MODELING OF THREE-DIMENSIONAL PROPAGATION ON A COASTAL WEDGE WITH A SEDIMENT SUPPORTING SHEAR

Transcription

1 MODELING OF THREE-DIMENSIONAL PROPAGATION ON A COASTAL WEDGE WITH A SEDIMENT SUPPORTING SHEAR Piotr Borejko Vienna University of Technology, Karlsplatz 13/E26/3, 14 Vienna, Austria Fax: Electronic mail: piotr.borejko@tuwien.ac.at Abstract: In this paper, a generalized-ray based approach, which includes shear in the sediment, is applied to demonstrate that cross-slope acoustic propagation on a coastal wedge may significantly be affected by the horizontal refraction [bending of the acoustic ray paths (as viewed from above) due to the sloping bathymetry], and thus may indeed be threedimensional (3-D). The approach shows that some acoustic signals coming in along paths with multiple seafloor interactions, thus propagating up the slope and back to the receiver, may dominate those coming in along paths with only a few seafloor interactions, thus traveling near the straight source-to-receiver path that is parallel to the wedge apex. (Intuitively, a signal coming in along a path with multiple bottom interactions is expected to be weaker than that coming in along a path with only a few bottom interactions.) These theoretical predictions are consistent with measurements of 3-D acoustic propagation on the Florida shelf [Kevin D. Heaney and James J. Murray, J. Acoust. Soc. Am. 125(4), (29)], nearly shaped like the canonical wedge. Keywords: 3-D Propagation, Shallow Water Propagation, Penetrable Wedge

2 1. INTRODUCTION The traditionally used physical model of the propagation of sound in a shallow water environment is that of an acoustic wave mode in a flat layer of fluid over a fluid or elastic (shear supporting) bottom (the Pekeris model). In a flat layer, the acoustic wave is multiply reflected and the propagating wave mode is a result of interference of the reflected waves, whose acoustic ray paths are in the vertical plane containing the source and receiver [viewed from above as straight (without a bend) paths]. In a Pekeris-type shallow water environment, the acoustic propagation is thus two-dimensional (2-D). On a coastal wedge, the cross-slope propagation of sound may be three-dimensional (3-D) due to sloping bathymetry, and the Pekeris model may no longer be accurate. A more accurate model of a coastal wedge environment is that of the canonical wedge (the penetrable wedge model). In 1986, three benchmark wedge problems [1,2] were proposed, and, since then, there has been a significant research activity on the 3-D acoustic propagation in the canonical wedge, which has been analyzed by applying a variety of theoretical [3 5] and numerical [6,7] modeling approaches. Phenomena predicted include: the horizontal refraction, a modal shadow zone that is bounded by a caustic, multiple mode arrivals, the mode wave-front curvature, the mode capture, as well as the selective cut-off of up-slope propagating modes. As reviewed in Ref. [8], there has been limited application of these approaches to measured data, primarily due to the lack of experimental observation of the 3-D propagation of sound on a coastal wedge. Only recently the 3-D sound was first observed in a set of experiments on the Florida shelf [9]; and later accurately modeled by applying a novel hybrid approach [8], thus achieving excellent agreement with data. The propagation feature of interest in a coastal wedge environment is the horizontal refraction. In this paper, a generalized-ray based approach, which includes shear in the sediment, is applied to acoustic wave field calculations in a canonical wedge to demonstrate that the recently measured 3-D sound on the Florida shelf [9], nearly shaped like the canonical wedge, may be affected by the horizontal refraction, and thus may indeed be 3-D. 2. INTEGRAL REPRESENTATIONS OF THE 3-D ACOUSTIC WAVE FIELD Consider a canonical wedge of homogeneous lossless fluid of density ρ and sound speed c, bounded above by a horizontal plane and below by a sloping plane interface with an infinite homogeneous lossless solid elastic medium of density ρ 2 and P and S wave speeds c P and c S, respectively. The boundary planes intersect along the line of apex (the wedge apex), and the apex angle (the wedge angle) is α. A point source is located in the wedge where the wedge thickness is h, thus the distance of the source from the apex is then d = h/tanα. The boundary condition at the pressure-release (impenetrable) horizontal plane ensures the vanishing of acoustic pressure; and those at the penetrable sloping fluid-solid interface ensure the continuity of normal stress and normal velocity, and the vanishing of shearing stresses. The present problem is to determine time series of the acoustic pressure p at a point receiver placed on the sloping bottom directly cross slope of a point source at a large range of r = 2h. This problem may be treated by applying the method of generalized ray [1,11], and the solution of 3-D acoustic wave field thus obtained is then represented by

