ACOUSTIC SCATTERING IN WAVE-COVERED
SHALLOW WATER. THE COHERENT FIELD
B.J. USCINSKI
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Silver St., Cambridge U.K. CB3 9EW
E-mail:[email protected]
The coupled mode approach to acoustic propagation in shallow water with rough surface
and bottom fails when the number of modes becomes too large. A method is presented,
based on a set of coupled integral equations, that allows the acoustic field to be calculated
for large surface and bottom roughness. Ensemble average forms of these equations are
derived for the mean field and a solution sought using Laplace transforms. The method
allows us to study how the coherent component of an acoustic field in shallow water is
affected by surface waves.
1 Introduction
Sound propagating in shallow water is reflected from both surface and bottom which are
then the boundaries of an effective waveguide. Irregular structures in these boundaries,
such as surface waves and bottom roughness affect the acoustic field. They impose
random modulations on what might otherwise be a regular waveguide diffraction pattern.
One method of calculating the acoustic field that can take boundary roughness into account
consists of representing the field as a sum of modes [1]. Roughness in the upper and
lower boundaries leads to coupling and energy exchange between the modes [2]. This
approach can be used successfully provided that the number of modes is not large. It fails
in the case of high acoustic frequencies when the wavelength is short and surface waves
and bottom roughness produce large scattering effects. The number of modes required in
this case is so large that it would be impossible to compute them. This paper describes a
non-modal method that allows us to deal with high-frequency acoustic propogation in the
shallow water case. The amount of back-scatter is assumed to be small so that the forward
propagating parabolic approximation can be used. A set of coupled integral equations
describes the propagation [3]. When the ensemble average of the acoustic field is sought
these equations can be solved using Laplace transforms. The resulting expressions allow
us to study how the mean, or coherent, component of an acoustic field propagating in
shallow water is affected by the rough wave-covered sea surface.
2
The shallow water model
The simplified model used to dscribe the shallow water ”wave guide” is discussed in
detail elsewhere [3, 4]. The upper boundary is a pressure release surface and has a
reflection coefficient pX =d. The reflection properties of the ocean bottom are, in general,
329
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 329-336.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
330
B.J. USCINSKI
X
ds
Z1 = S1(x)
Z0
Z 2 = d + S2(x)
ds
Z
Figure 1. The coordinate system showing the rough surface and bottom together with the source
location.
complicated. We thus restrict our treatment to the special cases where the shear-waves
propagate at low speed and are radiated away from the liquid-solid interface. The effective
acoustic reflection coefficient of such a bottom at low grazing angles has an amplitude
somewhat less than unity and a phase that varies between 120◦ and 180◦ . The simplest
first approximation would thus be to take a bottom reflection coefficient RB = −1 in such
cases. This allows the general principles of the method to be demonstrated conveniently
and results checked against modal solutions in specific cases. The shallow water channel
is shown in Fig. 1. Let (x, z) be a Cartesian system of coordinates. The sea surface and
bottom follow the contours
Z1 = S1 (x), Z2 = d + S2 (x)
(1)
respectively. The source is situated at (0, z0 ). Let p(r) be the acoustic pressure at the
point r = (x, z) while r" = (x" , z(x" )) lies on the bounding surface. We assume that the
acoustic field propagates predominantly in the forward x direction allowing us to write,
in the parabolic approximation
p(x, z) = E(x, z)eikz
(2)
where E(x, z) is the slowly varying envelope of the acoustic pressure field p(x, z) and
k is the acoustic wave-number. In the parabolic approximation the integral form of the
Helmholtz equation for E(x,z) becomes
!
E(x, z) = {G(r, r" )Ez! (r" ) − E(r" )Gz ! (r, r" )} ds
(3)
where G is the parabolic form of the Green’s function and the subscript z " denotes the
vertical derivative, i.e. with respect to z.
Consider the surface over which Eq. (3) is to be integrated and note that ds is the
outward facing surface element. It consists of four separate sections as shown in Fig. 2.
Thus we can write
!
!
!
!
[ ]ds +
[ ]ds +
[ ]ds +
[ ]ds
(4)
E(x, z) =
S(1)
S(2)
S(3)
S(4)
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SCATTERING IN WAVE-COVERED SHALLOW WATER
Z1
S(1)
X0
X
S(4)
S(3)
e
S(2)
Z2
Z
Figure 2. The four surfaces S(1), S(2), S(3) and S(4).
Now consider S(4) over x0 = x + !. The contribution from this surface is zero, since
there is no backward propagation from x0 to x. Next, let both surfaces S(1) and S(2)
move to ±∞ respectively. The contributions from S(1) and S(2) then vanish leaving
!
