#### 期刊菜单

Study on Approximate Dispersion Relation of Nonlinear Hydroacoustic Wave
DOI: 10.12677/IJFD.2023.113009, PDF, HTML, XML, 下载: 480  浏览: 1,122

Abstract: Currently, in the research of nonlinear hydroacoustic waves, it is commonly assumed that the dispersion relation is linear in order to simplify calculations. However, the linear dispersion relation is not suitable for large-amplitude waves. So, this study investigates the dispersion relation of nonlinear hydroacoustic waves in a finite depth compressible single-layer fluid covered by an elastic plate. The fluid is assumed to be inviscid, compressible, and irrotational. We construct the governing equations and boundary conditions that represent the relationships between hydrodynamic, elastic, and inertial forces, and obtain an approximate nonlinear dispersion relation for hydroacoustic waves. The characteristics of hydroacoustic wave modes are analyzed, and the effects of important physical factors such as the thickness of the elastic plate on wave propagation properties are discussed. This research provides theoretical references for engineering practical problems such as polar marine resource detection, underwater target detection, seafloor seismic and tsunami warning.

1. 引言

Mikhin和Morozov (2008) [1] 分析了线性水声波在冰盖覆盖下的传播特性。S. Michele (2020) [2] 等人利用线性薛定谔方程来描述大的时空间尺度上水声波的传播特性。Zhang等(2015) [3] 对冰洋环境中水声波传播进行了数值模拟和理论分析。刘亚东(2022) [4] 探究表面弹性冰层与受扰动影响的可压缩流体之间的声弹性问题，利用Green函数得到波形的积分形式解。线性模型将海洋中的冰层视为弹性板，并运用水弹性理论来分析波浪作用下冰层的响应。这种线性模型在描述水声波与冰层的相互作用行为方面提供了有价值的结果。它能够有效预测波浪在冰层上的传播特性，并为水声工程设计和应用提供重要的参考。

2.数学模型

${\nabla }^{2}\phi -\frac{1}{{c}^{2}}\frac{{\partial }^{2}\phi }{\partial {t}^{2}}=0,\text{}\left(-h\le z\le 0\right),$ (1)

Figure 1. Coordinate diagram of nonlinear interaction between underwater sound and infinite elastic plate in a single layer fluid

$\frac{\partial \zeta }{\partial t}+\frac{\partial \phi }{\partial x}\frac{\partial \zeta }{\partial x}-\frac{\partial \phi }{\partial z}=0,$ (2)

$\frac{\partial \phi }{\partial t}+\frac{1}{2}{|\nabla \phi |}^{2}+\frac{p}{\rho }+g\zeta =0,$ (3)

$\frac{\partial \phi }{\partial z}=0,\text{}\left(z=-h\right).$ (4)

3. 近似的非线性色散关系

${\omega }^{2}=-\frac{g{\lambda }_{n}\mathrm{tan}\left({\lambda }_{n}h\right)}{1-\stackrel{˜}{d}{\lambda }_{n}\mathrm{tan}\left({\lambda }_{n}h\right)}\left(1+\stackrel{˜}{D}{k}_{n}^{4}-\stackrel{˜}{T}{k}_{n}^{2}\right),$ (5)

$\frac{\Omega }{\omega }=\epsilon ,$ (6)

${\Omega }^{2}=-\frac{{\epsilon }^{2}g{\lambda }_{n}\mathrm{tan}\left({\lambda }_{n}h\right)}{1-\stackrel{˜}{d}{\lambda }_{n}\mathrm{tan}\left({\lambda }_{n}h\right)}\left(1+\stackrel{˜}{D}{k}_{n}^{4}-\stackrel{˜}{T}{k}_{n}^{2}\right),$ (7)

${K}_{a}\left({K}_{a}\equiv \frac{g}{\stackrel{˜}{d}{\omega }^{2}}\right),$ (8)

${\lambda }_{n}h=\left(n-1\right)\pi +\Delta ,$ (9)

$\Delta$ 是一个无限接近于0的量。

${\frac{\Omega }{{\epsilon }^{2}}}^{2}=-\frac{g{\lambda }_{n}\mathrm{tan}\left[\left(n-1\right)\pi +\Delta \right]}{1-\stackrel{˜}{d}{\lambda }_{n}\mathrm{tan}\left[\left(n-1\right)\pi +\Delta \right]}\left(1+\stackrel{˜}{D}{k}_{n}^{4}-\stackrel{˜}{T}{k}_{n}^{2}\right),$ (10)

${\frac{\Omega }{{\epsilon }^{2}}}^{2}=-\frac{g{\lambda }_{n}\mathrm{tan}\Delta }{1-\stackrel{˜}{d}{\lambda }_{n}\mathrm{tan}\Delta }\left(1+\stackrel{˜}{D}{k}_{n}^{4}-\stackrel{˜}{T}{k}_{n}^{2}\right),$ (11)

