32 2 Vol 32 2 20 2 Journal of Harbin Engineering University Dec 20 doi 0 3969 /j issn 006-7043 20 2 004 5000 2 TB566 A 006-7043 20 2-544-05 A method of seeking eigen-rays in shallow water with an irregular seabed ZHANG Wei YANG Shi'e TANG Yunfeng HUANG Yiwang Science and Technology on Underwater Acoustic Laboratory Harbin Engineering University Harbin 5000 China Abstract For three dimensional 3D shallow water sound propagation if the horizontal deflexion of a ray introduced by reflection on an irregular seabed is not taken into account there will be a serious error in the source bearing Therefore the research of an eigen-ray model for shallow 3D water is of great significance for underwater a- coustics In this paper a compensation and scanning method was applied to seek eigen-rays and respective transmission time was calculated based on the ray model in shallow water with an irregular seabed The results contrasting with the theoretical transmission time indicate that the method of seeking eigen-rays proposed in this paper is feasible In the computation program reflection points were directly derived from ray equations avoiding ray tracing step by step and nearly doubling computing speed Keywords eigen ray 3D shallow water transmission time horizontal deflexion -3 4 5 6-7 8 9-0 200-09-28 04045 984- E-mail zhangwei667@ 63 com 93- x 0 y 0 z 0 x r y r z r c z x y
2 545 Z = h x y Snell s 6 2 μ 2 v 2 s 2 s 2 = ncos θ 2 c z 0 c z = cos θ 0 = n z cos θ 2 z θ 2 θ 2 Snell zj+ μ x j+ - x j - j dz = 0 { z j 槡 n 2 - μ 2 j - v 2 j 3 zj+ v y j+ - y j - j dz = 0 z j 槡 n 2 - μ 2 j - v 2 j θ θ 0 z z 0 x j y j z j x j + y j + z j + j μ a = μ 2 v a = v 2 s a = - s 2 u j v j s j n x y z I = n μ j v j s j R = n μ j + v j + s j + 3 I = N I N - N N I tan π t = g ln 4 + θ 0 2 R = - N I N - N N I 5 tan π 4 + θ' 8 0 2 W = s j - b μ j - b 2 v j μ j+ = b W - b 2 b v j - b 2 μ j + μ j + b s j /F v j+ = b 2 W + v j + b 2 s j + b b v j - b 2 μ j /F { s j+ = - W + b 2 v j + b 2 s j + b μ j + b s j /F x y z μ v z u x y z 3 x u y u z u x = x u x y z x a y a z a μ a v a s a m αu j v j s j x x m x r x r x u u j = ncos θcos α x m y m z m x r v j = ncos θsin α 4 x { r y y r ' z s j = nsin θ z r ' x m - y m - z m - x r 3 z z h x y b b 2 F = + b 2 + b 2 z 2 N = t = F b b 2 - c z sinθ z dz 7 z 7 sinθ 0 θ 0 θ' g 7 6 g 8 6 7 0 2 2 2 x r z r 2 z z rn ' z rn + ' 3 4 2 x y z x y z
546 32 3 α = atan y r - y 0 ± β x r - x 0 dβ β β 2 β n β n β n + 2 x r y y rn ' y rn + ' 2 y rn ' y rn + ' Δy n = y rn ' - y r α = α ± arctan Δy n / x rn - x 0 0 5 0 Δy n εy α 0 Fig The trajectory of eigen rays β n ' β n + ' 3 β n ' β n + ' dβ' Table The transmission time of eigen rays and error β ' β 2 ' β m ' 2 α 0 min z r - z r ' /s /s /s /ms /ms εz 2 635 57 2 635 59 2 635 58 0 02 0 0 2 2 628 2 2 628 2 2 628 2 0 0 3 2 626 20 2 628 05 2 626 9 85-0 0 2 4 2 630 79 2 630 75 2 630 85-0 04 0 06 5 5 2 635 57 2 635 56 2 635 57-0 0 0 5 5 5 4 2 0 06 ms 3 3 3 8 7 3 2 Z = 25 + 5 0-3 x + 3 0-3 y c z = 500 + 5 0-4 Z m /s 0 m 0 m 5 m 00 m c = 3 000 m 0 m 5 m - 0 c 0 + gz c 0 = 530 m /s g = - 0 ~ 0 50 m 6 4 km 0 m 5 0 0 m 2 3 7 8
2 547 Table 2 2 The angles and transmission time of eigen rays for 3D / / / / /s Fig 2-9 42-0 88 59 4 868 4 0 30 2 038 7 2 2-8 57-0 53 4-3 654 7 0 50 2 2 038 4 The projection in the vertical direction of eigen rays 3 7 47-0 52 44 2 05 2 0 23 3 2 032 4 4 7 94-0 63 82 2 98 7 0 2 8 2 03 4 5 8 33-0 79 2 3 22 5 0 2 6 2 03 0 6 9 64-0 222 82-4 962 0 57 7 2 036 8 3 Table 3 /s /s Fig 3 The projection in the horizontal direction of eigen rays 2 026 8 2 038 7 4 2 5 3 2 2 026 4 2 038 4 3 2 5 3 2 024 2 2 032 4 3 2 6 3km 4 2 023 8 2 03 4-2 98 9 m 5 2 024 3 2 03 0 2 3 4 5 5 6 2 028 7 2 036 8 0 5 5 0 825 s 2 356 s 2 2 Bellhop 3 2 4 3 3 3 The contrast of transmission time between 2D and 3D congeneric eigen rays 4
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