Research ArticleDirect FVM Simulation for Sound Propagation inan Ideal Wedge

Hongyu Ji Xinhai Xu Xiaowei Guo Shuai Ye Juan Chen and Xuejun Yang

State Key Laboratory of High Performance Computing College of Computer National University of Defense TechnologyChangsha 410073 China

Correspondence should be addressed to Xinhai Xu xuxinhainudteducn

Received 8 February 2016 Accepted 21 April 2016

Academic Editor Carlo Rainieri

Copyright copy 2016 Hongyu Ji et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The sound propagation in a wedge-shaped waveguide with perfectly reflecting boundaries is one of the few range-dependentproblems with an analytical solution This provides a benchmark for the theoretical and computational studies on the simulationof ocean acoustic applications We present a direct finite volume method (FVM) simulation for the ideal wedge problem and bothtime and frequency domain results are analyzed We also study the broadband problem with large-scale parallel simulations Theresults presented in this paper validate the accuracy of the numerical techniques and show that the direct FVM simulation couldbe applied to large-scale complex acoustic applications with a high performance computing platform

1 Introduction

The research on the sound propagation problem has amultidisciplinary and practical significance over oceanologybiology shipbuildingmilitary affairs andmany other scienceand engineering subjects [1ndash3] Field survey and physicalexperiments are main approaches to the research of theunderwater sound propagation However the high costs lowdata rates and difficulties in arranging experiments restrictthe applications of experimental approach With the rapiddevelopment of high performance computing technologycomputational solution has become a much more importantapproach for the studying of underwater sound propagationproblems

The construction of a computational solution for thisproblem has been extensively investigated over the lastdecades Underwater sound propagation can be mathe-matically described by the wave equation thus essentiallydifferent computational models use different approximationsor discretization methods In overall there are seven types ofcomputationalmethods to solve the sound propagation prob-lem the spectral or fast field program (FFP) [4] the normal-mode solution (NM) [5ndash7] the parabolic equation solution(PE) [8ndash10] the ray modeling and ray solution [11ndash13] the

wave-number integration method [14] and finite-differencemethod (FDM) [15 16] or finite element method (FEM)[17 18] In order to improve the computational efficiencymost of themethods (except FDMand FEM) reduce the waveequation into an approximate one However the efficiencyof this type of approach is obtained by sacrificing generalitythrough the applied assumptions and approximations Refer-ence [19] has had an in-depth discussion over the capabilitiesand limitations of those sound propagation models

Running benchmark problems and comparing thenumerical results are important and necessary steps forvalidating a computational solution The ideal wedgeproblem describes the sound propagation in a wedge-shapewaveguide and it is one of the benchmarks proposedby Acoustical Society America (ASA) [20 21] It is oneof the few range-dependent problems with an analyticalsolution [22 23] Hence the ideal wedge problem has beenextensively studied for validating the solutions calculatedby various numerical approaches [18 24 25] Luo et al [26]completed the original benchmark problem and provided ananalytical solution for an ideal wedge problem with a rigidor a pressure-release bottom

Although these fast computational methods were suc-cessfully used in explaining many observed ocean acoustic

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3703974 9 pageshttpdxdoiorg10115520163703974

2 Shock and Vibration

phenomena there are still lots of scientific issues which can-not be accurately addressed by the approximated approachFor realistic problems we usually have to solve a two-waywave equation with complex geometry For this purpose thenumerical approach which directly discretizes the governingequations should be adopted Direct FDM and FEM sim-ulations [15 17] are used to model the acoustic problemsNevertheless since the direct discrete solutions must beable to represent the actual spatial and temporal variationof the acoustic field all these methods are much morecomputationally intensive compared to the approximationmethods Therefore the direct methods are rarely used tosimulating the ocean acoustic propagation applications in thepast

Over the past few decades high performance comput-ing (HPC) techniques have made great achievements andthe simulation time is possible to be significantly reducedthrough large-scale parallel computing whichmeans that thetraditional direct simulation approach may become applica-ble for realistic ocean acoustic problems on a modern HPCplatform

In this paper we design an acoustic numerical solverbased on the finite volume method (FVM) and present thefull numerical results of the direct FVM simulations forsound propagation in an ideal wedge Both time domainand frequency domain results are analyzed furthermore wealso model the broadband problem The broadband probemis relatively difficult to simulate through the approximatedapproach therefore it is seldom studied over the last decadesThe method proposed in this paper has its advantages forcomplex applications such as irregular boundary problems orbroadband problems

The remainder of the paper is organized as follows thenumerical techniques used in this paper are introduced inSection 2 Also a presentation of the analytical solution to theideal wedge problem from the Acoustical Society of America(ASA) is included in Section 2 The 2D numerical resultsare presented and analyzed in Section 3 Accuracy analysisand large-scale parallel simulations are provided too Theconclusion follows in Section 4

2 Model and Numerical Method

In this section we give a synopsis of the ASA wedgebenchmark and an introduction to the numerical techniquesused in our simulations

21 The Ideal Wedge Problem In the ideal wedge probleminvestigated by this paper a pressure-release sea surfaceand a rigid or pressure-release bottom constitute the mainboundaries of the 2Dwedge Both the surface and the bottomare perfectly reflecting boundaries and in the opposite theleft boundary has to be set to an nonreflecting (absorbing)boundary Although in the real ocean environment thewedge angle is very small in this paper we set it to 45∘ forsimplicity As shown in Figure 1 there is a line source situatedin the wedge

The 2D wedge problem is a vertical section of the 3Dproblem The 3D wedge problem may have two types of

Source

0

400

800

Dep

th (m

)

0 400 800Range (m)

c = 1500ms120588 = 10 gcm3

Pressure-

releas

e or r

igid botto

m

Figure 1 The sketch of the ideal wedge environment used in thenumerical experiments The line source is located at 400m in rangeand 200m in depth The source frequency is 25Hz

wave sources One is a spatial point locating in the fieldthe other takes a form of horizontal infinite straight lineacross the section parallel to the apex [26] The latter is anideal theoretical model not describing any entities directlyHowever it plays an elemental and significant role like the2D airfoil in flight vehicle aerodynamics and is widely usedfor experimental researches in computational acoustics

According to Luo et al [5] even in 2-dimensionalsimulation the problem with a point source generally hasa cylindrical geometry although the problem with a linesource is usually described with cartesian coordinates Thesolutions of these two problems take different forms For theconvenience of setting up the coordinates and meshing thecomputing domain we choose the line source

Luo et al [26] proposed an analytical solution for the idealwedge problem which can be used as a reference for accuracyvalidating We summarized the solution in this section

Under the 2D cylindrical coordinates the ideal wedgeproblem could be expressed as

119888 (r) 120588 (r) nabla sdot (1

120588 (r)nabla119901 (119905 r)) +

1205972

119901 (119905 r)1205971199052

= 120575 (119903) 119878 (119905 r) (1)

where r = (119903 120579) 119888 120588 and 119878 denote wave speed massdensity and sound source Applying themethod of separationof variables the wave equation could be reduced to theHelmholtz equation

1

119903

120597

120597119903(119903

120597119875

120597119903) +

1

1199032

1205972

119875

1205971205792

+ 1198962

119875 = minus120575 (119903 minus 119903

119904)

119903120575 (120579 minus 120579

119904) (2)

The analytical solution to the Helmholtz equation withrigid andor pressure-release bottom boundary conditionsbecomes

119875 (119903 120579)

=119894120587

1205790

infin

sum

119899=1

sin (120574119899120579119904) sin (120574

119899120579) 119869120574119899

(119896119903lt)119867(1)

120574119899

(119896119903gt)

(3)

Shock and Vibration 3

in which

120574119899=

(119899 minus 12) 120587

1205790

rigid bottom119899120587

1205790

pressure-release bottom

119903lt= min (119903 119903

119904)

119903gt= max (119903 119903

119904)

(4)

In order to obtain the exact value of the pressure level ateach spatial point this analytical solution also needs somenumerical computations for the expansion coefficients of theBessel and Hankel functions [23]

22 Numerical Technique The normal-mode methods andother frequency domain methods focus on the solution ofthe Helmholtz equation These approaches solve the soundpropagation process in the frequency domain Neverthelessthrough the direct FVM simulations we can directly obtainthe computational solution of the original wave equation

To numerically solve the wave equation we use an opensource CFD toolbox released by the OpenCFD Ltd namedOpenFOAM The equation is discretized through the finitevolume method which locally satisfied the physical conser-vation laws through the integral over a control volume Forthe spatial discretization terms various predefined schemesare selectively applied and the temporal terms are discretizedusing a simple Euler scheme Finally the wave equation isreduced to a linear system thus using the iterative solversin OpenFOAM we can get the solutions for the equationsat every time step Solvers in the toolbox include the con-jugate (PCG) and biconjugate gradient (PBiCG) methodsMore details are presented in the OpenFOAM Manual Theboundary conditions for sea surface bottom and the leftvertical numerical boundary are described numerically [27]

(i) The rigid bottom

120597119901

120597n= 0 (5)

(ii) The pressure-release bottom

119901 = 0 (6)

(iii) The nonreflecting boundary

120597119901

120597119905+ nabla (119901120601) = 0 (7)