3 N p = p + p ±k, (1) k=1 p = p c R f (t R c ) = p t 2 c H (t t ) f (t τ ) π Re q(τ ) 1 S dg dτ dq dτ, p t c = ρ 4πc 2, (2) t 1 p ±k = p c H (t t ±k ) f (t τ ) 2π Re q(τ ) 1 SΠ dg ±k ±k dτ dq dτ, k = 1, 2, 3,..., N. (3) t ± k This solution is exact other than the omission of wave diffracted at the wedge apex, and it does not account for absorption neither in the wedge nor in the bottom. In Eq. (1); p is the pulse due to the th incident (source) wave field; p ±k is that due to the ±kth multi-reflected wave field undergoing k successive reflections, where the +/ sign means that the first reflection is off the sloping/horizontal boundary; and N = π/α, where α is an integer submultiple of π. In Eq. (2), R is the source-to-receiver distance, and f (t) is the know time function of the source pulse p. In Eqs. (2) and (3); H (t) is the Heaviside step function; and t,t ±k are the minimum arrival times of the th, ±kth fields, respectively. In Eqs. (2) and (3), f (t) = f (t)/ t is convoluted with the complex ray integrals, whose real parts (thus Re stands for real part ) represent the received wave fields. In the ray integrals; S is the known source function; Π ±k is the sum of four products of the known 3-D plane-wave reflection coefficients; g 1,g 1 ±k are the known inverse phase functions of the th, ±kth wave fields, respectively; and [, q(τ )] is an interval in the wave slowness q. Hence; the ray integral in Eq. (2) represents the th spherical wave field, which may be regarded as a superposition of plane waves; and that in Eq. (3) represents the ±kth multireflected wave field, which consists of the following wave-forms (enumerated in ascending order according to their arrival times at the receiver): a multiplicity of the critically refracted (lateral) P and S waves and the pseudo-rayleigh interface waves, a spherical wave of k multiple reflections off the boundary planes, and a multiplicity of the Scholte interface waves. 3. LARGE-RANGE CROSS-SLOPE PROPAGATION EXAMPLE In a canonical wedge, acoustic propagation from a point source to a cross-slope receiver exhibits horizontal refraction as the pulses [Eq. (1)] interact with the sloping bottom. A pulse undergoing only a few bottom interactions propagates along an acoustic ray path with a small azimuthal arrival angle (arrival direction vs. source-to-receiver direction, as measured in the horizontal boundary plane), and a pulse undergoing multiple bottom interactions propagates along a ray path with a large azimuthal arrival angle. Only the source and surface reflected pulses arrive along 2-D ray paths which are in the vertical plane, thus viewed from above as straight (without a bend) ray paths parallel to the line of apex. All the other pulses interact with the bottom, and thus arrive along 3-D paths, propagating up the slope and back to the receiver, since the change in both the plane of incidence and the incidence angle upon each bottom reflection introduces a curvature into the projection of a ray path onto the horizontal boundary plane.

