A Boundary Element Model for Underwater Acoustics in Shallow Water

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1 A Boundary Eleent Model for Underwater Acoustics in hallow Water J.A.F. antiago,l.c.wrobel Abstract: This work presents a boundary eleent forulation for two-diensional acoustic wave propagation in shallow water. It is assued that the velocity of sound in water is constant, the free surface is horizontal, and the seabed is irregular. The boundary conditions of the proble are that the sea botto is rigid and the free surface pressure is atospheric. For regions of constant depth, fundaental solutions in the for of infinite series can be eployed in order to avoid the discretisation of both the free surface and botto boundaries. When the seabed topography is irregular, it is necessary to divide the fluid region using the subregions technique. In this case, onlyirregular bottoboundariesand interfaces between regions of different depth need to be discretised. Nuerical siulations of several probles are included, ranging fro sooth to abrupt variations of the seabed. The results are verified by coparison with a ore standard BEM forulation in which the coplete seabed is discretised and truncated at a large distance. keyword: Boundary eleent ethod, subregions, underwater acoustics, shallow water, waveguides Introduction Due to the advent of high-speed coputers and the recent developents of nuerical physics, sound propagation in the ocean can be studied and quantitatively described in greater detail with the ore exact wave theory. Increasing concern for coastal areas has, in recent years, focussed studies of ocean acoustic wave propagation on shallow water environents. The ost coon nuerical techniques used to odel underwater acoustic wave propagation are ray ethods, noral ode ethods, and parabolic equation ethods [Jensen, Kuperan, Porter and chidt (99)]. Ray ethods are ost coonly used in deep water and are restricted to high frequencies; noral ode ethods are best suited for low frequencies but experience difficulties with doains that are both range and depth dependent; parabolic equation ethods neglect backscattering effects which are likely to be iportant in very shallow water and near the shore [Grilli, Pedersen and tepanishen (998)]. Brunel University, Departent of Mechanical Engineering, Uxbridge, Middlesex, UB8 3PH, UK, on leave fro COPPE/Federal University of Rio de Janeiro, Departent of Civil Engineering, Rio de Janeiro, Brazil Brunel University, Departent of Mechanical Engineering, Uxbridge, Middlesex, UB8 3PH, UK The present paper proposes a novel boundary eleent forulation for the nuerical odelling of shallow water acoustic propagation, in the frequency doain, over irregular botto topography. The odel assues a two-diensional geoetry, representative of coastal regions, which have little variation in the long shore direction. A recent application of the boundary eleent ethod (BEM) using a hybrid odel which cobines a standard BEM in an inner region with varying bathietry and an eigenfunction expansion in the outer region of constant depth was presented by Grilli, Pedersen and tepanishen (998). An iportant earlier work on acoustic scattering in the open ocean was presented by Dawson and Fawcett (99), in which the waveguide surfaces were taken to be flat except for a copact area of deforation where the acoustic scattering takes place. The BEM odel presented here akes use of two odified Green s functions, one of which satisfies the free surface boundary condition while the other directly satisfies the boundary conditions on the free surface and the horizontal part of the botto boundary. Alternatively, a Green s function in the for of eigenfunction expansions is eployed to iprove the convergence characteristics of the latter. Therefore, only bottoirregularitiesand interfaces need to be discretised. Results of the propagation and scattering of underwater acoustic waves in a region containing a vertical step-up, in a region of constant depth containing a botto deforation in the for of a cosine bell, and in a region representative of the seabed close to shore, are included to assess the accuracy of the nuerical solutions. Governing Equations of the Proble Consider the proble of acoustic wave propagation in a region Ω of infinite extent, shown in Fig.. Assuing that this ediu in the absence of perturbations is quiescent, the velocity of sound is constant and the source of acoustic disturbance is tie-haronic, the proble is governed by the Helholtz equation [Kinsler, Frey, Coppens and anders (98)]: u + k u = Nes B α δ (E α ;) in Ω () where u is the velocity potential, B α is the agnitude of the acoustic source E α located at (x Eα ;y Eα ), is the source point located at (x ;y ), Nes is the nuber of acoustic sources, δ (E α ;) is the Dirac delta generalised function and k = w=c