[ ]ds = Einc (x, z)
(5)
E(x, z) =
S(3)
Thus the surface S(3) represents the incident field, i.e. that would have been expected
from the source in the absence of any surface. We can now rewrite Eq. (4) as
!
G(r; r" )Ez ! (r" ) − E(r" )Gz! (r; r" )dx"
E(x, z) − Einc (x, z) =
S(2)
!
−
G(r; r" )Ez! (r" ) − E(r" )Gz ! (r; r" )dx"
(6)
S(1)
2.1 Case for a Bottom with RB = −1
Now E(r" ) is zero on both S(2) and S(1) so Eq. (6) becomes
! x
E(x, z) − Einc (x, z) =
G(r; x" , z2 (x" ))Ez! (x" , z2 (x" ))dx"
0!
x
−
G(r; x" , z1 (x" ))Ez! (x" , z1 (x" ))dx"
(7)
0
On specifying E(x, z) to surfaces z2 and z1 respectively, we have
! x
Einc (x, z2 (x)) = −
(G(x, z2 (x); x" , z2 (x" ))Ez! (x" , z2 (x" ))
0
−G(x, z2 (x); x" , z1 (x" ))Ez ! (x" , z1 (x" ))) dx"
(8)
and
Einc (x, z1 (x)) = −
!
0
x
(G(x, z1 (x); x" , z2 (x" ))Ez! (x" , z2 (x" ))
−G(x, z1 (x); x" , z1 (x" ))Ez ! (x" , z1 (x" ))) dx"
(9)
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B.J. USCINSKI
Equations (8) and (9) are coupled equations for the unknowns Ez! (x, z2 (x)),
Ez! (x, z1 (x)). These equations can be solved for the two surface derivatives which,
when used in Eq. (7), allow us to obtain E(x, z) anywhere in the space between z1 , z2 .
3
The mean field problem
If we wish to obtain the mean or average scattered field by numerical simulations we
would have to compute it for many statistically independent realisations of the rough
surface S(x) and take the ensemble average. An alternative, and more efficient, approach
would be to derive equations for the mean field itself. In order to do this we need to
analyse the stochastic properties of both the scattered acoustic field and the rough surface
S(x) and see how they are related to each other. First consider the rough surface S(x).
The irregular features in S(x) have a certain correlation distance in the x direction of
the order of the scale size L, the length of a typical feature in the rough surface. As
the acoustic field propogates in the x direction it interacts with the rough surface and its
complex amplitude E(x, z) is gradually modulated in an irregular manner. Now consider
E(x" , z) at some range x" after it has traversed several correlation lengths. The structure
in E(x" , z) has been caused by the accumulated interactions with the surface up to that
range. Now suppose that the surface contour S(x) changed only in the vicinity of x" over
a distance of about one correlation length. The change to E(x" , z) will be small because
its structure has been determined by all the preceding irregularities, so that the effect of
the change in S(x) in the final step will be negligible by comparison.
3.1 Statistical Independence of Surface and Field
From the above reasoning we can draw the important conclusion that in this type of low
angle propagation the complex acoustic amplitude E(x" , z) is statistically independent
of the local form of the surface S(x) at any range x" , provided that several correlation
lengths have been traversed. The following are therefore also valid:
1. Both E(x" , z) and Ez ! (x" , z) are statistically independent of G(x, z; x" , S(x" )) since
it is a function of S(x" ).
2. Likewise E(x" , z) and
G(x, S(x); x" , S(x" )).
Ez! (x" , z)
are
statistically
independent
of
3. However, in G(x, S(x); x" , S(x" )) the surface heights S(x), S(x" ) are not statistically
independent of each other but are correlated. This needs to be taken into account
when carrying out the statistical averaging.
3.2 The Statistics of the Rough Surface
It is assumed that the surface height S(x) is a stationary random process with Gaussian
statistics having zero mean and variance µ2 . The normalised spatial autocorrelation
function of S(x) is
ρ(x − x" ) =
< S(x)S(x" ) >
µ2
(10)
SCATTERING IN WAVE-COVERED SHALLOW WATER
333
These conditions are not unreasonable since many natural surfaces, including that of the
wave-covered sea surface, have statistics that are very close to Gaussian. The autocorrelation function ρ(x − x" ) can be of arbitrary form. In the case of the sea surface it would
be that corresponding to a Pierson-Moscowitz wave-height spectrum, for example. The
one point probability distribution of surface height S is then
"
#
S2
1
√
exp − 2
(11)
P1 (S) =
2µ
2πµ
while the two-point joint probability distribution of
Sa = S(x" ),
Sb = S(x)
(12)
is
"
#
1
Sa2 − 2ρSa Sb + Sb2 )
$
P2 (Sa , Sb ) =
exp −
.
