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Variability of phase and amplitude fronts due to horizontal refraction in shallow water

Boris G. Katsnelson, Valery A. Grigorev, and James F. Lynch

Citation: The Journal of the Acoustical Society of America 143, 193 (2018);

View online: https://doi.org/10.1121/1.5020274

View Table of Contents: http://asa.scitation.org/toc/jas/143/1

Published by the Acoustical Society of America

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Variability of phase and amplitude fronts due to horizontal

refraction in shallow water

Boris G. Katsnelson

a)

Leon Charney School of Marine Sciences, University of Haifa, Mount Carmel, Haifa 3498838, Israel

Valery A. Grigorev

Voronezh State University, Universitetskaya sq 1, Voronezh 394018, Russia

James F. Lynch

Woods Hole Oceanographic Institution, 98 Water Street, MS No.12, Woods Hole, Massachusetts 02543, USA

(Received 16 July 2017; revised 8 December 2017; accepted 13 December 2017; published online

16 January 2018)

The variability of the interference pattern of a narrow-band sound signal in a shallow water wave-

guide in the horizontal plane in the presence of horizontal stratiﬁcation, in particular due to linear

internal waves, is studied. It is shown that lines of constant phase (a phase front) and lines of con-

stant amplitude/envelope (an amplitude front) for each waveguide mode may have different direc-

tions in the spatial vicinity of the point of reception. The angle between them depends on the

waveguide’s parameters, the mode number, and the sound frequency. Theoretical estimates and

data processing methodology for obtaining these angles from experimental data recorded by a hori-

zontal line array are proposed. The behavior of the angles, which are obtained for two episodes

from the Shallow Water 2006 (SW06) experiment, show agreement with the theory presented.

V

C2018 Acoustical Society of America.https://doi.org/10.1121/1.5020274

[JAC] Pages: 193–201

I. INTRODUCTION

A variability of the oceanic waveguide’s parameters in

the horizontal plane (due to bathymetry or the sound speed

proﬁle) lead to the set of effects in sound propagation, which

are called “horizontal refraction,” or more generally “3D

effects.” These effects have been studied theoretically and

experimentally both in the deep ocean (Collins et al., 1995)

and in shallow water (Heaney and Murray, 2009;

Katsnel’son et al., 2007). As has been shown in numerous

past papers, there are a number of acoustical manifestations

of horizontal refraction due to coastal wedges. The ﬁrst

observation and measurement of horizontal angle effects

was carried out in a coastal wedge by (Doolittle et al., 1988).

Subsequently, the following phenomena were studied: focus-

ing/defocusing of the sound ﬁeld in the horizontal plane in

the presence of nonlinear internal waves (Katsnelson and

Pereselkov, 2000;Badiey et al., 2005); variations of the

interference pattern in the horizontal plane in the area of a

coastal wedge in (Deane and Buckingham, 1993;Katsnelson

et al., 2013), or in the area of the underwater canyon in

(Duda et al., 2011;Y.-T. Lin, 2013); multipath propagation

in the horizontal plane in an experiment in the Florida Strait

(Heaney and Murray, 2009); and many others. In the papers

(Bender et al., 2014;Ballard et al., 2012), the importance of

taking horizontal refraction into account for inverse prob-

lems was discussed as well.

The quantitative description of sound propagation in

shallow water in the presence of horizontal inhomogeneities

can be based upon the decomposition of the complex

amplitude of the sound ﬁeld Pðr;zÞat some frequency xinto

waveguide modes (vertical modes) wnðz;rÞwhich depend

upon the horizontal coordinates r¼ðx;yÞ, as well as on ver-

tical parameters (zis the vertical coordinate directed down-

ward, and z¼0 corresponds to the surface). Modal

amplitudes Anðr;xÞin the sound ﬁeld decomposition satisfy

a wave equation which can be solved either in the ray

approximation (horizontal rays and vertical modes)

(Weinberg and Burridge, 1974) or using the parabolic equa-

tion method (Collins et al., 1995;Colosi, 2016).

One of the key features of these phenomena is the fre-

quency dependence of the modal amplitude Anðr;xÞ(in par-

ticular, the amplitude and phase of the horizontal rays) and,

in turn, the interference pattern of the sound ﬁeld in the hori-

zontal plane. In other words, after separating out the vertical

coordinate z, a two-dimensional (x-y plane) wave equation is

obtained, which describes wave propagation in a two-

dimensional dispersive medium.

