THREE-DIMENSIONAL WIDE-ANGLE AZIMUTHAL PE SOLUTION TO MODIFIED BENCHMARK PROBLEMS

Transcription

1 THREE-DIMENSIONAL WIDE-ANGLE AZIMUTHAL PE SOLUTION TO MODIFIED BENCHMARK PROBLEMS LI-WEN HSIEH, YING-TSONG LIN, AND CHI-FANG CHEN Department of Engineering Science and Ocean Engineering, National Taiwan University, No., Sec. 4, Roosevelt Rd., Taipei, Taiwan 6, R.O.C. In predicting wave propagation, the size of the angle of propagation plays an important role; thus, the concept of wide-angle is introduced. Most existing acoustic propagation prediction models do have the capability of treating the wide-angle but the treatment, in practice, is vertical propagation angle. This is desirable for solving D (r-z) problems. Typically, 3D problems are dealt with an N by D approximation. To deal with problems possessing 3D effects, the azimuthal coupling terms have to be considered in PE approximation. Moreover, in extending the D treatment to 3D, the wide-angle capability is maintained in most 3D models, but it is still vertical. Hence the concept of wide-angle is introduced in azimuthal direction to enhance the capability of predicting the azimuthal coupling and thus the 3D effects. A truncated wedge-shaped ocean which is modified from ASA benchmark problems is used in this study. The results show apparent 3D effects and validate the 3D wide-angle azimuthal PE model, a wide-angle version of FOR3D. Introduction In the mid 97 s, Tappert [] introduced the Parabolic Equation (PE) approximation method in solving underwater acoustic propagation problems to the acoustic community, which was called the standard PE, also recognized as narrow-angle PE. When the PE approximation is introduced to solve the underwater acoustic wave propagation problem, there are many approaches to apply the rational function approximations to the square-root operator. The approximations pose limitations in the propagation angles. A comparison of angular limitations of some rational function approximations is given in Ref. [9]. Energy outside the angle of propagation is neglected, therefore the acoustic solution inaccurately propagated which leaded to phase errors accumulated over ranges. Phase errors in parabolic approximations were analyzed with normal-mode theory [] and also investigated by Tappert et al. []. However, these results only considered the vertical propagation angle. To correct this type of error, the wide-angle equation must be considered. Other authors []-[7] also contributed to its development. For 3D development, the Lee-Saad-Schultz (LSS) 3D Wide-angle Model clearly and explicitly possesses the vertical wide-angle capability [6]. The same development was followed to formulate the azimuthal wide-angle capability using the LSS 3D Wide-angle Model [7]. In some 3D PE models, the approximations include both vertical and azimuthal expansions, yet only the vertical wide angle capability is taken in to account. Chen et al. [7] modified FOR3D [], [7] to include wide-angle capability in azimuthal direction. This model is used in this paper to discuss the propagation angle limitation in azimuthal direction. Munk et al., [7] investigated the refraction of sound by islands and seamounts using the concept of vertical mode and horizontal ray. They clearly showed the refraction of modal rays on horizontal plane, thus in this paper we took this problem as our first numerical test case. The other test case we use is a modified ASA benchmark problem which has been used by Fawcett [3], [3] to discuss the azimuthal wide angle capability. Typically, 3D problems are dealt with an NxD approximation, which treats a 3D field as a fan-like composition of many rz - plane without the θ -coupling terms. Two- dimensional wide angle capability of PE in vertical direction has been discussed by Lee et al. []. Three-dimensional effects on underwater acoustic transmission have attracted considerable interests recently [6]-[5], [8]. The primary cause of 3D effects is the bottom topographical variations [9]. In order to deal with problems likely to possess 3D effects, the θ -coupling terms are required in PE approximation [6]. In the benchmark problem of 3D wedge, the 3D effect is due to the horizontal refraction of the bottom-reflecting sound ray, shown in Fig.. The azimuthal wide-angle 3D PE code [7] (AWA3D) is based on 3D parabolic equation, which is an initial-boundary values problem. The boundaries imposed on the sidewall are usually generated by

