n + m ) ϕ } + ( - 1 ) m J n - m ( k ρ 0 sin γ ) × { ± cos n ϕ 0 sin cos ( n - m ) ϕ + sin n ϕ 0 cos sin ( n - m ) ϕ } ] .
A n ( k ρ 0 sin γ ) ½ ( 2 - δ o n ) H n ( 1 ) ( k ρ 0 sin γ ) ,
A n ( k ρ sin γ ) ½ ( 2 - δ o n ) H n ( 1 ) ( k ρ sin γ ) .
ψ e o m ( 3 ) ( r , γ ) = n = 0 ( - 1 ) n A n ( k ρ 0 sin γ ) × [ cos n ϕ 0 ψ e o n + m ( 1 ) ( r , γ ) ± sin n ϕ 0 ψ e o n + m ( 1 ) ( r , γ ) ± ( - 1 ) m cos n ϕ 0 ψ e o n - m ( 1 ) ( r , γ ) + ( - 1 ) m sin n ϕ 0 ψ e o n - m ( 1 ) ( r , γ ) ] ( ρ < ρ 0 )
ψ e o m ( 3 ) ( r , γ ) = n = 0 ( - 1 ) n A n ( k ρ sin γ ) [ cos n ϕ 0 ψ e o n + m ( 1 ) ( r 0 , γ ) ± ( - 1 ) m cos n ϕ ψ e o n - m ( 1 ) ( r 0 , γ ) + ( - 1 ) m sin n ϕ 0 ψ e o n - m ( 1 ) ( r 0 , γ ) ] ( ρ > ρ 0 ) .
M e o m ( 3 ) ( r , γ ) = n = 0 ( - 1 ) n A n ( k ρ 0 sin γ ) [ cos n ϕ 0 M e o n + m ( 1 ) ( r , γ ) ± sin n ϕ 0 M o e n + m ( 1 ) ( r , γ ) + ( - 1 ) m cos n ϕ 0 M e o n - m ( 1 ) ( r , γ ) + ( - 1 ) m sin n ϕ 0 M o e n - m ( 1 ) ( r , γ ) ]             ( ρ < ρ 0 )
M e o m ( 3 ) ( r , γ ) = n = 0 ( - 1 ) n A n ( k ρ sin γ ) [ cos n ϕ 0 M e o n + m ( 1 ) ( r 0 , γ ) ± sin n ϕ 0 M o e n + m ( 1 ) ( r 0 , γ ) + ( - 1 ) m cos n ϕ 0 M e o n - m ( 1 ) ( r 0 , γ ) + ( - 1 ) m sin n ϕ 0 M o e n - m ( 1 ) ( r 0 , γ ) ]             ( ρ > ρ 0 ) .
N e o m ( 3 ) ( r , γ ) = n = 0 ( - 1 ) n A n ( k ρ 0 sin γ ) [ cos n ϕ 0 N e o n + m ( 1 ) ( r , γ ) ± sin n ϕ 0 N o e n + m ( 1 ) ( r , γ ) + ( - 1 ) m cos n ϕ 0 N e o n - m ( 1 ) ( r , γ ) + ( - 1 ) m sin n ϕ 0 N o e n - m ( 1 ) ( r , γ ) ]             ( ρ < ρ 0 ) ,
N e o m ( 3 ) ( r , γ ) = n = 0 ( - 1 ) n A n ( k ρ sin γ ) [ cos n ϕ 0 N e o n + m ( 1 ) ( r 0 , γ ) ± sin n ϕ 0 N o e n + m ( 1 ) ( r 0 , γ ) + ( - 1 ) m cos n ϕ 0 N e o n - m ( 1 ) ( r 0 , γ ) + ( - 1 ) m sin n ϕ 0 N o e n - m ( 1 ) ( r 0 , γ ) ]             ( ρ > ρ 0 ) .
M e o m ( 3 ) ( r , γ ) = n , μ , j ( - 1 ) n A n ( e o m j , μ ) M j , n + μ ( 1 ) ( r , γ )             ( ρ < ρ 0 ) ,
M e o m ( 3 ) ( r , γ ) = n , μ , j ( - 1 ) n A n ( e o m j , μ ) M j , n + μ ( 1 ) ( r 0 , γ )             ( ρ > ρ 0 ) ,
N e o m ( 3 ) ( r , γ ) = n , μ , j ( - 1 ) n A n ( e o m j , μ ) N j , n + μ ( 1 ) ( r , γ )             ( ρ < ρ 0 ) ,
N e o m ( 3 ) ( r , γ ) = n , μ , j ( - 1 ) n A n ( o e m j , μ ) N j , n + μ ( 1 ) ( r 0 , γ )             ( ρ > ρ 0 ) ,
A n ( e , m e , m ) = A n ( k ρ 0 sin γ ) cos n ϕ 0 , A n ( e , m e , - m ) = ( - 1 ) m A n ( k ρ 0 sin γ ) cos n ϕ 0 , A n ( e , m o , m ) = A n ( k ρ 0 sin γ ) sin n ϕ 0 , A n ( e , m o , - m ) = ( - 1 ) m A n ( k ρ 0 sin γ ) sin n ϕ 0 , A n ( o , m o , m ) = A n ( k ρ 0 sin γ ) cos n ϕ 0 , A n ( o , m o , - m ) = ( - 1 ) m A n ( k ρ 0 sin γ ) cos n ϕ 0 , A n ( o , m e , m ) = - A n ( k ρ 0 sin γ ) sin n ϕ 0 , A n ( o , m e , - m ) = ( - 1 ) m A n ( k ρ 0 sin γ ) sin n ϕ 0 ,
A n ( e , m e , m ) = A n ( o , m o , m ) = { ( - 1 ) n / 2 A n ( 2 k h sin γ ) n even 0 n odd ,
A n ( e , m e , - m ) = - A n ( o , m o , - m ) = { ( - 1 ) ( n / 2 ) + m A n ( 2 k h sin γ ) n even 0 n odd ,
A n ( e , m o , m ) = - A n ( o , m e , m ) = { 0 n even ( - 1 ) n - 1 2 A n ( 2 k h sin γ ) n odd ,
A n ( e , m o , - m ) = - A n ( o , m e , - m ) = { 0 n even ( - 1 ) n - 1 2 + m A n ( 2 k h sin γ ) n odd .

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