Given this background, the goal of this paper is to study

the speciﬁc physical features of the interference pattern due

to its frequency dependence or, more speciﬁcally, the behav-

ior of the phase and amplitude fronts of a sound pulse propa-

gating in a shallow water waveguide. Such frequency

dependence in the presence of horizontal refraction plays

signiﬁcant role in shallow water acoustics (Katsnelson and

Pereselkov, 2004;Dong et al., 2015).

II. PHASE AND AMPLITUDE FRONTS IN THE

INTERFERENCE PATTERN

Let us consider a narrow-band sound signal propagating

from a point source in shallow water with parameters (for

example, the sound speed proﬁle and/or bathymetry)

a)

Electronic mail: bkatsnels@univ.haifa.ac.il

J. Acoust. Soc. Am. 143 (1), January 2018 V

C2018 Acoustical Society of America 1930001-4966/2018/143(1)/193/9/$30.00

depending on the horizontal (x-y) coordinates, and being

received by some receiving system. The complex sound

ﬁeld, which depends on both time and the spatial point ðr;zÞ

has the form

Pðr;z;tÞ¼ðX

n

Anðr;xÞwnðz;r;xÞexp ðixtÞdx;

(1)

where wnðz;r;xÞare the waveguide modes, which depend

weakly on the horizontal coordinates as adiabatic modes and

smoothly upon the frequency (excluding the region near the

cutoff frequency). The corresponding modal eigenvalues qn

(propagation constants) depend on rand xas well:

qn¼qnðr;xÞ.

If the spectrum of the signal is narrow enough, i.e.,

x1<x<x2, where Dx¼x2x1x, then the eigen-

function, due to the smooth frequency dependence of the

mode shape, can be taken outside the integral (1) at some

central frequency x1<x0<x2,

Pðr;z;tÞ¼X

n

wnðz;r;x0ÞðAnðr;xÞexp ðixtÞdx

¼X

n

wnðz;r;x0ÞPnðr;tÞ;(2)

where we have introduced the modal amplitude, which

depends on the horizontal coordinates and time, as

Pnðr;tÞ¼ðAnðr;xÞexp ðixtÞdx:(3)

Neglecting mode coupling, Anðr;xÞsatisﬁes the two-

dimensional Helmholtz equation where the wave number is

qnðr;xÞ. Using standard techniques, it is possible to ﬁnd the

spectral amplitude Anðr;xÞin the ray approximation

Anðr;xÞ¼anðr;xÞexp isnðr;xÞ½;(4)

where snðr;xÞis the eikonal in the horizontal plane, corre-

sponding to the mode n, which can be found from the two

dimensional eikonal equation

r?snðr;xÞ½

2¼q2

nðr;xÞ;(5)

where r?¼ð@=@x;@=@yÞis the gradient in the horizontal

plane.

Due to the frequency dependence of the eigenvalues on

the right side of Eq. (5), the horizontal rays have different

paths for different frequencies On the basis of ray theory, it is

possible to ﬁnd horizontal rays joining the source and receiver

(the so-called eigenrays). The frequency dependence of the

ray path is the main peculiarity of the ray pattern in our given

case, in other words eigenrays corresponding to different fre-

quencies have different trajectories [Fig. 1(a)].

Next, we will analyze the sound ﬁeld of a particular

mode with index number n; for brevity, we will omit show-

ing this index number explicitly in the following discussion.

If we introduce a vector q:(jqj¼q), tangent to the horizon-

tal ray path, then at the locations of the source and receiver,

the vectors qðxÞ, corresponding to different frequencies,

will be different both in modulus and direction. In the Fig.

1(a), the vectors q1and q2are shown for frequencies x1and

x2at the location of the receiver.

It is seen that, for the sound ﬁeld of a narrowband signal

in the neighborhood of the receiver, an interference pattern

is formed. For this pattern, a rather simple illustration can be

presented for the superposition of two horizontal rays

(locally plane waves). To illustrate the interference pattern

in the horizontal plane formed by two such rays with wave

vectors q1and q2, corresponding to two close frequencies

x1and x2[Fig. 1(a)], also with equal amplitudes for sim-

plicity, coming into the receiver, we write the sound ﬁeld as

Pðr;tÞ¼Aexp iðq1rx1tþu1Þ½

þAexp iðq2rx2tþu2Þ½

¼Aðpr tÞexp iðqr xtþuÞ½;(6)

where u1and u2are some phases, q¼ðq2þq1Þ=2,

x¼ðx2þx1Þ=2, p¼Dq=Dx,u¼ðu2þu1Þ=2, Dq¼q2

q1,Dx¼x2x1,Du¼u2u1, and Aðpr tÞ¼2A

cos ½ðDxðpr tÞþDuÞ=2.

