The analysis of the field excited in a waveguide by a point noise source is performed using the phase space representation of this field given by the distribution of its amplitude in the depth–angle–time space. The transition from the traditional description of the field amplitude as a function of depth and time to phase space representation is performed using the coherent state expansion developed in quantum mechanics. In this paper, the correlation function of noise signals arriving at different points of the phase plane depth–angle is investigated. Numerical simulation data show that measurements of signal correlations in phase space, performed with the help of a receiving vertical antenna, can be used as input data in solving the problem of source localization and reconstruction of unknown parameters of the sound speed profile. It is shown that in phase space there is an analog of the classical interference pattern observed in the distribution of sound intensity in the distance–frequency plane. The slopes of striations in this interference pattern, as in the conventional one, are given by the Chuprov waveguide invariant.

1.
L. M.
Brekhovskikh
and
Y.
Lysanov
,
Fundamentals of Ocean Acoustics
(
Springer-Verlag
,
New York
,
2003
), p.
278
.
2.
W.
Munk
and
C.
Wunsch
, “
Ocean acoustic tomography: A scheme for large scale monitoring
,”
Deep-Sea Res.
26
,
123
161
(
1979
).
3.
ATOC Consortium
, “
Ocean climate change: Comparison of acoustic tomography, satellite altimetry, and modeling
,”
Science
281
,
1327
1332
(
1998
).
4.
P. F.
Worcester
,
B. D.
Cornuelle
,
M. A.
Dzieciuch
,
W. H.
Munk
,
M.
Howe
,
A.
Mercer
,
R. C.
Spindel
,
J.
Colosi
,
A.
Metzger
,
T.
Birdsall
, and
A. B.
Baggeroer
, “
A test of basin-scale acoustic thermometry using a large-aperture vertical array at 3250-km range in the eastern north pacific ocean
,”
J. Acoust. Soc. Am.
105
,
3185
3201
(
1999
).
5.
F. B.
Jensen
,
W. A.
Kuperman
,
M. B.
Porter
, and
H.
Schmidt
,
Computational Ocean Acoustics
(
Springer
,
New York
,
2011
).
6.
S. D.
Chuprov
, “
Interference structure of a sound field in a layered ocean
,” in
Ocean Acoustics, Current Status
, edited by
L. M.
Brekhovskikh
and
I. B.
Andreyeva
(
Nauka
,
Moscow
,
1982
).
7.
G. L.
D'Spain
and
W. A.
Kuperman
, “
Application of waveguide invariants to analysis of spectrograms from shallow water environments that vary in range and azimuth
,”
J. Acoust. Soc. Am.
106
(
5
),
2454
2468
(
1999
).
8.
C. H.
Harrison
, “
The relation between the waveguide invariant, multipath impulse response, and ray cycles
,”
J. Acoust. Soc. Am.
129
(
5
),
2863
2877
(
2011
).
9.
A. L.
Virovlyansky
, “
Stable components of sound fields in the ocean
,”
J. Acoust. Soc. Am.
141
(
2
),
1180
1189
(
2017
).
10.
A. L.
Virovlyansky
,
A.
Kazarova
, and
L.
Lyubavin
, “
Matched field processing in phase space
,”
J. Ocean Eng.
45
(
4
),
1583
1593
(
2020
).
11.
R. J.
Glauber
,
Quantum Theory of Optical Coherence. Selected Papers and Lectures
(
Wiley-VCH
,
Weinheim
,
2007
), p.
639
.
12.
J. R.
Klauder
and
E. C. G.
Sudarshan
,
Fundamentals of Quantum Optics
(
W.A. Benjamin
,
New York
,
1968
), p.
304
.
13.
W. P.
Schleich
,
Quantum Optics in Phase Space
(
Wiley-VCH
,
Berlin
,
2001
), p.
685
.
14.
A. L.
Virovlyansky
, “
Isolation of the field component formed by a given beam of rays at the aperture of a receiving antenna in an inhomogeneous environment
,”
Phys. Usp.
66
(
9
),
951
960
(
2023
).
15.
A. L.
Virovlyansky
and
A.
