DTIC ADA506808: An Energy-Conserving One-Way Coupled Mode Propagation Model

9 Pages·1.5 MB·English

by  Defense Technical Information Center

Form Approved
REPORT DOCUMENTATION PAGE
OMB No. 0704-01-0188
i ne puD4ic reporting rjuraen tor tnis collection of information is estimated 10 average i nour per response, including ine time lor reviewing instructions, searcning exisung oata sources, gatnenng
and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,
including suggestions for reducing the burden to Department of Defense, Washington Headquarters Services Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson
Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to
comply with a collection of information if it does not display a currently valid OMB control number.
PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.
1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE
3. DATES COVERED (From - To)
05-2002
Technical
4. TITLE AND SUBTITLE
5a. CONTRACT NUMBER
AN ENERGY-CONSERVING ONE-WAY COUPLED MODE PROPAGATION
5b. GRANT NUMBER
MODEL
5c. PROGRAM ELEMENT NUMBER
0601152N
6. AUTHORS
5d PROJECT NUMBER
A. T. Abawi
5e. TASK NUMBER
5f WORK UNIT NUMBER
8. PERFORMING ORGANIZATION
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
REPORT NUMBER
SSC San Diego
San Diego, CA 92152-5001
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S)
Office of Naval Research
11. SPONSOR/MONITOR'S REPORT
800 North Quincy Street
Arlington, VA 22217-5000
20090803042
12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution is unlimited.
13. SUPPLEMENTARY NOTES
This is a work of the United States Government and therefore is not copyrighted. This work may be copied and disseminated
without restriction. Many SSC San Diego public release documents are available in electronic format at
http://www.spawar.navy.mil/sti/publications/pubs/index.html
14. ABSTRACT
The equations of motion for pressure and displacement fields in a waveguide have been used to derive an energy-conserving , one-
way coupled mode propagation model. This model has three important properties: First, since it is based on the equations of
motion, rather than the wave equation, instead of two coupling matrices, it only contains one coupling matrix. Second, the
resulting coupling matrix is anti-symmetric, which implies that the energy among modes is conserved. Third, the coupling matrix
can be computed using the local modes and their depth derivatives. The model has been applied to two range-dependent cases:
Propagation in a wedge, where range dependence is due to variations in water depth and propagation through internal waves,
where range dependence is due to variations in water sound speed. In both cases the solutions are compared with those obtained
from the parabolic equation (PE) method.
Published in Journal of the Acoustical Society of America, vol. 111, no. 1, Pt. 1, 160-167
15. SUBJECT TERMS
Mission Area: Surveillance
coupling matrix equation of motion
waveguide
wave equation
parabolic equation method propagation model
17. LIMITATION OF
18. NUMBER
16. SECURITY CLASSIFICATION OF:
19a. NAME OF RESPONSIBLE PERSON
ABSTRACT OF
A. T. Abawi
a. REPORT b. ABSTRACT
c. THIS PAGE
PAGES
19B. TELEPHONE NUMBER (Include area code)
uu
u
8
U
[phone]
U
Standard Form 298 (Rev. 8/98)
Prescribed by ANSI Std. 239 18
An energy-conserving one-way coupled mode propagation model
Ahmad T. Abawi
SPA WAR Systems Center, San Diego, California 92152-5001
(Received 18 December 2000; revised 14 August 2001; accepted 24 August 2001)
The equations of motion for pressure and displacement fields in a waveguide have been used to
derive an energy-conserving, one-way coupled mode propagation model. This model has three
important properties: First, since it is based on the equations of motion, rather than the wave
equation, instead of two coupling matrices, it only contains one coupling matrix. Second, the
resulting coupling matrix is anti-symmetric, which implies that the energy among modes is
conserved. Third, the coupling matrix can be computed using the local modes and their depth
derivatives. The model has been applied to two range-dependent cases: Propagation in a wedge,
where range dependence is due to variations in water depth and propagation through internal waves,
where range dependence is due to variations in water sound speed. In both cases the solutions are
compared with those obtained from the parabolic equation (PE) method. © 2002 Acoustical
Society of America. [DOI: 10.1121/1.1419088]
PACS numbers: 43.30.Bp [SAC-B]
I. INTRODUCTION
acoustic wavelength only one of the two coupling matrices
has significant contribution. By neglecting one of the cou-
The coupled mode theory commonly used in acoustics
pling matrices and the horizontal derivative of the density,
2
was originally derived by Pierce' and Milder from the wave
McDonald was able to derive an expression for the remain-
equation for the velocity potential. In this formulation the
ing coupling matrix in terms of local modes and their depth
field in a range-dependent waveguide is expanded in terms
derivatives. The expression for the coupling matrix derived
local modes with range-dependent coefficients (mode ampli-