So, in the vicinity of the receiver the sound ﬁeld is mod-

ulated in space and time, with carrier frequency xand com-

plex amplitude Aðpr tÞ. The scales of this modulation in

the horizontal plane, i.e., the interference pattern similar to

Fig. 1(b), depends upon the parameters of the waveguide.

Let us now introduce a phase front (lines of constant phase)

qr xt¼constant, an amplitude front (line of constant

amplitude) pr t¼constant, and normals to these lines,

i.e., the vectors q(the direction of propagation of the phase

front) and p(the direction of propagation of the amplitude

front).

If the spectrum of the source is contained in the band

x1<x0<x2, then the wave vectors of the horizontal rays

coming into the receiver will be located in the sector between

FIG. 1. (a) Two horizontal rays for dif-

ferent frequencies coming into a

receiver. (b) Directions of the ampli-

tude and phase fronts (wave vectors p

and q). (c) Spatial Fourier transform in

the horizontal plane.

194 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.

q1and q2. We can get an expression for the received signal

by integration of Eq. (3) assuming a narrow spectrum where

we can use a linear decomposition of the wave vector: q0

¼qðx0Þ¼qðxÞþpðxÞðx0xÞ¼qþpðx0xÞ,where

xis the carrier frequency, for example x¼ðx2þx1Þ=2,

pðxÞ¼dq=dx. So the received signals have the form

Pðr;tÞ¼ðx2

x1

Aðr;x0Þexp iðq0rx0tþu0Þ

dx0

¼Aðpr tÞexp iðqr xtÞ½;(7)

where the amplitude is Aðpr tÞ¼Ðx2

x1Aðr;x0Þexp

½iðx0xÞðpr tÞþiu0dx0. In particular, assuming that

within this band the spectral amplitude is constant, i.e.,

Aðr;x0Þ¼A, we get

Apr t

ðÞ

¼2ADxsin Dxpr t

ðÞ

Dxpr t

ðÞ

:(8)

The time duration of the envelope of this signal is 2p=Dx,

and the spatial scale of the envelope in the direction of prop-

agation is 2p=jDqj. The interference pattern of the signal

in the horizontal plane has rather complex character when

the lines of constant amplitude have a different direction

than the lines of constant phase. Waves of this type are

called inhomogeneous (Born and Wolf, 1968), and are simi-

lar to those presented in Fig. 1(b). [A similar interference

pattern has been observed for a wedge model by Katsnelson

et al. (2013).]

We note that if the spectral components in Eqs. (6) and

(7) have different amplitudes, then the result of their summa-

tion has a more complex form than in the right side of Eqs.

(7) or (8), and in that case we will deﬁne the amplitude front

as a line (or some area in the horizontal plane) of maximal

amplitude.

III. THEORETICAL ESTIMATES OF AMPLITUDE AND

PHASE FRONT VARIATIONS

Let us consider theoretical and numerical estimations of

the possible angles for the amplitude front (AF) and phase

front (PF) directions within the framework of a simpliﬁed

model for typical shallow water conditions.

As an example, consider sound propagation in the

Shallow Water 2006 experiment (Newhall et al., 2007)in

the presence of linear internal waves with small amplitude,

having some anisotropy, and propagating mainly toward the

coast. This situation has been seen to take place before and

after the passage of a train of intense nonlinear internal

waves [see, for example, Sabinin and Serebryani (2005)].

Let the source and receiver be placed at a distance Drel-

ative to each other along the raxis (Fig. 2), which is approxi-

mately parallel to the coastal line. For SW06, D2104

m. The unperturbed sound speed proﬁle cðzÞis characterized

by a rather sharp thermocline, with thickness much less than

the water depth, that is htH(for SW06 ht=H0:1). The

displacement of the isodensity surface from its equilibrium

shape (plane) fðr;zÞcharacterizes the internal waves and the

corresponding variation of the sound speed proﬁle cðr;zÞ

¼cðzÞþdcðr;zÞ. The correction to the sound speed proﬁle

can be written (Flatte, 1979)

dcr;z

ðÞ

cz

ðÞ ¼QN2z

ðÞ

fr;z

ðÞ

;(9)

where NðzÞis the Brunt-Vaisala frequency (buoyancy fre-

quency), and the parameter Qdepends on the water column

properties. In shallow water Q2:4s

2

/m.