2 7 Copyright cfl Tech cience Press CME, vol., no.3, pp.73-8, is the wave nuber, in which w is the frequency and c is the velocity of sound in water. y F free surface Telles and Valenti (999)]. In this case the doain Ω is divided into several regions, as depicted in Fig.. Two types of subregions can be seen in the figure, for which different fundaental solutions are eployed. In the first (Ω and Ω )the region has irregular botto while in the other (Ω and Ω 3 )it has a constant depth. botto B u F I3 free surface 3 I3 B Figure : General ocean section for D acoustic propagation probles in shallow water x CD I B botto CD3 x The proble is subject to the following boundary conditions: a) Atospheric pressure at the free surface u (X) = on Γ F () b) Zero penetration velocity at the (rigid) seabed u (X) = on Γ B (3) c) oerfeld radiation condition at infinity u (X) =iku (X) () in which Γ F is the free surface, Γ B is the irregular botto, and n is the outward noral vector. According to Green s second identity, Eq. () can be transfored into the following boundary integral equation [Chen and Zhou (99); Lacerda (997)] Z C ()u() = Γ Z Nes G (;X) u Γ (X) (X)dΓ(X)+ G (;X) u (X)dΓ(X) B α G (E α ;) (5) where Γ is equal to Γ F [Γ B, X is the field point located at (x;y) and G(;X) is the free-space Green s function. The functions u(x) and u=(x) represent thevelocitypotentialanditsnoral derivative. The coefficient C() depends on the boundary geoetry at the source point. It is noted that the Green s function iplicitly satisfies the oerfeld condition, therefore no discretisation of the boundary at infinity is necessary. 3 Nuerical Analysis 3. Boundary eleent ethod General propagation probles with irregular seabed topography can be dealt with by the subregions technique [antiago, Figure : Regions dividing a typical shallow water section Instead of using the fundaental solution of the Helholtz equation for the free-space, Green s functions which directly satisfy the boundary conditions either on Γ F or on Γ F, Γ CD and Γ CD3 are adopted. Therefore, only the irregular parts of the botto boundaries (Γ B and Γ B ) and interfaces (Γ I, Γ I3 and Γ I3 ) need to be discretised. Introducing the appropriate boundary conditions into Eq. (5) yields the following integral equation for a region Ω χ with irregular botto: G f (;X) C ()u() = u (X)dΓ (X)+ ZΓBχ Z Γ I (X)» G u f (;X) (X) G f (;X) (X) u (X) dγ (X)+ Nes χ B α G f (E α ;) (6) and for a region Ω χ with constant depth: C ()u() = Z Γ I G D (;X) u (X) (X)» G u D (;X) (X) Nes χ dγ (X)+ B α G D (E α ;) (7) where Nes χ is the nuber of acoustic sources in the region Ω χ, Γ Bχ is the specific irregular botto and Γ I is the union of all interfaces, taking into account the direction of integration, belonging to region Ω χ. In order to solve Eq. (6) and (7) nuerically, the external boundary and interfaces are discretised into a nuber of el-

3 A Boundary Eleent Model for Underwater Acoustics in hallow Water 75 eents whose geoetries are odelled through shape functions and geoetrical nodal points. Over these eleents, the velocity potential and its noral derivative are interpolated as functions of the eleent nodal points. Constant eleents with linear geoetry have been used in this work. Applying the collocation ethod to Eq. (6) and (7) gives, in ters of an intrinsic coordinate η, foraregionω χ with irregular botto: Neb χ C ( p )u( p )= Nei χ» L q q= Nes χ + q= u q L q v q G f ( p ;X q )dη u q G f ( p ;X q ) dη + (X q ) G f ( p ;X q ) dη (X q ) B α G f (E α ; p ) (8) and for a region Ω χ with constant depth: Nei χ» L q C ( p )u( p )= v q G D ( p ;X q )dη q= Nes G D ( p ;X χ q ) u q dη + B α G D (E α ; p ) (9) where p ranges fro to Nf χ, with Nf χ the total nuber of functional nodes, Neb χ, Nei