b
by McDonald was used by Abawi et al who derived a sys-
tudes). The application of the continuity of pressure and the
tem of one-way coupled mode equations for the mode am-
vertical component of particle velocity allows a partial sepa-
plitudes. Although this method is a practical computational
ration of the depth and range variables and yields a system of
method for solving range-dependent problems, it still suffers
second order coupled differential equations for the mode am-
from approximations made by neglecting one of the two cou-
plitudes. However, as is pointed out by Rutherford and
pling matrices. More importantly, since this method is based
3
Hawker, while the boundary condition of continuity of ver-
on the same boundary conditions as those used by Pierce and
tical component of particle velocity is correct for horizontal
Milder, the energy among modes is not conserved.
boundaries, when applied to problems with nonhorizontal
The coupled mode model that is presented in this paper
boundaries, this boundary condition is only an approxima-
is derived, not from the wave equation, as is the case for the
tion to the correct boundary condition of the continuity of the
Picrce-Milder method, but from the equations of motion for
normal component of particle velocity. Rutherford and
the pressure and displacement fields. This method, which
Hawker showed that one consequence of this approximation
7
was first used by Shcvchenko, has common use in seismol-
is nonconscrvation of energy. They used the WKBJ method
x 9 10
ogy and geophysics, Odom, Maupin and Tromp. The
to obtain a solution which satisfies both the proper boundary
derivation in this paper follows the derivation of Tromp.
condition and conserved energy to first order in the slope of
While the model derived by Tromp is for the general clastic
the nonhorizontal boundaries and interfaces. This problem
waveguide, this model is derived for a waveguide consisting
was also addressed by Fawcett, who derived a system of
of fluid layers. Since the equations of motion constitute a
coupled mode equations which satisfies the correct boundary
system of two first order coupled differential equations, the
conditions. However, in addition to the two coupling matri-
coupled mode equations resulting from them only contain a
ces, which is typical of all coupled mode theories derived
single coupling matrix. Furthermore, this method provides a
from the wave equation, the equations derived by Fawcett
natural framework for applying the correct boundary and in-
also contain two other so-called interface matrices. These
terface conditions without adding any more complexity to
matrices require a knowledge of the range derivatives of the
the numerical solution of the equations. In fact, it is shown in
local modes, which can only be computed approximately.
this paper that the proper application of the boundary and
The inaccuracy resulting from this along with the complexity
interface conditions not only simplifies the numerical com-
involved in solving the system of differential equations make
putation of the coupling matrix by allowing it to be ex-
this an impractical computational method for solving range-
pressed in terms the local modes and their depth derivatives,
dependent problems.
it also makes it possible to show that the resulting coupling
In an attempt to reduce the complexity involved in com-
matrix is anti-symmetric, which guarantees the conservation
puting the coupling matrices, McDonald' used the original
3 6 0
of energy among modes. ' ''
Picrce-Milder equations to argue that for a waveguide
whose horizontal length scales are much larger than the
The method presented in this paper and the references
160 J.Acoust. Soc. Am. 111 (1), Pt. 1, Jan. 2002 0001-4966/2002/111(1)/160/8/$19.00
cited in the above fall in the category of continuous coupled
where [£]* = £+-£-, the +/- indicate the value of the
mode theory where the solution of the wave equation in a
parameter f just above/below the interface and n is the unit
range-dependent waveguide is obtained by solving of a set of
vector normal to the interface.
coupled differential equations. The solution for a range-
The field quantities can be expressed as the sum of nor-
dependent waveguide can also be obtained by the discrete
mal modes
coupled mode method." In this method the range-dependent
waveguide is approximated by range-independent stair steps
/>(x,*)-2 />„(*)«'*"*. •(*,*)=2 •„(*,*)«'*•*,
and the coupled mode solution is obtained by matching the
n n
solutions of the wave equation for neighboring stair steps at
where p„ and u„ denote the normal modes. Substituting the
their common boundary. This method is easy to implement
above expressions into the equations of motion results in the
numerically and has wide application in ocean acoustics.
following relationships for the modes
However, the method that is presented in this paper has two
2
ik p„=poi u„,
advantages over the discrete coupled mode method. The first
n
advantage is it expresses the coupling matrix in terms of
physical parameters and thus provides insight into the pro-
cess of mode coupling by clearly showing what is respon-

