This paper deals with accuracy assessment for cross-derivative (depth and azimuth) term corrections to a three-dimensional (3D) parabolic equation (PE). The focus is on local errors, involving comparison of the two sides of the PE at insertion of (scaled) Helmholtz-equation reference solutions. For media with a particular type of lateral sound-speed variation, mode expansion together with wavenumber integration to compute modal expansion coefficients produces very accurate Helmholtz solutions for the field and its spatial derivatives. There are explicit expressions for the wavenumber integrands in terms of Airy and exponential functions, and accuracy improvements by PE cross-derivative terms are easy to assess. For a relevant example, with 3D effects similar to those in a 3D Acoustical Society of America wedge benchmark, inclusion of a leading-order as well as an additional, higher-order, cross-derivative term in the PE is favorable. The additional term provides a fourth-order accurate approximation of the PE square-root operator. A fifth-order accurate Padé approximation yields further improvement, approaching the numerical accuracy limit set by the PE method itself. The adiabatic approximation is exact for the particular media under study, but the local PE errors are similar for related 3D wedge examples with mode coupling.

1.
W. L.
Siegmann
and
D.
Lee
, “
Aspects of three-dimensional parabolic equation computations
,”
Comput. Math. Appl.
11
,
853
862
(
1985
).
2.
D.
Lee
and
M. H.
Schultz
,
Numerical Ocean Acoustic Propagation in Three Dimensions
(
World Scientific
,
Singapore
,
1995
), pp.
1
207
.
3.
E. K.
Westwood
, “
Broadband modeling of the three-dimensional penetrable wedge
,”
J. Acoust. Soc. Am.
92
,
2212
2222
(
1992
).
4.
G. B.
Deane
and
M. J.
Buckingham
, “
An analysis of the three-dimensional sound field in a penetrable wedge with a stratified fluid or elastic basement
,”
J. Acoust. Soc. Am.
93
,
1319
1328
(
1993
).
5.
F.
Sturm
, “
Numerical study of broadband sound pulse propagation in three-dimensional oceanic waveguides
,”
J. Acoust. Soc. Am.
117
,
1058
1079
(
2005
).
6.
Y.-T.
Lin
and
T. F.
Duda
, “
A higher-order split-step Fourier parabolic-equation sound propagation solution scheme
,”
J. Acoust. Soc. Am.
132
,
EL61
EL67
(
2012
).
7.
F. B.
Jensen
,
W. A.
Kuperman
,
M. B.
Porter
, and
H.
Schmidt
,
Computational Ocean Acoustics
, 2nd ed. (
Springer
,
New York
,
2011
), pp.
1
794
.
8.
Y.-T.
Lin
,
J. M.
Collis
, and
T. F.
Duda
, “
A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximants
,”
J. Acoust. Soc. Am.
132
,
EL364
EL370
(
2012
).
9.
F.
Sturm
, “
Leading-order cross term correction of three-dimensional parabolic equation models
,”
J. Acoust. Soc. Am.
139
,
263
270
(
2016
).
10.
P. S.
Petrov
and
F.
Sturm
, “
An explicit analytical solution for sound propagation in a three-dimensional penetrable wedge with small apex angle
,”
J. Acoust. Soc. Am.
139
,
1343
1352
(
2016
).
11.
W. L.
Siegmann
,
G. A.
Kriegsmann
, and
D.
Lee
, “
A wide-angle three-dimensional parabolic wave equation
,”
J. Acoust. Soc. Am.
78
,
659
664
(
1985
).
12.
D.
Lee
and
W. L.
Siegmann
, “
A mathematical model for the 3-dimensional ocean sound propagation
,”
Math. Modell.
7
,
143
162
(
1986
).
13.
M. D.
Collins
, “
The adiabatic mode parabolic equation
,”
J. Acoust. Soc. Am.
94
,
2269
2278
(
1993
).
14.
J. A.
Fawcett
and
T. W.
Dawson
, “
Fourier synthesis of three-dimensional scattering in a two-dimensional oceanic waveguide using boundary integral equation methods
,”
J. Acoust. Soc. Am.
88
,
1913
1920
(
1990
).
15.
L. M.
Brekhovskikh
and
O. A.
Godin
,
Acoustics of Layered Media II
(
Springer
,
Berlin
,
1992
), pp.
1
395
.