For linear internal waves, the gradient of the displace-

ment in the horizontal plane c¼r

?fðr;zÞhas the value

jcj0:01 and the perturbation of the sound speed in the area

of the thermocline due to the displacement is jdcj1m/s.

The eigenvalue of the nth acoustic mode qðr;xÞin the

presence of the internal waves can be found using perturba-

tion theory

qðr;xÞ¼q0ðxÞþdqðr;xÞ;(10)

where the ﬁrst order correction dqðr;xÞhas the form

(Katsnelson et al., 2012)

dqr;x

ðÞ

¼Qk2

q0f0r

ðÞ

ðH

0

w0z

ðÞ

hi

2

N2z

ðÞ

Uz

ðÞ

dz;(11)

where fðr;zÞ¼f0ðrÞUðzÞ,f0ðrÞis the displacement of the

point corresponding to maximal amplitude, UðzÞis the ﬁrst

mode of the internal gravity wave normalized to its maximal

value maxUðzÞ¼1, w0ðzÞand q0are the eigenfunction and

the eigenvalue of the mode nin the absence of an internal

wave (mode number omitted), and k¼x=~

c, where ~

cis

some average representative value of the sound speed in

water.

dqðr;xÞ=q0ðxÞis the ﬁrst order correction to the refrac-

tive index in the horizontal plane, determining the trajectory

of the ray paths from Eq. (5), and depending on mode num-

ber and frequency.

Let us next suppose that the displacement fdepends on

a coordinate s, transverse to the acoustic track (Fig. 2).

Further, in the area of the ray paths (which have small hori-

zontal angles in our problem) this displacement depends

only linearly on s:fðr;zÞ¼cs, in the sense of a Taylor

expansion. In this case, the correction to the refraction index

in horizontal plane has the form

FIG. 2. Eigenrays joining source S and receiver R for two frequencies. We

deﬁne positive angles in the counterclockwise direction. IW denotes the direc-

tion of motion of an internal wave. Estimates based upon references

(Katsnelson et al.,2012;Katsnelson et al., 2013;Weinberg and Burridge, 1974)

give: jujjaj12;jDbj14;jDuj0:1for Dx10 Hz.

J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al. 195

dqr;x

ðÞ

q0x

ðÞ ex

ðÞ

s;(12)

e¼cQk2

q0

ðÞ

2ðH

0

w0z

ðÞ

hi

2

N2z

ðÞ

Uz

ðÞ

dz:(13)

In some cases, in particular for SW06, it is possible to

approximate that the function under integral (13) is concen-

trated within the thermocline of thickness ht. Rough estima-

tion of the integral in Eq. (13) gives N2

0ht=H, where N0is the

value of the Brunt-Vaisala frequency inside the thermocline

layer (its maximal value). The Brunt-Vaisala frequency can

be expressed via the vertical gradient of temperature T:N2ðzÞ

¼ðg=qÞðdq=dzÞ¼bTgðdT=dzÞ, where g10 m/s

2

,qis

the density of water, and bT1:3104. 1/K is the water

thermal compressibility coefﬁcient. The variation of salinity

in our situation is negligible.

So, in the area of the thermocline, where the temperature

gradient can be 1 K/m, the maximal value of the squared

Brunt-Vaisala frequency is estimated as: N2

0ð11:5Þ

103Hz

2

. Neglecting the frequency dependence, which is

small for a narrow-band signal, jejjcjQN2

0ht=Hð2:54Þ

106v

1

.

All the angles we deal with are deﬁned in Fig. 2.

Positive values of angles are in the counterclockwise direc-

tion. It is possible to estimate the horizontal angle uof an

arriving ray relative to the acoustic track (Fig. 2) using a

well known expression for the ray cycle distance D

(Katsnelson et al., 2012), assuming a linear dependence of

the refractive index on the coordinate sonly Eq. (12),

D¼2 tan u

e2u

e:(14)

Thus, as a result

uDe

2:(15)

For the distance 20 km, juj1–2. The sign of this angle

is determined by the gradient of the soundspeed along the s

axis. So, in Fig. 2we have u<0 for @v=@s>0 (if e>0).

