Low-frequency sound in shallow seas propagates as discrete normal modes whose amplitudes and arrival times fluctuate strongly when internal gravity waves — a stochastic background plus intermittent trains of large-amplitude nonlinear solitary waves — displace the summer thermocline. Zhou, Zhang and Rogers (1996) showed, using measured Yellow Sea and Massachusetts Bay internal wave records and numerical simulation, that individual modes differ sharply in sensitivity to internal waves: in a strong-thermocline environment one group of modes has arrival-time variations far more sensitive to the internal wave field, while the first mode's arrival time is the most stable. Their analysis, however, rested on a two-dimensional deterministic model with a Fresnel-zone validity restriction and no horizontal refraction. Since 1996, three-dimensional propagation modeling and stochastic coupled-mode transport theory for background internal waves have matured as largely separate lines; the surviving open core is a validated quantitative statistical description of individual-mode fluctuations (amplitude and arrival time or phase) under combined realistic forcing — intermittent nonlinear wave packets superposed on a stochastic background, with three-dimensional effects included — together with the environmental conditions (thermocline strength, water depth, frequency, wave-field statistics) under which the observed mode-sensitivity grouping holds or breaks down.

Background

Low-frequency sound (tens of Hz to a few kHz) in shallow seas (water depths of order tens of meters) is trapped between the sea surface and the seabed and propagates as a sum of discrete normal modes — the waveguide's standing-wave solutions in depth, each with its own depth profile and phase and group speeds. In summer, solar heating creates a thermocline, a thin layer across which temperature and hence sound speed change rapidly; this density stratification supports internal gravity waves, i.e., waves of the density interface below the sea surface. The linear, background part of the internal wave field behaves as a random medium conventionally described by the Garrett–Munk spectrum, an empirical universal spectral model of internal wave displacements, while the strongly nonlinear part organizes into trains of large-amplitude internal solitary waves (solitons) that displace the thermocline by many meters as they propagate in packets. Because the sound-speed structure moves with the thermocline, internal waves scatter sound and redistribute energy among modes (mode coupling), producing large, anisotropic, frequency-selective fluctuations of received intensity and phase. Zhou, Zhang and Rogers (1991) first explained the anomalously large summer transmission loss in coastal zones as mode-coupling loss caused by internal soliton packets, and Zhou and Zhang (1995) reported from Yellow Sea data that certain acoustic modes are much more sensitive to internal waves than others, with the interaction depending on frequency, thermocline profile, seabed properties, and source/receiver geometry.

The 1996 paper of Zhou, Zhang and Rogers drove numerical simulations with measured internal wave records from the Yellow Sea and Massachusetts Bay and found: fluctuations depend strongly on receiver depth and range and are anisotropic; individual acoustic modes have clearly different sensitivities to the internal wave field; in shallow water with a strong thermocline there is a group of modes whose arrival-time variation is far more sensitive to internal waves than that of others, while the arrival time of the first (lowest) mode is the most stable; and the fluctuation spectra of intensity and phase are much broader than the internal wave spectrum itself. That analysis rested on a two-dimensional (range–depth) deterministic internal wave model, assumed the characteristic length of the internal wave field always exceeds the first Fresnel zone (the transverse coherence region of the acoustic wavefield), restricting validity to relatively short ranges and higher frequencies, and neglected horizontal refraction — bending of the sound path in the horizontal plane — and azimuthal scattering by curved internal wave fronts. A full three-dimensional acoustic/internal wave model, a statistical description of modal fluctuations under the stochastic internal wave field (random background plus intermittent soliton packets), and the quantitative conditions under which the modal categorization holds for other thermocline strengths and water depths therefore remained open directions; the authors' follow-on program treated validation of the predicted modal coupling and fluctuations as an explicit objective, the primary purpose of the dedicated summer-1996 Yellow Sea experiment reported in Zhou et al. (1997).

