Dyson, Munk & Zetler (1976) showed that the phase and intensity scintillations of long-range 406-Hz continuous-wave (CW) acoustic multipath transmissions (Eleuthera to Bermuda, about 1250 km; the MIMI, Miami-Michigan, records) are governed by a single parameter, the mean-square rate of single-path phase ν2\nu^{2}; they verified parameter-free saturated-regime predictions (log-intensity variance π2/6\pi^{2}/6, i.e. 5.57 dB rms; intensity/phase rate correlation 2/π2/\pi), and computed ν\nu from the Garrett-Munk internal-wave spectrum and ray geometry with no adjustable parameters (agreement at the 25-40% level), assessing internal tides at about 10% of ν2\nu^{2} ("significant but not dominant"). The work left open the mechanistic completion of this interpretation: (1) how the semidiurnal acoustic phase energy splits among coherent barotropic surface-tide modulation along the path, incoherent internal-tide modulation distributed along the path, and locally generated internal tides near the terminals - the coherent-tide explanation matches the observed ~2.2-cycle semidiurnal amplitude only under an assumed uniform 1 cm/s tidal current, contradicted by independent current measurements of ~0.3 cm/s; (2) the origin of the low-frequency excess of observed phase spectra over the composite random-walk-plus-fade model; (3) the robustness of the parameter-free ray-geometric prediction to realistic ocean structure; (4) validation of fade statistics and spectral rolloffs against adequately sampled (~1 min cadence) records, since the 5-minute-averaged MIMI data undercount fades by factors of 3-11; and (5) the spectral redistribution of discrete tidal lines by the random internal-wave continuum. No published work closes the attribution for CW multipath scintillation spectra, the low-frequency excess, or the tide-continuum interaction.

Background

Low-frequency sound in the deep ocean travels within the SOFAR (sound fixing and ranging) channel, the waveguide formed by the sound-speed minimum at roughly 1000-1200 m depth in the mid-latitude open ocean; this trapping allows a 406-Hz continuous-wave (CW) signal to propagate coherently over ranges of order 1000 km. In the MIMI (Miami-Michigan) experiment, such transmissions from a source near Eleuthera, Bahamas, were received at Bermuda (about 1250 km range) and at a midstation (about half that range), with the received signal's slowly varying phase and intensity recorded as 5-minute averages. Because the signal arrives as the sum of many ray paths (a multipath field), the phase and intensity fluctuate strongly - acoustic scintillations - including deep fades in which the interfering phasors nearly cancel.

Dyson, Munk & Zetler (1976) modeled this with a random-walk-plus-fade picture: each single-path phase performs a Gaussian random walk (phase increments over lag τ\tau have variance ν2τ2\nu^{2}\tau^{2}, where ν2\nu^{2} is the mean-square rate of single-path phase), and many comparable paths sum to a complex Gaussian field. The model predicts that the spectra of the Cartesian components X, Y (the slowly varying quadrature components of the received pressure) are Gaussian in frequency with width set by ν\nu, and, in the saturated regime of fully developed random interference among many comparable arrivals, gives parameter-free statistics: a log-intensity variance π2/6\pi^{2}/6 (5.57 dB rms) and a correlation 2/π2/\pi between the absolute rates of intensity and phase change. Both were confirmed by the MIMI records to within about 12%. Spectral fitting gave ν−1\nu^{-1} of 270 s (Eleuthera-Bermuda) and 357 s (Eleuthera-midstation). Starting from the Garrett-Munk (GM) spectrum - the empirical canonical model of the ocean's internal gravity-wave field; internal waves are buoyancy-driven oscillations of the stratified ocean interior that perturb sound speed and hence ray phase - and from ray geometry along the path, the same authors computed theoretical values ν−1\nu^{-1} of 192 s and 286 s with no adjustable parameters, i.e. oceanographic measurements alone accounted for the observed fluctuation rates at the 25-40% level. Internal tides - internal waves at tidal frequencies, generated where the barotropic (surface) tide's currents flow over sloping topography - were assessed at about 10% of the total ν2\nu^{2}; the observed semidiurnal peak in the multipath phase spectrum (amplitude about 2.2 cycles at Bermuda, a small fraction of the total fortnightly phase variance) agreed with a coherent surface-tide estimate of about 2.3 cycles based on an assumed uniform 1 cm/s tidal current along the whole path, while the Cartesian spectra showed no tidal peak because the coherent rotation is averaged over the many paths. The companion theory paper of Munk & Zachariasen (1976) appeared in the same volume, and this line of work seeded the internal-wave fluctuation program based on path-integral methods (statistical treatments that sum phase and amplitude over ensembles of ray paths), consolidated in the 1979 Cambridge monograph edited by Flatte and, with broadband sources, ocean acoustic tomography (Munk & Wunsch 1979), as reviewed by Worcester & Munk (2009).