Let us next ﬁnd the angle between horizontal eigenrays

corresponding to different frequencies for the same vertical

mode. Vectors qðxÞand qðxþDxÞcome to the location of

the receiver at different angles uðxÞand uðxþDxÞ

¼uðxÞþDu. The angle Ducan be estimated as

Du¼D

2

de

dxDx:(16)

To estimate the value de=dx, we use the expression

(Katsnelson et al., 2012) for the connection between the

phase and group velocities of a waveguide mode, which in

simpliﬁed form is

q

x

@q

@x1

~

c2:(17)

For the Pekeris model, Eq. (17) is an exact expression where

in the right hand side one sees the sound speed in the water

layer. In our more general case, the right side is an average

value of the sound speed over the depth and Eq. (17) is an

estimate. Next, using Eqs. (12) and (17) we see

de

dx¼ d

dx

1

q

@q

@s

¼1

q2

@q

@x

@q

@s1

q

@

@s

@q

@x

2xe

q2~

c22e

x:(18)

Thus we get for Du, from Eq. (16) corresponding to the fre-

quency interval Dx,

Du¼De

xDx¼2u

xDx:(19)

It follows from Eq. (19) that for horizontal eigenrays which

are close to straight lines, the arriving and outgoing angles

decrease with increasing frequency.

For a distance of 20 km, a main pulse frequency f¼x=

ð2pÞ¼300Hz, and a frequency band Df¼Dx=ð2pÞ¼10Hz,

we estimate that jDuj0:1, and we also can write Du=Df

0:01 deg/Hz.

Next, let us estimate the angle Dbbetween the ampli-

tude and phase fronts (Fig. 2), or equivalently between the

vectors p¼Dq=Dxand q, using the triangle formed by the

vectors qand qþDqfor frequencies xand xþDx. In this

case, taking into account the smallness of /,

Dbq

DqDu2uq

Dq

Dx

x:(20)

The factor containing ðq=DqÞðDx=xÞis close to the ratio of

the modal group velocity to the modal phase velocity. If we

estimate the group velocity to phase velocity ratio roughly as

1, then Db2u, which leads to the estimate of the angle

between the phase and amplitude fronts as jDbj14.

The angle a(Fig. 2), determining the direction of the

amplitude front (vector p) relative to the direction of the

acoustic track, is

a¼uþDb;(21)

or roughly au. It is interesting to note that the normal

vector to the amplitude front is directed to the other side, rel-

ative to the acoustic track, than the normal vector to the

phase front. We remark that this is the result of the rather

idealized model used for our estimates; in a more realistic

situation the ray pattern, and in turn the connection between

angles, can be much more complex.

We should also note that the above-mentioned esti-

mates, which display increasing effects of horizontal refrac-

tion with decreasing frequency, are obtained assuming a

linear dependence of the displacement fand are valid for ray

paths in this assumed region.

IV. A METHOD TO FIND THE ANGLE BETWEEN THE

AMPLITUDE AND PHASE FRONTS FROM

EXPERIMENTAL DATA, AND ANALYSIS OF SW06

RESULTS

Let us now consider how to ﬁnd the values of the afore-

mentioned angles on the basis of experimental acoustical

196 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.

data. Information about the details of the interference pattern

in the horizontal plane (for example, the directions of the

normals to the amplitude front and the phase front) can be

obtained using a spatial two-dimensional Fourier transform

(FT). For ﬁxed time and depth, for example near the bottom,

the amplitude of the sound ﬁeld is Pðr;H;t0Þ¼Pðx;yÞ(sup-

pressing depth Hand time t

0

) and the FT has the form

Gðqx;qyÞ¼ððPðx;yÞexp iðqxxþqyyÞ

dxdy

:

(22)

A typical spatial FT in the plane ðqx;qyÞfor the sound

ﬁeld formed by close horizontal rays is shown in Fig. 1(c),

where the parameters of the problem (vectors qand p) and

the corresponding angles are denoted.

The spatial spectrum produced by Eq. (22) is a result of

integrating the sound ﬁeld in the horizontal plane, but by

using dispersion relationships for the waveguide modes it is

possible to get Gðqx;qyÞdirectly from experimental data

measured by an L-shaped array.

Let us denote direction of the horizontal line array

(HLA) as the xaxis, which means that the set of observa-

tional data can be represented as the function Pðx;tÞ. As the

ﬁrst step, we construct the space-time spectrum of the

received signal depending on the component qxand the fre-

quency x,

Gðqx;xÞ¼ððPðx;tÞexp iðqxxxtÞ½dxdt

:(23)

Let us next use the dispersion relation giving the con-

nection between the components of the vector q,

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q2

xþq2

y

q¼qðxÞ;(24)

where qðxÞ, as a function of frequency, can be obtained from

the Sturm-Liouville eigenvalue problem for the vertical cross-

section at the receiver position. From Eq. (24), we can express

frequency as a function of the components ðqx;qyÞand after

substitution into Eq. (23), it is possible to get Gðqx;qyÞand in

turn to estimate all the aforementioned angles.

