Root finding in the complex plane for seismo-acoustic propagation scenarios with Green s function solutions

Transcription

1 Root finding in the complex plane for seismo-acoustic propagation scenarios with Green s function solutions Brittany A. McCollom a) and Jon M. Collis Department of Applied Mathematics and Statistics, Colorado School of Mines, 15 Illinois Street, Golden, Colorado 841 (Received 17 June 13; revised 19 July 14; accepted 3 July 14) A normal mode solution to the ocean acoustic problem of the Pekeris waveguide with an elastic bottom using a Green s function formulation for a compressional wave point source is considered. Analytic solutions to these types of waveguide propagation problems are strongly dependent on the eigenvalues of the problem; these eigenvalues represent horizontal wavenumbers, corresponding to propagating modes of energy. The eigenvalues arise as singularities in the inverse Hankel transform integral and are specified by roots to a characteristic equation. These roots manifest themselves as poles in the inverse transform integral and can be both subtle and difficult to determine. Following methods previously developed [S. Ivansson et al., J. Sound Vib. 161 (1993)], a root finding routine has been implemented using the argument principle. Using the roots to the characteristic equation in the Green s function formulation, full-field solutions are calculated for scenarios where an acoustic source lies in either the water column or elastic half space. Solutions are benchmarked against laboratory data and existing numerical solutions. VC 14 Acoustical Society of America. [ PACS number(s): 43.3.Bp, 43.3.Ma, 43.3.Zk [TFD] Pages: I. INTRODUCTION The primary focus of this work is the waveguide propagation problem of a point source that emits a compressional wave within a range-independent environment featuring a water column overlying an elastic sediment; this environment is termed the elastic Pekeris waveguide. The water column is assumed to have a pressure release surface above and to be bounded below by a semi-infinite isospeed elastic half space. Problems of this type and their solution formulation have a basis in the original work of Pekeris in which a separable solution to the elliptic wave equation is assumed and solutions are summed over all possible wave numbers, 1 where individual terms in the summation are referred to as modes. The original work of Pekeris considered the bottom to be a fluid half space. The elastic Pekeris waveguide problem was subsequently considered by Press and Ewing (EP), and their solution was recently benchmarked against laboratory data to high accuracy. 3 To treat the point source singularity, the EP solution assumes an artificial interface at the source, and applies continuity conditions across the interface. An alternative to this source treatment is to use a delta function as a forcing term, resulting in a Green s function source representation. 4 More recently, solutions have been derived for a Green s function formulation for the problem of a compressional wave point source in the water column. 5 Either solution approach leads to a complex valued equation, referred to as the characteristic equation, the roots of which are the horizontal wave numbers to the waveguide propagation problem. The characteristic equation is transcendental a) Author to whom correspondence should be addressed. Electronic mail: b.mccollom1@gmail.com and its roots are complex valued; this motivates the use of a numerical root finding routine in the complex plane. In Sec. II, solutions are developed for the cases where a compressional wave point source is assumed in either the water or the sediment in the aforementioned environment. An algorithm for root finding within the complex plane is presented in Sec. III, based on methods previously developed by S. Ivansson et al. 6 The algorithm uses the argument principle, Romberg integration, a rectangle halving strategy, and the Newton-Raphson method. With a routine capable of determining horizontal wave numbers, the problem of the source lying within the water column is benchmarked against experimental data in Sec. IV. The solution of