χ are the nuber of external boundary eleents and the nuber of eleents of all interfaces in the region Ω χ,respectively; p are collocation points and L q is the length of eleent Γ q. Finally, u q and v q are the velocity potential and its noral derivative, respectively, at the point X q which is the id-point of eleent Γ q. In the above equations, the first odified Green s function G f ( p ;X q ) identically satisfies the boundary condition on the free surface Γ F while the second function G D ( p ;X q ) identically satisfies the boundary condition on the free surface Γ F and seabed Γ B. Eq. (8) and (9) are coputed for each specific region separately, and are then coupled by iposing continuity of velocity potential and its noral derivative over the interfaces. Hence, applying these equations to all functional nodal points and considering that all boundary conditions are iplicitly satisfied leads to Ay = b () where the syste atrix A contains the influence coefficients, vector b contains the contribution of the acoustic sources, and vector y contains the unknown values of velocity potentials u at the irregular botto boundaries, and velocity potentials u and their noral derivatives v at the interfaces. 3. Fundaental solutions The fundaental solutions entioned in the previous section can be developed by two different eans. The first is the ethod of iages using either a single source point reflection or ultiple source point reflections, while the second uses eigenfunctions (noral odes) of the depth-separated Helholtz equation [Jensen, Kuperan, Porter and chidt (99)]. 3.. Irregular botto For the first subregion type (see Fig. ) the odified Green s function G f (;X), obtained by eans of the ethod of iages with a single source, is eployed in the following for: G i h f (;X) = H () G f (;X) = ik» H () r (kr) (X) kr (F) r (F) # H () (kr) H () kr (F) i () () where H () ( ) and H () ( ) are Hankel functions of the first kind of order and, respectively, r and r (F) are the distances fro the field point X to the source point and its reflection with respect to the free surface, respectively. These distances can be written as: r = jx j = q (x x ) +(y y ) X =(x;y) =(x ;y ) (3) fi fi fix (F)fi fi = r(x x ) + y y (F) (F) = x ;y (F) () r (F) = in which y (F) = y F y (5) where y F is the y-coordinate of the free surface. 3.. Constant depth For the second subregion type, the function G D (;X) is used in the for of two infinite series, the odified Greens function G F (;X) or G B (;X) and the Greens function G (;X). The first series, obtained through the ethod of iages with ultiple source point reflections, is given by [Jensen, Kuperan, Porter and chidt (99); antiago and Wrobel

4 76 Copyright cfl Tech cience Press CME, vol., no.3, pp.73-8, (999a)]: G i F (;X) = i () + = j= H () i (( j+)f) kr G i h B (;X) = H () i = h H () (kr) H () kr (F) i + h () j+ H () ( jf) kr () 5 j= (kr)+h () kr (B) i + H () kr ( jb) (6) (7) G f (,X) (F) r (F) r X free surface (F) botto (B) G(,X) F (5F) (F) r r r (F) (F) r (F) (5F) r (F) r (3F) (F) (3F) X u G(,X) B (3B) (B) r (B) r (3B) r (B) r (B) (B) r r (5B) (B) X with noral derivative: " G F (;X) = ik ik H () = G B (;X) ik H () r (kr) H() " () + () j+ j= # (( j+)f) r (( j+)f) kr = = ik " () 5 j= H () kr (F) r (F) # kr ( jf) r ( jf) H () r (kr) + H() kr (B) r (B) H () kr ( jb) ( jb) r # (8) (9) where the superindices (F), ( jf), (( j + )F), (B) and ( jb) identify the reflected source points. The above expressions (6) and (7) represent the sae infinite series of Hankel functions and their iages, and only differ in the way the series are truncated. The odified Green s function G F (;X) exactly satisfies the boundary condition on the free surface, but its noral derivative produces a very sall non-zero value at the botto boundary. Alternatively, the odified Green s function G B (;X) produces a very sall residual at the free surface but its noral derivative exactly satisfies the boundary condition at the botto. The distances fro the field point X to the reflections of source point (see Fig. 3) with respect to free surface(f) and botto(b) are denoted as r (F) ( jf) (( j+)f), r and r for G F (;X) and r (B) ( jb) and r for G B (;X) [antiago and Wrobel (999b)]. The distances r and r can be written ( jf) (( j+)f) as: r (JF) = fi fi fix (JF)fi