We next apply this methodology to data processing and

analysis of the SW06 results.

As an example, consider two episodes from the Shallow

Water 2006 experiment. The layout of the experiment is

shown in Fig. 3. NRL300 is the Naval Research Laboratory

source, ﬁxed near the bottom. It radiated linear frequency

modulated (LFM) signals: the duration of each one was

2.048 s, the frequency band was 270330 Hz, and the time

period of radiation was 4.096 s. The receiver was an L-

shaped array: the horizontal part of it (the horizontal line

array, or HLA) was of length L¼465 m and contained 32

hydrophones spaced by 15 m. Its direction is the xaxis, as

seen in Fig. 3, with L=2xL=2. The vertical part of

the array (the vertical line array, or VLA) contained 16

hydrophones, and its horizontal coordinates are given by (L/

2, 0). The angles in the horizontal plane determining the ori-

entation of the HLA relative to the acoustic track are shown

in Fig. 3(a):b1¼25:75,b0¼26:04,b2¼26:34. The

depth of the sea along the acoustic track was 80 m.

A strong characteristic of the experimental region is the

existence of nonlinear internal waves (NIW) with amplitudes

525 m, propagating approximately toward the coast (perpen-

dicular to the acoustic track). These arise at the M2 tidal period

(roughly two times per day) and form the dominant anisotropy

of the internal wave (IW) ﬁeld. In this paper, two transmission

time intervals from 19 of August 2006 are considered:

10:01–10:08 Greenwich mean time (GMT) (Fig. 4)and

08:30–08:38 GMT. Variations of the sound speed proﬁle are

determined by three thermistor strings, denoted SW45, SW54,

and SW32, deployed approximately along the acoustic track.

Let us ﬁrst consider the time period 10:01–10:08 GMT.

In Fig. 5(a), thermistor records for this period are shown,

with the instruments placed in the lower part of the thermo-

cline. As we can see, vertical oscillations of the thermocline

layer in area of the source and receiver are seen with ampli-

tude 1 m; in the middle part of acoustic track, their ampli-

tude is 3 m. In Fig. 3(b), the amplitude of one of the pulses

jPðx;tÞj received by the HLA is shown after matched ﬁlter-

ing. (“Matched ﬁltering” means that the spectrum of the

received signals is multiplied by the complex conjugate

spectrum, which leads to the compression of the received

LFM pulse from 2 to 0.03 s.) As a result, the signal-to-

noise-ratio (SNR) is increased by about 70 times, and this

makes it possible to separate modes using arrival times. In

Fig. 3(b) we can see two dark lines corresponding to modes

3 and 4. The shape of the intensity distribution on the VLA

FIG. 3. (a) Scheme of the SW06 exper-

iment. (b) Amplitude of one pulse,

received by the HLA after matched

ﬁeld compression, showing arrival

time dependence.

J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al. 197

can be used to determine the mode number: for mode 3,

there are 3 maxima, and for mode 4, there are 4 maxima.

We note that the maximal compression for the 60 Hz

band, which is 1=60 0:02 s, was 1.5 times less than that

observed in the experiment, probably due to different phase

shifts for different frequency components provided by intra-

modal dispersion, and due to different phase shifts along dif-

ferent horizontal rays corresponding to different frequencies.

FIG. 4. Data processing. (a) Space-time spectrum of mode 3; the amplitude is shown in Fig. 3(b). (b) Filtering of spectral maxima for frequencies in the band

290 65 Hz. (c) Calculation of the spatial spectrum in coordinates ðqx;qyÞusing the space-time spectrum.

FIG. 5. Episode 10:01–10:08 GMT. (a) Thermistor records (in degrees Centigrade) on SW45, SW32, and SW54. These were located at: the NRL300, in the

middle of acoustic track and at the receiver (VLA). (b) Temporal variations of the amplitude front and the phase front. The three dotted lines correspond to

angles b1,b2, and b0, giving the directions from the source to the ends and middle point of HLA [Fig. 3(a)]. Values of angles for amplitude and phase fronts

are denoted by numbers for the frequencies shown in the ﬁgure.

198 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.

Let us consider the following analysis for mode 3. In Fig.