Press and Ewing is also compared against the experimental data to compare the different solution approaches. In Sec. IV, the related problem of a compressional wave point source assumed in the elastic medium is considered; solutions are benchmarked against a wavenumber integration solution. II. GREEN S FUNCTION FORMULATIONS Consider the problem of a time-harmonic acoustic point source in a seismo-acoustic environment, that of the elastic Pekeris waveguide, where a compressional wave point source is allowed to be anywhere within the environment. The point source is represented using a spatial delta function in the frequency domain. Assume a water column with compressional wave speed c 1 and constant density q 1 overlies a semi-infinite elastic half space of constant density q, with compressional wave speed c and shear wave speed c s.a pressure release boundary above the water column is assumed at z ¼, and a fluid-elastic interface at z ¼ H. An azimuthally symmetric cylindrical geometry is assumed, with depth z oriented positively downward, r the radial distance from the z-axis, and the boundaries planar and parallel; 136 J. Acoust. Soc. Am. 136 (3), September /14/136(3)/136/1/$3. VC 14 Acoustical Society of America

2 a range-independent environment. It is assumed that the point source has angular frequency x and strength S x and lies on the z-axis at ðr; zþ ¼ð; z s Þ, with time dependence of exp ðixtþ. Solutions for the compressional and shear displacement potentials, / and w, are given in terms of inverse Hankel transformations / 1 r; z ð Þ ¼ 1 ð 1 1 / ðr; zþ ¼ 1 ð 1 1 ð Þ / 1ð k r; zþh 1 ðk r rþk r dk r ; (1) ð Þ / ð k r; zþh 1 ðk r rþk r dk r ; () w ðr; zþ ¼ 1 ð 1 w ð k r; zþh ð1þ ðk r rþk r dk r ; (3) 1 for k r the horizontal wavenumber, H ð1þ the Hankel function of the first kind, and subscripts (1) and () denoting solutions in the water column and bottom, respectively. The potentials / 1, /, and w are defined based on the location of the source. Vertical wavenumbers are given p in terms of medium and horizontal wavenumbers: k z;1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 k r ; and, to satisfy radiation conditions in the half space, 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi >< kj kr ; jk r j<jk j j k z;j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4) >: i kr k j ; jk r j>jk j j for j ¼, s, the subscript (s) denoting a quantity associated with the shear wave field. The relationships between the horizontal and vertical displacements and the range and depth-dependent displacement potentials are given by 7 u 1 ; w 1 ; (5) ; þ k s w : (6) Boundary conditions are pressure release at the sea surface; and continuity of vertical displacement, normal stress, and tangential stress at the bottom interface. Boundary conditions are expressed as / 1 ¼ ; z ¼ ; (7) w 1 ¼ w ; z ¼ H; (8) ðr zz Þ 1 ¼ðr zz Þ ; z ¼ H; (9) ðr zz Þ ¼ ; z ¼ H; (1) for the normal and tangential stresses r zz ¼ kr / þ and r zr ¼ lð@u=@z with Lame constants k and l. The compressional and shear wave speeds are related to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the Lame constants by c j ¼ ðk j þ l j Þ=q j, for j ¼ 1,, p and c s ¼ ffiffiffiffiffiffiffiffiffiffiffiffi l =q. Loss in the bottom is included by using complex wave speeds C ¼ c =ð1 þ iga p Þ and C s ¼ c s =ð1 þ iga s Þ, 8 where a p and a s are the compressional and shear wave attenuations in decibels per wavelength, and g ¼ð4p log 1 eþ 1. The compressional and shear wave fields have corresponding medium wavenumbers, k 1 ¼ x=c 1, k ¼ x=c, and k s ¼ x=c s. A Green s function representation is used to treat the source singularity. Normal mode solutions for / 1 ðr; zþ, / ðr; zþ, and w ðr; zþ are derived in the following sections. Two scenarios are considered for benchmarking: a compressional wave point source in either the water column or the elastic half space. The case of a source in the water column has been considered before,,3,5 and fluid layer seafloor compressional sources have also been considered previously. 