The numerical schemes used to descretize the waveequation and the boundary conditions are listed in Table 1The solver and preconditioner used for solving 119901 from thelinear algebraic equations are listed in Table 2

3 Results

In the following section we first give a description to theplatform and the parameters of the numerical experiments

Table 1 Numerical schemes used for discretizing the wave equationand the boundary conditions

Differential operators Numerical schemes119889119889119905 Euler1198892

1198891199052 Euler

119892119903119886119889 Gauss linear119889119894V Gauss linearInterpolation schemes Linear

Table 2 Solver and preconditioner used for solving 119901 of the linearalgebraic equations

119901 items SettingsSolver PCGPreconditioner DIC

Then we give the numerical results for the periodic singlefrequency source problem with different bottom boundariesTheproblemwith a broadbandpulse source is also consideredand tested Finally the large-scale parallel simulation isapplied and analyzed

31 Platform and Parameters

The Platform In the experiments we use an HPC clustersituated in the State Key Laboratory of High PerformanceComputing This computing platform consists of hundredsof computing nodes and each node contains 12 Intel Xeon21 GHz E5-2620 CPU cores and a total memory of 16GB

The Parameters The parameters include the sound speed inthe waveguide 119888 = 1500ms the density of the media 120588 =

10 gm3 the wedge angle of 45∘ and the source locationof 400m in range and 200m in depth The periodic sourceproblem uses a source with 25Hz frequency The broadbandproblem employs a square pulse lasting 01 s

32 Accuracy Analysis The accuracy of the numerical tech-niques used in this paper is measured with the ASA idealwedge problem The specific features of this problem areshown in Figure 1

321 Convergence Analysis Generally the accuracy of thenumerical simulation is dependent on its mesh density Aconvergence analysis with differentmesh desities is illustratedin Figure 2 In this sectionwe use three sets ofmesh data for atotal of 127200 508800 and 2035200 mesh cells respectivelyThe original data are collected from the numerical pressurefield 119901 at 150-m depth as shown in Figure 2(a) with thedashed line The bottom boundary condition is set rigid Theanalytical solution uses 60 modes Its section at 150-m depthis plotted as a control group

In this paper the analytical solution is shown in terms oftransmission loss (TL) in units of dB re 1m which is definedas

TL = 20 log10

10038161003816100381610038161003816100381610038161003816

119901 (119903)

1199010(1)

10038161003816100381610038161003816100381610038161003816

(8)

4 Shock and Vibration

Source

Rigid botto

m

00

400 800

150

400

800

Section lineD

epth

(m)

(a) The section at 150-m depth

AnalyticalFVM

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(b) 127200 cells

Range (m)0 100 200 300 400 500 600

0

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(c) 508800 cells

Range (m)

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(d) 2035200 cells

Figure 2 Comparison of the Fourier transform of the FVM result 119901 at 150-m depth and the section of the analytical solution to theHelmholtzequation at the same depth (a)The section line (b) TL result in a mesh of 127200 cells (c) TL result in a mesh of 508800 cells (d) TL result ina mesh of 2035200 cells The continuous line indicates the analytical solution and the descrete dots indicate the result of the FVM simulation

where the 119903 is the location of a field point and the referencepressure takes the form of

1199010(119909) =

119894

4119867(1)

0

(1198960119909) (9)

In the equation of 1199010(119909) 1198960is defined as 119896

0= 21205871198911500 with

119891 denoting the source frequency [28]The analytical solution to the Helmholtz equation is

a frequency domain function The direct FVM simulationoutputs the time series of the pressure field 119901 at every cellcenter of themesh In order to compare the results of the FVMsimulation to the analytical solution we take the Fouriertransform of the FVM results from 119905 = 20 s to 119905 = 50 s andwe consider 2 receiver lines in depth of 30m and 150m TheFVM results are also shown in terms of TL defined as

TL1015840 = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119901 (119909119899)

1199010

100381610038161003816100381610038161003816100381610038161003816

(10)

where the reference pressure 1199010is set to the same value with

the amplitude of the source waveThe locations of ridges and troughs of the FVM results are

same with the analytical solution and remain invariant whilethe number of cells is increasing This analysis shows that amesh with 508800 cells is sufficient for the convergence

The troughs of the curves in 2 indicate the destructiveinterference of the wave coming from the source and eachreflected waves Looking into the details we may find thatsome numerical troughs (ie the one marked with an ldquoXrdquoin Figure 2(c)) go deeper than the analytical solution whilesome others (ie the one marked with a ldquoΔrdquo in Figure 2(c))stop at a high position And some troughs (ie the onesmarked with ldquoXrdquos in Figures 2(b) and 2(c)) go deeper whenthe field is divided into more cells

Comparing those figures with Figure 3(f) it is clearthat the troughs are horizontally located at the root of thepressure-range function That is because (10) maps the 119901(119903)near 0 to minus infinity This becomes the origin of the deep

Shock and Vibration 5

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Range (m)

Pres

sure

minus20

minus40

(d) 119901 section at 01 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(f) 119901 section at 20 s

Figure 3 The overview of the pressure field 119901 of the rigid bottom problem at time 01 s and 10 s and the spatial section of the pressure field119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show the snapshot at119905 = 20 s

dips of the curves and of the differences of the numericalresults and the analytical solution mentioned above

322 Results for the Rigid Bottom Here we consider theideal wedge problem drawn in Figure 1 with a rigid bottomTwo snapshots of the complete spatial result of the directFVM simulation for the rigid bottom problem are plotted inFigure 3 We also plot the the pressure field 119901 at the time of01 s 10 s and 20 s over the line of 150m in depth

The asymptotic stability of the FVM result is evident inFigure 3 In the initial 1 s time interval the numerical solutionpresents the propagation procedure of the sound wave Thehigh similarity of the results in snapshot of 119905 = 10 s and119905 = 20 s indicates the wave has reached its stable status in thewaveguideThis fact rationalizes the usage of the time intervalfrom 119905 = 20 s to 119905 = 50 s in the Fourier transformof the FVMresults

In Figure 4 the TL results in different depth are illus-trated The Fourier transform of multiple section-resultscould outline the complete features of the 119901 field bothspatially and temporally Figures 4(a) and 4(b) show thecomparison at 30-m depth and 150-m depth For both theFourier transform and the analytical solution their TL formsare defined as that in (8) and (10)

Over both of the two receiver lines the locations ofridges and troughs in range are the same between the FVM

result and the analytical solution Also the error of the FVMsimulation at most ridges has been controlled within 1 dB

323 Results for the Pressure-Release Bottom In this experi-ment we consider the ideal wedge problem with a pressure-release bottom Two snapshots of the spatial result of thedirect FVM simulation to the problem are shown in Figure 5Figure 5 also shows the spatial section of the pressure field 119901at the time of 01 s 10 s and 20 s along the line of 150m indepth

The results of the pressure-release bottom problem alsoshow a behavior of temporal convergence The wave prop-agation has reached the stable status after the time of 10 sObserving the curvesrsquo right ends we find a different reflectionmode with that in the simulation for the rigid bottomproblem

In Figure 6 the TL results of the pressure-release bottomproblem are illustrated The results are selected at 30-m and150-m depth The Fourier transform of multiple section-results outline the complete features of the 119901 field bothspatially and temporally Figures 6(a) and 6(b) show thecomparison at 30-m depth and 150-m depth

From the results shown in Figure 6 we can see thatmostlythe horizontal locations of the ridges and troughs in the FVMresults meet a good agreement with those of the analyticalsolution The errors at the ridges are controlled well

6 Shock and Vibration

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

Range (m)

(b) 150-m depth result

Figure 4 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the rigid bottom problem (a) TL resultat depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

Range (102 m)

(d) 119901 section at 01 s

0 100 200 300 400 500 600

0204060

Range (m)

Pres

sure

minus20

minus40

minus60

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0204060

Pres

sure

minus20

minus40

minus60

(f) 119901 section at 20 s

Figure 5 The overview of the pressure field 119901 of the pressure-release bottom problem at time 01 s and 10 s and the spatial section of thepressure field 119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show thesnapshot at 119905 = 20 s

33 Large-Scale Parallel Simulations In this section themethod of the direct FVM simulation is applied to thebroadband problem which is seldom studied in the researchon the approximated approach during the past few dacadesThe ideal wedge waveguide is also used and the results areanalyzed after the experiments

331 Broadband Problem The broadband problem solved inthis paper uses the initial and boundary conditions from theASAbenchmark It has a single square pulse source lasting for01 s in the same ideal wedge waveguide as shown in Figure 1Each subfigure in Figure 7 presents a snapshot of the wavepropagation in the pressure field 119901 Considering the relation

Shock and Vibration 7

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(b) 150-m depth result

Figure 6 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem(a) TL result at depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119905 = 003 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119905 = 006 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119905 = 009 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(d) 119905 = 012 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(e) 119905 = 015 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(f) 119905 = 018 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(g) 119905 = 021 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(h) 119905 = 024 s

Figure 7 The simulation for the single pulse source problem in the ideal wedge geometry The bottom boundary is set rigid The t value ofeach snapshot is listed in the subtitles of the figures The small blue rings around the source in (g) and (h) are the secondary reflections onthe edge of the source

between the sound speed 119888 and the waveguidersquos span thesnapshots are selected from 119905 = 003 s to 119905 = 024 s