pc \P<-0"
sible for it. The other, more important advantage is that this
(1)
method in principle can be extended to handle propagation in
Z
-U„ = zdj)„,
three dimensions, where the discrete coupled mode method
pco~
is designed for propagation in two dimensions and there is
no obvious way to modify it to handle propagation in three
[p„]- = 0, [z„u„]! = 0.
dimensions.
In the above equations u denotes the jr component of the
n
This paper is organized in the following way. In Sec. II
displacement.
the coupled mode model is derived, where some of the de-
In a range-dependent environment the pressure and the
tails of the derivation are given in the appendices. In Sec. Ill
displacement vector may be expressed as a sum of local nor-
the model is applied to two range-dependent propagation
mal modes with range-dependent coefficients, c„(x)
scenarios. The first one is propagation in a wedge, where
range-dependence is entirely due to variations in water
ik x
p(x,z) = 2 c„{x)p„{z;x)e "
depth. The parameters used in this example are scaled to
match those used by Coppens and Sanders in a model tank
(2)
experiment. The second example is propagation through in-
*„>
u(x,z) = 2 c„(x)u„(z;x)e
ternal waves, where range-dependence is entirely due to the
variations in water sound speed. The parameters used in this
In this notation the parametric range-dependence of the local
example were those used in a test case in the Shallow Water
modes at range x is indicated by the semicolon separating z
13
Acoustics Modeling workshop. In both of the above ex-
and x.
amples the results obtained from the coupled mode model
Substitution of the above expansion into the equations of
are compared with those obtained from the parabolic equa-
motion gives
14
tion (PE) method.
V 2 lk
<?*2 c P„e' =2 pc» c u„e »\
B n
II. DERIVATION OF THE COUPLED-MODE EQUATIONS
ik x ik x
^2 c u„e - ='2 -p„-z-a u„\c„e " .
Consider the equations of motion with the x-axis in the
n l :
" " \ pc f
direction of propagation
k x
Multiplying the first equation by u„e~' "< and the second
d p = pio~u .
x x
equation by p e~ "", adding the two equations and inte-
m
grating along the depth of the waveguide gives
pc'
2 {(u„p +p„u )f) c„ + c {p d u„ + u d p„)
m m x n m x m x
n Jo
iik k )x
+ ic„k„(u„p +p„u )}e "- - dz
Z'U = d-p.
m m
po)
In the above equations/; is the pressure and u is the displace-
n JO
I""
ment vector. The pressure and the normal component of the
displacement are continuous across any interface. This may
(3)
be expressed as
[p]Z = 0, [n-u]! = 0,
2
Since according to Eq. (1) u„ = ik p„/(p(D ), we find
n
J. Acoust. Soc. Am., Vol. 111, No. 1, Pt. 1, Jan. 2002 Ahmad T. Abawi: Energy-conserving one-way coupled mode model 161
2d (c„)k +c d k = 2 A „c„.
(4)
(k„ + k„) -p a p„ dz
x m m x m m
m x
Jo P
h P
The details of the above derivation is given in Appendix B.
\k ~k )
dzPnlP
n m
In the above equation, A „ is the coupling matrix given by
m
ris