2 D calculation which is not the actual boundary, and then the acoustic field computed is doubtful. Smith [] has commented that computing 36 o azimuth range in the beginning ranges, then narrowing the angular width of the computing azimuth range when marching out in the radial ranges, to avoid the weak boundary effect. Chen et al. [] proposed the overlap (computed field)/iteration (sidewall boundary) scheme to reduce the error due to the boundary effect. Three benchmark problems involving acoustic propagation in range-dependent media are recently proposed for numerical consideration at a special session of the 4 st Meeting of the Acoustical Society of America in Chicago. These problems are indeed challenging for a numerical scheme. In this paper, the first benchmarking, 3-D test subcase (a) is calculated by a three-dimensional azimuthal wide-angle model (AWA3D) with real 3D side-wall boundaries. This paper is organized as follows. Sec. discusses the selected benchmark of 3D wedge. Sec. 3 outlines the numerical scheme used to solve the benchmark, including the foundation of the AWA3D code and overlap/iteration scheme. Then the derivation of the modal decomposition of Helmholtz equation and illustration of the propagation angles in horizontal plane in Sec. 4. The narrow and wide angle PE approximations are presented in Sec. 5 along with the phase error estimation that demonstrates the azimuthal propagation angle limitation. Sec. 6 presents numerical solutions with discussions. Sec. 7 summarizes this paper. The Benchmark Problem This benchmark problem is the first subcase (a) in the 3D wedge benchmark problem of the Benchmarking Shallow Water Range-Dependent Acoustic Propagation Modeling problems recognized by the Acoustical Society of America (ASA), and has been addressed in a special session of the ASA 4st Meeting in Chicago. The scenario of this case is a 3D wedge environment analogous to the benchmark ASA wedge [5] but with a point source instead of a line source, such that this is a 3D problem instead of a D wedge in the previous ASA benchmark. The geometry of this wedge is given in Table and Fig.. The environment consists of a homogeneous water column (c w = 5 m/s, ρ w = g/cm 3 ), a sloping, penetrable and lossy bottom (c b = 7 m/s, α b =.5 db/λ, ρ w =.5 g/cm 3 ), and a flat pressure release surface. The water depth at the source position is m. The angle of the slope is approximately.86 o, i.e. the horizontal distance from the apex to the source is 4km. A 3D field solution is sought for a 5Hz point source placed at m depth, and outputs of TL vs. range or TL vs. depth respectively corresponding to the four different receivers shown in Table (R~R4) are required in the session, that will be shown in Figs. 4, 5, 7, and 8. 3 The Numerical Scheme We use AWA3D, an azimuthal wide-angle 3D PE code [7], and an overlap (computed field)/iteration (sidewall boundary) scheme for generating real sidewall boundaries [6] to calculate the 3D field. The following subsections briefly discuss the above numerical calculations. 3. The AWA3D code The one-way, outgoing 3D wave equation, in an operator form, can be expressed as follows: ( ur = ik + + X + Y ) u, () where ρ X = ( n (, r θ, z) ) +, Y. = () k z ρ z k r θ A rational function was applied to approximate the above square-root operator, i.e.,

