An energy-conserving parabolic equation for elastic media

An energy-conserving parabolic equation for elastic media Michael D. Collins

Naval Research Laboratory, Washington, DC 20375

(Received 11 January 1993; revised 22 March 1993; accepted 8 April 1993)

The accuracy of the parabolic equation (PE) method is improved for range-dependent elastic media. The energy-conserving elastic PE is a generalization of the energy-conserving acoustic PE. A range-dependent waveguide is approximated by a sequence of range-independent regions. A linear approximation is derived for the nonlinear condition corresponding to conservation of compressional energy flux at the vertical interfaces between range-independent regions. The rotated elastic PE is applied to generate reference solutions for demonstrating accuracy. The energy-conserving elastic PE is applied to solve problems involving the propagation of an interface wave along a sloping ocean bottom and a problem involving mode cutoff and coupling into shear wave beams in the ocean bottom. The latter problem is analogous to a cutoff problem that was first solved with the acoustic PE [F. B. Jensen and W. A. Kuperman, J. Acoust. Soc. Am. 67, 1564-1566 (1980)].

PACS numbers: 43.30.Ma, 43.20. Gp

INTRODUCTION

The parabolic equation (PE) method is useful for solving sound propagation problems in range-dependent ocean environments. 1 As discussed in Ref. 2, the PE method has undergone a few decades of development to reduce its limitations. In its present stage of development, the PE method is both accurate 3 and etticient 4 for most problems involving fluid sediments and some problems involving elastic sediments. SPerhaps the most important re- maining limitation of the PE method is its lack of accuracy for many range-dependent problems involving elastic sediments. As in Ref. 3, we define range-dependent PEs to be those PEs designed to achieve improved accuracy for range-dependent problems and range-independent PEs to be all other PEs. The existing range-dependent PEs for elastic media include the rotated elastic PE, 5 which is re- stricted to a small class of problems, and the two-way elastic PE, 2 which is presently limited to purely solid waveguides.

In this paper, we derive the energy-conserving elastic P E, which is a generalization of the energy-conserving acoustic PE of Ref. 3. This range-dependent elastic PE is accurate for and applicable to a large class of problems involving waveguides consisting of both fluid and solid layers. A range-dependent ocean environment is approximated by a sequence of range-independent regions. The range-independent elastic PE derived in Sec. I is used to propagate the solution through the range-independent regions. The other part of the energy-conserving elastic PE solution, which is derived in Sec. II, involves an amplitude correction that is applied at the vertical interfaces between range-independent regions. In Sec. III, we present benchmark solutions to illustrate the accuracy of the energy- conserving elastic PE and generate solutions for problems involving sloping ocean bottoms. In Sec. IV, we present a complete energy-conservation correction for the acoustic PE that suggests an approach for improving the accuracy of the energy-conserving elastic PE.

I. THE RANGE-INDEPENDENT ELASTIC PE

A range-dependent elastic waveguide is divided into a sequence of range-independent regions. In each of the range-independent regions, the wave equation may be solved either by separation of variables or with the PE method. In this section, we derive the range-independent elastic PE of Ref. 5, which is used to propagate the solution through the range-independent regions. A method for approximating transmitted fields at the vertical interfaces between regions is derived in Sec. II.

The depth z is the distance below the ocean surface and the range x is the horizontal distance from a line or point source of circular frequency w. The elastic properties of the medium are assumed to depend only on x and z. The factor exp(--iwt) is removed from the horizontal displacement u and the vertical displacement w. The spreading factor x- •/2 is also removed for the case of a point source. In each range-independent region, the elastic properties are piecewise continuous functions of depth, and the following equations of motion 6 are valid away from the source:

a2u a2u aA at• au at• aw (z+t,) +Tzz Tzz+ ax -ø,

a2w a2w a/•

where the dilatation A is defined by

azaz

(1)

--0,

(2)

au aw

A=•xx+ Oz' (3) This formulation is convenient for problems involving piecewise continuous depth variations in the elastic parameters. Loss is handled by using the complex compressional and shear speeds Cp=cp/( 1 +i,lfip) and C•=c/( 1 +i,lfi•), where cp and c• are the real wave speeds, fie and fi• are the compressional and shear attenuations in decibels per wave-

975 J. Acoust. Soc. Am. 94 (2), Pt. 1, August 1993 975

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.102.42.98 On: Sat, 22 Nov 2014 11:12:18

length, and ,/=(40rrlog10e) -1. The Lam• constants /t and/z are related to the wave speeds and the density p by ,;[ = p( C'• 2 -- 2Cs 2 ) and/x = pCs 2.