Both enabling capabilities have advanced substantially since. On the modeling side, three-dimensional parabolic equation models (a standard range-marching method for wave propagation in range-dependent environments) and three-dimensional coupled-mode models now reproduce observed horizontal refraction, ducting between internal wave fronts, and mode-specific beam effects (Frank, Badiey, Lynch and Siegmann 2005; Duda, Lin and Lynch 2008; Lin, Duda and Lynch 2009; Dossot 2011; Turgut, Mignerey and Orr 2013; Zhang, Luo and Zhang 2018). On the statistical side, coupled-mode transport theory — evolution equations for the cross-mode coherence matrix that treat the internal wave field as a random medium — predicts mode amplitude and intensity moments in shallow water (Colosi, Duda and Morozov 2012; Raghukumar and Colosi 2014) and has been validated against Shallow Water 2006 (SW06) experiment data at 175 Hz and 1.2 kHz (Raghukumar et al. 2014); Colosi (2008) derived analytic resonance conditions governing mode coupling by internal solitary waves and its sensitivity to environmental parameters. What later work has not delivered is the combination the source program asked for: a validated statistical account of individual-mode fluctuations — including arrival time and phase, not only intensity — under combined forcing by intermittent nonlinear packets and a stochastic background with three-dimensional effects included, and a quantitative map of the environmental conditions under which the mode-sensitivity grouping holds or fails.

Problem Statement

For low-frequency sound in shallow-water waveguides with a summer thermocline, establish a quantitative, validated statistical description of the fluctuations of individual acoustic normal modes — amplitude and arrival time or phase — induced by realistic internal wave fields that combine intermittent nonlinear soliton packets with a stochastic Garrett-Munk-like background, including three-dimensional effects (horizontal refraction and azimuthal scattering by curved internal wave fronts), and determine the quantitative environmental conditions — thermocline strength, water depth, acoustic frequency, source/receiver geometry, and internal wave space-time statistics — under which the modal sensitivity categorization observed by Zhou, Zhang and Rogers (1996) holds, namely that one group of modes is far more sensitive in arrival time to internal waves while the first mode's arrival time remains the most stable. Because that categorization was stated only qualitatively, the second part of the task requires fixing it operationally: the answer must state the per-mode arrival-time fluctuation measure on which sensitivity is judged and the separation that defines the high-sensitivity group; the categorization is then counted as holding in an environment when the first mode has the smallest such fluctuation measure among all modes and a proper subset of the remaining modes forms a distinctly more sensitive group, and as breaking down wherever either condition fails. Two capabilities surrounding the 1996 analysis — the availability of three-dimensional acoustic/internal wave propagation models and statistical (as opposed to purely deterministic) treatment of modal fluctuations under background random fields — have been substantially filled by later literature and are therefore supporting tools here rather than open ends; the open core is their integration into a mode-level statistical description that includes intermittent nonlinear packets and three-dimensional effects, and the delineation of the categorization's validity domain across environments. An answer may take the form of an analytic or transport-type statistical theory, a validated three-dimensional simulation study with ensemble statistics and a regime map, a dedicated field experiment with mode-resolved acoustics and co-located three-dimensional environmental characterization, or a combination, delivering quantitative predictions of individual-mode fluctuation statistics and explicit environmental validity conditions for the mode-sensitivity grouping.