CB\
2
+ d\-d (d p )\p +-p p d {k )
(k„ + k ) -p fl {p„)dz + k„ p„f.
: x n m n m x
m m x

Jo P Jo
z
~k„
+ (k -k<)d \-jp p -d \-d jd p„)dz. (8)
x n m z zPm x
Xd \-\dz +
(5)
7'(*(i-*».)-»
-PnPmdxh
r
We would like to transfer all terms involving the range de-
rivatives of the modes from inside the integral to the bound-
The above equation is not yet in the desired form, as it con-
ary term by using integration by parts. The first term can be
tains the range derivative of the modes, which is difficult to
written as
compute accurately. In the remainder of this section we will
use the modal equations and the boundary and interface con-
ditions to convert the above equation into one which only
\oMp-
a dz
zPn\Pm
contains the local modes and their depth derivatives.
Consider the mode equations for mode n and mode m
f * / M
d \-)(dj> )(dj>„)dz.
M~j^zP )Pn
x m
n
Jo \pl
2 2
i\^d „) + -{k -k „)p„=o, (6)
zP
Similarly, the second and the fifth terms can be written as

\P IP
I di-d&tP,,}
JlPn, dz
2
(7)
dz\~dj>m \ + -{k -kl)p = 0.
Jo \P
m

\ p IP
B\
The modal equation is obtained by substituting u„
-dz(dxPn)Pn
-#z(Pn)dz(Pn,)dz,
0 P
1
= ik p„l{p«> ) into the second equation in Eq. (1). Next
n
evaluate
and

[» /I \
d \-d p \d (p )dz
B
\ {p a [Eq. (6)]-[Eq. {l)^ }dz.
z z m x n
m x xPn

Jo \p I
Jo
This gives
-Sz{p„)d {p )dz.
•d (Pn)(dzPn
z m
x
Jo P
Substituting these into Eq. (8) gives
flz azPn P m + rlz i)z xP ) P
J \* \~p) ) \p ^ " ) "
r'
(k + k„) -p„, d p„dz
m x
1 , - , /l
Jo P
+
~PnPm <?*(* ) + (k -k-„)9 \-\p„p
x n
]
= (k„-k r j"\ ( -a\-^(s p ){a )
m : m :Pn
+ k k
( ~n,- n)-Pm #xPn~ <?zl ~ #zPm I #xPn <•
2 2
+ -PnPm *»(* ) + (* -*5)*,(-W„
dz
The fifth term in the above equation can be written as
CB]
h d
(k -k„)(k + k„) -p„, djt„ dz
x\~ \^ p„)p + -d {d p )p„
m m
: m x z n
Jo P
(9)
d d d +
-dx(Pn)(dzPm)
z\ *\ -} zPnjPm d \-d (dxP„) \Pn
: :
The boundary term in the above equation may be written as
2 2
+ -PnPm d {k ) + {k -kl)d \-\p„p„
x x

/ 1 \ 1 1
d S
*\ pj^P^P^ ~ x(dzPn)Pm- ~dx(Pn)(dzPn
-3 \-d p \d p \dz,
z z m x n
)
#x\ -dzPn\Pm-'' x(P„)-(<'>zP„,)
or
162 J. Acoust. Soc. Am., Vol. 111, No. 1, Pt. 1, Jan. 2002
Ahmad T. Abawi: Energy-conserving one-way coupled mode model
Since the derivative along the interface of a continuous func-
2
[{k -k„k )p„p
\k k„)A <
tion/is continuous, we have
m n
m mn
PnPn
dz
1J= a
[T-V/lt = 0, where T=x+—z.
-0d»»a«p«]*xH+ ^*,(* )
y