4(a), the space-time spectrum Gðqx;xÞis shown, calculated

using Eq. (23) for mode 3 and for one of received pulses

Pðx;tÞ. The next step is to get the spatial spectrum, similar to

Fig. 1(c), using Eq. (24), and to ﬁnd the direction of the vector

pin coordinates ðqx;qyÞusing a linear interpolation of the

points corresponding to maxjGðqx;qyÞj.InFig.4(b),themax-

ima of the spectrum Gðqx;xÞare shown, obtained for fre-

quencies in the band 290 65 Hz, which correspond to the

neighborhood of the main maximum at 290Hz. Using the val-

ues of xand qxfor each maximum, we ﬁnd qyfrom Eq. (24),

where the dispersion law qðxÞand in turn xðqÞwere obtained

from the SturmLiouville problem using experimental data

about the waveguide parameters at the location of the

receiver. So, the positions of the maxima are found at the

coordinates ðqx;qyÞ, and also the straight line of maxima for

the spatial spectrum Gðqx;qyÞ[Fig. 4(c)]. Angles bam and

a¼b0bam, characterizing the angle of the amplitude front

with the HLA and the acoustic track (the angle between pand

the axes xand r), were found from the slope of the line of

maxima relative to the axis qx. Using the slope of the lines

from the origin to the points corresponding to frequencies

285, 290, and 295 Hz, we have found the angles bph and

u¼b0bph, which determine the angles between the phase

front and the HLA and the acoustic track (angle between q

and axes xand r) for the given frequencies [Fig. 4(c)]. A fre-

quency band of 10 Hz (290 65 Hz) was chosen for calcula-

tions to insure the condition of linearity of the variation of the

wave vector in this interval. Geometrically, this means that

the vectors q¼ðqx;qyÞin this frequency interval are placed

in a straight line.

For the time interval 10:01–10:08 GMT, 98 pulses were

processed. The results are shown in Fig. 5(b), where the tem-

poral dependence of the angles characterizing the amplitude

front and the phase front are presented. On the vertical axis

on the left side, the values of bph and bam are placed, and on

the right side uand a. These angles are shown for frequen-

cies 285 and 295 Hz.

Analysis of the angles obtained above leads to a good

correspondence with the theoretical estimates (Fig. 2): the

average value of the angle between the directions of the nor-

mal to the phase fronts for frequencies 285 and 295 Hz is seen

to be jDuj0:1in accordance with the estimate from Eq.

(19); the average modulus of the angle between the directions

of the amplitude front and the phase front for frequency

290 Hz is jDbj¼jbph bam j2:9[Eq. (20)]; the average

value of angle between the direction of the normal to the

phase front and the acoustic track at frequency 290Hz is

juj0:63[Eq. (15)]; the average value of the angle between

the direction of the normal to the amplitude front and the

acoustic track is jaj2:7. We note that the ﬂuctuations of

the steering angle in SW06 that were reported in the paper by

(Duda et al., 2012) for signals at 200 Hz, using the coherence

function of the horizontal line array in the presence of nonlin-

ear internal waves, were of the same order.

Next, let us consider the following. At 10:05 GMT, the

angle of the amplitude front aand two angles of the phase

front uat the frequencies 285 and 295Hz are equal to each

other. In accordance with the simpliﬁed theory, this can only

take place for straight-line trajectories of the horizontal eigen-

rays where bam ¼bph ¼b0¼26:04. In the experiment, at

10:05 GMT we have bam bph 26:8,thatisgreaterthan

b0almost by 1. This can be explained either by a systematic

error in the measurement of angles, or by a more complex

shape of the real IW than was supposed in the simpliﬁed the-

ory. In the latter case, in spite of real horizontal refraction, the

directions of the amplitude front and the phase front are the

same in the neighborhood of the receiver. The systematic

error can be produced by the deviations from a straight-line

shape of HLA, as was discussed in (Duda et al., 2012).

Next, we note that, in accordance with the theory pre-

sented above, if direction of horizontal gradient of the sound

speed proﬁle is changed, then the signs of angles uand aare

changed also. This leads to the following inequalities for

angles bph1 and bph2, corresponding to 285 and 295 Hz, and

angle bam:bam <bph2 <bph1 if the gradient is directed

toward positive values of s, and bam >bph2 >bph1, in the

case of the opposite gradient direction.

In the experiment at time 10:05 GMT, a change of the

sign of the angles takes place that can be interpreted as a

change of the direction of gradient of the sound speed or a

change of the thermocline slope (e.g., the forward face or

back face of an internal wave).