Kazarova
, “
The sound field intensity distribution in the deep sea in the depth-angle-time phase space
,”
Acoust. Phys.
69
(
5
),
604
615
(
2023
).
16.
H.
Goldstein
,
C. P.
Poole
, and
J. L.
Safko
,
Classical Mechanics
(
Addison-Wesley
,
San Francisco, CA
,
2000
), p.
647
.
17.
A. L.
Virovlyansky
,
A. Y.
Kazarova
, and
B. G.
Katsnelson
, “
Phase space representation of sound fields in Lake Kinneret
,”
IEEE J. Oceanic Eng.
48
(
4
),
1174
1186
(
2023
).
18.
D. V.
Makarov
, “
On measurement of acoustic pulse arrival angles using a vertical array
,”
Acoust. Phys.
63
(
6
),
673
680
(
2017
).
19.
D. V.
Makarov
and
L. E.
Kon'kov
, “
Angular spectrum of acoustic pulses at long ranges
,”
J. Mar. Sci. Eng.
11
,
29
(
2023
).
20.
J. A.
Colosi
,
Sound Propagation through the Stochastic Ocean
(
Cambridge University Press
,
New York
,
2016
), p.
420
.
21.
S. M.
Flatte
,
R.
Dashen
,
W. M.
Munk
,
K. M.
Watson
, and
F.
Zakhariasen
,
Sound Transmission Through a Fluctuating Ocean
(
Cambridge University Press
,
Cambridge, UK
,
1979
).
22.
J. A.
Colosi
and
M. G.
Brown
, “
Efficient numerical simulation of stochastic internal-wave-induced sound-speed perturbation field
,”
J. Acoust. Soc. Am.
103
,
2232
2235
(
1998
).
23.
J. A.
Colosi
,
T. F.
Duda
, and
A. K.
Morozov
, “
Statistics of low-frequency normal-mode amplitudes in an ocean with random sound-speed perturbations: Shallow-water environments
,”
J. Acoust. Soc. Am.
131
,
1749
1761
(
2012
).
24.
C. W.
Gardiner
,
Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences
(
Springer-Verlag
,
New York
,
1985
).
25.
D.
Makarov
,
S.
Prants
,
A.
Virovlyansky
, and
G.
Zaslavsky
,
Ray and Wave Chaos in Ocean Acoustics
(
Word Scientific
,
Hackensack, NJ
,
2010
), p.
389
.
26.
M. A.
Alonso
, “
Rays and waves
,” in
Phase-Space Optics
, edited by
M.
Testorf
,
B.
Hennely
, and
J.
Ojeda-Castaneda
(
McGraw-Hill
,
New York
,
2010
), Chap. 8, pp.
237
277
.
27.
L. D.
Landau
and
E. M.
Lifshitz
,
Mechanics
, 3rd ed. (
Pergamon Press
,
Oxford
,
1976
).
28.
L. D.
Landau
and
E. M.
Lifshitz
,
Quantum Mechanics
(
Pergamon Press
,
Oxford
,
1977
).
29.
H. P.
Bucker
, “
Use of calculated sound fields and matched-field detection to locate sound sources in shallow water
,”
J. Acoust. Soc. Am
59
(
2
),
368
373
(
1976
).
30.
M. J.
Hinich
, “
Maximum-likelihood signal processing for a vertical array
,”
J. Acoust. Soc. Am.
54
(
2
),
499
503
(
1973
).
31.
A. B.
Baggeroer
,
W. A.
Kuperman
, and
P. N.
Mikhalevsky
, “
Matched field signal processing in underwater sound channels (review)
,”
J. Ocean. Eng.
18
(
4
),
401
424
(
1993
).
32.
A.
Sazontov
and
A.
Malekhanov
, “
An overview of matched field methods in ocean acoustics
,”
Acoust. Phys.
61
(
2
),
213
230
(
2015
).
33.
A. L.
Virovlyansky
, “
Beamforming and matched field processing in multipath environments using stable components of wave fields
,”
J. Acoust. Soc. Am.
148
(
4
),
2351
2360
(
2020
).
You do not currently have access to this content.