The tone scale of the figures indicates the sound pressure119901 We can see that the sound wave has become flatter withits expansion The width of the wave packet becomes largerover time Intuitively speaking the numerical simulationperforms well on portraying the dissipation and dispersionof the sound wave in its propagation

332 Parallel Efficiency The computation of the direct FVMsimulation for the ideal wedge problem has been parallelizedThe experiments use the broadband pulse sound source andthe rigid bottom condition The time step is set as 1 times 10minus4 sand the number of steps is set 104 In this paper up to 384processors are employed for the computation The numberof processors (NoP) is increased by 6 times 2

119899 so as to satisfythe structure of the experimental platform mentiond in the

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

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2 Shock and Vibration

phenomena there are still lots of scientific issues which can-not be accurately addressed by the approximated approachFor realistic problems we usually have to solve a two-waywave equation with complex geometry For this purpose thenumerical approach which directly discretizes the governingequations should be adopted Direct FDM and FEM sim-ulations [15 17] are used to model the acoustic problemsNevertheless since the direct discrete solutions must beable to represent the actual spatial and temporal variationof the acoustic field all these methods are much morecomputationally intensive compared to the approximationmethods Therefore the direct methods are rarely used tosimulating the ocean acoustic propagation applications in thepast

Over the past few decades high performance comput-ing (HPC) techniques have made great achievements andthe simulation time is possible to be significantly reducedthrough large-scale parallel computing whichmeans that thetraditional direct simulation approach may become applica-ble for realistic ocean acoustic problems on a modern HPCplatform

In this paper we design an acoustic numerical solverbased on the finite volume method (FVM) and present thefull numerical results of the direct FVM simulations forsound propagation in an ideal wedge Both time domainand frequency domain results are analyzed furthermore wealso model the broadband problem The broadband probemis relatively difficult to simulate through the approximatedapproach therefore it is seldom studied over the last decadesThe method proposed in this paper has its advantages forcomplex applications such as irregular boundary problems orbroadband problems

The remainder of the paper is organized as follows thenumerical techniques used in this paper are introduced inSection 2 Also a presentation of the analytical solution to theideal wedge problem from the Acoustical Society of America(ASA) is included in Section 2 The 2D numerical resultsare presented and analyzed in Section 3 Accuracy analysisand large-scale parallel simulations are provided too Theconclusion follows in Section 4

2 Model and Numerical Method

In this section we give a synopsis of the ASA wedgebenchmark and an introduction to the numerical techniquesused in our simulations

21 The Ideal Wedge Problem In the ideal wedge probleminvestigated by this paper a pressure-release sea surfaceand a rigid or pressure-release bottom constitute the mainboundaries of the 2Dwedge Both the surface and the bottomare perfectly reflecting boundaries and in the opposite theleft boundary has to be set to an nonreflecting (absorbing)boundary Although in the real ocean environment thewedge angle is very small in this paper we set it to 45∘ forsimplicity As shown in Figure 1 there is a line source situatedin the wedge

The 2D wedge problem is a vertical section of the 3Dproblem The 3D wedge problem may have two types of

Source

0

400

800

Dep

th (m

)

0 400 800Range (m)

c = 1500ms120588 = 10 gcm3

Pressure-

releas

e or r

igid botto

m

Figure 1 The sketch of the ideal wedge environment used in thenumerical experiments The line source is located at 400m in rangeand 200m in depth The source frequency is 25Hz

wave sources One is a spatial point locating in the fieldthe other takes a form of horizontal infinite straight lineacross the section parallel to the apex [26] The latter is anideal theoretical model not describing any entities directlyHowever it plays an elemental and significant role like the2D airfoil in flight vehicle aerodynamics and is widely usedfor experimental researches in computational acoustics

According to Luo et al [5] even in 2-dimensionalsimulation the problem with a point source generally hasa cylindrical geometry although the problem with a linesource is usually described with cartesian coordinates Thesolutions of these two problems take different forms For theconvenience of setting up the coordinates and meshing thecomputing domain we choose the line source

Luo et al [26] proposed an analytical solution for the idealwedge problem which can be used as a reference for accuracyvalidating We summarized the solution in this section

Under the 2D cylindrical coordinates the ideal wedgeproblem could be expressed as

119888 (r) 120588 (r) nabla sdot (1

120588 (r)nabla119901 (119905 r)) +

1205972

119901 (119905 r)1205971199052

= 120575 (119903) 119878 (119905 r) (1)

where r = (119903 120579) 119888 120588 and 119878 denote wave speed massdensity and sound source Applying themethod of separationof variables the wave equation could be reduced to theHelmholtz equation

1

119903

120597

120597119903(119903

120597119875

120597119903) +

1

1199032

1205972

119875

1205971205792

+ 1198962

119875 = minus120575 (119903 minus 119903

119904)

119903120575 (120579 minus 120579

119904) (2)

The analytical solution to the Helmholtz equation withrigid andor pressure-release bottom boundary conditionsbecomes

119875 (119903 120579)

=119894120587

1205790

infin

sum

119899=1

sin (120574119899120579119904) sin (120574

119899120579) 119869120574119899

(119896119903lt)119867(1)

120574119899

(119896119903gt)

(3)

Shock and Vibration 3

in which

120574119899=

(119899 minus 12) 120587

1205790

rigid bottom119899120587

1205790

pressure-release bottom

119903lt= min (119903 119903

119904)

119903gt= max (119903 119903

119904)

(4)

In order to obtain the exact value of the pressure level ateach spatial point this analytical solution also needs somenumerical computations for the expansion coefficients of theBessel and Hankel functions [23]

22 Numerical Technique The normal-mode methods andother frequency domain methods focus on the solution ofthe Helmholtz equation These approaches solve the soundpropagation process in the frequency domain Neverthelessthrough the direct FVM simulations we can directly obtainthe computational solution of the original wave equation

To numerically solve the wave equation we use an opensource CFD toolbox released by the OpenCFD Ltd namedOpenFOAM The equation is discretized through the finitevolume method which locally satisfied the physical conser-vation laws through the integral over a control volume Forthe spatial discretization terms various predefined schemesare selectively applied and the temporal terms are discretizedusing a simple Euler scheme Finally the wave equation isreduced to a linear system thus using the iterative solversin OpenFOAM we can get the solutions for the equationsat every time step Solvers in the toolbox include the con-jugate (PCG) and biconjugate gradient (PBiCG) methodsMore details are presented in the OpenFOAM Manual Theboundary conditions for sea surface bottom and the leftvertical numerical boundary are described numerically [27]

(i) The rigid bottom

120597119901

120597n= 0 (5)

(ii) The pressure-release bottom

119901 = 0 (6)

(iii) The nonreflecting boundary

120597119901

120597119905+ nabla (119901120601) = 0 (7)

The numerical schemes used to descretize the waveequation and the boundary conditions are listed in Table 1The solver and preconditioner used for solving 119901 from thelinear algebraic equations are listed in Table 2

3 Results

In the following section we first give a description to theplatform and the parameters of the numerical experiments

Table 1 Numerical schemes used for discretizing the wave equationand the boundary conditions

Differential operators Numerical schemes119889119889119905 Euler1198892

1198891199052 Euler

119892119903119886119889 Gauss linear119889119894V Gauss linearInterpolation schemes Linear

Table 2 Solver and preconditioner used for solving 119901 of the linearalgebraic equations

119901 items SettingsSolver PCGPreconditioner DIC

Then we give the numerical results for the periodic singlefrequency source problem with different bottom boundariesTheproblemwith a broadbandpulse source is also consideredand tested Finally the large-scale parallel simulation isapplied and analyzed

31 Platform and Parameters

The Platform In the experiments we use an HPC clustersituated in the State Key Laboratory of High PerformanceComputing This computing platform consists of hundredsof computing nodes and each node contains 12 Intel Xeon21 GHz E5-2620 CPU cores and a total memory of 16GB

The Parameters The parameters include the sound speed inthe waveguide 119888 = 1500ms the density of the media 120588 =

10 gm3 the wedge angle of 45∘ and the source locationof 400m in range and 200m in depth The periodic sourceproblem uses a source with 25Hz frequency The broadbandproblem employs a square pulse lasting 01 s

32 Accuracy Analysis The accuracy of the numerical tech-niques used in this paper is measured with the ASA idealwedge problem The specific features of this problem areshown in Figure 1

321 Convergence Analysis Generally the accuracy of thenumerical simulation is dependent on its mesh density Aconvergence analysis with differentmesh desities is illustratedin Figure 2 In this sectionwe use three sets ofmesh data for atotal of 127200 508800 and 2035200 mesh cells respectivelyThe original data are collected from the numerical pressurefield 119901 at 150-m depth as shown in Figure 2(a) with thedashed line The bottom boundary condition is set rigid Theanalytical solution uses 60 modes Its section at 150-m depthis plotted as a control group

In this paper the analytical solution is shown in terms oftransmission loss (TL) in units of dB re 1m which is definedas

TL = 20 log10

10038161003816100381610038161003816100381610038161003816

119901 (119903)