3 3 + X + Y = + X X + Y Y +Ο ( X, Y 3 ). (3) 8 8 Then, Eq. () can be rewritten in the form ur ik = + + X X + Y Y u 8 8. (4) Eq. (4) accommodates the wide-angle capability in the azimuth direction [7] since the higher order approximation term ( nd order) of the operator Y is introduced. As used in FOR3D [6], a local solution to Eq. (4) can be written as + X X + Y Y 8 8 (,, δ u r+ r θ z) = e e u( r, θ, z). (5) Following the same technique to split the two exponential operators as in FOR3D [6] δ + X X δ Y Y 8 8 δ ( + θ ) = ( θ ) u r r,, z e e e u r,, z. Then the second exponential operator is treated similarly to the first exponential operator. Should the reader desires more details about the numerical treatment for AWA3D, please refer to Ref. [7]. The accuracy of the schemes comes from the following sources of errors: 3 E X Y 3 = X X + Y Y =Ο( X Y 8 8, ). (7) δ δ + + X Y X Y 4 4 δ δ + + δ δ 3 E = e e = XY Y + O( X, Y ) (8) δ δ 4 8 X Y where δ δ + + X Y 4 4 δ δ δ δ = + X + ( δ ) X Y + ( δ ) Y +... δ δ 8 8 X Y (9) The total error can be viewed as δ δ δ δ ( ) ( δ ) ( δ ) = + X + Y + XY + X + Y ( ) ( ) ( )( ) ( ) + + = = 8 δ δ X Y δ δ Error e e X Y δ X XY Y O X Y (6),. () 3. The true sidewall boundary The modified equation of the AWA3D code, one-way, outgoing 3D wave equation, is a mathematically classified parabolic partial differential equation. So the initial and boundary values must be imposed while calculating the three-dimensional sound field. Lee et al. [6] recommended a D PE solution as the sidewall boundary condition. But this D PE solution is not the true boundary since it does not include the azimuthal coupling effect, and will contaminate the computed acoustic fields []. Therefore, it is necessary to seek a true sidewall boundary by computing a 3D solution instead of a D solution. Chen et al. [] proposed the iteration scheme in overlapped computed field to create a true sidewall boundary. The iteration continues until the convergence of the 3D solutions shows the π periodicity in the overlapped region. This scheme applied in this paper is described in the next paragraph, and the diagram is shown in Fig. 3. 3

4 Since the upslope propagation path meets the apex at range of 4 km, the luxury of computing a full 36 o or larger azimuth region does not exist beyond 4-km range. Therefore, the computation is divided into two phases. The fist phase is to calculate the acoustic field in the vicinity of the source of the 4-km circle. The iteration scheme in overlapped region is applied to the 4-kim circle with the source at the center, which is shown as the shaded circle in Fig. 3. The second phase is that the solutions at 4-km range are used as the starter to compute the 3D solution in a full 98º azimuth sector up to 5-km range. The 4-km range calculation of phase I is as follows: The port sidewall is set at 9º azimuth ( o azimuth is parallel to the apex.), and the starboard is set at 9º azimuth, i.e., the starboard sidewall is down slope. But since we use the overlap/iteration scheme, the starboard sidewall is 45º azimuth, which is the same at 9º because of π periodicity; such that the total computed azimuth range will be 459º, while the overlapped range is from 35º azimuth to 45º azimuth, 99º in total. At each range marching step, the sidewall boundaries at 9º and 45º azimuth are computed first with a D PE model. After calculating the field (at the same range), the computed field at 35º replaces the 9º port boundary, and the one at 9º replaces the 45º starboard boundary. Then the sound field is re-computed with these new boundary conditions. This replacement procedure (iteration) is repeated until the field at 9º/45º azimuth takes the same as (or converge to) that at 35º/9º azimuth. After the above iteration procedure, we match to the next range and repeat the overlap and iteration scheme. For the desired TL at 5-km range, phase II is used to obtain the solution. We choose 9º azimuth to be the angle of the port sidewall boundary, at which angle the radial track will have enough horizontal distance (5 km) not to hit the apex (p.s. exact angle is 9.7 o ). So the azimuth range of 9º to 89º is the computation domain of the AWA3D code behind 4 km range (shown in Fig. 3(b)), i.e. the port sidewall boundary is set at 9º azimuth and the starboard is set at 89º azimuth behind 4 km range. And the initial field at 4 km range in this domain (98º, out 4 km range) is imposed by the iterated solution at the range limits (4 km) of the iteration scheme. Since we couldn t overlap the field behind 4 km, so we implement the AWA3D code only with the D PE solution boundaries but with the real 3D initial field (iterated solution in 4 km range). 4 Modal Decomposition in Horizontal Plane The 3D Helmholtz equation for harmonic source in cylindrical coordinates is p p p p kn ( r, θ, z) p=, r r r z r θ () where p( r, θ, z) is the acoustic pressure field, n( r, θ, z) = c / c( r, θ, z) is the index of refraction, c is the reference sound speed, c( r, θ, z) is the sound speed, and k is the reference wave number such that k = π f / c, with f being the source frequency. Since the physics on the horizontal plane is of our interests in this study, we may neglect the z plane and consider it as a D problem where the spatial variables are r and θ. Thus the Helmholtz equation written in polar coordinates ( r, θ ) can be deduced to the far-field wave equation of the form urr + ikur + u θθ + k n ( r, θ ) u = r, () by letting p ( r, θ) = u( r, θ) v( r), (3) where ( ) π () i k r 4 ikr ( ) ~, π kr r v r = H k r = e e k r (4) is the zeroth-order Hankel function of the first kind for the outgoing wave representation. Equation () may be solved by the method of separation of variables, 4