7 Suppose that a compressional wave point source lies in the elastic seafloor at depth z s ¼ H þ d, withh < z s < 1 and d the depth of the source below the interface. The depth-dependent potentials / 1ðk r ; zþ, / ðk r ; zþ,andw ðk r ; zþ are defined as / 1ðk r; zþ¼ae ik z;1z þ Be ik z;1z ; (11) / e ik z;jzzsj ðk r; zþ ¼ S x þ Ce ik z;ðzh Þ ; (1) 4pik z; w ðk r;zþ ¼De ik z;sðzhþ ; (13) where A, B, C, and D are horizontal wavenumber-dependent amplitude coefficients, and the elastic medium vertical wavenumbers are defined by Eq. (4). Applying the boundary conditions, Eqs. (7) (1), to the potentials given by Eqs. (11) (13) yields a system of equations of the form Mx ¼ b, where ik z;1 e ik z;1h ik z;1 e ik z;1h ik z; kr M ¼ q 1 x e ik z;1h q 1 x e ik z;1h ðl kr q x Þ il kr k B z;s A ; (14) ik z; ðkr k z;s Þ 1 1 A B x ¼B C A ; and b ¼S xe ikz;d ik z; 4pik z; l kr q x C A : (15) D ik z; J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions 137

3 The solution to this linear system has singularities when the determinant of the coefficient matrix M disappears, giving rise to poles in the inverse Hankel transform. 4 This occurs when det ðmþ ¼ q 1ks 4k z; tanðhk z;1 Þ iq k z;1 h þ 4kr k z;k z;s þ kr i k s ¼ : (16) Equation (16) is referred to as the characteristic equation for this problem, and it is transcendental in k r, where the vertical wavenumbers all depend on k r. Despite the differing source treatments, this equation is the same characteristic equation as was found by Ewing, Jardetsky, and Press (EJP). 7 By row reducing the augmented matrix ½MjbŠ, the coefficients A, B, C, and D are found, and the potentials / 1ðk r ; zþ, / ðk r ; zþ, and w ðk r ; zþ are determined to be / 1ðk r; zþ ¼ S xiq x kr k s e ik z; d sinðk z;1 zþ ; (17) pfðk r Þ / ðk r; zþ ¼ S xe ik z;jzz z j 4pik z; S xk z;1 c s q kr k s cos ð kz;1 HÞ S x fðk r Þ pk z; fðk r Þ e ik z;ðdh Þ e ikz;z ; (18) and w ð k r;zþ ¼ S xiq c s k z;1 kr k s e ik z; dk z;s H ð pfðk r Þ Þ cosðk z;1 H Þe ik z;sz ; (19) with f ðk r Þ :¼ q 1 x ks k z; sinðk z;1 HÞþicðk r Þ cosðk z;1 HÞ; () and cðk r Þ¼q c s k z;1½4k r k z;k z;s þðk r k s Þ Š. The function in () is the right hand side of the characteristic equation multiplied by iq c s k z;1 cos Hk z;1 so that it does not have poles, and it is this function that is used to determine horizontal wavenumbers in the complex plane. Using the residue theorem, and the EJP integration contour and branch cuts, 4,9 modal sum representations for / 1 ðr; zþ, / ðr; zþ, and w ðr; zþ are found to be / 1 ðr; zþ ¼ 4p X1 / r; z ð Þ ¼ 4pi k s X 1 h i kr ðnþ k s k ðþ n kr ðþ n e ik ðnþ sinðk z;1 zþ f k n ; (1) z; d H ðþ 1 ðnþ r ð Þ h i z;1 k r ðþ n k cos k ðþ n s z;1 H kr ðnþ e ik ðþ n ð z; dhþ H ðþ 1 e ðþ n ik ðnþ z; z k ðnþ z; f kr ðn ; () Þ and w r; z ð Þ ¼ 8p k s X 1 h i z;1 k r ðþ n k s k ðnþ e i k ðþ n z; dk ðþ n z;s H e cos k ðnþ z;1 H kr ðnþ H ð1þ ðþ n ik ðþ n z;s z f k ðn ; (3) Þ r for f ðk r Þ, the first derivative of f ðk r Þ, and k ðnþ z;1, kðnþ z;, and kðnþ z;s the vertical wave numbers in terms of the medium wave number and the nth horizontal wave number kr ðnþ satisfying the characteristic equation, Eq. (). Note that in developing this solution, branch line integrals that would make the integral approximation exact, have been ignored; these integrals decay as 1=r p, whereas the residues decay as 1= ffiffi r,sothe only contribution to the field would be near to the source. 7 Note that Aki and Richards, in their discussion on leaky modes, show that the branch line integrals can also be expressed as a sum of residues. However, as differs from the normal mode components, these leaky mode residues will decay exponentially in the half space. 1 In contrast, when the compressional wave point source lies within the water column, i.e., < z s < H, the potentials are found to be 138 J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions

4 / 1 ðr; zþ ¼ is x X 1 / ðr; zþ ¼ S xq 1 x ak ðþ n sin X 1 r kr ðþ n k ðnþ z;1 z s sin k ðnþ z;1 z kr ðþ n H ðþ 1 ðnþ z;1 f kr ðn ; (4) Þ k ðnþ k n sin k n ðþ z;s e ik n ðþ z;1 z s kr ðnþ H ð1þ ðþ n ðþ ð z; zhþ f k ðþ n ; (5) r and w ðr; zþ ¼ is x q 1 x X1 z; sin k n k ðþ n e ik n ðþ z;1 z s kr ðnþ H ð1þ ðþ n ðþ z;s ðzhþ f k ðþ n ; (6) r where aðk r Þ :¼ q 1 x k s k z; cosðk z;1 HÞþicðk r Þsinðk z;1 HÞ: (7) These potential solutions also depend on the roots to Eq. (). These roots lie in the complex plane and to find these analytically is an arduous task, if not impossible. Further, there is a possibility of repeated roots to the characteristic equation depending on the amount of attenuation present in the waveguide. 1,11 In order to realize all of these roots, the horizontal wavenumbers, a numerical root finding routine capable of finding roots within the complex plane is required. In the next section, a root finding algorithm capable of finding every root, even when repeated, is presented. III. ROOT FINDING IN THE COMPLEX PLANE A hybrid root finding algorithm is presented, based on a procedure developed by S. Ivansson et al., with improvements based on procedures developed by T. Johnson et al., and M. Dellnitz et al. 6,1,13 Routines for root finding in the complex plane are generally based on the use of the argument principle. Consider a function f ðzþ that is analytic within a simple, closed, positively oriented contour C that lies in the complex plane C (a meromorphic function). Let N and P be the number of complex roots and poles, respectively, of f ðzþ within C. The argument principle states that ð 1 f ðþ z dz ¼ N P; (8) pi fðþ z C where each zero and pole is counted as many times as its multiplicity. 9 The integral in Eq. (8) is referred to as the winding integral. For the problems considered in this work, f, given by Eq. (), has been arranged so that there are not any poles of f within C, i.e., P ¼. With such an arrangement, the winding integral gives the number of complex roots within C. Using this fact, the number of roots within a given contour can be determined, and the contour manipulated until each root is isolated. Once there is a single root within a contour there are many methods that can be used to locate the root, e.g., the Newton-Raphson or secant methods. The winding integral-based method determines repeated roots and their multiplicity, which may arise physically within certain geoacoustic parameter regimes. 11 For the problems considered in this work repeated roots were not encountered. A first step in the root finding routine is to define the contour C. Any simple closed contour can be chosen, and rectangles and discs are the most common; 6,1 15 in the current work rectangular contours are used. Given a contour, the winding integral can be evaluated either by directly applying any acceptable quadrature method, or by first integrating by parts once and then numerically integrating. Once the number of roots have been computed, the contour can be dissected into smaller sub-contours until only one root lies within each sub-contour. At that point, any root finding routine that works within the complex plane can be used to find the root within the contour. The root finding algorithm begins with a single rectangular contour containing all of the desired roots, where rectangle size is determined by geophysical parameters. Using Romberg s method for numerical quadrature, the winding integral is evaluated in order to determine the number of roots within the rectangle. Upon iterating, if a rectangle contains no roots, it is discarded. If the rectangle contains only one root, then the Newton-Raphson method is used to find the root. If the rectangle contains more than one root, n say, then the length of the rectangle s diagonal is compared with some prescribed accuracy tolerance. If the diagonal is smaller than, then the midpoint of the rectangle is accepted as the root, with multiplicity n, otherwise, the rectangle is split in half along its longest sides and the procedure described above is performed on each sub-rectangle. IV. EXAMPLES The accuracy of the derived Green s function solutions are demonstrated through benchmark comparisons for two scenarios, those of a compressional wave point source in the water or in the sediment. After benchmarking, characteristics of the waveguide are discussed in terms of eigenvalues for two cases termed soft and hard bottom propagation scenarios. First, the Green s function and EJP solutions for the source in the water are compared against laboratory experimental data. 16 Second, the Green s function solution for a compressional source in the sediment is benchmarked J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions 139