1199010(1)

10038161003816100381610038161003816100381610038161003816

(8)

4 Shock and Vibration

Source

Rigid botto

m

00

400 800

150

400

800

Section lineD

epth

(m)

(a) The section at 150-m depth

AnalyticalFVM

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(b) 127200 cells

Range (m)0 100 200 300 400 500 600

0

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(c) 508800 cells

Range (m)

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(d) 2035200 cells

Figure 2 Comparison of the Fourier transform of the FVM result 119901 at 150-m depth and the section of the analytical solution to theHelmholtzequation at the same depth (a)The section line (b) TL result in a mesh of 127200 cells (c) TL result in a mesh of 508800 cells (d) TL result ina mesh of 2035200 cells The continuous line indicates the analytical solution and the descrete dots indicate the result of the FVM simulation

where the 119903 is the location of a field point and the referencepressure takes the form of

1199010(119909) =

119894

4119867(1)

0

(1198960119909) (9)

In the equation of 1199010(119909) 1198960is defined as 119896

0= 21205871198911500 with

119891 denoting the source frequency [28]The analytical solution to the Helmholtz equation is

a frequency domain function The direct FVM simulationoutputs the time series of the pressure field 119901 at every cellcenter of themesh In order to compare the results of the FVMsimulation to the analytical solution we take the Fouriertransform of the FVM results from 119905 = 20 s to 119905 = 50 s andwe consider 2 receiver lines in depth of 30m and 150m TheFVM results are also shown in terms of TL defined as

TL1015840 = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119901 (119909119899)

1199010

100381610038161003816100381610038161003816100381610038161003816

(10)

where the reference pressure 1199010is set to the same value with

the amplitude of the source waveThe locations of ridges and troughs of the FVM results are

same with the analytical solution and remain invariant whilethe number of cells is increasing This analysis shows that amesh with 508800 cells is sufficient for the convergence

The troughs of the curves in 2 indicate the destructiveinterference of the wave coming from the source and eachreflected waves Looking into the details we may find thatsome numerical troughs (ie the one marked with an ldquoXrdquoin Figure 2(c)) go deeper than the analytical solution whilesome others (ie the one marked with a ldquoΔrdquo in Figure 2(c))stop at a high position And some troughs (ie the onesmarked with ldquoXrdquos in Figures 2(b) and 2(c)) go deeper whenthe field is divided into more cells

Comparing those figures with Figure 3(f) it is clearthat the troughs are horizontally located at the root of thepressure-range function That is because (10) maps the 119901(119903)near 0 to minus infinity This becomes the origin of the deep

Shock and Vibration 5

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Range (m)

Pres

sure

minus20

minus40

(d) 119901 section at 01 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(f) 119901 section at 20 s

Figure 3 The overview of the pressure field 119901 of the rigid bottom problem at time 01 s and 10 s and the spatial section of the pressure field119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show the snapshot at119905 = 20 s

dips of the curves and of the differences of the numericalresults and the analytical solution mentioned above

322 Results for the Rigid Bottom Here we consider theideal wedge problem drawn in Figure 1 with a rigid bottomTwo snapshots of the complete spatial result of the directFVM simulation for the rigid bottom problem are plotted inFigure 3 We also plot the the pressure field 119901 at the time of01 s 10 s and 20 s over the line of 150m in depth

The asymptotic stability of the FVM result is evident inFigure 3 In the initial 1 s time interval the numerical solutionpresents the propagation procedure of the sound wave Thehigh similarity of the results in snapshot of 119905 = 10 s and119905 = 20 s indicates the wave has reached its stable status in thewaveguideThis fact rationalizes the usage of the time intervalfrom 119905 = 20 s to 119905 = 50 s in the Fourier transformof the FVMresults

In Figure 4 the TL results in different depth are illus-trated The Fourier transform of multiple section-resultscould outline the complete features of the 119901 field bothspatially and temporally Figures 4(a) and 4(b) show thecomparison at 30-m depth and 150-m depth For both theFourier transform and the analytical solution their TL formsare defined as that in (8) and (10)

Over both of the two receiver lines the locations ofridges and troughs in range are the same between the FVM

result and the analytical solution Also the error of the FVMsimulation at most ridges has been controlled within 1 dB

323 Results for the Pressure-Release Bottom In this experi-ment we consider the ideal wedge problem with a pressure-release bottom Two snapshots of the spatial result of thedirect FVM simulation to the problem are shown in Figure 5Figure 5 also shows the spatial section of the pressure field 119901at the time of 01 s 10 s and 20 s along the line of 150m indepth

The results of the pressure-release bottom problem alsoshow a behavior of temporal convergence The wave prop-agation has reached the stable status after the time of 10 sObserving the curvesrsquo right ends we find a different reflectionmode with that in the simulation for the rigid bottomproblem

In Figure 6 the TL results of the pressure-release bottomproblem are illustrated The results are selected at 30-m and150-m depth The Fourier transform of multiple section-results outline the complete features of the 119901 field bothspatially and temporally Figures 6(a) and 6(b) show thecomparison at 30-m depth and 150-m depth

From the results shown in Figure 6 we can see thatmostlythe horizontal locations of the ridges and troughs in the FVMresults meet a good agreement with those of the analyticalsolution The errors at the ridges are controlled well

6 Shock and Vibration

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

Range (m)

(b) 150-m depth result

Figure 4 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the rigid bottom problem (a) TL resultat depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

Range (102 m)

(d) 119901 section at 01 s

0 100 200 300 400 500 600

0204060

Range (m)

Pres

sure

minus20

minus40

minus60

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0204060

Pres

sure

minus20

minus40

minus60

(f) 119901 section at 20 s

Figure 5 The overview of the pressure field 119901 of the pressure-release bottom problem at time 01 s and 10 s and the spatial section of thepressure field 119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show thesnapshot at 119905 = 20 s

33 Large-Scale Parallel Simulations In this section themethod of the direct FVM simulation is applied to thebroadband problem which is seldom studied in the researchon the approximated approach during the past few dacadesThe ideal wedge waveguide is also used and the results areanalyzed after the experiments

331 Broadband Problem The broadband problem solved inthis paper uses the initial and boundary conditions from theASAbenchmark It has a single square pulse source lasting for01 s in the same ideal wedge waveguide as shown in Figure 1Each subfigure in Figure 7 presents a snapshot of the wavepropagation in the pressure field 119901 Considering the relation

Shock and Vibration 7

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(b) 150-m depth result

Figure 6 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem(a) TL result at depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119905 = 003 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119905 = 006 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119905 = 009 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(d) 119905 = 012 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(e) 119905 = 015 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(f) 119905 = 018 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(g) 119905 = 021 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(h) 119905 = 024 s

Figure 7 The simulation for the single pulse source problem in the ideal wedge geometry The bottom boundary is set rigid The t value ofeach snapshot is listed in the subtitles of the figures The small blue rings around the source in (g) and (h) are the secondary reflections onthe edge of the source

between the sound speed 119888 and the waveguidersquos span thesnapshots are selected from 119905 = 003 s to 119905 = 024 s

The tone scale of the figures indicates the sound pressure119901 We can see that the sound wave has become flatter withits expansion The width of the wave packet becomes largerover time Intuitively speaking the numerical simulationperforms well on portraying the dissipation and dispersionof the sound wave in its propagation

332 Parallel Efficiency The computation of the direct FVMsimulation for the ideal wedge problem has been parallelizedThe experiments use the broadband pulse sound source andthe rigid bottom condition The time step is set as 1 times 10minus4 sand the number of steps is set 104 In this paper up to 384processors are employed for the computation The numberof processors (NoP) is increased by 6 times 2

119899 so as to satisfythe structure of the experimental platform mentiond in the

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

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Shock and Vibration

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International Journal of

Shock and Vibration 3

in which

120574119899=

(119899 minus 12) 120587

1205790

rigid bottom119899120587

1205790

pressure-release bottom

119903lt= min (119903 119903

119904)

119903gt= max (119903 119903

119904)

(4)

In order to obtain the exact value of the pressure level ateach spatial point this analytical solution also needs somenumerical computations for the expansion coefficients of theBessel and Hankel functions [23]

22 Numerical Technique The normal-mode methods andother frequency domain methods focus on the solution ofthe Helmholtz equation These approaches solve the soundpropagation process in the frequency domain Neverthelessthrough the direct FVM simulations we can directly obtainthe computational solution of the original wave equation

To numerically solve the wave equation we use an opensource CFD toolbox released by the OpenCFD Ltd namedOpenFOAM The equation is discretized through the finitevolume method which locally satisfied the physical conser-vation laws through the integral over a control volume Forthe spatial discretization terms various predefined schemesare selectively applied and the temporal terms are discretizedusing a simple Euler scheme Finally the wave equation isreduced to a linear system thus using the iterative solversin OpenFOAM we can get the solutions for the equationsat every time step Solvers in the toolbox include the con-jugate (PCG) and biconjugate gradient (PBiCG) methodsMore details are presented in the OpenFOAM Manual Theboundary conditions for sea surface bottom and the leftvertical numerical boundary are described numerically [27]

(i) The rigid bottom

120597119901

120597n= 0 (5)

(ii) The pressure-release bottom

119901 = 0 (6)

