ACOUSTIC FLUCTUATIONS AND THEIR
HARMONIC STRUCTURE
R. FIELD, J. NEWCOMB, J. SHOWALTER, J. GEORGE AND Z. HALLOCK
Naval Research Laboratory, Stennis Space Center, MS 39529, USA
E-mail: [email protected]
Acoustic fluctuations are experimentally and theoretically shown to be harmonically
related to the sound speed. The acoustic data analyzed here is shown to fluctuate
predominately at the first and second harmonics of the M2 tide. Theoretical results based
on the Pekeris waveguide show that a large-scale, slowly fluctuating sound speed like the
M2 can cause more rapid fluctuations in the acoustic field in the form of acoustic
harmonics. The theoretical results compare favorably with the acoustic data. The
theoretical results also show the impact of the water column sound speed on the
horizontal wave numbers and possibly on acoustic inversion methods that utilize them.
1
Introduction
From September 26 through November 2, 2000 the U. S. and Japan conducted a joint
experiment in the New Jersey Bight off the U.S. East Coast. The experiment was
conducted in three legs, the first of which investigated low frequency acoustic
fluctuations in the presence of a shelf/slope front and internal waves. Legs II and III
investigated acoustic scattering at 5.5 kHz and methods of acoustic inversion for sea
floor properties, respectively. This paper focuses on a subset of the experimental results
of Leg I.
Leg I was conducted from September 26 through October 5. During this leg, while
the oceanographic measurements were underway, an autonomous, bottom-moored
acoustic source continually transmitted signals in the 800 to 1250 Hz frequency range.
The acoustic data was received on two acoustic sensors for a period of 24 hours. Toward
the end of Leg I, a lower frequency acoustic source was suspended and transmitted 50,
125, 275 and 475 Hz signals simultaneously and continuously for twelve hours.
Depending on frequency, the data shows 15–18 dB fluctuations over periods of 7–10
min. Of the four source/receiver geometries instrumented, only one is discussed here.
Section 2 lays out the experimental geometry for the acoustic track that was parallel
to the shelf break and reviews the experimental acoustic and sound speed data. It is
shown that the acoustic fluctuations are harmonically related to the M2 tide. Section 3
lists a new set of fluctuation equations for the Pekeris waveguide. These equations, along
with those of the rigid waveguide, were derived in a paper currently under review [1].
The equations derived for these simple waveguides predict the harmonic nature observed
in the acoustic fluctuation measurements. Section 4 compares a “back-of-the-envelope”,
Pekeris waveguide simulation with the measured fluctuations. Since acoustic bottom
inversion may utilize the horizontal wavenumber spectrum, the effects of sound speed
271
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 271-278.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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R. FIELD ET AL.
dynamics in the water column on the horizontal wavenumber spectrum is shown in Sect.
5. The effects can be accounted for from the fluctuation equations listed in Sect. 3.
Section 6 summarizes and concludes.
2
2.1
Experimental geometry and data
Experimental Geometry
Ne
w
Je
rs
ey
Figure 1 shows the general location of the SWAT experiments and the specific location
of one of the four source/receiver geometries used during the acoustic propagation
experiment.
Receiver
Source
Figure 1. The general location of the SWAT experiments is shown in the open circle. One of the
four source-receiver geometries is shown for the acoustic propagation part of SWAT.
2.2
Acoustic Data
The acoustic signals were transmitted from an autonomous, bottom-moored source. The
water depth at the source position was 132 meters. The source was 9 meters off the sea
floor making the source depth 123 meters. The ocean depth over most of the acoustic
path and at the receiver was 139 meters. The receiver depth was 50 meters and its range
from the source was 9715 meters. The transmitted signals were linear sweeps from 800
Hz to 1250 Hz. The duration of each sweep was 10 s for a sweep rate of 45 Hz/s. The
sweeps were repeated every 20 s resulting in 10 s of ambient noise between each sweep.
Because of limitations in the acoustic recording system, only sweep frequencies up to
1107 Hz were processed.
The data were spectrally processed using a contiguous series of non-overlapped 256
point Hann weighted FFTs resulting in a frequency bin width of 8.7 Hz and an effective
noise bin width of 13 Hz. Thus, each FFT spanned a time period of only 0.115 s that
corresponded to about 5 Hz of the received sweep. These processing parameters were
chosen to ensure that each FFT (i.e., time period) would have the minimum possible
number of bins containing a portion of the received sweep.