5 TABLE I. Geometric and geoacoustic values used in experimental data comparisons. Parameter against a solution produced by the wavenumber integration model OASES. 17 When evaluating modal sum solutions the root finding algorithm described in Sec. III is used to numerically compute the roots to f ðk r Þ¼, where f ðk r Þ is given by Eq. (). The roots to Eq. () are then used in the solutions presented in the Sec. II, and also in the EJP solutions given in Ref. 7 (pp ). In acoustics, a standard metric of comparison is the transmission loss (TL), defined by TLðr; zþ ¼ log pr; ð z Þ ; p Value Solid Depth, H (cm) 14.5 Liquid Density, q 1 (g/cm 3 ) 1. Solid Density, q (g/cm 3 ) Compressional Speed Liquid, c 1 (m/s) 148 Compressional Speed Solid, c (m/s) 9 Shear speed, c s (m/s) 15 Compressional attenuation, a p (db/m/khz).33 Shear attenuation, a s (db/m/khz) 1. where p is the reference pressure 1 m from the source, and pðr; zþ is the acoustic pressure. For comparisons, the source is normalized so that the reference pressure amplitude is unity by assuming the source strength to be S x ¼4p=ðx qþ. The units of this particular source strength are m =Pa, representing the volume injection amplitude necessary to produce a pressure amplitude of 1 Pa at 1 m from the source. A. The NRL experiment In order to test the validity of various seismo-acoustic models, a series of scale-model tank experiments were performed at the U.S. Naval Research Laboratory in Washington D.C. in A large fresh water tank was used to represent the ocean, and an elastic bottom was modeled using a PVC slab (1 cm 1 cm 1 cm) suspended in the water by cables attached to the corners. A robotic apparatus was used to position the acoustic source and hydrophone receiver for accurate positioning. The source was fixed, while the receiver was moved in mm increments away from the source to produce a virtual aperture. Experimental data was found to be of extremely high quality. A detailed explanation of the model and experiment can be found in the reference. FIG. 1. Transmission loss vs range for propagation in a soft bottom ðc s < c 1 Þ range-independent environment at a near-bottom receiver depth of z r ¼ 13:71 cm. Comparisons show data (solid curve) and calculations from the elastic Pekeris waveguide Green s function solution (dashed curve) for source frequencies and positions: (a), (b) 18 khz, and (c), (d) 8 khz; (a), (c) mid-depth at 6.91 cm, and (b), (d) deep at cm. 14 J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions

6 B. Source in the water Consider a compressional wave point source in the water column. Using the physical parameters given in Table I, the horizontal wavenumbers are computed using the algorithm presented in Sec. III. This case is referred to as the soft bottom case, where c s < c 1. The horizontal wavenumbers are then used to compute modal sums for the Green s function and EJP solutions. Figure 1 displays comparisons of the transmission loss of experimental data (solid curve) versus the Green s function solution s loss (dashed curve) at a receiver depth of z r ¼ 13:71 cm. Figure displays an analogous comparison, but for the EJP solution against the experimental data. Comparisons are given for shallow and deep source depths of z s ¼ 6:91 cm [Figs. 1(a), 1(c), (a), and(c)] andz s ¼ 13:69 cm [Fig. 1(b), 1(d), (b), and (d)] at frequencies of 18 Hz [Figs. 1(a), 1(b), (a), and (b)] and 8 Hz [Figs. 1(c), 1(d), (c), and(d)]. There is very good agreement between the two analytic solutions and the data. At both frequencies, solutions provide a close match in both pattern phase and amplitude to the experimental data. The fields do differ between the two source depths, with a more complicated pattern for the deep source case and slightly poorer, though still good agreement. There are points at which the solutions and data disagree, however agreement is generally good with neither producing a clearly better result. Note that the propagation track extended beyond the edge of the PVC slab, at 1.5 m, and this can be seen in the comparisons as differences are more pronounced. C. Source in the sediment Consider a compressional point source in the sediment bottom. Figure 3 shows the solution from the Green s function formulation (dashed curve) compared to a solution computed using a wave number integration model, OASES (solid curve), 17 that is known to give accurate results for range-independent seismo-acoustic problems, including those with compressional sources. 