2µ2 (1 − ρ2 )
2πµ2 1 − ρ2
(13)
where ρ = ρ(x − x" )
3.3 Ensemble Averages
We now take the statistical average of Eqs. (7), (8) and (9) using the distributions (11)
and (13) for the rough surfaces S1 and S2 . Remembering that the scattered field
Es (x, z) = E(x, z) − Einc (x, z)
(14)
and introducing the notation
< Einc (x, zi (x)) > = fi (x)
< Es (x, z) > = fs (x, z)
< Ez ! (x, zi ) > = φi (x)
< G(x, zi (x), x" , zj (x" )) > = Gij (x − x" )
< G(r; x" , zi (x" )) > = Gi (x − x" , z)
(15)
we obtain
fs (x, z) =
!
f2 (x) = −
x
[φ2 (x" )G2 (x − x" ; z) − φ1 (x" )G1 (x − x" ; z)] dx"
0
!
f1 (x) = −
(16)
x
0
!
0
[φ2 (x" )G22 (x − x" ) − φ1 (x" )G21 (x − x" )] dx"
(17)
[φ2 (x" )G12 (x − x" ) − φ(x" )G11 (x − x" )] dx"
(18)
x
334
B.J. USCINSKI
3.4 Scaling
Since it is convenient to work with dimensionless variables the following scaling is now
introduced. In the vertical we use the quantity
%
L
f=
(19)
k
which is seen to be very close to the Fresnel radius for an observer at a range L, the
correlation distance of the rough surface. In the horizontal, L itself is the scaling factor.
The scaled horizontal and vertical distances are thus
x
z
X= , Z=
(20)
L
f
and
γ2 =
µ2 k
L
(21)
which is a variance of the scaled surface height.
4
Solution by Laplace transforms
On taking the Laplace transforms of Eqs. (16)-(19) we obtain
Fs (λ) = Φ2 (λ)K2 (λ) − Φ1 (λ)K1 (λ)
(22)
F2 (λ) = −Φ2 (λ)K22 (λ) + Φ1 (λ)K21 (λ)
(23)
F1 (λ) = −Φ2 (λ)K12 (λ) + Φ1 (λ)K11 (λ)
(24)
where
F (λ) =
Φ(λ) =
K(λ) =
!
∞
!0 ∞
! 0∞
f (X)e−λX dX
φ(X)e−λX dX
G(X)e−λX dX
(25)
0
the unknown quantities Φ1 , Φ2 can be obtained by solving equations (23) and (24) and
set in (22) to give
Fs (λ, z) =
K1 (F2 K12 − F1 K22 ) − K2 (F2 K11 − F1 K21 )
K11 K22 − K21 K12
(26)
Finally, the coherent, or mean scattered field < Es (X, Z) > can be found by taking the
inverse Laplace transform of Eq. (26)
< Es (X, Z) >= LT −1 [Fs (λ, Z)]
(27)
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SCATTERING IN WAVE-COVERED SHALLOW WATER
1
0.1
5
0.5
0.05
0.01
0
10
20
30
X
Figure 3. The attenuation factor β(X, Z) for a channel with a flat bottom and a surface with an
exponential autocorrelation function. Here γ0 = 0.05, 0.5, 5.0, Z0 = 16; d = 32
5
Attenuation factor
We now illustrate the use of the above general results in the special case of a rough
surface when the bottom is flat. In order to simplify the calculations we use a surface
with an exponented spatial autocorrelation function, ρ = exp{−X}. This allows us to
investigate the overall effect of surface waves without the more complicated algebra that
results when a Pierson-Moscowitz autocorrelation function is employed. The effect of
different wave heights can be tested by adjusting the scaled variance γ. The inverse
Laplace transform was evaluated numerically for a number of different wave heights.
The effect of the wave covered surface on < Es (x, z) > the ”coherent” scattered field,
is best illustrated by considering the ratio
β(X, Z) =
< Es (X, Z) >
< Es0 (X, Z) >
(28)
where < Es0 (X, Z) > is the mean scattered field in the absence of surface waves.
The attenuation factor β is shown in Fig. 3 as a function of range X at the centre of
the channel for several values of γ 2 corresponding to different surface roughness. We
see that the coherent component is progressively attenuated by surface scattering as the
range increases, and the higher the waves, the greater this effect.
This paper sets out the general reasoning behind the present approach and the methods
used to implement it. The specific forms obtained for the ensemble averages Eq. (15) and
their Laplace transforms Eq. (25) are not given here because of space restrictions. They
are, however, quite analogous to the expressions obtained in the case of a single rough
surface which are set out fully in [5].
References
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field propagating in a wave guide with statistically rough walls, izu V.U.Z. Radiofizika
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336
B.J. USCINSKI
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533–547 (1978).
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