Let us consider the variations of angles bam and bph

together with the thermistor records. It is seen from Fig. 5(a)

that, on average in the time interval 10:01–10:04 GMT, the

forward front of the moving IW is passing through the acous-

tic track (SW45 shows an approximately constant level of

thermocline displacement, whereas SW32 and SW54 show an

increase in displacement). In this case, we should have deﬂec-

tion of the eigenrays toward the area s>0, and so the

inequality bam <bph2 <bph1 takes place (Fig. 2). In the inter-

val 10:04–10:08 GMT, the character of the IW motion is

changed; speciﬁcally, the thermistor string SW32 shows

mainly a decrease of the thermocline’s displacement, whereas

SW45 and SW54 show an approximately constant level of

displacement. Here, it is probable that the back edge of the

IW is passing through the acoustic track, and the values of the

angles correspond to: bam >bph2 >bph1 [Fig. 5(b)].

Similar work was done for another time interval,

08:30–08:38 GMT, where 110 pulses were processed.

Thermistor records are shown in Fig. 6(a). The variations of

the angles determining the directions of the amplitude front

and the phase front are shown in the Fig. 6(b).

It is seen that the properties of the angle variability have

the same character as the ﬁrst example. In particular, at

08:37 GMT, the directions of the amplitude front and the

phase front are the same. Average values of the angles in

this period are jDuj0:07,jDbj2:1,juj0:67,

jaj2:7, in accordance with theoretical estimations.

Next, let us examine the angles bph and bam for the ﬁrst

and second periods for frequencies 290 and 305 Hz, taking

into account our conﬁdence/error interval. In the ﬁrst case:

bph ¼26:760:2,bam ¼2663; in the second case:

bph ¼26:760:1,bam ¼28:861:5. These conﬁdence/

error intervals correspond to the mean square deviations. As

we can see, the mean square ﬂuctuations of the amplitude

front’s angle are bigger than those of the phase front by

J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al. 199

approximately by 15 times. In other words, the phase front is

essentially more stable with respect to variations of the

waveguide parameters in the horizontal plane than the ampli-

tude front. This is a standard and expected result.

Finally, we should note that, in our theory, we neglect the

coupling between vertical modes of different number which is

determined by the ratio of matrix element Vmn Ðwm

ð@wn=@rÞdz to the difference jqmqnj. In our case this ratio

jVnm=jqmqnj103for the closest pair of modes: 3–2

and 3–4. Thus, neglecting mode coupling is reasonable.

V. CONCLUSIONS

Let us present our conclusions as a simple list.

(1) It has been shown that the frequency dependence of tra-

jectories of horizontal rays (or of the modal amplitude

distribution in horizontal plane) leads to a speciﬁc inter-

ference pattern in the horizontal plane in the area of

reception of the sound pulse. This pattern is interpreted

as a combination of two structures: phase fronts (lines of

constant phase) and amplitude fronts (lines of constant/

maximal amplitude or envelopes). These lines have

different directions and different spatial scales of vari-

ability. We note that a similar interference pattern takes

place in nonlinear and nano-optics for the propagation of

two frequency beams in a crystal (Bakunov et al., 2012).

(2) On the basis of analytical estimations, it is shown that in a

typical situation, if horizontal refraction is due to internal

waves of small amplitude (1 m), then the angle of revo-

lution of the phase front (direction of the horizontal ray)

is about u1–2for an acoustical track of typical

length 20 km. The angle Dubetween horizontal rays, cor-

responding to different frequencies, depends on the differ-

ence between the frequencies. Using the estimate

Du=Dx0:01 deg/Hz, we get for a narrow-band pulse

with bandwidth 10 Hz that this angle is approximately

Du0:1We note that our estimates should be modiﬁed

in the presence of nonlinear internal waves (Duda et al.,

2012) where experimental measurement of these angles is

more difﬁcult.

(3) The angle bdetermining the direction of the normal to

the amplitude front can be essentially greater than the

angle determining the normal to the phase front. It

implies a more remarkable manifestation of horizontal

FIG. 6. Results of data processing for the period 08:30–08:38 GMT.

200 J. Acoust. Soc. Am. 143 (1), January 2018 Katsnelson et al.

refraction than one would suppose using the trajectory of

the horizontal rays. The variation of the direction of the

amplitude front (or its sensitivity in the direction to

waveguide parameter variations) is essentially higher

than that for the phase front.

ACKNOWLEDGMENT

The authors are grateful to Professor O. Godin for

helpful discussions. This work was supported by the Israel

Science Foundation, Grant No. 565/15, and the Ministry of

Education and Sciences of the Russian Federation, Grant

No. 14.Z50.31.0037.

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