18 Assuming the bottom depth is H ¼ 5 m, the source depth is z s ¼ H þ d ¼ 6 m (i.e., d ¼ 1 m), and using the geoacoustic parameters given in Table II, normal mode solutions are determined. In Figs. 3(a) and 3(c) the two solutions are compared at a frequency of 15 Hz, for receiver depths of z r ¼ m and z r ¼ 499 m; and in Figs. 3(b) and 3(d) analogous results are presented for a 5 Hz source. The comparison plots of the Green s function solution and OASES transmission loss all exhibit the same behaviors: Near the source, the shape of the curves are similar; however, the numerical values of the FIG.. Transmission loss vs range for propagation in a soft bottom ðc s < c 1 Þ range-independent environment at a near-bottom receiver depth of z r ¼ 13:71 cm. Comparisons show data (solid curve) and calculations from the elastic Pekeris waveguide EJP solution (dashed curve) for source frequencies and positions: (a), (b) 18 khz, and (c), (d) 8 khz; (a), (c) mid-depth at 6.91 cm, and (b), (d) deep at cm. J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions 141

7 FIG. 3. Transmission loss curves from OASES (solid curve) and the elastic Pekeris waveguide Green s function solution (dashed curve) for a compressional wave point source located in the elastic medium 1 m below the fluid-elastic interface z ¼ 6 m. Curves are shown for a receiver depth of z ¼ m for (a) 15 Hz and (b) 5 Hz sources, as well as near the interface at depth z r ¼ 499 m for (c) 15 Hz and (d) 5 Hz sources. losses differ (with greater differences when the receiver depths are closer to the interface). At ranges farther from the source, the shapes of the curves and computed losses move closer to each other and become in excellent agreement. Differences near the source are most likely due to either the neglect of the continuous spectrum or the branch line integrals. The curves demonstrate excellent agreement across frequencies and receiver depths, in particular for a receiver located near to the interface. D. Eigenmodes for soft and hard ocean bottoms The propagation scenarios in the previous examples were for problems in which the shear wavespeed in the TABLE II. Geoacoustic values for comparisons against OASES. seafloor was less than the speed of sound in the overlying water layer, giving rise to so-called Leaky modes. 1 Consider a scenario, e.g., a basalt seafloor in the Western Atlantic, in which the shear wave speed is greater than the acoustic wave speed in the water. This scenario is referred to as the hard bottom case, where c s > c 1. Using the geophysical parameters given in Table III, and assuming the same geometric parameters as in the experimental data comparison for the deep source case, horizontal wave numbers are found from Eq. (). Horizontal wave numbers are given in Fig. 4 for both the soft, Fig. 4(a), and hard, Fig. 4(b), bottom scenarios. Wave number lines corresponding to the medium wave numbers are plotted vertically, representing the TABLE III. Geoacoustic values used in hard-ocean bottom simulations. Parameter Value Parameter Value Liquid Density, q 1 (g/cm 3 ) 1. Solid Density, q (g/cm 3 ) Compressional Speed Liquid, c 1 (m/s) 15 Compressional Speed Solid, c (m/s) 17 Shear speed, c s (m/s) 8 Compressional attenuation, a p (db/m/khz).6 Shear attenuation, a s (db/m/khz).5 Liquid Density, q 1 (g/cm 3 ) 1. Solid Density, q (g/cm 3 ).58 Compressional Speed Liquid, c 1 (m/s) 148 Compressional Speed Solid, c (m/s) 45 Shear speed, c s (m/s) 4 Compressional attenuation, a p (db/m/khz).3 Shear attenuation, a s (db/m/khz).7 14 J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions

8 FIG. 4. Horizontal wave numbers plotted in the complex plane for the soft and hard ocean bottom cases: (a) c s < c 1 ; and (b) c s > c 1. compressional wave numbers in the water and sediment, and the shear wave number, denoting regimes of propagation. Transmission loss curves are given in Fig. 5 for three different source depths for a 18 khz deep sound source at cm. The curves are noticeably different than those obtained for the soft-bottom case. For the soft and hard bottom examples, 36 and 37 modes were found. Modes are counted from the right of the figure, the first mode referred to as mode zero, representing the Scholte mode. In either case, the Scholte mode is found to have a real component of the horizontal wavenumber greater than that of the wavenumber in the water column. Also in both cases, mode one is seen to be very close to the water wave number with subsequent modes following to the left. In terms of percentage of shear wave speeds, for the soft and hard bottom examples, the Scholte wave speeds, determined from horizontal wave number values, are found to be 8% and 61% of the shear wave speeds. Mode shape functions in terms of displacement potentials are plotted for select modes in Figs. 6 and 7. Shape functions correspond to the soft, Fig. 6, and hard, Fig. 7, bottom scenarios considered in the previous paragraph. In Fig. 6, the Scholte mode shape function (corresponding to mode zero), is plotted along with the final mode shape function in the water, representing a water-borne mode, and the first shape function in the elastic bottom, representing energy propagated in the elastic medium. Note that the amplitude of the shape function is non-negligible in the sediment for modes 8 through 35, explaining why, for propagation scenarios of this type, significant energy is propagated in the ocean s sediment. In Fig. 7, again the mode zero shape function is plotted along with the final mode shape function for the water and the first shape function for the elastic bottom. Of interest, the final shape function in the water has significant amplitude nearer to the ocean bottom interface, in fact more than double the amplitude of all other modes, due to significant imaginary component of the horizontal wave FIG. 5. Transmission loss vs range for propagation in a range-independent environment over a hard ocean bottom ðc s > c 1 Þ for a 18 khz sound source. Plots show the elastic Pekeris waveguide Green s function solution for a deep source at z ¼ 13:69 cm, and receiver depths: (a).5 cm; (b) 7.5 cm; and (c) cm. number. The final two modes in the sediment have significant amplitude, and so field contributions. V. DISCUSSION Normal mode solutions to the elastic Pekeris waveguide problem using a Green s function formulation for a compressional wave point source were derived and benchmarked for scenarios featuring an acoustic source within the water column or within the elastic sediment. The derivation and implementation of a normal mode solution using a Green s function formulation to the elastic Pekeris waveguide problem with the source assumed in the J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions 143

9 FIG. 6. Select mode shape functions for the soft bottom case: (a) mode (Scholte mode); (b) mode 7; and (c) mode 8. sediment is newly implemented work. The Green s function solution gives promising results, which is apparent when benchmarked against the wavenumber integration solution produced by OASES. Although there are differences in the comparison curves near the source, there is excellent agreement in the far field. The cause of differences near the source is probably a result of the evanescent field or boundary line integrals that are neglected by the normal mode solution. These results demonstrate the validity of a Green s function representation for a compressional source within the sediment. FIG. 7. Select mode shape functions for the hard bottom case: (a) mode (Scholte mode); (b) mode 34; and (c) mode 35. An algorithm for root finding within the complex plane was detailed, and used to compute the roots of the characteristic equation for the elastic Pekeris waveguide problem. The root finding algorithm presented in Sec. III gives a method for finding the roots to a meromorphic function within the complex plane; this method has been numerically