5 u = R( r) Θ ( θ ). (5) By substituting Eq. (5) into Eq. (), we can derive Rrr Rr r + ikr + k( n ) r = R R Θ, (6) which may be separated into two differential equations with l being the separation constant, Θ θθ + l Θ= (7) and l Rrr + ikrr + [ k( n ) ] R=. r (8) A simple solution can satisfy Eq. (7), il Θ= e θ. (9) For Eq. (8), we can assume its solution of a simple form imr R = e. () With this assumption, Eq. (8) can be rewritten as l m + km+ k + k n =, r () whose solution is kr Θ θθ l m= k ± k + ( n ). () By means of Eqs. (4), (5), (9), and (), we can write the solution of the propagating wave as () i( m+ k ) r± ilθ P = H ( k r ) RΘ e. (3) r From the above, we can also deduce the dispersion relation on the horizontal plane being k = k r + k θ, (4) where the radial and the azimuthal wavenumbers respectively are k = m+ k, k = l. (5) r θ Fig. shows the relation between X Y and r θ coordinates. The components of wave number in r and θ directions, respectively, are dr kr = k = k cos ξ, (6) ds dθ kθ = k r = k sin ξ, (7) ds where ξ is the bending angle of the acoustic ray, i.e., the propagation angle. If we consider only the outgoing wave and rewrite Eq. () as m= k + k + ( n ) β, (8) where l β =. (9) kr Taking the square of Eq. (8) results in ( m+ k) = k( n β), (3) or kn = k = ( m+ k) + k β. (3) From Eqs, (5), (7), and (3), it yields k β = sinξ = n sin ξ, (3) k or 5

6 β = n sin ξ. (33) 5 PE Approximation and Error Estimation in Horizontal Plane Equation () represents a two-way propagating wave and can be expressed in an operator form + ik + k n ( r, θ ) u = r r r θ. (34) If only the outgoing wave is considered, it results in the operator form as u ( ), r = ik + + X + Y u (35) where X = ( n ) (36) and Y =. (37) kr θ There are several ways to expand the square root in Eq. (35) [9]. In the following we will present the azimuthally narrow and wide angle expansions []. In Eq. (35), considering azimuthally narrow angle, one can write 3 + X + Y + X X + Y + O( X, Y ). (38) 8 Applying Eq. (38) to Eq. (8), we have m+ k k + ( n ) ( n ) β. (39) 8 In Eq. (35), when considering azimuthally wide angle, one can write 3 + X + Y + X X + Y Y + O( X, Y 3 ). (4) 8 8 With Eq. (4), Eq. (8) can be written as m+ k k + ( n ) ( n ) β β. (4) 8 8 Next we follow McDaniel [] by considering a single normal mode propagation in the PE approximation and comparing the results with the solution of Helmholtz equation (3). We know from the solution of the Helmholtz equation (3) that the correct modal phase is exp i( m+ k ), or r exp( ikrr ) from Eq. (5). Also, from Eqs. (8) and (33), the exact phase velocity relates to propagation angles as m+ k = kr = k + ( n ) n sin ξ. (4) The same result also can be directly derived from Eq. (6) kr = ki cosξ = kn sin ξ. (43) Comparing Eq. (4) to the rational expansion of the square-root operator in PE approximations reveals the inherent phase error. In the following, we will examine the phase errors in the narrow angle and the wide angle approximations. Meanwhile, a case referring to an NxD calculation in 3D problem is also included to present the effect when the θ -coupling is omitted. For simplifying the results, we choose k = and consider the property of the square-root approximation n, (44) i.e., let n. Then the exact modal phase in Eq. (4) now reduces to m+ k = k = ξ (45) r sin. 6