dx
\pl
P

1 1 ,
This gives
Z
+ d h
-d;P„ d P„,+ -(k
r
z

P P
.,i{k„-k„,)x
(10)
' k„k )p„p„
m
Since bo\hp„ and d p„lp are continuous across the interface
z
z=/z(.ic), the above boundary term can be written as
The expression for the coupling matrix given by the above
equation is the main result of this paper. It shows the effect
d (h)d (p„)-{dj> )- d h d\-dj>„ \p„
of mode coupling due to contribution from volumetric and
x z m x
bathymetric variations in the waveguide separately. The first
part of the coupling matrix containing the integral is due to
With the help of the wave equation, Eq. (6), this may be
contribution from volumetric variations in the waveguide
written as
such as variations in sound speed and density as a function of
range. The second part is due to contribution from bathymet-
ric variations in range, i.e., variations in water depth, as is
2 2
dx{h)d,(p )-{d,p ) + d h-{k -k )p„p
n m x n n
evident from the presence of d h.
x
The coupling matrix has two important properties. First,
it is anti-symmetric, i.e., A = —A' . This implies that en-
Substituting this into Eq. (9) and the result into Eq. (5) yields
mn nm
Depth Slice at 30 m
Coupled Mode Solution
5 7.5 10 12.5
5 10
Range (km)
Range (km)
Depth Slice at 90 m
Adiabatic Mode Solution
r.
2.5 5 7.5 10 12.5
5 10
Range (km)
Range (km)
FIG. I. Propagation in an oceanic wedge: The water depth is constant at 200 m for the first 5 km and it slowly decreases to zero in the next 7.5 km resulting
in a wedge angle of 1.55 dcg. The top left panel shows the acoustic field due a 25 Hz source placed at 180 m as a function range and depth computed using
the coupled mode model. The bottom left panel shows the acoustic field computed using the adiabatic mode model. The top and bottom panels on the right
show a comparison of the transmission loss as a function of range computed using the PE model (solid) the coupled mode model (dotted) for two receiver
depths.
Ahmad T. Abawi: Energy-conserving one-way coupled mode model 163
J. Acoust. Soc. Am., Vol. 111, No. 1, Pt. 1, Jan. 2002
Sound Speed Fluctuations
The Acoustic Field
1520
100
1515
~200
1510
300
•B
1505
EL
Q
400
1500
500
1495
'l490 600
5 10 15
10 15
Range (km)
Range (km)
F = 25 Hz
F = 100Hz
17 18
17 18
Range (km)
Range (km)
FIG. 2. Propagation through internal waves: The ocean environment consists of a 200 m layer over a bottom half space. The sound speed in the water is
modeled to simulate fluctuations due to internal waves. The top left panel shows the sound speed fluctuations in the water. The top right panel shows the
acoustic field in this environment due to a 100 Hz source placed at 100 m computed using the coupled mode model. The bottom two panels show a comparison
of the transmission between the PE model (solid), the coupled mode model (dotted), and the adiabatic coupled mode model (dashed). The transmission loss
is computed at a receiver depth of 100 m and for source frequencies of 25 and 100 Hz.
ergy is conserved among modes. Second, it only contains the resulting in a wedge angle of approximately 1.55 deg. To
modes and their depth derivatives. This means that the cou- approximate the branch cut integral in the modal representa-
pling matrix can easily be computed using the local modes
tion of the field, a 1000 m deep false bottom is used. The
and their depth derivatives, which can be obtained from any
waveguide consists of two isovelocity layers: a water layer
normal mode code such as KRAKEN.
over a bottom layer. The water sound speed is 1500 m/s and
3
its density is 1.0 g/cm . The bottom sound speed is 1700 m/s
3
with a density of 1.15 g/cm . The attenuation in the bottom
III. EXAMPLES
is 0.5 dB/\. A 25 Hz source is placed at 180 m. These
environmental parameters are chosen to correspond to those
In this section the one-way coupled mode model devel-
used by Coppens and Sanders.
oped in the previous section is applied to two range-
The acoustic field in the waveguide is computed using
dependent cases. In the first example we use the one-way
Eq. (2) with the modal coefficients, c , obtained from the
coupled model to compute acoustic propagation an oceanic
m
solution of Eq. (4). The first order differential equation for
wedge wherein range-dependence is due to variations in the
the modal coefficients, Eq. (4), is solved by using fourth-
water depth. In the second example propagation through an
order Runge-Kutta integration. To obtain the modes and the
ocean with internal waves is computed where the ocean en-
coupling matrix as a function of range, the wedge is divided
