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The sound radiation field of a circular piston source in an infinite plane rigid baffle can be approximated as an axisymmetric directional spherical-wave. The interface response expressions of the axisymmetric directional spherical-wave for the piston parallel with the interface has already been obtained in previous studies. On condition that the distance from the piston center to the interface is much greater than the piston radius, we first derive the conical wave expansion of the axisymmetric directional spherical-wave which is obtained by using the conical wave expansion of the homogeneous spherical-wave and the formula of the axisymmetric directional spherical-wave excited by a circular piston in an rigid infinite plane, and then derive the expression of the interface reflection wave of the axisymmetric directional spherical-wave for the piston non-parallel to the interface. The expression of the interface reflection wave is simplified into a simplified expression by saddle point method on condition that the source distance is much larger than the acoustic wavelength. The simplified expression is not only simple in the calculation, but also clear in the physical meaning. The simplified expression shows that the interface reflection wave of the axisymmetric directional spherical-wave can be regarded as the product of the axisymmetric directional spherical-wave excited by the piston mirror image and the reflection coefficient. The calculations show that when the piston radius is smaller than the acoustic wavelength, the reflected wave is less sensitive to the angle included between the piston and the interface and the azimuth of the receiving point, and the directivity of the reflected wave is weak. When the piston radius is greater than the acoustic wavelength, the reflected wave is very sensitive to the angle included between the piston and the interface and the azimuth of the receiving point, and the directivity of the reflected wave is strong. Increasing the angle included between the piston and the interface, the reflected wave and its directivity both first increase and then decrease. The reflected wave is largest and the directivity of the reflected wave is strongest when the angle included between the piston and the interface is equal to that between the interface normal and the connecting line between the mirror image of the piston center and the receiving point.
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Keywords:
- acoustic wave /
- reflected wave /
- directivity /
- piston
[1] 腾舵, 杨虎, 李道江 2016 水声换能器基础 (西安: 西北工业大学出版社) 第1—3页
Ten D, Yang H, Li D J 2016 Underwater Acoustic Transducer Foundation (Xi’an: Northwestern Polytechnical University Press) pp1–3 (in Chinese)
[2] 杨坤德, 段睿, 李辉, 马远良 2019 水下声源定位理论与技术 (北京: 电子工业出版社) 第1—3页
Yang K D, Duan R, Li H, Ma Y L 2019 Theory and Technology of Underwater Sound Source Location (Beijing: Publishing House of Electronics Industry) pp1–3 (in Chinese)
[3] 莫喜平 2012 应用声学 31 171
Google Scholar
Mo X P 2012 Appl. Acoust. 31 171
Google Scholar
[4] Hall D E 1987 Basic Acoustics (New York: Harper & Row Publishers) pp161–177
[5] 朱中锐 2015 声呐障板下矢量水听器应用引论 (哈尔滨: 哈尔滨工程大学出版社) 第29页