7 With Eq. (44) and With Eq. (44) and k =, the narrow angle approximation Eq. (39) is reduced to + β = ξ (46) k =, the wide angle approximation Eq. (4) is reduced to m k sin. 4 m+ k β β = sin ξ sin ξ. (47) 8 8 In an NxD calculation, the θ -coupling is omitted, i.e., the operator Y is deduced to zero in Eq. (37) and the square-root expansion. The phase is thus m+ k k + ( n ) ( n ). (48) 8 With Eq. (44) and k =, the above equation is reduced to m+ k. (49) Comparing the phases for the various approximation Eq. (46), (47), and (49) to the exact form Eq. (45), we plotted the absolute values of phase errors as a function of propagation angle in Fig.. 4 Err = sin ξ sin ξ sin ξ, (5) 3W 8 Err N 3 = sin ξ sin ξ, (5) Err sin ξ. Nx D = (5) When the acceptable error has been set to. which corresponds to TL difference about db [], as indicated by the horizontal dashed line, the approximate angle limitations to NxD, narrow- and wide-angle approximations can be seen as,.5, and. The angular limitations for both narrow-angle and wide-angle approximations are about the same as those in vertical direction [], [9]. 6 Numerical Results This section presents numerical calculations of propagation loss (transmission loss, TL) obtained by using the azimuthal wide-angle 3D PE code [7] (AWA3D code) described in Section II. The ASA 3D wedge benchmark problem was designed to study the effects of the 3D range-dependent environment. This section deals with the upslope propagation and the cross-slope propagation in the wedge described in Sec.. 6. R propagation loss The 3D and NxD solutions for the R case of the 3D wedge benchmark are shown in Fig. 4. The result shows there is no significant difference between the 3D and NxD solution that means there is no 3D effect in the upslope direction at 3 m depth. Jensen et al. [5] discussed the drop-down at 3.5 km which is also shown in Fig.4. The radiation into the bottom is particularly evident around 3.5 km where the fundamental mode is cut off, i.e. energies are penetrating into the sloping, penetrable, and lossy bottom at that range, and the pattern of the normal mode propagation is totally distorted. They also discovered the IFD PE solution exhibited a ~3 db greater than that of the one-way coupled mode solution. Since AWA3D adopted similar numerical techniques in dealing with the one-way, square-root operator, the -3 db discrepancy also present in Fig. 4 at 3-km to 3.5-km ranges. 7