(iii) The nonreflecting boundary

120597119901

120597119905+ nabla (119901120601) = 0 (7)

The numerical schemes used to descretize the waveequation and the boundary conditions are listed in Table 1The solver and preconditioner used for solving 119901 from thelinear algebraic equations are listed in Table 2

3 Results

In the following section we first give a description to theplatform and the parameters of the numerical experiments

Table 1 Numerical schemes used for discretizing the wave equationand the boundary conditions

Differential operators Numerical schemes119889119889119905 Euler1198892

1198891199052 Euler

119892119903119886119889 Gauss linear119889119894V Gauss linearInterpolation schemes Linear

Table 2 Solver and preconditioner used for solving 119901 of the linearalgebraic equations

119901 items SettingsSolver PCGPreconditioner DIC

Then we give the numerical results for the periodic singlefrequency source problem with different bottom boundariesTheproblemwith a broadbandpulse source is also consideredand tested Finally the large-scale parallel simulation isapplied and analyzed

31 Platform and Parameters

The Platform In the experiments we use an HPC clustersituated in the State Key Laboratory of High PerformanceComputing This computing platform consists of hundredsof computing nodes and each node contains 12 Intel Xeon21 GHz E5-2620 CPU cores and a total memory of 16GB

The Parameters The parameters include the sound speed inthe waveguide 119888 = 1500ms the density of the media 120588 =

10 gm3 the wedge angle of 45∘ and the source locationof 400m in range and 200m in depth The periodic sourceproblem uses a source with 25Hz frequency The broadbandproblem employs a square pulse lasting 01 s

32 Accuracy Analysis The accuracy of the numerical tech-niques used in this paper is measured with the ASA idealwedge problem The specific features of this problem areshown in Figure 1

321 Convergence Analysis Generally the accuracy of thenumerical simulation is dependent on its mesh density Aconvergence analysis with differentmesh desities is illustratedin Figure 2 In this sectionwe use three sets ofmesh data for atotal of 127200 508800 and 2035200 mesh cells respectivelyThe original data are collected from the numerical pressurefield 119901 at 150-m depth as shown in Figure 2(a) with thedashed line The bottom boundary condition is set rigid Theanalytical solution uses 60 modes Its section at 150-m depthis plotted as a control group

In this paper the analytical solution is shown in terms oftransmission loss (TL) in units of dB re 1m which is definedas

TL = 20 log10

10038161003816100381610038161003816100381610038161003816

119901 (119903)

1199010(1)

10038161003816100381610038161003816100381610038161003816

(8)

4 Shock and Vibration

Source

Rigid botto

m

00

400 800

150

400

800

Section lineD

epth

(m)

(a) The section at 150-m depth

AnalyticalFVM

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(b) 127200 cells

Range (m)0 100 200 300 400 500 600

0

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(c) 508800 cells

Range (m)

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(d) 2035200 cells

Figure 2 Comparison of the Fourier transform of the FVM result 119901 at 150-m depth and the section of the analytical solution to theHelmholtzequation at the same depth (a)The section line (b) TL result in a mesh of 127200 cells (c) TL result in a mesh of 508800 cells (d) TL result ina mesh of 2035200 cells The continuous line indicates the analytical solution and the descrete dots indicate the result of the FVM simulation

where the 119903 is the location of a field point and the referencepressure takes the form of

1199010(119909) =

119894

4119867(1)

0

(1198960119909) (9)

In the equation of 1199010(119909) 1198960is defined as 119896

0= 21205871198911500 with

119891 denoting the source frequency [28]The analytical solution to the Helmholtz equation is

a frequency domain function The direct FVM simulationoutputs the time series of the pressure field 119901 at every cellcenter of themesh In order to compare the results of the FVMsimulation to the analytical solution we take the Fouriertransform of the FVM results from 119905 = 20 s to 119905 = 50 s andwe consider 2 receiver lines in depth of 30m and 150m TheFVM results are also shown in terms of TL defined as

TL1015840 = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119901 (119909119899)

1199010

100381610038161003816100381610038161003816100381610038161003816

(10)

where the reference pressure 1199010is set to the same value with

the amplitude of the source waveThe locations of ridges and troughs of the FVM results are

same with the analytical solution and remain invariant whilethe number of cells is increasing This analysis shows that amesh with 508800 cells is sufficient for the convergence

The troughs of the curves in 2 indicate the destructiveinterference of the wave coming from the source and eachreflected waves Looking into the details we may find thatsome numerical troughs (ie the one marked with an ldquoXrdquoin Figure 2(c)) go deeper than the analytical solution whilesome others (ie the one marked with a ldquoΔrdquo in Figure 2(c))stop at a high position And some troughs (ie the onesmarked with ldquoXrdquos in Figures 2(b) and 2(c)) go deeper whenthe field is divided into more cells

Comparing those figures with Figure 3(f) it is clearthat the troughs are horizontally located at the root of thepressure-range function That is because (10) maps the 119901(119903)near 0 to minus infinity This becomes the origin of the deep

Shock and Vibration 5

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Range (m)

Pres

sure

minus20

minus40

(d) 119901 section at 01 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(f) 119901 section at 20 s

Figure 3 The overview of the pressure field 119901 of the rigid bottom problem at time 01 s and 10 s and the spatial section of the pressure field119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show the snapshot at119905 = 20 s

dips of the curves and of the differences of the numericalresults and the analytical solution mentioned above

322 Results for the Rigid Bottom Here we consider theideal wedge problem drawn in Figure 1 with a rigid bottomTwo snapshots of the complete spatial result of the directFVM simulation for the rigid bottom problem are plotted inFigure 3 We also plot the the pressure field 119901 at the time of01 s 10 s and 20 s over the line of 150m in depth

The asymptotic stability of the FVM result is evident inFigure 3 In the initial 1 s time interval the numerical solutionpresents the propagation procedure of the sound wave Thehigh similarity of the results in snapshot of 119905 = 10 s and119905 = 20 s indicates the wave has reached its stable status in thewaveguideThis fact rationalizes the usage of the time intervalfrom 119905 = 20 s to 119905 = 50 s in the Fourier transformof the FVMresults

In Figure 4 the TL results in different depth are illus-trated The Fourier transform of multiple section-resultscould outline the complete features of the 119901 field bothspatially and temporally Figures 4(a) and 4(b) show thecomparison at 30-m depth and 150-m depth For both theFourier transform and the analytical solution their TL formsare defined as that in (8) and (10)

Over both of the two receiver lines the locations ofridges and troughs in range are the same between the FVM

result and the analytical solution Also the error of the FVMsimulation at most ridges has been controlled within 1 dB

323 Results for the Pressure-Release Bottom In this experi-ment we consider the ideal wedge problem with a pressure-release bottom Two snapshots of the spatial result of thedirect FVM simulation to the problem are shown in Figure 5Figure 5 also shows the spatial section of the pressure field 119901at the time of 01 s 10 s and 20 s along the line of 150m indepth

The results of the pressure-release bottom problem alsoshow a behavior of temporal convergence The wave prop-agation has reached the stable status after the time of 10 sObserving the curvesrsquo right ends we find a different reflectionmode with that in the simulation for the rigid bottomproblem

In Figure 6 the TL results of the pressure-release bottomproblem are illustrated The results are selected at 30-m and150-m depth The Fourier transform of multiple section-results outline the complete features of the 119901 field bothspatially and temporally Figures 6(a) and 6(b) show thecomparison at 30-m depth and 150-m depth

From the results shown in Figure 6 we can see thatmostlythe horizontal locations of the ridges and troughs in the FVMresults meet a good agreement with those of the analyticalsolution The errors at the ridges are controlled well

6 Shock and Vibration

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

Range (m)

(b) 150-m depth result

Figure 4 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the rigid bottom problem (a) TL resultat depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

Range (102 m)

(d) 119901 section at 01 s

0 100 200 300 400 500 600

0204060

Range (m)

Pres

sure

minus20

minus40

minus60

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0204060

Pres

sure

minus20

minus40

minus60

(f) 119901 section at 20 s

Figure 5 The overview of the pressure field 119901 of the pressure-release bottom problem at time 01 s and 10 s and the spatial section of thepressure field 119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show thesnapshot at 119905 = 20 s

33 Large-Scale Parallel Simulations In this section themethod of the direct FVM simulation is applied to thebroadband problem which is seldom studied in the researchon the approximated approach during the past few dacadesThe ideal wedge waveguide is also used and the results areanalyzed after the experiments

331 Broadband Problem The broadband problem solved inthis paper uses the initial and boundary conditions from theASAbenchmark It has a single square pulse source lasting for01 s in the same ideal wedge waveguide as shown in Figure 1Each subfigure in Figure 7 presents a snapshot of the wavepropagation in the pressure field 119901 Considering the relation

Shock and Vibration 7

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(b) 150-m depth result

Figure 6 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem(a) TL result at depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119905 = 003 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119905 = 006 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119905 = 009 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(d) 119905 = 012 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(e) 119905 = 015 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(f) 119905 = 018 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(g) 119905 = 021 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(h) 119905 = 024 s