fi = r(x x ) + (JF) = x ;y (JF) y y (JF) () (a) Figure 3 : Distance fro field point X to source point and its reflections with respect to free surface and botto in which J ranges fro to 5, and y (F) = ( )y F + y B y () y (3F) = ( y F + y B )+y () y (F) = (y F y B )+y (3) y (5F) = ( + )y F y B y () where y B is the y-coordinate of the botto. Eq. (), (5) and () to () are also used for G B (;X), with the indices F replaced by B, and vice-versa. The second series, in ters of noral odes, ay be expressed as [Jensen, Kuperan, Porter and chidt (99); Pedersen (996)]: G = i H e ik xjx x j (b) (5B) sin [k y (y F y )]sin [k y (y F y)] = k x (5) Its derivatives with respect to x and y are: G x G y = jx x j H (x x ) = sin [k y (y F y)]e ik xjx x j = i H = k y sin [k y (y F y )] k x sin [k y (y F y )] cos [k y (y F y)]e ik xjx x j (6) (7) where H is the depth. The paraeters k x and k y are horizon-

5 A Boundary Eleent Model for Underwater Acoustics in hallow Water 77 tal and vertical wavenubers, respectively: k y = π (8) H q k x = k ky (9) 3..3 Rearks A recent paper by Linton (998) acknowledges that standard representations of the Green s function in ters of infinite sus of iages, such as in (6) and (7), usually contain series which converge very slowly, and so are unsuitable for nuerical work. The calculation of the Green s function G given by (5) can be done uch ore efficiently due to the exponential ter in the series. It is clear that when k x becoes an iaginary nuber, i.e. when k y is greater than k, the power of the exponential in (5) becoes real and negative, and the series will decrease very rapidly for jx x j > : Nevertheless, when coordinate x of the source and field points is the sae, i.e. x x = ; the exponential ter of the function G is equal to one and convergence also becoes very slow. Another disadvantage of the Green s function G is that the singularity as y! y is not explicit. The influence coefficients of eleents on the interfaces of regions of constant depth and containing acoustic sources are calculated, for regular integration ( p 6= X q ), as: H L q pq = G L q pq = G ( p ;X q ) dη (3) (X q ) G ( p ;X q )dη (3) and for singular integration ( p = X p )as: H L p pp = G L p pp = Nes b p = G B ( p ;X p ) dη + (X p ) (3) G F ( p ;X p )dη (33) B α G (E α ; p ) (3) For regions with irregular botto, the influence coefficients and contribution of the acoustic sources are obtained as H L q pq = G L q pq = Nes b p = G f ( p;x q )dη + δ pq (35) G f ( p ;X q )dη (36) B α G f (E α ; p ) (37) where δ pq is the Kronecker delta. The integrals in Eq. (3) and (33) are coputed nuerically using Gaussian quadrature for either the coplete series or ter by ter. In the first case, the nuber of Gauss points ust be the sae for ters with sall and large r, increasing the coputer tie. A ter by ter integration with different nubers of Gauss points has been used. In addition, the asyptotic for of the Hankel function is eployed to integrate ters with source point reflection far fro the botto and free surface. Applications All the exaples analysed here siulate the propagation of acoustic waves in shallow water due to an acoustic source of unit agnitude, with the sound velocity c taken to be 5=s.. Region containing a vertical step-up The first exaple, shown in Fig., deals with a region containing a vertical step-up located at fro the origin. The deeper and shallower ends are equal to : and :5, respectively. The frequency f is taken to be Hzand the acoustic source is located at x E = : and y E = :5. y acoustic source.5 free surface botto...5 Figure : Geoetry for vertical step-up on the seabed (h =.) The boundary eleent discretisations are depicted in Fig. 5. In type (a), only Γ and the step-up Γ are discretised, each with 96 constant eleents of graded length, concentrated close to point (vv9). In type (b), the interface Γ was oved to a vertical line 5: fro the step and the external boundary Γ was extended along the botto of the shallower depth. The interface and vertical line of the boundary are discretised as in type (a), while the horizontal line is discretised with 9 constant eleents of graded length, concentrated close to points and3(vh38). In order to assess the accuracy of the results, a coparison was ade with a solution obtained using an alternative BEM forulation in which the botto boundary is fully discretised with 888 graded eleents, but the free surface is eliinated using a single iage source. The infinite botto boundary was truncated at the distances 78:5 and +8:5. Itwas x