vironment is chosen such that range-dependence is entirely
into range-independent stair steps. The local modes and the
due to variations in sound speed. The results are compared
M
local coupling matrix using Eq. (10) are computed in each
with those obtained using the parabolic equation PE
stair step and updated in the differential equation. The step
method.
size in the two examples that arc presented in this paper is 10
A. Propagation in a wedge
m. However, a step size of 50 m gives identical results.
The ocean environment in this example is scaled to cor- The results of the above computation are shown in Fig.
respond to the model tank experiment reported by Coppens 1. The top left panel in Fig. 1 shows the acoustic field com-
12
puted using the one-way coupled mode model described in
and Sanders. The water depth is initially 200 m for the first
5 km and then it slowly decreases to zero in the next 7.5 km this paper. It can be seen that as the water depth decreases,
164 J. Acoust. Soc. Am., Vol. 111, No. 1, Pt. 1, Jan. 2002 Ahmad T. Abawi: Energy-conserving one-way coupled mode model
the water modes (there arc three water modes in this ex-
normal mode solution does not agree with the parabolic
ample) cutoff in the form of discrete beams radiating into the
equation solution at all. This is more evident at the higher
bottom. The experimental data obtained by Coppens and
frequency where the effects of the internal waves, and thus
Sanders show the exact same behavior. Jensen and
the mode coupling due to them, is stronger.
16
Kuperman used the parabolic equation method to model
the acoustic propagation in this example and found results
ACKNOWLEDGMENTS
identical to those shown in the top left panel of Fig. 1. They
inteipreted the slow disappearance of the discrete water
This work was supported in part by the Office of Naval
modes into the bottom as an indication that energy contained
Research Contract No. N0001496WX30305 and in part by
in a given mode does not couple into the next lower mode
the SPAWAR Systems Center Internal Research (IR) pro-
but couples almost entirely into the continuous mode spec-
gram Contract No. ZU548R8A01. The author would like to
trum. While this effect, which is a consequence of mode
thank Dr. Bob Odom of the Applied Physics Laboratory at
coupling, is implicitly accounted for in the parabolic equa-
the University of Washington for useful discussions and for
tion (PE) formulation, the coupled mode model explicitly
providing relevant references for this paper. The author also
accounts for it through the coupling matrix. The two panels
thanks the anonymous referees for their useful comments.
on the right in Fig. 1 show a comparison of the transmission
loss computed using the one-way coupled model and the PE
APPENDIX A: THE MODE ORTHOGONALITY
model for two receiver depths. The close agreement between
RELATIONSHIP
the two models clearly demonstrates that the one-way
coupled mode model correctly accounts for mode coupling. The mode orthogonality equation is obtained by multi-
If the coupling matrix in the one-way coupled mode model
plying the mode equation (6) by p,„ and Eq. (7) by p„ sub-
are set equal to zero, the one-way coupled mode model re-
tracting the resulting equations and integrating to give
duces to the adiabatic mode model. The bottom left panel in
Fig. 1 shows the acoustic field computed using the adiabatic
</_-
z| -JzPnj-Pn <?J-<?zP»
mode solution. Observe that the adiabatic mode solution
does not have the correct field behavior near cutoff. While in
+(*i-*2) -
the coupled mode solution modes gradually radiate their en-
PmPn dz=0.
Jo p
ergy to the bottom near cutoff, in the adiabatic mode solution
this process occurs abruptly because there is no mechanism
Integrating the first integral by parts results in boundary
for the transfer of energy between modes.
terms
1
-Pn-^zPm
Pm-diPn
B. Propagation in internal waves
The ocean environment in this example is one of the test
Since either the mode or its derivative is zero at the bound-
13
cases used at the Shallow Water Acoustic workshop. It con-
aries, there is no contribution from the above interface terms.
sists of 200 m of water over a 400 m, isovclocity bottom.
What remains is
3
The bottom density 1.5 g/cm and its sound speed was 1700
m/s. The sound speed profile and the velocity fluctuations
ik -k„){k + k„) -p„,p„dz = 0.
m m
due to internal waves in the water column are modeled ac-
'o P
L
cording to
For m + n,
c{z,r) = c(z) + Ac(z,r),
f* 1