16.
F.
Sturm
and
A.
Korakas
, “
Comparisons of laboratory scale measurements of three-dimensional acoustic propagation with solutions by a parabolic equation method
,”
J. Acoust. Soc. Am.
133
,
108
118
(
2013
).
17.
Y.-T.
Lin
,
T. F.
Duda
,
C.
Emerson
,
G.
Gawarkiewicz
,
A. E.
Newhall
,
B.
Calder
,
J. F.
Lynch
,
P.
Abbot
,
Y.-J.
Yang
, and
S.
Jan
, “
Experimental and numerical studies of sound propagation over a submarine canyon northeast of Taiwan
,”
IEEE J. Ocean. Eng.
40
,
237
249
(
2015
).
18.
J. D.
Sagers
and
M. S.
Ballard
, “
Testing and verification of a scale-model acoustic propagation system
,”
J. Acoust. Soc. Am.
138
,
3576
3585
(
2015
).
19.
S.
Calvo
,
M.
Nicholas
,
J. M.
Fialkowski
,
R.
Gauss
,
D. R.
Olson
, and
A. P.
Lyons
, “
Scale-model scattering experiments using 3D printed representations of ocean bottom features
,” in
Proceedings of OCEANS-IEEE
, Washington, DC (October 19–22,
2015
), pp.
1
7
.
20.
O. A.
Godin
, “
Parabolic equation in the theory of the sound propagation in three-dimensionally heterogeneous media
,”
Dokl. Phys.
45
,
367
371
(
2000
).
21.
O. A.
Godin
, “
Reciprocity and energy conservation within the parabolic approximation
,”
Wave Motion
29
,
175
194
(
1999
).
22.
A.
Bamberger
,
B.
Engquist
,
L.
Halpern
, and
P.
Joly
, “
Higher order paraxial wave equation approximations in heterogeneous media
,”
SIAM J. Appl. Math.
48
,
129
154
(
1988
).
23.
D.
Yevick
and
D.
Thomson
, “
Split-step/finite-difference and split-step/Lanczos algorithms for solving alternative higher-order parabolic equations
,”
J. Acoust. Soc. Am.
96
,
396
405
(
1994
).
24.
G.
Dahlquist
,
Å.
Bjőrck
, and
N.
Anderson
,
Numerical Methods
(
Prentice-Hall
,
New York
,
1974
), pp.
1
573
.
25.
F.
Sturm
and
N.
Kampanis
, “
Accurate treatment of a general sloping interface in a finite-element 3-D narrow-angle PE model
,”
J. Comput. Acoust.
15
,
285
318
(
2007
).
26.
S.
Ivansson
, “
Sound propagation modeling
,” in
Applied Underwater Acoustics
, edited by
L.
Bjørnø
,
T.
Neighbors
, and
D.
Bradley
(
Elsevier
,
Amsterdam, the Netherlands
,
2017
), Chap. 3, pp.
185
272
.
27.
S.
Ivansson
and
I.
Karasalo
, “
A high-order adaptive integration method for wave propagation in range-independent fluid-solid media
,”
J. Acoust. Soc. Am.
92
,
1569
1577
(
1992
).
28.
S.
Ivansson
, “
Simple test cases with accurate numerical solutions for 3-D sound propagation modelling
,” in
Proceedings of the 4th Underwater Acoustics Conference and Exhibition
, Skiathos, Greece (September 3–8,
2017
), pp.
609
616
.
29.
J. A.
Fawcett
, “
Modeling three-dimensional propagation in an oceanic wedge using parabolic equation methods
,”
J. Acoust. Soc. Am.
93
,
2627
2632
(
1993
).
30.
T. L.
Foreman
, “
An exact ray theoretical formulation of the Helmholtz equation
,”
J. Acoust. Soc. Am.
86
,
234
246
(
1989
).
31.
R. M.
Fitzgerald
, “
Helmholtz equation as an initial value problem with application to acoustic computation
,”
J. Acoust. Soc. Am.
57
,
839
842
(
1975
).
32.
J.-X.
Qin
,
W.-Y.
Luo
,
R.-H.
Zhang
, and
C.-M.
Yang
, “
Three-dimensional sound propagation and scattering in two-dimensional waveguides
,”
Chin. Phys. Lett.
30
,
114301
(
2013
).
You do not currently have access to this content.