What the 1976 analysis left open - the problem posed here - is the mechanistic completion of that interpretation: which oceanographic process actually produces which part of the observed fluctuation spectra, whether the first-principles prediction survives realistic ocean structure, and whether the model's statistical predictions survive adequately sampled observations. The paper's own evidence made the tidal attribution genuinely open: the coherent surface-tide explanation matched the observed semidiurnal amplitude to within 5% but required a uniform 1 cm/s current, while independent current measurements from the MODE (Mid-Ocean Dynamics Experiment) moorings indicated only about 0.3 cm/s along the relevant azimuth, and the acoustic phase epochs varied inconsistently between fortnights.

Problem Statement

Complete, quantitatively and mechanistically, the interpretation of long-range deep-ocean acoustic multipath scintillation spectra in terms of internal waves and tides, as posed but left open by Dyson, Munk & Zetler (1976) on the 406-Hz MIMI Eleuthera-Bermuda and Eleuthera-midstation transmissions. For the same physical setting (long-range deep-ocean CW multipath transmission, with the MIMI records and any comparable new records as the observational base), the open questions are:

(1) Tidal attribution: a quantitative decomposition of the semidiurnal acoustic phase energy among the three candidate mechanisms - (a) coherent modulation by the barotropic surface tide along the entire path, (b) incoherent modulation by internal (baroclinic) tides distributed along the path, and (c) strong local internal-tide generation near the terminals - with uncertainty estimates, tested against independent oceanographic constraints (measured tidal currents; internal-tide amplitudes, phases, and coherence scales). The 1976 evidence leaves this open: the coherent surface-tide model reproduces the observed semidiurnal amplitude (about 2.2 cycles) only under an assumed uniform 1 cm/s tidal current, whereas independent current measurements indicate about 0.3 cm/s along the path azimuth and the acoustic phase epochs vary inconsistently between fortnights.

(2) Low-frequency excess: an identification and quantitative model of the excess phase energy observed at low frequencies in the multipath phase spectra relative to the composite random-walk-plus-fade spectral forms - for example coherent modulations by large-scale ocean features or other unresolved low-frequency dynamics - or a demonstration that the excess is an artifact of the sampling/analysis protocol.

(3) Robustness of the first-principles prediction: a test of how the parameter-free prediction of the single-path phase-rate parameter ν\nu from the internal-wave spectrum and ray geometry changes when the canonical ocean model (exponential stratification, canonical sound channel, Garrett-Munk internal-wave spectrum, idealized ray mix) is replaced by realistic structure - measured stratification and internal-wave statistics, actual water-mass structure along the path, and a realistic ray mix - given later evidence for order-unity Garrett-Munk violations in the Bermuda region.

(4) Observational validation: a test of the model's fade statistics (occurrence and duration) and spectral rolloffs (including the high-frequency ω−3\omega^{-3} tails) against multipath records sampled at a cadence fine enough (of order 1 min or finer, per the νδt≈0.1\nu\delta t\approx 0.1 criterion) and long enough to avoid the fade undercount (factors of 3-11) and the high-frequency aliasing inherent in the 5-minute-averaged MIMI records.

(5) Tide-continuum interaction: a quantitative treatment of how a discrete tidal line in the acoustic modulation is redistributed spectrally by the random internal-wave field - the interaction of a periodic tidal input with the stochastic internal-wave continuum - which the source acknowledged but did not treat.

These five results jointly support one overarching objective: a mechanism-resolved, validated account of long-range multipath scintillation spectra. The attributed contributions are internally consistent only if they sum, within stated uncertainties, to the observed phase and intensity spectra, including the semidiurnal band and the low-frequency behavior, for the long-range deep-ocean CW multipath setting; the robustness test (3), the observational validation (4), and the tide-continuum treatment (5) enter that consistency claim rather than standing outside it.