4 In this section, the results of wave field calculations for a canonical wedge model are presented to demonstrate that the recently measured 3-D sound on the Florida shelf [9], nearly shaped like a canonical wedge, may significantly be affected by the horizontal refraction. We thus place a receiver on the sloping bottom directly cross slope of a point source at a large range of 4 [km] in a 3 [deg] wedge. The point source is submerged in water of depth of 2 [m], its distance from the wedge apex is then d = [m]; and the source and receiver depths are then 1 and 2 [m], respectively. The geoacoustic parameters assumed for the wedge of isospeed lossless water over an infinite homogeneous lossless seabed are those of the southeast Florida continental shelf [12]: c = 15 [m/s], c P = 3 [m/s], c S = 146 [m/s], ρ = 1 [g/cm 3 ], and ρ 2 = 2.4 [g/cm 3 ]. We thus consider the case of a slow-speed shear-supporting bottom (c S < c), in which both the critically refracted (lateral) S and pseudo-rayleigh interface waves are absent. In the computed examples, the range r was normalized by the water depth at the source location h, the time t was normalized by the characteristic time t c = h/c, and each pressure pulse was normalized by the constant p c [Eqs. (2) and (3)]. The generalized-ray solution, as given in Eqs. (1) (3), was then used to compute time series of the acoustic pressure at a receiver placed on the sloping bottom at a cross-slope range of r/h = 2 from the source. The time function of the source pulse f (t) [Eq. (2)] used for the examples was a Gaussian cosine broadband pulse of center frequency c f = 25 [Hz] and bandwidth w =.5 [Hz]. Figure 1 shows the time series of the entire wave field (as represented by the pressure p in Eq. (1), where N = 6 for a 3 [deg] wedge) received at a bottom location of cross-slope range of r/h = 2 from the source. The series exhibits three distinct pulses (enumerated in ascending order according to their appearance times on the pressure curve): an early time pulse, due to multiple arrivals of the critically refracted P wave-fronts coming in along the refracted ray paths; an intermediate time pulse, due to multiple arrivals of the spherical wavefronts coming in along the direct ray paths; and a late time pulse, due to multiple arrivals of the Scholte interface waves [exhibiting (opposite to the critically refracted and spherical waves) no definite wave-fronts, and thus no definite propagation paths and arrival times]. Note that the intensity (the amplitude) of both the refracted- and spherical-waves pulses is of many orders of magnitude less than that of the Scholte-waves pulse. A thorough inspection of the time series of each multi-reflected wave field [as represented by the ray integral in Eq. (3)] shows that: the intensity of both a multiplicity of the critically refracted P waves and a spherical wave decreases as the index k (indicating the number of multiple reflections) increases from 1 to 6 ; but the intensity of a multiplicity of the Scholte interface waves first increases as k increases from 1 to about 19, to attain its maxima for k 2, 21, 22,..., 4, and it then decreases as k increases from about 41 to 6 ; and, notably, in the wave fields of index k 1, 2, 3,..., 19 and k 41, 42, 43,..., 6, the intensity of a multiplicity of the Scholte interface waves is of a few orders of magnitude less than that in those of index k 2, 21, 22,..., 4. We may then sort out wave fields into three groups; each group includes fields identified by the index k, the azimuthal arrival angle, and the Scholte-waves intensity. Thus; the first group includes fields of small k 1, 2, 3,..., 19, thus coming in along paths of small arrival angles, and exhibiting low Scholte-waves intensities; the second one includes those of intermediate k 2, 21, 22,..., 4, thus coming in along paths of intermediate arrival angles, and exhibiting high Scholte-waves intensities; and the third one includes those of large k 41, 42, 43,..., 6, thus coming in along paths of large arrival angles, and exhibiting low Scholte-waves intensities.

5 (a).1 r/h=2 (b) r/h= t P /t c t Sph /t c t Sch /t c Fig. 1: Time series of the entire wave field received at r/h = 2 ; (a) the refracted- and spherical-waves pulses; (b) the Scholte-waves pulse ( r is the range; h is the water depth at source/receiver location; t is the time; p is the pressure; t c and p c are the normalizing constants; t P is the beginning of the refracted-waves pulse; t Sph is the beginning of the spherical-waves pulse; t Sch is the beginning of the Scholte-waves pulse). Figure 2 shows the time series of the second group of fields (as given by a superposition of pulses p ±2, p ±21, p ±22,..., p ±4 ). This series [like that of the entire field (Fig. 1)] also exhibits low intensity refracted- and spherical-waves pulses, and a high intensity Scholtewaves pulse. As anticipated in the above wave-fields analysis, both the refracted- and spherical-waves pulses of the second group of fields are significantly weaker than those of the entire field, but the either Scholte-waves pulse exhibits the same intensity. (a).1 r/h=2 (b) r/h= t P /t c t Sph /t c t Sch /t c Fig. 2: Time series of the second group of wave fields received at r/h = 2 ; (a) the refracted- and spherical-waves pulses; (b) the Scholte-waves pulse ( r is the range; h is the water depth at source/receiver location; t is the time; p is the pressure; t c and p c are the normalizing constants; t P is the beginning of the refracted-waves pulse; t Sph is the beginning of the spherical-waves pulse; t Sch is the beginning of the Scholte-waves pulse). Then; in the time series of the entire wave field (Fig. 1); the critically refracted P and spherical waves of the first group of fields of small arrival angles dominate in the early and intermediate time pulses, respectively; and the Scholte interface waves of the second group of fields of intermediate arrival angles dominate in the late time pulse.