273
ACOUSTIC FLUCTUATIONS AND THEIR HARMONIC STRUCTURE
15
0.5
a
10
5
0
-5
-10
-15
274.8
b
0.4
AMPLITUDE
RECEIVED LEVEL-AVG (dB)
A time series for each frequency bin was created for frequencies from 800 Hz to
1107 Hz. Since the signal arrived at the recording system every 20 s, a time series of the
received signal was created with an effective sampling rate of 4320 samples/day. Figure
2a displays the calibrated received level with the average removed at a frequency of
854.4922 Hz. The time series is 24-hours long and has been low pass filtered to 360
cycles per day (cpd) to correspond to the two-minute time sampling of the temperature
and salinity data upon which the sound speed is constructed. Figure 2b displays its
amplitude spectrum in cpd. From the spectrum, it can be seen that most of the high
amplitude fluctuation energy is below 50 cpd.
0.3
0.2
0.1
275
275.2
275.4
275.6
275.8
0
276
0
TIME (DAYS)
50
100
150 200 250
CYCLES/DAY
300
350
400
Figure 2. a) Acoustic received level at 854.4922 Hz with average removed over a 24-hour period.
b) The amplitude spectrum of the acoustic received level time series.
Figure 3a shows the acoustic time series of Fig. 2a low pass filtered to 20 cpd and
Fig. 3b displays the amplitude spectrum of Fig. 2b out to only 50 cpd. Figure 3b shows
what appears to be fluctuations close to the M2 tide, F0 , and its first and second
5
4
3
2
1
0
-1
-2
-3
-4
-5
274.8
0.5
a
2.1094 cpd, F0
b
0.4
AMPLITUDE
RECEIVED LEVEL-AVG (dB)
harmonics, F1 and F2 , respectively. To show that this is in fact the case, the sound
speed data will be analyzed.
4.2188 cpd, F1
0.3
6.3281 cpd, F2
0.2
0.1
275
275.2
275.4
275.6
TIME (DAYS)
275.8
276
0
0
5
10
15
20 25 30 35
CYCLES/DAY
40
45
50
Figure 3. a) Acoustic time series low pass filtered to 20 cpd. b) The amplitude spectrum of the
acoustic time series out to 50 cpd showing the M2 tidal fluctuation and its first and second
harmonics.
274
2.3
R. FIELD ET AL.
Sound Speed Data
Figure 4a shows a time series of sound velocity recorded just above the acoustic receiver
at a depth of 45 meters. Figure 4b displays its amplitude spectrum. The two largest
peaks in Fig. 4b are the Inertial Period and the semi-diurnal tide, M2. The nominal value
of the M2 is 1.9322 cpd. For this relatively short time series the measured value is
1.9336 cpd.
1.2
a
1525
1520
1515
1510
1505
M2, 1.9336 cpd
0.8
0.6
0.4
0.2
1500
1495
272
b
Inertial Period, 1.3184 cpd
1
AMPLITUDE
SOUND VELOCITY (M/S)
1530
0
274
276
278
280
0
2
TIME (DAYS)
4
6
8
10
CYCLES/DAY
Figure 4. a) Sound speed time series at 45 meters just above the acoustic receiver. b) The
amplitude spectrum of the sound speed time series showing the Inertial Period and the M2 tide.
Figure 5a shows the sound speed time series of Fig. 4a band pass filtered to pass only
the M2 component of the spectrum. The vertical lines in the figure denote the time period
during which the acoustic transmissions shown in Fig. 2a took place. During this time
period the M2 fluctuates about an average of approximately 1517.5 m/s with an amplitude
of ± 1.625 m/s. Figure 5b displays the amplitude spectrum of the M2 during the acoustic
transmission time. Because of the short time window, the M2 takes on its maximum
value at 2.1094 cpd. The acoustic transmissions “see” the M2 fluctuating at this rate,
which is the value of F0 shown in Fig. 3b. The highest amplitude acoustic fluctuations are
at the first and second harmonics, F1 and F2 , respectively, of the M2.
0.6
a
1519
1518
1517
1516
1515
b
Windowed M2 , 2.1094 cpd
0.5
AMPLITUDE
SOUND VELOCITY (M/S)
1520
0.4
0.3
0.2
0.1
Time period of acoustic transmission
1514
272
0
274
276
TIME (DAYS)
278
280
0
2
4
6
8
10
CYCLES/DAY
Figure 5. a) Sound speed time series band-pass filtered to pass only the M2. b) The amplitude
spectrum of the M2 sound speed time series during the time of acoustic transmission.
ACOUSTIC FLUCTUATIONS AND THEIR HARMONIC STRUCTURE
3
275
Fluctuation equations for the Pekeris waveguide
In order to show that the measured acoustic fluctuations shown in Figs. 3a and 3b are
theoretically expected, results from a paper currently under review will be given [1].
From Frisk [2] the expression for the complex acoustic pressure in a Pekeris waveguide is
given by
p (r, z ) =
P
2π eiπ
4 N max
∑Γ
ρ
P
2
n
P
zn 0
P
zn
sin( k z )sin( k z )
n =1
eikn r
knP r
(1)
where Γ n is the amplitude, k zn and kn are the vertical and horizontal wavenumbers,
respectively,
written as
ρ is the density and r is the range.
If the time dependent sound speed is
C (t ) = C0 + Ae(t ) cos(Ω0t )
(2)
where C0 is the average sound speed, A, is the amplitude of the sound speed fluctuation,
e(t ) is a slow-varying envelope, Ω0 = 2π F0 is the radial “frequency” and F0 is the
fundamental frequency of the sound speed variation, it can be shown that Eq. (1) can be
written in the time dependent form
p P ( r , z, t ) =
2π eiπ
ρ
4
  N max