References

  1. F. Dyson, W. Munk, and B. Zetler, Interpretation of multipath scintillations Eleuthera to Bermuda in terms of internal waves and tides, The Journal of the Acoustical Society of America 59(5), 1121-1133 (1976). DOI: 10.1121/1.380966, https://doi.org/10.1121/1.380966 (primary source)
  2. W. H. Munk and F. Zachariasen, Sound propagation through a fluctuating stratified ocean: Theory and observation, The Journal of the Acoustical Society of America 59(4), 818-838 (1976). DOI: 10.1121/1.380933, https://doi.org/10.1121/1.380933
  3. W. Munk, B. Zetler, J. Clark, S. Gill, D. Porter, J. Spiesberger, and R. Spindel, Tidal effects on long-range sound transmission, Journal of Geophysical Research 86(C7), 6399-6410 (1981). DOI: 10.1029/JC086IC07P06399, https://doi.org/10.1029/JC086IC07P06399
  4. B. D. Dushaw, Mode-1 internal tides in the western North Atlantic Ocean, Deep-Sea Research Part I 53(3), 449-473 (2006). DOI: 10.1016/j.dsr.2005.12.009, https://doi.org/10.1016/j.dsr.2005.12.009
  5. R. B. Stoughton, S. M. Flatte, and B. M. Howe, Acoustic measurements of internal wave rms displacement and rms horizontal current off Bermuda in late 1983, Journal of Geophysical Research 91(C6), 7721-7732 (1986). DOI: 10.1029/JC091IC06P07721, https://doi.org/10.1029/JC091IC06P07721
  6. J. A. Colosi, A. B. Baggeroer, B. D. Cornuelle, M. A. Dzieciuch, W. H. Munk, P. F. Worcester, B. D. Dushaw, B. M. Howe, J. A. Mercer, R. C. Spindel, T. G. Birdsall, K. Metzger, and A. M. G. Forbes, Analysis of multipath acoustic field variability and coherence in the finale of broadband basin-scale transmissions in the North Pacific Ocean, The Journal of the Acoustical Society of America 117(3), 1538-1564 (2005). DOI: 10.1121/1.1854615, https://doi.org/10.1121/1.1854615
  7. A. Fredricks, J. A. Colosi, J. F. Lynch, G. Gawarkiewicz, C.-S. Chiu, and P. Abbot, Analysis of multipath scintillations from long range acoustic transmissions on the New England continental slope and shelf, The Journal of the Acoustical Society of America 117(3), 1038-1057 (2005). DOI: 10.1121/1.1835510, https://doi.org/10.1121/1.1835510
  8. K. B. Yoo, Shallow water temporal coherence of broadband acoustic signals in the presence of internal waves, IEEE Journal of Oceanic Engineering 30(2), 391-405 (2005). DOI: 10.1109/JOE.2005.850867, https://doi.org/10.1109/JOE.2005.850867
  9. T. C. Yang, Temporal coherence of sound transmissions in deep water revisited, The Journal of the Acoustical Society of America 124(1), 113-127 (2008). DOI: 10.1121/1.2932337, https://doi.org/10.1121/1.2932337
  10. P. F. Worcester and W. H. Munk, Sound transmission through a fluctuating ocean (1979) and ocean acoustic tomography (1995): An intertwined history, The Journal of the Acoustical Society of America 126(4), 2157 (2009). DOI: 10.1121/1.3248383, https://doi.org/10.1121/1.3248383
  11. B. Cho and N. C. Makris, Predicting the Effects of Random Ocean Dynamic Processes on Underwater Acoustic Sensing and Communication, Scientific Reports 10, 4525 (2020). DOI: 10.1038/s41598-020-61043-w, https://doi.org/10.1038/s41598-020-61043-w
  12. B. Cho, Small-sample unbiased linear coherence estimators for a complex Gaussian random process, The Journal of the Acoustical Society of America 159, 5707-5728 (2026). DOI: 10.1121/10.0044185, https://doi.org/10.1121/10.0044185
  13. J. A. Colosi, Sound Propagation through the Stochastic Ocean, Cambridge University Press (2016). DOI: 10.1017/cbo9781139680417, https://doi.org/10.1017/cbo9781139680417
  14. W. Munk and C. Wunsch, Ocean acoustic tomography: a scheme for large scale monitoring, Deep-Sea Research 26(2), 123-161 (1979). DOI: 10.1016/0198-0149(79)90073-6, https://doi.org/10.1016/0198-0149(79)90073-6
  15. W. Kuperman, review of Sound Transmission Through a Fluctuating Ocean (edited by S. M. Flatte, Cambridge University Press, 1979), The Journal of the Acoustical Society of America 68, 1009 (1980). DOI: 10.1121/1.384887, https://doi.org/10.1121/1.384887
  16. A. Shmelev, Three-Dimensional Acoustic Propagation Through Shallow Water Internal, Surface Gravity and Bottom Sediment Waves, doctoral thesis, MIT/WHOI Joint Program in Oceanography, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution (2011). DOI: 10.1575/1912/4860, https://doi.org/10.1575/1912/4860