(k,„+ -p„p dz = 0,
tj -1
where
n
Jo P
1515.0+0.016z z<26
and for m = n we choose to normalize the modes such that
c(z)--
/
1490(1.0+0.25(e~ ' + />-1.0)) z>26
r« i
and
(**+*„) -PmPn dz=2k„S „.
m
Jo P
z,2i
kc(z,r)= —e cos2irr.
APPENDIX B: DETAILS LEADING TO EQ. (4)
In the above b = (z — 200)/500 and r is measured in km. The
We start by multiplying the first equation in Eq. (1) by
top left panel in Fig. 2 shows the sound speed profile for this
u,„ and the second equation in Eq. (1) by p , adding the two
m
example. The top right panel shows the acoustic field in the
equations and integrating to get
waveguide for a 100 Hz source place at 30 m. The bottom
two panels in Fig. 2 show a comparison of the adiabatic
ik„\ (p„u,„ + u p )dz
n m
Jo
normal mode and the coupled mode solutions with the PE
solution for two source frequencies. In both cases the re-
1
PnP»
ceiver depth is at 70 m. The coupled mode solution agrees
d,\ -fl p \p + poy-u u \dz
: n m n m
2
/:
well with PE solution at both frequencies while the adiabatic pa)~
PC
Ahmad T. Abawi: Energy-conserving one-way coupled mode model 165
J. Acoust. Soc. Am., Vol. 111, No. 1, Pt. 1, Jan. 2002
Integrating the middle term on the right-hand side by parts
where the coupling matrix, A,„„ is defined by Eq. (10). For
m = n Eq. (B3) becomes,
yields
1
2 d c„k„+c„ d k„ + c k„ 2{a p„)—+p
Pm<l~-
x x n x n
;r>zPn\P
1,1
U

Jo \ p
l
Jo
~P)
pa)
0 \ ()C I
^)^r= .
( 0
+ c,
d-.Pn dzPm dz.
Jo \p
This can be written as
Since the quantities inside the square brackets are continuous
across the interface, the boundary term is zero. This results in
d c„k„+c„ d k„+c„k„ J d \ -jj dz
x x x
fl PnPm

•u \\ w f ( + ' \
(A
k +U +
' n\ {P U nPmW= \ T l^zPn)
n m

Jo Jo \ pc pu>
k
„Pn
+ c,
(3 h)
= 0.
x
i
X(/) pJ + paj u„u )dz. (Bl)
z m
It is shown in Appendix C that
Next consider the following term in Eq. (1), which can be
integrated to give,
(B4)
•A
+
z(d \i„)p dz = [z-\x p ) _-\ (z-u„ d p )dz.
which gives
z m n m z m

Jo Jo
(B2)
2d c„k„+c„ d k„=0.
x x
The continuity condition for the normal component of the
displacement can be written as
APPENDIX C: DERIVATION OF EQ. (B4)
[n-u„]: = [(-z+^x)-u„]! = 0.
The integral across the waveguide can be written as,
This gives,
[i-u„]t = {d h)u„.
x
Since p„, is continuous across the interface, Eq. (B2) reduces
Each one of the above integrals can be written as
to

.2\ <.. /_2
i )
JM?M:*$*-«**
z-{d u„)p,„ clz = [(d h)u Y_- \ (z-u„ dj> )dz.
z x nPm m