sin(k znP 0 z0 )sin(k znP 0 z ) iknP01r
  ∑ Γ n2 0

e J 0 ( I nP )  +
P
  n=1


kn 0 r


 (3)
  ∞

P
P
N max

P1
sin(
k
z
)
sin(
k
z
)
l
2
zn 0 0
zn 0
eikn 0 r J 2l ( I nP )  −
 2∑ (−1) cos(2l Ω 0t ) ∑ Γ n 0


n =1
knP0 r
  l =1



P
P
N max

 ∞
l
P
2 sin( k zn 0 z0 ) sin( k zn 0 z ) iknP01r
e J 2l +1 ( I n )  
 i 2∑ ( −1) cos[(2l + 1)Ω 0t ] ∑ Γ n 0
 
n =1
knP0 r
  l = 0
with
knP r ≡ knP01r − I nP cos Ω0t
(4)
ϕ  ϕ

knP01 ≡ knR0 +  k znR − n 0  n 0R
4h  2hkn 0

 
ϕ n 0  ϕ n0′ ω 2  rAe(t )
P
R

+
I n ≡  −  k zn −
2h  2h C03  knR0
 

(5)
(6)
In Eq. (3), the J n terms are Bessel functions of the first kind and Eq. (4) defines the time
dependent horizontal wavenumbers. In Eq. (5),
ϕn0
is the bottom phase evaluated at the
R
n0
average sound speed, h is the water depth and k is the horizontal wavenumber of the
rigid waveguide evaluated at the average sound speed. In Eq. (6),
derivative of the bottom phase and
ω
ϕ n0'
is the first
is the acoustic frequency. From Eq. (3) it can be
276
R. FIELD ET AL.
seen that the only allowed fluctuations in the acoustic field are those at the sound speed
fundamental and its harmonics. Equation (2) can be generalized to a real sound speed
field by expressing it as a series of sound speed components with different amplitudes
and envelopes. For the real sound speed case, the acoustic field will fluctuate at the
fundamental and the harmonics of each sound speed component.
4
“Back-of-the-envelope” calculations vs. data
This section compares time dependent Pekeris waveguide calculations with the data
shown in Figs. 3a and 3b. It is meant to be only a “back-of-the-envelope” calculation that
accounts for the low frequency part of the acoustic fluctuation spectrum. Accurate
modeling would, of course, require a range and depth dependent model to account for
higher frequency fluctuations found in the acoustic spectrum shown in Fig. 1b. For the
Pekeris simulation the following parameters are chosen: water depth = 132 meters,
bottom sound speed = 1800 m/s, bottom density = 2.0 g/cm3 , range = 9715 meters,
source depth = 123 meters, receiver depth = 50 meters and acoustic frequency =
854.4922 Hz. The water depth at the source was chosen. The bottom sound speed and
density were chosen based on average coring values within the first 100 cm of the bottom
sediment. The range, source depth, receiver depth and acoustic frequency are measured
values. The Pekeris waveguide time series was computed with a “frozen ocean”
approach. Equation (2) was used to generate a synthetic M2 fluctuation in sound speed.
From Figs. 5a and 5b, the following sound speed parameters for Eq. (2) were chosen: A
= 1.625 m/s, C0 = 1517.5 m/s and F0 = 2.1094 cpd. The envelope function, e(t ) , was
5
4
3
2
1
0
-1
-2
-3
-4
-5
274.8
0.5
a
Data
AMPLITUDE
RECEIVED LEVEL-AVG (dB)
set equal to one over the 24-hour period. The sampling used for the simulation was the
same as the data, 4320 cpd (every 20 s) over a period of 24 h. The results are shown in
Figs. 6a and 6b.
Model
0.4
F1
0.3
F0
Data (solid)
Model (dashed)
b
F2
F5
0.2
0.1
275
275.2
275.4
275.6
TIME (DAYS)
275.8
276
0
0
5
10
15
20 25 30 35
CYCLES/DAY
40
45
50
Figure 6. a) Acoustic time series over 24 hours. Solid line is data low pass filtered to 20 cpd.
The dashed line is the Pekeris M2 simulation at 2.1094 cpd. b) The amplitude spectrum of each
time series. The M2 fundamental and its harmonics that are labeled refer to the Pekeris simulation.
Figures 6a and 6b show that the Pekeris simulation gives a reasonable answer for the
fluctuations caused by the M2 tide. The simulation shows that the two largest acoustic
fluctuation components are the first and second harmonics of the M2 tide. In Fig. 6b the