Figure 7 The simulation for the single pulse source problem in the ideal wedge geometry The bottom boundary is set rigid The t value ofeach snapshot is listed in the subtitles of the figures The small blue rings around the source in (g) and (h) are the secondary reflections onthe edge of the source

between the sound speed 119888 and the waveguidersquos span thesnapshots are selected from 119905 = 003 s to 119905 = 024 s

The tone scale of the figures indicates the sound pressure119901 We can see that the sound wave has become flatter withits expansion The width of the wave packet becomes largerover time Intuitively speaking the numerical simulationperforms well on portraying the dissipation and dispersionof the sound wave in its propagation

332 Parallel Efficiency The computation of the direct FVMsimulation for the ideal wedge problem has been parallelizedThe experiments use the broadband pulse sound source andthe rigid bottom condition The time step is set as 1 times 10minus4 sand the number of steps is set 104 In this paper up to 384processors are employed for the computation The numberof processors (NoP) is increased by 6 times 2

119899 so as to satisfythe structure of the experimental platform mentiond in the

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

Source

Rigid botto

m

00

400 800

150

400

800

Section lineD

epth

(m)

(a) The section at 150-m depth

AnalyticalFVM

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(b) 127200 cells

Range (m)0 100 200 300 400 500 600

0

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(c) 508800 cells

Range (m)

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(d) 2035200 cells

Figure 2 Comparison of the Fourier transform of the FVM result 119901 at 150-m depth and the section of the analytical solution to theHelmholtzequation at the same depth (a)The section line (b) TL result in a mesh of 127200 cells (c) TL result in a mesh of 508800 cells (d) TL result ina mesh of 2035200 cells The continuous line indicates the analytical solution and the descrete dots indicate the result of the FVM simulation

where the 119903 is the location of a field point and the referencepressure takes the form of

1199010(119909) =

119894

4119867(1)

0

(1198960119909) (9)

In the equation of 1199010(119909) 1198960is defined as 119896

0= 21205871198911500 with

119891 denoting the source frequency [28]The analytical solution to the Helmholtz equation is

a frequency domain function The direct FVM simulationoutputs the time series of the pressure field 119901 at every cellcenter of themesh In order to compare the results of the FVMsimulation to the analytical solution we take the Fouriertransform of the FVM results from 119905 = 20 s to 119905 = 50 s andwe consider 2 receiver lines in depth of 30m and 150m TheFVM results are also shown in terms of TL defined as

TL1015840 = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119901 (119909119899)

1199010

100381610038161003816100381610038161003816100381610038161003816

(10)

where the reference pressure 1199010is set to the same value with

the amplitude of the source waveThe locations of ridges and troughs of the FVM results are

same with the analytical solution and remain invariant whilethe number of cells is increasing This analysis shows that amesh with 508800 cells is sufficient for the convergence

The troughs of the curves in 2 indicate the destructiveinterference of the wave coming from the source and eachreflected waves Looking into the details we may find thatsome numerical troughs (ie the one marked with an ldquoXrdquoin Figure 2(c)) go deeper than the analytical solution whilesome others (ie the one marked with a ldquoΔrdquo in Figure 2(c))stop at a high position And some troughs (ie the onesmarked with ldquoXrdquos in Figures 2(b) and 2(c)) go deeper whenthe field is divided into more cells

Comparing those figures with Figure 3(f) it is clearthat the troughs are horizontally located at the root of thepressure-range function That is because (10) maps the 119901(119903)near 0 to minus infinity This becomes the origin of the deep

Shock and Vibration 5

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Range (m)

Pres

sure

minus20

minus40

(d) 119901 section at 01 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(f) 119901 section at 20 s

Figure 3 The overview of the pressure field 119901 of the rigid bottom problem at time 01 s and 10 s and the spatial section of the pressure field119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show the snapshot at119905 = 20 s

dips of the curves and of the differences of the numericalresults and the analytical solution mentioned above

322 Results for the Rigid Bottom Here we consider theideal wedge problem drawn in Figure 1 with a rigid bottomTwo snapshots of the complete spatial result of the directFVM simulation for the rigid bottom problem are plotted inFigure 3 We also plot the the pressure field 119901 at the time of01 s 10 s and 20 s over the line of 150m in depth

The asymptotic stability of the FVM result is evident inFigure 3 In the initial 1 s time interval the numerical solutionpresents the propagation procedure of the sound wave Thehigh similarity of the results in snapshot of 119905 = 10 s and119905 = 20 s indicates the wave has reached its stable status in thewaveguideThis fact rationalizes the usage of the time intervalfrom 119905 = 20 s to 119905 = 50 s in the Fourier transformof the FVMresults

In Figure 4 the TL results in different depth are illus-trated The Fourier transform of multiple section-resultscould outline the complete features of the 119901 field bothspatially and temporally Figures 4(a) and 4(b) show thecomparison at 30-m depth and 150-m depth For both theFourier transform and the analytical solution their TL formsare defined as that in (8) and (10)

Over both of the two receiver lines the locations ofridges and troughs in range are the same between the FVM

result and the analytical solution Also the error of the FVMsimulation at most ridges has been controlled within 1 dB

323 Results for the Pressure-Release Bottom In this experi-ment we consider the ideal wedge problem with a pressure-release bottom Two snapshots of the spatial result of thedirect FVM simulation to the problem are shown in Figure 5Figure 5 also shows the spatial section of the pressure field 119901at the time of 01 s 10 s and 20 s along the line of 150m indepth

The results of the pressure-release bottom problem alsoshow a behavior of temporal convergence The wave prop-agation has reached the stable status after the time of 10 sObserving the curvesrsquo right ends we find a different reflectionmode with that in the simulation for the rigid bottomproblem

In Figure 6 the TL results of the pressure-release bottomproblem are illustrated The results are selected at 30-m and150-m depth The Fourier transform of multiple section-results outline the complete features of the 119901 field bothspatially and temporally Figures 6(a) and 6(b) show thecomparison at 30-m depth and 150-m depth

From the results shown in Figure 6 we can see thatmostlythe horizontal locations of the ridges and troughs in the FVMresults meet a good agreement with those of the analyticalsolution The errors at the ridges are controlled well

6 Shock and Vibration

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

Range (m)

(b) 150-m depth result

Figure 4 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the rigid bottom problem (a) TL resultat depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

Range (102 m)

(d) 119901 section at 01 s

0 100 200 300 400 500 600

0204060

Range (m)

Pres

sure

minus20

minus40

minus60

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0204060

Pres

sure

minus20

minus40

minus60

(f) 119901 section at 20 s

Figure 5 The overview of the pressure field 119901 of the pressure-release bottom problem at time 01 s and 10 s and the spatial section of thepressure field 119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show thesnapshot at 119905 = 20 s

33 Large-Scale Parallel Simulations In this section themethod of the direct FVM simulation is applied to thebroadband problem which is seldom studied in the researchon the approximated approach during the past few dacadesThe ideal wedge waveguide is also used and the results areanalyzed after the experiments

331 Broadband Problem The broadband problem solved inthis paper uses the initial and boundary conditions from theASAbenchmark It has a single square pulse source lasting for01 s in the same ideal wedge waveguide as shown in Figure 1Each subfigure in Figure 7 presents a snapshot of the wavepropagation in the pressure field 119901 Considering the relation

Shock and Vibration 7

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(b) 150-m depth result

Figure 6 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem(a) TL result at depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119905 = 003 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119905 = 006 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119905 = 009 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(d) 119905 = 012 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(e) 119905 = 015 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(f) 119905 = 018 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(g) 119905 = 021 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(h) 119905 = 024 s

Figure 7 The simulation for the single pulse source problem in the ideal wedge geometry The bottom boundary is set rigid The t value ofeach snapshot is listed in the subtitles of the figures The small blue rings around the source in (g) and (h) are the secondary reflections onthe edge of the source

between the sound speed 119888 and the waveguidersquos span thesnapshots are selected from 119905 = 003 s to 119905 = 024 s

The tone scale of the figures indicates the sound pressure119901 We can see that the sound wave has become flatter withits expansion The width of the wave packet becomes largerover time Intuitively speaking the numerical simulationperforms well on portraying the dissipation and dispersionof the sound wave in its propagation

332 Parallel Efficiency The computation of the direct FVMsimulation for the ideal wedge problem has been parallelizedThe experiments use the broadband pulse sound source andthe rigid bottom condition The time step is set as 1 times 10minus4 sand the number of steps is set 104 In this paper up to 384processors are employed for the computation The numberof processors (NoP) is increased by 6 times 2

119899 so as to satisfythe structure of the experimental platform mentiond in the

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Range (m)

Pres

sure

minus20

minus40

(d) 119901 section at 01 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

(f) 119901 section at 20 s

Figure 3 The overview of the pressure field 119901 of the rigid bottom problem at time 01 s and 10 s and the spatial section of the pressure field119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show the snapshot at119905 = 20 s

dips of the curves and of the differences of the numericalresults and the analytical solution mentioned above

322 Results for the Rigid Bottom Here we consider theideal wedge problem drawn in Figure 1 with a rigid bottomTwo snapshots of the complete spatial result of the directFVM simulation for the rigid bottom problem are plotted inFigure 3 We also plot the the pressure field 119901 at the time of01 s 10 s and 20 s over the line of 150m in depth