Zhu Z R 2015 Introduction to the Application of Vector Hydrophone under Sonar Baffle (Harbin: Harbin Engineering University Press) p29 (in Chinese)
[6] 莫喜平, 于婧涵 2019 声学学报 44 751
Mo X P, Yu J H 2019 Acta Acustica 44 751
[7] 何正耀 2020 水声换能器及基阵建模与设计 (北京: 科学出版社) 第2页
Hei Z Y 2020 Modeling and Design of Underwater Acoustic Transducer and Array (Beijing: Science Press) p2 (in Chinese)
[8] New R, Becker R I, Wilhelmij P 1981 J. Acoust. Soc. Am. 70 1518
Google Scholar
[9] Mast T D, Yu F 2005 J. Acoust. Soc. Am. 118 3457
Google Scholar
[10] Mellow T 2006 J. Acoust. Soc. Am. 120 90
Google Scholar
[11] 莫喜平 2018 应用声学 37 671
Google Scholar
Mo X P 2018 J. Appl. Acoust. 37 671
Google Scholar
[12] 纪伟, 刘忠乐 2017 水雷战与舰船防护 25 7
Ji W, Liu Z L 2017 Mine Warf. Ship Def. 25 7
[13] 梅元贵, 许建林, 耿烽, 周朝晖, 李刚 2006 铁道学报 28 74
Google Scholar
Mei Y G, Xu J L, Geng F, Zhou C H, Li G 2006 J. China Rail. Soc. 28 74
Google Scholar
[14] 李禹志, 李昕珈, 王青东, 郭各朴, 马青玉 2019 声学技术 38 327
Li Y Z, Li X J, Wang Q D, Guo G P, Ma Q Y 2019 Tech. Acoust. 38 327
[15] Amédin C K, Berry A, Champoux Y, Allard J F 1995 J. Acoust. Soc. Am. 98 1757
Google Scholar
[16] Wang C N, Cho H M 2005 Appl. Acoust. 66 866
Google Scholar
[17] Schakel M D, Smeulders D, Slob E C, Heller H K 2011 J. Appl. Phys. 109 15678
[18] Schakel M D, Smeulders D, Slob E C, Heller H K 2011 Geophysics 76 N29
Google Scholar
[19] Schakel M D, Smeulders D, Slob E C, Heller H K 2012 Transport Porous Med. 93 1
Google Scholar
[20] Pride S R 1994 Phys. Rev. B 50 15678
Google Scholar
[21] Pride S R, Haartsen M W 1996 J. Acoust. Soc. Am. 100 1301
Google Scholar
[22] 王鑫 2020 化学工程与装备 285 115
Wang X 2020 Chem. Eng. Eq. 285 115
[23] Aki K, Richards P G 2002 Quantitative Seismology (California: University Science Books) pp194–207
[24] Morse P M, Feshbach H 1953 Methods of Theoretical Physics (Part II) (NewYork: McGraw-Hill) pp1323, 1457
[25] 裴彦良, 王揆洋, 李官保, 李西双, 刘晨光 2007 石油仪器 21 20
Pei Y L, Wang K Y, Li G B, Li X S, Liu C G 2007 Petrol. Instrum. 21 20
[26] 谈国君 2010 江苏水利 000 28
Google Scholar
Tang G J 2010 Jiangsu Water Resour. 000 28
Google Scholar
[27] 张虹斌, 周忠海, 刘军礼, 程广欣 2012 中国水运 12 211
Zhang H B, Zhou Z H, Liu J L, Cheng G X 2012 China Water Transport 12 211
[28] 奚定平 1998 贝塞尔函数 (北京: 高等教育出版社) 第13—16页
Xi D P 1998 Bessel Function (Beijing: Higher Education Press) pp13–16 (in Chinese)
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图 1 无限大平面刚性障板中的圆形活塞与界面的示意图
Figure 1. Schematic diagram of a piston in an infinite plane rigid baffle and an interface.
图 2
${p_{\text{f}}}\left( M \right)$ ,${p_{\text{f}}}\left( O \right)$ 和$ {p_{{\text{f0}}}} $ 的时域波形 (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ Figure 2. Time-domain waveforms of
${p_{\text{f}}}\left( M \right)$ ,${p_{\text{f}}}\left( O \right)$ and$ {p_{{\text{f0}}}} $ : (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ .
图 3 活塞面关于界面的镜像
Figure 3. Mirror image of the piston surface with respect to the interface.
图 4
$ {P_{{\text{ref}}}} $ 与$ {P_{{\rm{ref\_jian}}}} $ 的比较 (a)$ a = 0.1\;\text{m} $ 且快速地层; (b)$ a = 0.1\;\text{m} $ 且慢速地层; (c)$ a = 1\;\text{m} $ 且快速地层; (d)$ a = 1\;\text{m} $ 且慢速地层Figure 4. Comparison of
$ {P_{{\text{ref}}}} $ and$ {P_{{\rm{ref\_jian}}}} $ : (a)$ a = 0.1\;\text{m} $ and fast formation; (b)$ a = 0.1\;\text{m} $ and slow formation; (c)$ a = 1\;\text{m} $ and fast formation; (d)$ a = 1\;\text{m} $ and slow formation.