6 78 Copyright cfl Tech cience Press CME, vol., no.3, pp.73-8, interface boundary 3 boundary 5. 3 interface Figure 5 : Discretisation of the proble using graded eleents, (a) only along the vertical line (vv), (b) along vertical and horizontal lines (vh) velocity potential Re(888) Re(vv9) Re(vh38) I(888) I(vv9) I(vh38) y for x = 5. Figure 7 : Velocity potential at points along the vertical line x = 5: found by trial and error that increasing these truncation distances produced very little difference in the solution. Fig. 6 presents the velocity potential at internal and nodal points along the vertical line x =, for different types of discretisation. Fig. 7 shows the velocity potential either on the interface or at internal points along the vertical line x = 5. It can be seen that the results depict a good agreeent for all discretisations, confiring that the use of subregions is appropriate for this type of proble. Velocity potential points at the boundary points at the interface or internal points Re(888) Re(vv9) Re(vh38) I(888) I(vv9) I(vh38) jx x j < w d = (38) where a d and w d are the height and total width of the deforation, and x is the algebraic value of the distance fro the y-axis to the peak of the deforation... A siple cosine bell on the botto The application is a coastal area with a siple cosine bell deforation at the botto, shown in Fig. 8. The acoustic source is located at the position (., 5.), directly over the center of the peak. The height a d and width w d of the deforation are taken to be 5: and :, respectively. Only the deforation was discretised, using ore than eleents per wavelength for all frequencies. 5 5 acoustic source y for x=. y() 5 Figure 6 : Velocity potential at points along the vertical line x = :. ooth deforations on the seabed In the next two exaples, the frequency f was varied fro 5Hz to 5Hz in increents of :5Hz to observe the behaviour of the velocity potential along the botto and at selected horizontal and vertical lines, for a range of frequencies. The defored botto surface is described by the following sinusoidal function f (x): f a ρ» ff d π (x x ) (x) = + cos w d x() Figure 8 : Geoetry of a coastal area with a siple cosine bell The aplitude of the velocity potential along the cosine bell deforation is presented in Fig. 9, for five selected frequencies. The resulting velocity potential field is seen to be syetric about the peak, as would be expected for this centered source, and to becoe ore coplex as the frequency increases. It can be noticed that sall perturbations in the results

7 A Boundary Eleent Model for Underwater Acoustics in hallow Water 79 appear, at the ends of the cosine bell, for the highest frequencies of f = Hz and f = 5Hz...8 f=35hz.6 velocity potential f=5hz f = Hz f=5hz f = 5 Hz Figure 9 : Aplitude of the velocity potential along the cosine bell deforation Fig. depicts the variation of the velocity potential along the vertical line x = :, for the range of frequencies. It can be seen that the nuber of propagating odes increases as the frequency increases. x() Figure : Aplitude of the velocity potential along the horizontal line y = 5:, for a range of frequencies fro 5Hz to 5Hz deforation are presented in Tab.. Notice that the sloping surfaces are half cosine bells, and their width in Tab. are half the value of w d in Eq (38). Table : Height and width of slopes and deforation height () width () left slope 5 deforation right slope 5 acoustic source (-,75) y() 5-5 Figure : Aplitude of the velocity potential along the vertical line x = :, for a range of frequencies fro 5Hz to 5Hz The variation of the velocity potential along the horizontal line y = 5: is presented in Fig.. The presence of the source located at x = : can clearly be observed in the figure... Idealised seabed close to shore An idealised region close to shore, containing two sloping surfaces and a cosine bell deforation on the seabed, as depicted in Fig., was analysed in order to observe the behaviour of the velocity potential with depth and range. The deeper and shallowerregionsareequal to: and 