The asymptotic stability of the FVM result is evident inFigure 3 In the initial 1 s time interval the numerical solutionpresents the propagation procedure of the sound wave Thehigh similarity of the results in snapshot of 119905 = 10 s and119905 = 20 s indicates the wave has reached its stable status in thewaveguideThis fact rationalizes the usage of the time intervalfrom 119905 = 20 s to 119905 = 50 s in the Fourier transformof the FVMresults

In Figure 4 the TL results in different depth are illus-trated The Fourier transform of multiple section-resultscould outline the complete features of the 119901 field bothspatially and temporally Figures 4(a) and 4(b) show thecomparison at 30-m depth and 150-m depth For both theFourier transform and the analytical solution their TL formsare defined as that in (8) and (10)

Over both of the two receiver lines the locations ofridges and troughs in range are the same between the FVM

result and the analytical solution Also the error of the FVMsimulation at most ridges has been controlled within 1 dB

323 Results for the Pressure-Release Bottom In this experi-ment we consider the ideal wedge problem with a pressure-release bottom Two snapshots of the spatial result of thedirect FVM simulation to the problem are shown in Figure 5Figure 5 also shows the spatial section of the pressure field 119901at the time of 01 s 10 s and 20 s along the line of 150m indepth

The results of the pressure-release bottom problem alsoshow a behavior of temporal convergence The wave prop-agation has reached the stable status after the time of 10 sObserving the curvesrsquo right ends we find a different reflectionmode with that in the simulation for the rigid bottomproblem

In Figure 6 the TL results of the pressure-release bottomproblem are illustrated The results are selected at 30-m and150-m depth The Fourier transform of multiple section-results outline the complete features of the 119901 field bothspatially and temporally Figures 6(a) and 6(b) show thecomparison at 30-m depth and 150-m depth

From the results shown in Figure 6 we can see thatmostlythe horizontal locations of the ridges and troughs in the FVMresults meet a good agreement with those of the analyticalsolution The errors at the ridges are controlled well

6 Shock and Vibration

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

Range (m)

(b) 150-m depth result

Figure 4 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the rigid bottom problem (a) TL resultat depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

Range (102 m)

(d) 119901 section at 01 s

0 100 200 300 400 500 600

0204060

Range (m)

Pres

sure

minus20

minus40

minus60

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0204060

Pres

sure

minus20

minus40

minus60

(f) 119901 section at 20 s

Figure 5 The overview of the pressure field 119901 of the pressure-release bottom problem at time 01 s and 10 s and the spatial section of thepressure field 119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show thesnapshot at 119905 = 20 s

33 Large-Scale Parallel Simulations In this section themethod of the direct FVM simulation is applied to thebroadband problem which is seldom studied in the researchon the approximated approach during the past few dacadesThe ideal wedge waveguide is also used and the results areanalyzed after the experiments

331 Broadband Problem The broadband problem solved inthis paper uses the initial and boundary conditions from theASAbenchmark It has a single square pulse source lasting for01 s in the same ideal wedge waveguide as shown in Figure 1Each subfigure in Figure 7 presents a snapshot of the wavepropagation in the pressure field 119901 Considering the relation

Shock and Vibration 7

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(b) 150-m depth result

Figure 6 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem(a) TL result at depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119905 = 003 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119905 = 006 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119905 = 009 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(d) 119905 = 012 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(e) 119905 = 015 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(f) 119905 = 018 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(g) 119905 = 021 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(h) 119905 = 024 s

Figure 7 The simulation for the single pulse source problem in the ideal wedge geometry The bottom boundary is set rigid The t value ofeach snapshot is listed in the subtitles of the figures The small blue rings around the source in (g) and (h) are the secondary reflections onthe edge of the source

between the sound speed 119888 and the waveguidersquos span thesnapshots are selected from 119905 = 003 s to 119905 = 024 s

The tone scale of the figures indicates the sound pressure119901 We can see that the sound wave has become flatter withits expansion The width of the wave packet becomes largerover time Intuitively speaking the numerical simulationperforms well on portraying the dissipation and dispersionof the sound wave in its propagation

332 Parallel Efficiency The computation of the direct FVMsimulation for the ideal wedge problem has been parallelizedThe experiments use the broadband pulse sound source andthe rigid bottom condition The time step is set as 1 times 10minus4 sand the number of steps is set 104 In this paper up to 384processors are employed for the computation The numberof processors (NoP) is increased by 6 times 2

119899 so as to satisfythe structure of the experimental platform mentiond in the

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

International Journal of

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RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

TL (d

B)

0 100 200 300 400 500 600

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

Range (m)

(b) 150-m depth result

Figure 4 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the rigid bottom problem (a) TL resultat depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119901 field at 01 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119901 field at 10 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119901 field at 20 s

0 100 200 300 400 500 600

0

20

40

Pres

sure

minus20

minus40

Range (102 m)

(d) 119901 section at 01 s

0 100 200 300 400 500 600

0204060

Range (m)

Pres

sure

minus20

minus40

minus60

(e) 119901 section at 10 s

Range (m)0 100 200 300 400 500 600

0204060

Pres

sure

minus20

minus40

minus60

(f) 119901 section at 20 s

Figure 5 The overview of the pressure field 119901 of the pressure-release bottom problem at time 01 s and 10 s and the spatial section of thepressure field 119901 at depth of 150m (a) and (d) show the snapshot at 119905 = 01 s (b) and (e) show the snapshot at 119905 = 10 s (c) and (f) show thesnapshot at 119905 = 20 s

33 Large-Scale Parallel Simulations In this section themethod of the direct FVM simulation is applied to thebroadband problem which is seldom studied in the researchon the approximated approach during the past few dacadesThe ideal wedge waveguide is also used and the results areanalyzed after the experiments

331 Broadband Problem The broadband problem solved inthis paper uses the initial and boundary conditions from theASAbenchmark It has a single square pulse source lasting for01 s in the same ideal wedge waveguide as shown in Figure 1Each subfigure in Figure 7 presents a snapshot of the wavepropagation in the pressure field 119901 Considering the relation

Shock and Vibration 7

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(b) 150-m depth result

Figure 6 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem(a) TL result at depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119905 = 003 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119905 = 006 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119905 = 009 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(d) 119905 = 012 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(e) 119905 = 015 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(f) 119905 = 018 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(g) 119905 = 021 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(h) 119905 = 024 s

Figure 7 The simulation for the single pulse source problem in the ideal wedge geometry The bottom boundary is set rigid The t value ofeach snapshot is listed in the subtitles of the figures The small blue rings around the source in (g) and (h) are the secondary reflections onthe edge of the source

between the sound speed 119888 and the waveguidersquos span thesnapshots are selected from 119905 = 003 s to 119905 = 024 s

The tone scale of the figures indicates the sound pressure119901 We can see that the sound wave has become flatter withits expansion The width of the wave packet becomes largerover time Intuitively speaking the numerical simulationperforms well on portraying the dissipation and dispersionof the sound wave in its propagation

332 Parallel Efficiency The computation of the direct FVMsimulation for the ideal wedge problem has been parallelizedThe experiments use the broadband pulse sound source andthe rigid bottom condition The time step is set as 1 times 10minus4 sand the number of steps is set 104 In this paper up to 384processors are employed for the computation The numberof processors (NoP) is increased by 6 times 2

119899 so as to satisfythe structure of the experimental platform mentiond in the

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

AnalyticalFVM

0 200 400 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

(a) 30-m depth result

0 100 200 300 400 500 600

0

Range (m)

TL (d

B)

minus10

minus20

minus30

minus40

minus50

minus60

minus70

AnalyticalFVM

(b) 150-m depth result

Figure 6 Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem(a) TL result at depth of 30m (b) TL result at depth of 150m

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(a) 119905 = 003 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(b) 119905 = 006 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(c) 119905 = 009 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(d) 119905 = 012 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(e) 119905 = 015 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(f) 119905 = 018 s

0

2

4

6

8

0 200 400 600 800

Range (m)

Dep

th (102

m)

(g) 119905 = 021 s

0 200 400 600 8000

2

4

6

8

Range (m)

Dep

th (102

m)

(h) 119905 = 024 s

Figure 7 The simulation for the single pulse source problem in the ideal wedge geometry The bottom boundary is set rigid The t value ofeach snapshot is listed in the subtitles of the figures The small blue rings around the source in (g) and (h) are the secondary reflections onthe edge of the source

between the sound speed 119888 and the waveguidersquos span thesnapshots are selected from 119905 = 003 s to 119905 = 024 s

The tone scale of the figures indicates the sound pressure119901 We can see that the sound wave has become flatter withits expansion The width of the wave packet becomes largerover time Intuitively speaking the numerical simulationperforms well on portraying the dissipation and dispersionof the sound wave in its propagation

332 Parallel Efficiency The computation of the direct FVMsimulation for the ideal wedge problem has been parallelizedThe experiments use the broadband pulse sound source andthe rigid bottom condition The time step is set as 1 times 10minus4 sand the number of steps is set 104 In this paper up to 384processors are employed for the computation The numberof processors (NoP) is increased by 6 times 2

119899 so as to satisfythe structure of the experimental platform mentiond in the