8 6. R propagation loss The 3D and NxD solutions for the R case of the 3D wedge benchmark are shown in Fig. 5. The result shows there is no significant difference between the 3D and NxD solution as the R case that means there is also no 3D effect in the upslope direction at 5m depth. The fluctuation behind km, especially significant after km, reveals the unstable of the AWA3D code in dealing with the wave propagation in the sediment; nevertheless, it properly handles the water column. Jensen et al. [5] discovered the noise in IFD PE solutions at 5-m depth for ranges from km to 4 km. They suspected that it was due to the insufficient depth of the artificial absorbing floor. Figs. 5 and 6 exhibit the similar noise at the same region. The depth of the artificial absorbing floor lies at km beneath the sea floor which is even shallower than Jensen et al. [5] s 4-km depth. So the noise was caused by the same reason. Fig. 6 shows the 3D field solution generated by the AWA3D code. The result shows no significant difference between the 3D and D solutions presented in the previous article [5], i.e., there is no 3D effect in the upslope direction. 6.3 R3 propagation loss The 3D and NxD solutions for the R3case of the 3D wedge benchmark are shown in Fig. 7. The result shows there is robust phase difference between the 3D and NxD solutions that means there is the 3D effect in the cross slope direction at 3m depth. As the previous article [9] commented, the primary cause of 3D effects is the bottom topographical variations. In this case, the 3D effect is due to the horizontal refraction of the bottom-reflecting sound ray, which has been shown in Fig.. Moreover, we can see that the energy bends toward the slope from the TL contour and the 3D-NxD difference. We also observe the normal mode propagation pattern in the far field from the propagation loss curve. 6.4 R4 propagation loss The 3D and NxD solutions for the R4 case of the 3D wedge benchmark are shown in Fig. 8. It is evident that higher modes are cut off when far away from the source in the 3D calculation, since higher modes imply larger vertical propagating angles which would result in larger horizontal refraction shown in Fig Truncated Wedge Problem In this section we consider the truncated wedge problem [3], [3]. All the following numerical results have been obtained running FOR3D and its azimuthal wide-angle version Description of the Test Case The truncated wedge problem is a modified ASA benchmark problem, and investigated by Fawcett implementing the azimuthal operator with FFT [3] and higher-order finite difference schemes [3]. Fig. 3 shows the along-slope sectional view. The maximum range of computation is 4 km and angular span is 36 degrees. The source frequency is 5 Hz. We use mode 3 as the exciting mode. In all calculations, we set r = 6 m, z = m, and θ = /6. For the problem being symmetrical, in the following we will only show the results in the first quadrant, i.e., θ Transmission Loss and Comparisons In this test case, we have chosen the across-slope section at a fixed depth of 35 m to present the computational results, as shown in Fig. 4. 8

9 Fig. 4(a) plotted the transmission loss curves obtained by 3DWide-angle, 3DNarrow-Angle, and NxD schemes. All the computations agreed closely to a range of about 8 km. Beyond that distance, the 3D curves, including wide-angle and narrow-angle, started to drop, exhibiting the shadow zone property, whereas the NxD did not catch the behavior. When it went to the range of about km, the two 3D curves began to show some difference. From 9 km and further, the results by wide-angle and narrow-angle schemes mismatched in the form of phase shift. The differences can be seen in Fig. 4(b). Since the NxD calculation was not able to predict the shadow zone, the differences comparing to 3D calculations can be up to more than 5 db. As for the 3D wide-angle and narrow-angle calculations, the alternating behavior of the difference curve beyond 9 km represents the phase errors in the PE approximations of the schemes. 7 Concluding Remarks In this paper, the numerical solutions of the first subcase (a) in the 3D wedge benchmark problem of the Benchmarking Shallow Water Range-Dependent Acoustic Propagation Modeling problems and the azimuthal angular limitation of PE approximation has been investigated for 3D one-way wave propagation. We use AWA3D code, an azimuthal wide-angle 3D PE code, and an overlap (computed field)/iteration (sidewall boundary) scheme for 3D sidewall boundaries [] to calculate this 3D field. The numerical results show what we have anticipated. The 3D effect is present in the cross-slope case, since the primary cause of 3D effect is the bottom topographical variations, i.e. the horizontal refraction of the bottom-reflecting sound ray. Moreover, since the ray along the slope will not refracting horizontally, the 3D effect does not exist in the upslope and downslope direction. The angle limits exist in PE approximations of the square-root operators Eqs. (38) and (4). The limits are about the same for both azimuthal and vertical directions since the approximations are similar. From the plots of a simple analytic result it has been found that the limits for NxD, 3DNarrow-, and Wide-angle approximations are,.5, and when the tolerance is set as.. Two numerical test cases have been used to show such limitation. The differences between the schemes revealed in the form of phase shift or phase error indicate the capability and the necessity to catch the modal ray behavior correctly. It is also found that the azimuthal grid θ has to be considered carefully so as to predict the azimuthal coupling effects. Acknowledgement The authors are very grateful for the support of the National Science Council Research Project (NSC88-6-E--5, NSC89-6-E-44), Republic of China. The authors would like to thank Dr. Alex Tolstoy and Prof. Kevin Smith s effort in making the Benchmark session possible. We also are very grateful to Dr. Ding Lee s invaluable help and comments on 3D propagation. References [] F. D. Tappert, The parabolic approximation method, in Wave Propagation and Underwater Acoustics, edited by J. B. Keller and J. S. Papadakis (Springer, New York, 977). [] D. Lee and S. T. McDaniel, Ocean Acoustic Propagation by Finite Difference Methods (Pergamon Press, Oxford, 988). [3] G. Botseas, D. Lee, and K. E. Gilbert, IFD: Wide Angle Capability, U.S. Naval Underwater Systems Center TR No. 695, 983. [4] D. J. Thomson, Wide-angle parabolic equation solution to range-dependent benchmark problems, J. Acoust. Soc. Am. 87, 54-5 (99). [5] F. B. Jensen and C. M. Ferla, Numerical solutions of range-dependent benchmark problems in ocean acoustics, J. Acoust. Soc. Am. 87, (99). 9