图 5 不同夹角α下在M点
$( {20\sqrt 3 , \;0, \; - 10} )$ 处的反射波 (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ Figure 5. Reflected wave under different included angle α for point M at
$( {20\sqrt 3 , \;0, \; - 10} )$ : (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ .
图 6 不同夹角α下在M点
$( {20\sqrt 3 , \;45, \; - 10} )$ 处的反射波 (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ Figure 6. Reflected wave under different included angle α for point M at
$( {20\sqrt 3 , \;45, \; - 10} )$ : (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ .
图 7 不同夹角α下在M点
$( {20\sqrt 3 , \;90, \; - 10})$ 处的反射波 (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ Figure 7. Reflected wave under different included angle α for point M at
$( {20\sqrt 3 , \;90, \; - 10} )$ : (a)$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ .
图 8
$\alpha = {45^{\circ}}$ 时不同${\varphi _M}$ 下的反射波和其指向图 (a)$a = 0.1\;\text{m}$ 时的反射波; (b)$a = 0.1\;\text{m}$ 时的指向图; (c)$a = 1\;\text{m}$ 时的反射波; (d)$a = 1\;\text{m}$ 时的指向图Figure 8. Reflected waves and directional diagrams at different azimuths
${\varphi _M}$ for$\alpha = {45^{\circ}}$ : (a) Reflected waves for$ a = 0.1\;\text{m} $ ; (b) directional diagram for$ a = 0.1\;\text{m} $ ; (c) reflected waves for$ a = 1\;\text{m} $ ; (d) directional diagram for$ a = 1\;\text{m} $ .
图 9 不同夹角α下的反射波的指向图 (a)
$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ Figure 9. Directional diagrams under different angle α: (a)
$a = 0.1\;\text{m}$ ; (b)$a = 1\;\text{m}$ .
图 10
$a = 1\;\text{m}$ 时不同夹角α下的指向图 (a)$ {\beta _{M{E_{1}}}} = {20^{\circ}} $ ; (b)$ {\beta _{M{E_{1}}}} = {40^{\circ}} $ Figure 10. Directional diagrams under different angle α for
$a = 1\;\text{m}$ : (a)$ {\beta _{M{E_{1}}}} = {20^{\circ}} $ ; (b)$ {\beta _{M{E_{1}}}} = {40^{\circ}} $ .
图 11
$\left| B \right|$ 随$ {\beta _{M{E_{1}}}} $ 的变化曲线Figure 11. Variation curve of
$\left| B \right|$ with$ {\beta _{M{E_{1}}}} $ .表 1 流体和固体参数
Table 1. Parameters of fluid and solid.
密度/
(${\rm{ kg}} \cdot {{\rm{m}}^{ - 3} }$)纵波速度/
(${\text{m}} \cdot {{\text{s}}^{ - 1}}$)横波速度/
(${\text{m}} \cdot {{\text{s}}^{ - 1}}$)流体 1000 1500 — 快速地层 2600 4000 2300 慢速地层 2000 2200 1200 
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[1] 腾舵, 杨虎, 李道江 2016 水声换能器基础 (西安: 西北工业大学出版社) 第1—3页
Ten D, Yang H, Li D J 2016 Underwater Acoustic Transducer Foundation (Xi’an: Northwestern Polytechnical University Press) pp1–3 (in Chinese)
[2] 杨坤德, 段睿, 李辉, 马远良 2019 水下声源定位理论与技术 (北京: 电子工业出版社) 第1—3页
Yang K D, Duan R, Li H, Ma Y L 2019 Theory and Technology of Underwater Sound Source Location (Beijing: Publishing House of Electronics Industry) pp1–3 (in Chinese)