References

  1. Ji-Xun Zhou, Xue-Zhen Zhang, and Peter H. Rogers, 'Modal characteristics of acoustic signal fluctuations induced by shallow water internal waves', OCEANS 96 MTS/IEEE Conference Proceedings, vol. 1, pp. 1-8, 1996. DOI: 10.1109/oceans.1996.572431, https://doi.org/10.1109/oceans.1996.572431
  2. Ji-Xun Zhou and Xue-Zhen Zhang, 'Modal sensitivity of acoustic interactions with internal waves in shallow water', The Journal of the Acoustical Society of America, vol. 98, no. 5, Supplement, p. 2868, 1995. DOI: 10.1121/1.413209, https://doi.org/10.1121/1.413209
  3. Ji-xun Zhou, Xue-zhen Zhang, and Peter H. Rogers, 'Resonant interaction of sound wave with internal solitons in the coastal zone', The Journal of the Acoustical Society of America, vol. 90, no. 4, pp. 2042-2054, 1991. DOI: 10.1121/1.401632, https://doi.org/10.1121/1.401632
  4. Ji-Xun Zhou, Peter H. Rogers, Gary W. Caille, Renhe Zhang, Guoliang Jin, Liangying Lei, Peter H. Dahl, Robert C. Spindel, and Zijun Gan, 'Preliminary report on the summer '96 joint China–U.S. internal wave/acoustic wave experiment in the Yellow Sea', The Journal of the Acoustical Society of America, vol. 101, no. 5, Supplement, p. 3015, 1997. DOI: 10.1121/1.418529, https://doi.org/10.1121/1.418529
  5. J. X. Zhou, 'Modal Characteristics of Acoustic Interactions with Internal Waves in Shallow Water', technical report, Office of Naval Research / Georgia Institute of Technology, DTIC accession ADA628818, 1997. DOI: 10.21236/ADA628818, https://doi.org/10.21236/ADA628818 (extended technical report of the 1996 OCEANS conference paper)
  6. J. C. Preisig and T. F. Duda, 'Coupled acoustic mode propagation through continental-shelf internal solitary waves', IEEE Journal of Oceanic Engineering, 1997. DOI: 10.1109/48.585945, https://doi.org/10.1109/48.585945
  7. T. F. Duda and J. C. Preisig, 'A modeling study of acoustic propagation through moving shallow-water solitary wave packets', IEEE Journal of Oceanic Engineering, 1999. DOI: 10.1109/48.740153, https://doi.org/10.1109/48.740153
  8. Boris Katsnelson and Serguey Pereselkov, 'Horizontal refraction and sound fluctuations caused by internal waves packets in shallow water', The Journal of the Acoustical Society of America, 1999. DOI: 10.1121/1.427962, https://doi.org/10.1121/1.427962
  9. Scott D. Frank, Mohsen Badiey, James F. Lynch, and William L. Siegmann, 'Experimental evidence of three-dimensional acoustic propagation caused by nonlinear internal waves', The Journal of the Acoustical Society of America, 2005. DOI: 10.1121/1.1942428, https://doi.org/10.1121/1.1942428
  10. Mohsen Badiey, Jing Luo, Boris Katsnelson, Alexander Tskhoidze, James Lynch, and James Moum, 'Phase fluctuations and horizontal refraction of low-frequency sound signals in shallow water in presence of internal waves', The Journal of the Acoustical Society of America, 2008. DOI: 10.1121/1.4782570, https://doi.org/10.1121/1.4782570
  11. Timothy F. Duda, Ying-Tsong Lin, and James F. Lynch, 'Acoustic mode beam effects of nonlinear internal gravity waves in shallow water', The Journal of the Acoustical Society of America, 2008. DOI: 10.1121/1.2936032, https://doi.org/10.1121/1.2936032
  12. Ying-Tsong Lin, Timothy F. Duda, and James F. Lynch, 'Acoustic mode radiation from the termination of a truncated nonlinear internal gravity wave duct in a shallow ocean area', The Journal of the Acoustical Society of America, 2009. DOI: 10.1121/1.3203268, https://doi.org/10.1121/1.3203268