10 [6] D. Lee and M. H. Schultz, Numerical Ocean Acoustic Propagation in Three Dimensions (World Scientific, Singapore, 995). [7] C.-F. Chen, Y.-T. Lin, and D. Lee, A three-dimensional azimuthal wide-angle model for the parabolic wave equation, J. Comp. Acoust., 7 (4) (999), [8] D. Lee, Three-dimensional effects: Interface between the Harvard Open Ocean Model and a three-dimensional acoustic model, in Oceanography and Acoustics Prediction and Propagation Models, edited by A. R. Robinson and D. Lee (AIP Press, 994). [9] J.-J. Lin and C. Chen, Three-dimensional effect on acoustic transmission in Taiwan s Northeastern Sea, Proceeding of International Shallow-Water Acoustics, Beijing, 997. [] R.N. Baer, Propagation through a three-dimensional eddy including effects on an array, J. Acoust. Soc. Am. 69, 7-75 (98) [] J.S. Perkins and R.N. Baer, An approximation to the three-dimensional parabolic-equation method for acoustic propagation, [] D. Lee, G. Botseas and W. L. Siegmann, Examination of three-dimensional effects using a propagation model with azimuthal coupling capability, J. Acoust. Soc. Am. 9, 39 (99) [3] J. A. Fawcett, Modeling three-dimensional propagation in an oceanic wedge using parabolic equation methods, J. Acoust. Soc. Am., 93 (5) (993), [4] F.B. Sturm and J.A. Fawcett, Numerical simulation of the effects of bathymetry on underwater sound propagation using three-dimensional parabolic models, Rep. SM-34, SACLANTCEN ASW Research Centre, San Bartolomeo, Italy, March (998). [5] G.H. Brooke, D.J. Thomson and G.R. Ebbeson, PECan: A Canadian parabolic equation model for underwater sound propagation, J. Comput. Acoust. 9, 69 () [6] C.-F. Chen, J.-J. Lin, W.-S. Hwang, and T. W.-H. Sheu, The validity of two-dimensional solutions imposed as the side-wall boundaries on the underwater acoustic computation in a three-dimensional environment, in Theoretical and Computational Acoustics 97, ed. Yu-Chiung Teng, E.C. Shang, and Yih-Hsing Pao, Martin Shultz, and Alan Pierce (World Scientific Publishing, 999). [7] W. H. Munk and F. Zachariasen, Refraction of Sound by Islands and Seamounts, J. Atmosph. Ocean. Tech., 8 (99), [8] A. Tolstoy, 3-D propagation issues and models, J. Comp. Acoust. 4 (3) (996), [9] F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics (AIP Press, New York, 994). [] S. T. McDaniel, Propagation of normal mode in the parabolic approximation, J. Acoust. Soc. Am., 57 () (975), [] F. D. Tappert and M. G. Grown, Asymptotic phase errors in parabolic approximations to the one-way Helmholtz equation, J. Acoust. Soc. Am., 99 (3) (996), [] G. Botseas, D. Lee, and D. King, FOR3D: A computer model for solving the LSS three-dimensional wide angle wave equation, U.S. Naval Underwater Systems Center TR No. 7943, 987. [3] F. Sturm and J. A. Fawcett, On the use of higher-order azimuthal schemes in 3-D PE modeling, J. Acoust. Soc. Am., 3 (6) (3),