Jo Jo
and
Substituting this into Eq. (3) and using Eq. (Bl) gives
2, S c„\ (u„p +p u )dz + c„\ [p d u„
x m n m m x
J
A"
n Jo JO
Substituting for these quantities we find
+ ,(k k )x
+ u d p )dz + c„{(d h)ii } _e "- » = Q.
m x n x nPm
Next substituting for u„=ik„p„lu>-p gives
J:*$*-*J:®*-*
B
( '
2
id c„){k„+k ) ~p„p dz + c (k„ + k )
x m m n m
Jo P
The integral on the right-hand side is a constant which gives

f« p f« 1
m
I'AH^
' (^xPn)—dz+c„(d k„)\ -p„p dz
x m

Jo P Jo P
„dz
-C„k \ d \-\p p„
A. D. Pierce, "Extension of the method of normal modes to sound propa-
n x n
gation in an almost-stratificld medium," J. Acoust. Soc. Am. 37, 19-27
(1965).
k,,PnPn
J(k -k )x I
*D. M. Milder, "Ray and wave invariants for sofar channel propagation,"
_Q
(B3)
n m
(S h
x
J. Acoust. Soc. Am. 46. 1259-1263 (1969).
'Steven R. Rutherford and Kenneth E. Hawker, "Consistent coupled mode
theory of sound propagation for a class of nonscparablc problems," J.
Using the orthogonality of the modes we find,
Acoust. Soc. Am. 70, 554-564 (1981).
''John A. Fawcctt, "A derivation of the differential equations of coupled-
2 o c k + c f> « = 2u A c
x m m m A m mn n%
mode propagation," J. Acoust. Soc. Am. 92, 290-295 (1992).
Ahmad T. Abawi: Energy-conserving one-way coupled mode model
166 J. Acoust. Soc. Am., Vol. 111, No. 1, Pt. 1, Jan. 2002
5
B. Edward McDonald, "Bathymctric and volumetric contributions to
waveguide with stepwisc depth variations of a penetrable bottom," J.
ocean acoustic mode coupling," J. Acoust. Soc. Am. 100, 219-224
Acoust. Soc. Am. 74, 188-195 (1983).
(1996).
12
A. B. Coppcns and J. V. Sanders, "Transmission of sound into a fast fluid
''Ahmad T. Abawi, Michael D. Collins, and W. A. Kupcrman, "The coupled
bottom from an overlaying fluid wedge," Proceedings of Workshop on
mode parabolic equation," J. Acoust. Soc. Am. 102, 233 238 (1997).
Seismic Propagation in Shallow Water (Office of Naval Research, Arling-
7
V. V. Shevchcnko, "Irregular acoustic waveguides." Sov. Phys. Acoust. 7,
ton, VA, 1978).
389-397 (1962).
""Shallow water acoustic modeling workshop," September 1999, Naval
"Robert I. Odom "A coupled mode examination of irregular waveguides
Postgraduate School, Monterey, CA.
including the continuum spectrum," Gcophys. J. R. Astron. Soc. 86, 425-
14
Michael D. Collins, "A slipt-stcp pade solution for the parabolic equation
453 (1986).
method." J. Acoust. Soc. Am. 93, 1736-1742 (1993).
9
V. Maupin. "Surface waves across 2-d structures: A method based on
"Michael B. Porter, "The kraken normal mode program," SACLANT Un-
coupled local modes," Gcophys. J. 93, 173-185 (1988).
dersea Research Centre, La Spczia, Italy, Rep. SM-245, 1991.
'"Jcrocn Tromp, "A coupled local-mode analysis of surface-wave propaga-
F. B. Jensen and W. A. Kupcrman, "Sound propagation in a wedge shaped
tion in a laterally heterogeneous waveguide," Gcophys. J. Int. 117, 153
ocean with penetrable bottom." .1. Acoust. Soc. Am. 67, 1564-1566
161 (1994).
(1980).
" R. B. Evans, "A coupled mode solution for acoustic propagation in a
Ahmad T. Abawi: Energy-conserving one-way coupled mode model 167
J. Acoust. Soc. Am., Vol. 111, No. 1, Pt. 1, Jan. 2002