6 Thus, large-range cross-slope acoustic propagation in a canonical wedge of geoacoustic parameters of the southeast Florida continental shelf is indeed 3-D, although the horizontal refraction affects the propagation of both the critically refracted P and spherical waves (dominant refracted- and spherical-waves arrivals due to the first group of fields of small arrival angles) to lesser extent than that of the Scholte interface waves (dominant Scholte-waves arrivals due to the second group of fields of intermediate arrival angles). (A less detailed analysis of these 3-D propagation effects is presented in Ref. [13] for a set of geoacoustic parameters different from that of the southeast Florida shelf.) As reported in Ref. [9], in a set of experiments on the continental shelf off the east coast of Florida, 3-D propagation effects were not only observed, but dominated acoustic propagation. Two distinct signal arrivals, of different propagation paths and arrival angles, were observed: one coming in along a path of about 7 [deg] arrival angle, labeled the direct path; and the other, significantly stronger than the direct arrival in some cases, coming in later along a path of up to 3 [deg] arrival angle sometime, labeled the inshore path. REFERENCES [1] Session R, Underwater acoustics II: quality assessment of numerical codes, part 2: benchmarks, J. Acoust. Soc. Am. 8, S36 S38 (1986). [2] E. K. Westwood, Ray model solutions to the benchmark wedge problems, J. Acoust. Soc. Am. 87, (199). [3] A. D. Pierce, Guided mode disappearance during upslope propagation in variable depth shallow water overlying a fluid bottom, J. Acoust. Soc. Am. 72, (1982). [4] E. K. Westwood, Broadband modeling of the three-dimensional penetrable wedge, J. Acoust. Soc. Am. 92, (1992). [5] M. J. Buckingham, Theory of three-dimensional acoustic propagation in a wedge like ocean with a penetrable bottom, J. Acoust. Soc. Am. 82, (1987). [6] F. B. Jensen, C. M. Ferla, Numerical solutions of range-dependent benchmark problems in ocean acoustics, J. Acoust. Soc. Am. 87, (199). [7] F. Sturm, Numerical study of broadband sound pulse propagation in threedimensional oceanic wave-guides, J. Acoust. Soc. Am. 117, (25). [8] K. D. Heaney, R. L. Campbell, J. J. Murray, Comparison of hybrid threedimensional modeling with measurements on the continental shelf, J. Acoust. Soc. Am. 131, (212). [9] K. D. Heaney, J. J. Murray, Measurements of three-dimensional propagation in a continental shelf environment, J. Acoust. Soc. Am. 125, (29). [1] Y-H. Pao, F. Ziegler, Y-S. Wang, Acoustic waves generated by a point source in a sloping fluid layer, J. Acoust. Soc. Am. 85, (1989). [11] P. Borejko, C-F. Chen, Y-H. Pao, Application of the method of generalized ray to acoustic waves in a liquid wedge over elastic bottom, J. Comput. Acoust. 9, (21). [12] M. S. Ballard, Modeling three-dimensional propagation in a continental shelf environment, J. Acoust. Soc. Am. 131, (212). [13] P. Borejko, An analysis of cross-slope pulse propagation in a shallow water wedge, in Proceedings of the 1th European Conference on Underwater Acoustics, Edited by Tuncay Akal, Istanbul, (21).