11 Figures and Table Fig.. The horizontal refraction of the bottom-reflecting sound ray. Source is at m depth and km far from the apex. The launch angle is perfectly cross the slope and 7 o depressed. The blue line is the ray, and the red one is the projection in the plane at m depth. Subcase (a) Source/Receiver Characteristics: The following source receiver geometries are designed to produce outputs of TL vs. Range, TL vs. Depth, or TL vs. Time, as appropriate. S - One depth ( m), CW point source (5 Hz). R - One depth (3 m), all ranges ( - 4 km), upslope. R - One depth (5 m), all ranges ( - 4 km), upslope. R3 - One depth (3 m), all ranges ( - 5 km), cross-slope (along source position isobath). R4 - One range (5 km), cross-slope, all depths ( - m). Environmental Characteristics: ASA benchmark wedge parameters -- WITH attenuation slope, θ =.86 o starting depth, D = m at the source range to apex, R = 4 km sound speed in water, c w = 5 m/s density in water, ρ w =. g/cm 3 sound speed in bottom, c b = 7 m/s bottom attenuation, α b =.5 db/λ bottom density, ρ b =.5 g/cm 3 Parameters used in 3D calculation r= m θ= z= m Table The geometry list of the 3D wedge benchmark problem.

12 Fig.. The scenario of the 3D wedge benchmark problem. (a) (b) Fig. 3. (a) The overlap (computed field)/iteration (sidewall boundary) scheme within 4km range. (b) The top view of computed azimuth sector. The black circle is applied the overlap/iteration scheme. Fig. 4. Propagation loss versus range for the R TL curve of the 3D wedge benchmark. Note that there is no significant difference between AWA3D and NxD solution.

13 Fig. 5. Propagation loss versus range for the R TL curve of the 3D wedge benchmark. Note that there is no significant difference between AWA3D and NxD solution. Fig. 6. Propagation loss contour for the R and R TL curve of the 3D wedge 3

14 (a) Fig. 7. Propagation loss versus range for the R3 TL curve of the 3D wedge benchmark. Note that there is robust phase difference between AWA3D and NxD solution. The range interval is (a) ~4 km and (b) ~5 km. (b) 4

15 (a) Fig. 8. (a) The propagation loss contour at 3m depth resulted in the AWA3D code. (b) The difference (3D NxD) contour at the same depth (3m). (b) Fig. 9. Propagation loss versus range for the R4 TL curve of the 3D wedge benchmark. Fig.. The horizontal refraction of the bottom-reflecting sound ray. Source is at m depth and km far from the apex. The launch angles are both perfectly cross the slope, but 7 o depressed for blue ray and 5 o depressed for red ray. i.e., the larger vertical propagating angle more tend to refract the ray toward the slope. 5

16 Y e θ dx ds dθ r ds Fig.. X Y and r θ coordinate systems on a horizontal plane. θ r ξ dr ds X e r Fig.. Phase error vs. azimuthal propagation angle for no θ -coupling, narrow angle, and wide angle approximations where the nominal beamwidths for acceptable phase error are less than.. m 5 m/s.g/cm 3 m Modal Source m 38 m 7 m/s.5g/cm 3 36 m 36 m.5db/λ 45 m 7 m/s.5g/cm 3 64 m.5db/λ with.5db/λ/ m 7 m/s.5g/cm 3.dB/λ m Fig. 3. Schematic drawing (along-slope) of 3D penetrable truncated wedge problem. 6

17 (a) (b) Fig. 4. (a) Transmission loss and (b) differences curves plotted across-slope obtained by 3DWide-Angle, 3DNarrow-Angle, and NxD schemes. 7