[3] 莫喜平 2012 应用声学 31 171
Google Scholar
Mo X P 2012 Appl. Acoust. 31 171
Google Scholar
[4] Hall D E 1987 Basic Acoustics (New York: Harper & Row Publishers) pp161–177
[5] 朱中锐 2015 声呐障板下矢量水听器应用引论 (哈尔滨: 哈尔滨工程大学出版社) 第29页
Zhu Z R 2015 Introduction to the Application of Vector Hydrophone under Sonar Baffle (Harbin: Harbin Engineering University Press) p29 (in Chinese)
[6] 莫喜平, 于婧涵 2019 声学学报 44 751
Mo X P, Yu J H 2019 Acta Acustica 44 751
[7] 何正耀 2020 水声换能器及基阵建模与设计 (北京: 科学出版社) 第2页
Hei Z Y 2020 Modeling and Design of Underwater Acoustic Transducer and Array (Beijing: Science Press) p2 (in Chinese)
[8] New R, Becker R I, Wilhelmij P 1981 J. Acoust. Soc. Am. 70 1518
Google Scholar
[9] Mast T D, Yu F 2005 J. Acoust. Soc. Am. 118 3457
Google Scholar
[10] Mellow T 2006 J. Acoust. Soc. Am. 120 90
Google Scholar
[11] 莫喜平 2018 应用声学 37 671
Google Scholar
Mo X P 2018 J. Appl. Acoust. 37 671
Google Scholar
[12] 纪伟, 刘忠乐 2017 水雷战与舰船防护 25 7
Ji W, Liu Z L 2017 Mine Warf. Ship Def. 25 7
[13] 梅元贵, 许建林, 耿烽, 周朝晖, 李刚 2006 铁道学报 28 74
Google Scholar
Mei Y G, Xu J L, Geng F, Zhou C H, Li G 2006 J. China Rail. Soc. 28 74
Google Scholar
[14] 李禹志, 李昕珈, 王青东, 郭各朴, 马青玉 2019 声学技术 38 327
Li Y Z, Li X J, Wang Q D, Guo G P, Ma Q Y 2019 Tech. Acoust. 38 327
[15] Amédin C K, Berry A, Champoux Y, Allard J F 1995 J. Acoust. Soc. Am. 98 1757
Google Scholar
[16] Wang C N, Cho H M 2005 Appl. Acoust. 66 866
Google Scholar
[17] Schakel M D, Smeulders D, Slob E C, Heller H K 2011 J. Appl. Phys. 109 15678
[18] Schakel M D, Smeulders D, Slob E C, Heller H K 2011 Geophysics 76 N29
Google Scholar
[19] Schakel M D, Smeulders D, Slob E C, Heller H K 2012 Transport Porous Med. 93 1
Google Scholar
[20] Pride S R 1994 Phys. Rev. B 50 15678
Google Scholar
[21] Pride S R, Haartsen M W 1996 J. Acoust. Soc. Am. 100 1301
Google Scholar
[22] 王鑫 2020 化学工程与装备 285 115
Wang X 2020 Chem. Eng. Eq. 285 115
[23] Aki K, Richards P G 2002 Quantitative Seismology (California: University Science Books) pp194–207
[24] Morse P M, Feshbach H 1953 Methods of Theoretical Physics (Part II) (NewYork: McGraw-Hill) pp1323, 1457
[25] 裴彦良, 王揆洋, 李官保, 李西双, 刘晨光 2007 石油仪器 21 20
Pei Y L, Wang K Y, Li G B, Li X S, Liu C G 2007 Petrol. Instrum. 21 20
[26] 谈国君 2010 江苏水利 000 28
Google Scholar
Tang G J 2010 Jiangsu Water Resour. 000 28
Google Scholar
[27] 张虹斌, 周忠海, 刘军礼, 程广欣 2012 中国水运 12 211
Zhang H B, Zhou Z H, Liu J L, Cheng G X 2012 China Water Transport 12 211
[28] 奚定平 1998 贝塞尔函数 (北京: 高等教育出版社) 第13—16页
Xi D P 1998 Bessel Function (Beijing: Higher Education Press) pp13–16 (in Chinese)
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