5:, respectively. The acoustic source is located at ( :; 75:). Two subregions were eployed to siulate this proble, with an interface at x = :. The height and width of the slopes and x() Figure : Geoetry for region containing two slopes and a deforation on the seabed close to shore Fig. 3 and present the aplitude of the velocity potential along two vertical lines at x = 5: and x = 3:: Fig. 3 shows that, for the frequency of 5Hz, only one propagating ode can be seen in the shallower region of depth H = 5: while two propagating odes can clearly be seen in Fig. for the deeper region of depth H = 9:. The nuber of propagating odes substantially increases with frequency for both the shallower and deeeper regions. 5 Conclusions In this paper, we have presented a boundary eleent forulation for underwater acoustic wave propagation and scattering

8 8 Copyright cfl Tech cience Press CME, vol., no.3, pp.73-8, Acknowledgeent: The first author would like to thank CNPq, the Brazilian Research Council, for providing financial support to this research. References Figure 3 : Aplitude of the velocity potential along the vertical line x = 5:, for a range of frequencies fro 5Hz to 5Hz Figure : Aplitude of the velocity potential along the vertical line x = 3:, for a range of frequencies fro 5Hz to 5Hz by localised irregularities. ince the Green s functions adopted identically satisfy the boundary conditions at the free surface and horizontal seabeds, it is possible to use the subregions technique developed and validated here to siulate coplex probles with botto irregularities. By doing so, the boundary eleent discretisation is restricted to the localised irregularitiesand interfaces between regions of different depth. Although the series (5) in ters of noral odes draatically iproves the speed of convergence, it is still necessary to iprove the convergence of the infinite series when the source and field points are located along the sae vertical line. Nuerous techniques exist for accelerating slowly-convergent series, such as Euler s transforation for alternating series [Abraowitz and tegun (965)], hanks transforation [hanks (955)], Wynn s algorith [Wynn (966)], Kuer s transforation [Linton (998)] and Ewald s ethod [Linton (998)], aong others. The ipleentation and coparison of the efficiency of soe of these techniques is presently under way. Abraowitz, M.; tegun, I.A. (965): Handbook of Matheatical Functions. Dover, New York. Chen,G.;Zhou,J. (99): Boundary Eleent Methods. Acadeic Press, London. Dawson, T.W.; Fawcett, J.A. (99): A boundary integral equation ethod for acoustic scattering in a waveguide with nonplanar surfaces. J. Acoust. oc. Aer., vol. 87, pp Grilli,.; Pedersen, T.; tepanishen, P. (998): A hybrid boundary eleent ethod for shallow water acoustic propagation over an irregular botto. Eng. Anal. Boundary Eleents, vol., pp Jensen, F. B.; Kuperan, W. A.; Porter, M. B.; chidt, H. (99): Coputational Ocean Acoustics. Aerican Institute of Physics, Woodbury, NY. Kinsler, L.E.; Frey, A.R.; Coppens, A.B; anders, J.V. (98): Fundaentals of Acoustics. John Wiley, New York. Lacerda, L.A. (997): Boundary Eleent Forulations for Outdoor ound Propagation. PhD Thesis, Departent of Civil Engineering, COPPE / UFRJ, Brazil. Linton, C.M. (998): The Green s function for the twodiensional Helholtz equation in periodic doains. J. Eng. Math.,vol. 33, pp Pedersen, T.K. (996): Modelling hallow Water Acoustic Wave Propagation. Mc Thesis, Departent of Ocean Engineering, University of Rhode Island, UA. antiago, J.A.F.; Telles, J.C.F.; Valenti, V.A. (999): An optiized block atrix anipulation for boundary eleents with subregions. Adv. Eng. oft., vol. 3, pp antiago, J.A.F.; Wrobel, L.C. (999a): Nuerical odelling of the propagation of underwater acoustic waves. 5 th Brazilian Congress of Mechanical Engineering, Águas de Lindóia, Brazil. antiago, J.A.F.; Wrobel, L.C. (999b): D odelling of shallow water acoustic wave propagation using subregion techniques. International Conference on Boundary Eleent Techniques, Queen Mary and Westfield College, University of London, UK. hanks, D. (955): Non-linear transforations of divergent and slowly-convergent sequences. J. Math. Phys., vol. 3, pp.-. Wynn, P. (966): On the convergence and stability of the epsilon algorith. IAM J. Nuer. Anal., vol. 3, pp. 9-.