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

0 20 40 60 800

2

4

6

8

10

12

Number of processors

Cloc

k tim

e (103

s)

(a) 508800 cells

100 150 200 250 300 350

6

8

10

12

14

16

18

Number of processors

Cloc

k tim

e (103

s)

(b) 8140800 cells

Figure 8 The clock time versus the number of processors

Section 31 Here we use two sets of mesh data with 508800and 8140800 mesh cells each The basic set with 508800 cellsis employed for the scalability experiment on NoPs up to96 The other is used in the situation of NoPs of 96 andlarger than 96 These parallel tests use a scotch methodin the decomposition process Figure 8(a) shows the clocktime under different NoPs in the mesh with 508800 cellsFigure 8(b) shows the clock time under different NoPs in themesh with 8140800 cells The refined mesh has four timesNoPs in both range and depth dimensions The clock timeplotted in the figures is the average value of 3 repeated trials

From Figures 8(a) and 8(b) we find that the clock timeof the process reduced expotentially with the increasing ofthe NoP The clock time of NoP = 12 is 193067 s Thesame indicator in the larger scale experiment of NoP = 384

reads 521667 s (about 27 times that of the 12 processorsrsquoexperiment) The ratio of the two time values infers that thecontribution of the expansion of the problem scale to the timeexpanding is offset by adding processors

4 Conclusion

In this paper we propose the platform and techniques of adirect FVM simulation for the sound propagation problemThe techniques of this simulation are applied to an idealwedge problem characterized by a homogeneous water col-umn a pressure-release sea surface a rigid or pressure-releasebottom and a periodic single frequency or a broadband pulseline source The accuracy of the simulation is analyzed bycomparing the numerical results with an analytical solutionA series of parallel computing experiments on solving thebroadband problem is implemented

We have studied the time and frequency domain resultsof the FVM simulation for the ideal wedge problem and havecompared the results with the analytical solution proposedby Luo et al [26] These comparisons reveal the accuracy ofthe method proposed in this paper The numerical resultsobtain a good agreement with the analytical solution Themesh convergence of the FVM simulation is analyzed Theconvergence tests reveal that the mesh density does not havenotable influence on the accuracy of the simulation

Both the periodic single frequency source and the broad-band pulse source are simulated and analyzed We haveinvestigated the time and frequency domain results Theplatform and techniques of the simulation show a hugeadaptability to these two different sources By observing thephysical phenomena the asymptotic stability of the solutionsto the periodic problem is considered This fact helps usensure the integration interval of the Fourier transform inthe accuracy analysis The numerical pulse generated by andpropagated in the simulation for the broadband problembehaves well

The large-scale parallel tests are implementedThe resultson up to 384 processors show a good scalability The tech-niques of the direct FVM simulation could be easily appliedto a parallel environment with hundreds of processors andthis application could significantly reduce the CPU time inthe simulation

In summary a direct FVM simulation for the ideal wedgeproblem with a homogeneous wedge column a pressure-release surface a rigid or pressure-release bottom and a peri-odic or single pulse line source is proposed With the exper-iments and analysis this simulation method shows its appli-cation prospects in the complicated simulations on oceanacoustics with higher performance computing platforms

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (Grant nos 61303071 and 61120106005) andthe Open Fund from the State Key Laboratory of HighPerformance Computing (nos 501503-01 and 201503-02)

References

[1] R-H Zhang ldquoProgress in research of ocean acoustics in ChinardquoPhysics vol 23 no 9 pp 513ndash518 1994 (Chinese)

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

[2] X-L Jin ldquoThe development of research in marine geophysicsand acoustic technology for submarine explorationrdquo Progress inGeophysics vol 22 no 4 pp 1243ndash1249 2007

[3] W-Q Zhao Z-M Xi J-B Lu and C-M Sheng ldquoUnderwateracoustic simulation of radar targetsrsquo electromagnetic scatteringcharactersrdquo Ship Electronic Engineering vol 30 no 11 pp 102ndash105 2010 (Chinese)

[4] S J Franke and G W Swenson Jr ldquoA brief tutorial on the fastfield program (FFP) as applied to sound propagation in the airrdquoApplied Acoustics vol 27 no 3 pp 203ndash215 1989

[5] W Luo C Yang J Qin and R Zhang ldquoA numerically stablecoupled-mode formulation for acoustic propagation in range-dependent waveguidesrdquo Science China Physics Mechanics andAstronomy vol 55 no 4 pp 572ndash588 2012

[6] J-X Qin W-Y Luo and R-H Zhang ldquoTwo-dimensionalsound propagation over a continental sloperdquo Acta Acustica vol39 no 2 pp 145ndash153 2014 (Chinese)

[7] D N Maksimov A F Sadreev A A Lyapina and A SPilipchuk ldquoCoupled mode theory for acoustic resonatorsrdquoWave Motion vol 56 pp 52ndash66 2015

[8] F Sturm ldquoNumerical study of broadband sound pulse propaga-tion in three-dimensional oceanic waveguidesrdquo Journal of theAcoustical Society of America vol 117 no 3 I pp 1058ndash10792005

[9] J Senne A Song M Badiey and K B Smith ldquoParabolicequation modeling of high frequency acoustic transmissionwith an evolving sea surfacerdquo Journal of the Acoustical Societyof America vol 132 no 3 pp 1311ndash1318 2012

[10] S D Frank R I Odom and J M Collis ldquoElastic parabolicequation solutions for underwater acoustic problems usingseismic sourcesrdquo Journal of the Acoustical Society of Americavol 133 no 3 pp 1358ndash1367 2013

[11] E K Westwood ldquoBroadband modeling of the three-dimensional penetrable wedgerdquo Journal of the AcousticalSociety of America vol 92 no 4 pp 2212ndash2222 1992

[12] E KWestwood ldquoComplex ray solutions to the 3minusDwedge ASAbenchmark problemsrdquo The Journal of the Acoustical Society ofAmerica vol 109 no 5 p 2333 2001

[13] J M Hovem ldquoRay trace modeling of underwater sound prop-agationrdquo in Modeling and Measurement Methods for AcousticWaves and for Acoustic Microdevices InTech 2013

[14] C A Langston ldquoAcousticseismic wavenumber integrationusing the WKBJ approximationrdquo AGU Fall Meeting Abstractsvol 1 article 52 2011

[15] R A Stephen ldquoSolutions to range-dependent benchmark prob-lems by the finite-difference methodrdquo Journal of the AcousticalSociety of America vol 87 no 4 pp 1527ndash1534 1990

[16] T Maeda and T Furumura ldquoFDM simulation of seismic wavesocean acoustic waves and tsunamis based on tsunami-coupledequations of motionrdquo Pure and Applied Geophysics vol 170 no1-2 pp 109ndash127 2013

[17] M Petyt J Lea and G H Koopmann ldquoA finite elementmethod for determining the acoustic modes of irregular shapedcavitiesrdquo Journal of Sound and Vibration vol 45 no 4 pp 495ndash502 1976

[18] J E Murphy and S A Chin-Bing ldquoA finiteminuselement model forocean acoustic propagation and scatteringrdquo The Journal of theAcoustical Society of America vol 86 no 4 pp 1478ndash1483 1989

[19] A Tolstoy ldquo3-D propagation issues and modelsrdquo Journal ofComputational Acoustics vol 4 no 3 pp 243ndash271 1996

[20] L B Felsen ldquoQuality assessment of numerical codes part 2benchmarksrdquo The Journal of the Acoustical Society of Americavol 80 supplement 1 pp S36ndashS37 1986

[21] L B Felsen ldquoNumerical solutions of two benchmark problemsrdquoThe Journal of the Acoustical Society of America vol 81 supple-ment 1 1987

[22] M J Buckingham and A Tolstoy ldquoAn analytical solutionfor benchmark problem 1 the lsquoidealrsquo wedgerdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1511ndash1513 1990

[23] G B Deane and M J Buckingham ldquoAn analysis of the three-dimensional sound field in a penetrable wedge with a stratifiedfluid or elastic basementrdquo Journal of the Acoustical Society ofAmerica vol 93 no 3 pp 1319ndash1328 1993

[24] Y Desaubies and K Dysthe ldquoNormalminusmode propagation inslowly varying ocean waveguidesrdquoThe Journal of the AcousticalSociety of America vol 97 no 2 pp 933ndash946 1995

[25] D J Thomson ldquoWide-angle parabolic equation solutions totwo range-dependent benchmark problemsrdquo Journal of theAcoustical Society of America vol 87 no 4 pp 1514ndash1520 1990

[26] W-Y Luo C-MYang J-XQin andR-H Zhang ldquoBenchmarksolutions for sound propagation in an ideal wedgerdquo ChinesePhysics B vol 22 no 5 Article ID 054301 2013

[27] J Schmalz and W Kowalczyk ldquoImplementation of acousticanalogies in OpenFOAM for computation of sound fieldsrdquoOpen Journal of Acoustics vol 5 no 2 pp 29ndash44 2015

[28] W-Y Luo and R-H Zhang ldquoA coupled-mode method forsound propagation with multiple sources in range-dependentwaveguidesrdquo Acta Acustica vol 36 no 6 pp 568ndash578 2011(Chinese)

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of