validated through application to benchmark problems. Additionally, the root finding algorithm and related normal mode solutions demonstrate the approaches ability to find the wavenumber associated with the Scholte interface wave, with potential applications to classifying this poorly understood phenomenon. Application of the root finding algorithm presents a potentially powerful tool for characterizing seismic phenomena, certainly giving more information than horizontal wavenumber spectra from the Hankel transform of pressure data would give. The simplicity of the derivation of the Green s function solutions for the elastic Pekeris waveguide problems, and great results from comparisons against the experimental data and OASES indicate the Green s function representation of a compressional source is accurate for representing a compressional wave point source. This approach could be applied when considering multiple layers of different material, as well as multiple sources. A Green s function formulation for a shear wave source is also a scenario worthy of investigation, which combined with the representation of the compressional wave source could provide a realistic representation for a generic geophysical source. Although the implementation of the root finding routine gives desirable results, there is room for improvement: A robust implementation would require root detection on contours among other numerical considerations. 1 C. L. Pekeris, Theory of propagation of explosive sound in shallow water, in Propagation of Sound in the Ocean, Geol. Soc. Am., Mem. 7, 1 11 (1948). F. Press and M. Ewing, Propagation of explosive sound in a liquid layer over-lying a semi-infinite elastic solid, Geophysics. 15, (195). 3 J. D. Schneiderwind, J. M. Collis, and H. J. Simpson, Elastic Pekeris waveguide normal mode solution comparisons against laboratory data, J. Acoust. Soc. Am. 13, EL18 EL188 (1). 4 F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt, Computational Ocean Acoustics, nd ed. (Springer, New York, 11), Chap.. 5 D. D. Ellis and D. M. F. Chapman, A simple shallow water propagation model including shear wave effects, J. Acoust. Soc. Am. 78, (1985). 6 S. Ivansson and I. Karasalo, Computation of modal wave numbers using an adaptive winding-number integral method with error control, J. Sound Vib. 161, (1993). 7 M. Ewing, W. Jardetsky, and F. Press, Elastic Waves in Layered Media (McGraw-Hill, New York, 1957), Chap M. D. Collins, A higher-order parabolic equation for wave propagation in an ocean overlying an elastic bottom, J. Acoust. Soc. Am. 86, (1989). 9 J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, 5th ed. (Jones and Bartlett Publishers, Sudbury, MA, 6) pp. 93, K. Aki and P. G. Richards, Quantitative Seismology, nd ed. (University Science Books, Sausalito, CA, ), Sec R. B. Evans, The existence of generalized eigenfunctions and multiple eigenvalues in underwater acoustics, J. Acoust. Soc. Am. 9, 4 9 (199). 144 J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions

10 1 T. Johnson and W. Tucker, Enclosing all zeros of an analytic function-a rigorous approach, J. Comput. App. Math. 8, (9). 13 M. Dellnitz, O. Sch utze, and Q. Zheng, Locating all the zeros of an analytic function in one complex variable, J. Comput. App. Math. 138, (). 14 P. R. Brazier-Smith and J. F. M. Scott, On the determination of the roots of dispersion equations by use of winding number integrals, J. Sound Vib. 145, (1991). 15 B. Davies, Locating the zeros of an analytic function, J. Comput. Phys. 66, (1986). 16 J. M. Collis, W. L. Siegmann, M. D. Collins, H. J. Simpson, and R. J. Soukup, Comparison of simulations and data from a seismo-acoustic tank experiment, J. Acoust. Soc. Am. 1, (7). 17 H. Schmidt, OASES, Version 3.1, User Guide and Reference Manual, Massachusetts Institute of Technology, Boston, MA, 4, available at (Last viewed August 15, 1). 18 S. D. Frank, R. I. Odom, and J. M. Collis, Elastic parabolic equation solutions for underwater acoustic problems using seismic sources, J. Acoust. Soc. Am. 133, (13). J. Acoust. Soc. Am., Vol. 136, No. 3, September 14 B. A. McCollom and J. M. Collis: Normal Mode Green s Function Solutions 145