${\frac{\Omega }{{\epsilon }^{2}}}^{2}=-\frac{g{\lambda }_{n}\Delta }{1-\Delta \stackrel{˜}{d}{\lambda }_{n}}\left(1+\stackrel{˜}{D}\stackrel{˜}{D}{k}_{n}^{4}-\stackrel{˜}{T}{k}_{n}^{2}\right),$ (12)

${\frac{\Omega }{{\epsilon }^{2}}}^{2}=-\frac{g{\lambda }_{n}\left({\lambda }_{n}h-\left(n-1\right)\pi \right)}{1-\stackrel{˜}{d}{\lambda }_{n}\left({\lambda }_{n}h-\left(n-1\right)\pi \right)}\left(1+\stackrel{˜}{D}{k}_{n}^{4}-\stackrel{˜}{T}{k}_{n}^{2}\right),$ (13)

$g=9.8{\text{m/s}}^{\text{2}},\text{}h=1500\text{m},\text{}\rho =1024{\text{kg*m}}^{-3},\text{}{\rho }_{e}=917{\text{kg*m}}^{-3},\text{}c=1500\text{m/s},\text{}E=5\text{GPa},\text{}v=0.3,\text{}T=0$ ，代入色散方程(11)可以得到水声波的波数频率关系。

Figure 2. Wave number-frequency relation of the second mode hydroacoustic wave when the thickness of the elastic plate is 40 meters

Figure 3. Wave number-frequency relation of the second mode hydroacoustic wave when the thickness of the elastic plate is 60 meters

Figure 4. Wave number-frequency relation of the third mode hydroacoustic wave when the thickness of the elastic plate is 40 meters

Figure 5. Wave number-frequency relation of the third mode hydroacoustic wave when the thickness of the elastic plate is 60 meters

Figure 6. Wave number-frequency relation of water acoustic wave when $\epsilon =1$

Figure 7. Wave numbe-frequency relation of water acoustic wave when $\epsilon =1.01$

Figure 8. Influence of water acoustic wave modes on propagation characteristics for $\epsilon =1$

Figure 9. Influence of water acoustic wave modes on propagation characteristics for $\epsilon =1.01$

4. 结论

1) 非线性影响：引入一个参数来修正线性色散关系，以考虑大振幅波浪对水声波的非线性影响。频率较低时 $\epsilon$ 对水声波波数的影响较大，频率较高时 $\epsilon$ 对水声波波数影响较小。

2) 弹性板厚度影响：通过分析弹性板厚度对水声波传播特性的影响，发现弹性板的厚度增加会导致同一频率下的波数增加。在高频率下，波的传播受到冰层厚度的影响相对较小，其他因素可能更加显著地影响波的传播行为。

3) 水声波模态影响：研究了不同水声波模态对传播的影响。发现随着模态增加，同频率下的波数减小，频率较低时模态变化对波数影响较大，而高频时影响较小。

NOTES

*通讯作者。

 [1] Mikhin, M. and Morozov, A. (2008) Acoustic Wave Propagation in an Ice-Covered Ocean. The Journal of the Acoustical Society of America, 123, 951-963. [2] Michele, S. and Renzi, E. (2019) Effects of the Sound Speed Vertical Profile on the Evolution of Hydroacoustic Waves. Journal of Fluid Mechanics, 883, A28. https://doi.org/10.1017/jfm.2019.907 [3] Zhang, J., Li, Z. and Jiang, W. (2015) Theoretical and Numerical Inves-tigations of Acoustic Propagation in Ice-Covered Oceans. The Journal of the Acoustical Society of America, 137, 2520-2530. [4] 刘亚东. 海洋表面冰盖、密度分层、底部粘性对水声波传播特性的影响[D]: [硕士学位论文]. 上海: 上海大学, 2022. [5] 张世功, 吴先梅, 张碧星, 安志武. 一维非线性声波传播特性[J]. 物理学报, 2016, 65(10): 8. [6] Yang, X., Dias, F. and Liao, S. (2018) On the Steady-State Resonant Acoustic-Gravity Waves. Journal of Fluid Mechanics, 849, 111-135. https://doi.org/10.1017/jfm.2018.422 [7] Kadri, U. (2016) Generation of Hy-droacoustic Waves by an Oscillating Ice Block in Arctic Zones. Advances in Acoustics and Vibration, 2016, Article ID: 8076108, 1-7. https://doi.org/10.1155/2016/8076108 [8] Kadri, U. and Wang, Z. (2021) Approximate Solution of Nonlinear Triad Interactions of Acoustic-Gravity Waves in Cylindrical Coordinates. Communications in Nonlinear Sci-ence and Nu-Merical Simulation, 93, Article 105514. https://doi.org/10.1016/j.cnsns.2020.105514 [9] 胡晨彤. 考虑海水可压性的海底震动与海水运动关系及其在海啸生成中的应用[D]: [硕士学位论文]. 上海: 上海交通大学, 2020. [10] Abdolali, A., Kadri, U., Parsons, W. and Kirby, J.T. (2018) On the Propagation of Acoustic-Gravity Waves under Elastic Ice Sheets. Journal of Fluid Mechanics, 837, 640-656. https://doi.org/10.1017/jfm.2017.808