  13. Altan Turgut, Peter C. Mignerey, and Marshall H. Orr, 'Measured three-dimensional effects of mode-1 and mode-2 nonlinear internal waves on broadband acoustic wave propagation in shallow water', The Journal of the Acoustical Society of America, 2013. DOI: 10.1121/1.4830740, https://doi.org/10.1121/1.4830740
  14. Georges Albert Dossot, 'Acoustic fluctuations in shallow water due to nonlinear internal waves', Ph.D. dissertation, University of Rhode Island, 2011. DOI: 10.23860/diss-dossot-georges-2011, https://doi.org/10.23860/diss-dossot-georges-2011
  15. John A. Colosi, 'Acoustic mode coupling induced by shallow water nonlinear internal waves: Sensitivity to environmental conditions and space-time scales of internal waves', The Journal of the Acoustical Society of America, 2008. DOI: 10.1121/1.2956471, https://doi.org/10.1121/1.2956471
  16. John A. Colosi, Timothy F. Duda, and Andrey K. Morozov, 'Statistics of low-frequency normal-mode amplitudes in an ocean with random sound-speed perturbations: Shallow-water environments', The Journal of the Acoustical Society of America, 2012. DOI: 10.1121/1.3666002, https://doi.org/10.1121/1.3666002
  17. Kaustubha Raghukumar and John A. Colosi, 'High frequency normal mode statistics in a shallow water waveguide: The effect of random linear internal waves', The Journal of the Acoustical Society of America, 2014. DOI: 10.1121/1.4881926, https://doi.org/10.1121/1.4881926
  18. Kaustubha Raghukumar, John A. Colosi, Ying-Tsong Lin, Timothy F. Duda, Arthur Newhall, Kyle M. Becker, and Paul Hines, 'Comparison of shallow water mode transport theory with acoustic transmissions during Shallow Water Experiment 2006', The Journal of the Acoustical Society of America, vol. 135, no. 4, Supplement, p. 2260, 2014. DOI: 10.1121/1.4877408, https://doi.org/10.1121/1.4877408
  19. D. Benjamin Reeder, 'Statistics of acoustic intensity fluctuations caused by non-linear internal waves of elevation in shallow water in the South China Sea', The Journal of the Acoustical Society of America, 2009. DOI: 10.1121/1.3249369, https://doi.org/10.1121/1.3249369
  20. Fan Li, Xinyi Guo, Tao Hu, and Li Ma, 'Acoustic travel-time perturbations due to shallow-water internal waves in the Yellow Sea', Journal of Computational Acoustics, 2014. DOI: 10.1142/S0218396X14400037, https://doi.org/10.1142/S0218396X14400037
  21. Zhen Wang, Tao Hu, Qunyan Ren, Shengming Guo, and Li Ma, 'Analysis of acoustic-energy fluctuations caused by internal waves in the Yellow Sea', IEEE conference paper, 2020. DOI: 10.1109/IEEECONF38699.2020.9389180, https://doi.org/10.1109/IEEECONF38699.2020.9389180
  22. Ze-Zhong Zhang, Wen-Yu Luo, and Ren-He Zhang, 'An efficient three-dimensional coupled normal mode model and its application to internal solitary wave problems', Chinese Physics Letters, 2018. DOI: 10.1088/0256-307X/35/8/084301, https://doi.org/10.1088/0256-307X/35/8/084301
  23. Boris Katsnelson, Valery Petnikov, and James Lynch, 'Sound Field in Shallow Water with Random Inhomogeneities', in Fundamentals of Shallow Water Acoustics, Springer, 2011. DOI: 10.1007/978-1-4419-9777-7_5, https://doi.org/10.1007/978-1-4419-9777-7_5
  24. James Lynch, John A. Colosi, Timothy F. Duda, Ying-Tsong Lin, and Arthur Newhall, 'Acoustic intensity fluctuation studies in shallow water over the past quarter century', The Journal of the Acoustical Society of America, 2017. DOI: 10.1121/1.4987905, https://doi.org/10.1121/1.4987905
  25. Yanyu Jiang, Valery Grigorev, and Boris Katsnelson, 'Sound field fluctuations in shallow water due to nonlinear internal waves and determination of their parameters', Proceedings of Meetings on Acoustics, 2021. DOI: 10.1121/2.0001473, https://doi.org/10.1121/2.0001473