harmonics that are labeled refer to the model simulation.
ACOUSTIC FLUCTUATIONS AND THEIR HARMONIC STRUCTURE
5
277
Impact of water column dynamics on the horizontal wavenumber
spectrum
Since acoustic bottom inversion is a major component of the SWAT collaboration, the
Pekeris waveguide will be used to generate complex pressures as a function of range and
time in order to determine how changes in the water column sound speed affect the
horizontal wavenumber spectrum. The same Pekeris parameters will be used as in the
previous simulation except the frequency is changed to 50 Hz and the source and receiver
depths are 61 m (half the water depth). The procedure used to compute the horizontal
wavenumber spectrum is to calculate complex pressures as a function of range for a
constant sound velocity, then do a Hankel transform over the range aperture to get the
horizontal wavenumber spectrum. This procedure is repeated over a 24 h time period
where each Hankel transform is computed with a new but constant sound velocity over
the range interval. The range aperture is calculated in 1-meter intervals from 5000 meters
to 9046 meters.
Figure 7a shows the result of the calculation for a sound velocity that remains
constant over the 24 h period. Four modes are found and the spectrum remains constant.
Mode 1 has the largest amplitude. Figure 7b shows the same calculation using the sound
velocity time series measured at a depth of 45 meters. This is the same sound velocity
shown in Fig. 4a from day 274.8 to 275.8. The sound velocity time series is plotted to
the left of Fig. 7b. The maximum sound velocity, labeled 1, is 1524.5 m/s and the
minimum, labeled 2, is 1507.8 m/s. From Eqs. (4) and (6), it can be seen that the time
dependent part of the wavenumber adds energy to the wavenumber spectrum and makes
one wavenumber “bleed” into adjacent bins. Therefore, the water column will impose
limits to the extent one can unambiguously interpret changes in the wavenumber spectrum
in terms of changing geoacoustic parameters.
6
Summary and conclusions
It has been experimentally and theoretically shown that acoustic fluctuations are
harmonically related to the sound speed. For a “monochromatic ocean signal”, the only
fluctuations allowed in the acoustic field are the sound speed fundamental and its
harmonics. In general, the harmonic content is a function of the amplitude of the sound
speed fluctuation, the acoustic frequency, the number of modes, the bottom phase and
range. Because the time dependent horizontal wavenumber includes an additional term
on the order of ω 2 A C03 knR0 , each bin of the horizontal wavenumber spectrum can
accumulate energy over time causing the amplitudes of the wavenumbers to “bleed” from
one bin to another. This may pose significant problems for bottom inversion in cases
where changes in the horizontal wavenumber spectrum are interpreted as changing
geoacoustic conditions.
278
R. FIELD ET AL.
M od e N um b ers
4
3
2 1
(a)
Amplitude
TIME (DAYS)
0 .0
1 .0
K
n
M ode N um bers
4
3
2 1
(b)
0 .0
Amplitude
TIME (DAYS)
1
2
1 .0
Kn
Figure 7. a) The horizontal wavenumber spectrum computed with a constant sound speed. The
first four modes are shown. b) The horizontal wavenumber spectrum computed with the
measured sound speed time series shown on the left.
Acknowledgements
This work is supported by the Office of Naval Research, Program Element PE 62435N.
References
1. Field, R.L. and George J., Acoustic fluctuations in simple shallow water waveguides, J.
Acoust. Soc. Am. (submitted Jan. 2002).
2. Frisk, G.V., Ocean and Seabed Acoustics: A Theory of Wave Propagation (Prentice-Hall,
Inc., New Jersey, 1994) Chap. 5.