The formulation of elasticity given by Eqs. (1) and (2) is not convenient for the PE method, which is based on factoring the equations of motion. Differentiating Eq. (1) with respect to x and Eq. (2) with respect to z and sum- ming, we obtain

a2A a2A oq/t, a2w ap (A+2/•) •x2+(A+2/•)•---•+po2A+2-•zz•x2+W2-•zzW

+ •zz+2•zz •zz+•z •zz A +2•z •z•zz =0. (4)

Equations (2) and (4) provide the following generalization of the formulation of Ref. 7:

L• +M = (5) W •

where the matrices L and M contain depth operators. Since L and M commute with •/•x, Eq. (5) factors and has the outgoing solution,

• =i(L-•M) •/2 . (6) To derive the elastic PE, we rearrange Eq. (6) as follows:

•x =iko•i+k•2(L_lM_k• ) a w ' (7)

where k0 is a representative wave number and I is the identity operator. We apply a Pad6 approximation for the operator square root and obtain the range-independent elastic PE,

•x w

. (8) j=l , •

Choices for the coe•cients a•,n and b j, n are given in Refs. 2, 5, and 8. It was shown in Ref. 9 that complex coe•cients are required for stability. Numerical solutions of Eq. (8) with appropriate boundary and interface conditions are described in Refs. 2 and 5. Range-dependent problems can be solved (with varying levels of accuracy) by allowing the elastic parameters in the range-independent elastic PE to depend on range.

II. ENERGY CONSERVATION

In this section, we derive a method based on energy- flux conservation for approximating the field transmitted across a vertical interface. We apply the small-angle approximation used to derive the energy-conserving acoustic PE. 3 This approach was first applied for improving the accuracy of outgoing solutions in Ref. 10. Although the

energy-conserving elastic PE does not provide the reflected field, it is more accurate for outgoing energy than the range-independent elastic PE.

A vertical interface separates range-independent regions A and B. The subscripts A and B are used to denote evaluation in the respective regions. The subscripts i and t are used to denote incident and transmitted quantities. The following conditions • correspond to conservation of energy flux for both wave types across a vertical interface:

= , (9) WlYxz ] t WlYxz ] i

where the stresses are defined by

au

axx=Za+2u, au aw

&.

The two-way elastic PE conserves energy flux because it conserves both the displacements and the stresses across the vertical interface. 2

The following representation is obtained using Eqs. (3) and ( 10):

where

( x+2u -2u 0 1

(13)

The following representation is obtained using Eqs. (2), (3), and (11)'

-- =s (]4) C•X • O'xz '

where

S-= a a ' (15)

From Eqs. (6) and (14), we obtain

=--iS(L-•M) -1/2 A . • O'xz W

(16)

From Eqs. (9), (12), and (16), we obtain the following nonlinear boundary-value problem for the transmitted field:

(A) SB( LjIMB)_•/2(•w) RBwt t

--RA(•)i•A(L•IMA)-I/2(•w) i (17)

976 J. Acoust. Soc. Am., Vol. 94, No. 2, Pt. 1, August 1993 Michael D. Collins: Elastic energy conserving parabolic equation 976


There are three obvious approaches for solving Eq. (17): Derive an equivalent linear boundary-value problem that can be solved directly (a possible approach is discussed in Sec. IV); derive a convergent iteration formula for solving Eq. (17) exactly (this approach is effective for the two-way PE of Ref. 2); derive an approximate solution that is accurate for a large class of problems. Although the first two approaches should eventually be considered, we have cho- sen the third approach because it has proven to be robust and efficient for the acoustic case.

The energy-conserving acoustic PE was obtained by assuming that energy propagates nearly horizontally. This assumption is not valid when compressional energy couples into steeply propagating shear energy, a common oc- currence at horizontal interfaces. We therefore do not at-

tempt to satisfy both of the conditions in Eq. (9) and replace the second condition with conservation of w. The examples in Sec. III illustrate that accuracy may be achieved for many problems by enforcing only the first condition. On a sloping fluid-solid interface that is approximated by a sequence of stair steps, axz is forced to vanish on the runs and therefore vanishes on the rises in the limit

of a large number of stair steps. It follows that the second condition in Eq. (9) is automatically satisfied at a fluid- solid interface. For the rotated PE, a similar fortunate sit- uation occurs at the ocean surface, which is sloped in rotated coordinates. •2

Following the approach used to derive the energy- conserving acoustic PE, we neglect all depth derivatives of the dependent variables in the compressional energy flux condition in Eq. (17) and obtain the leading-order correction,

_3 x 1/2s ( pep 3) •2At= ( pcpl A l-•i' (18)

The accuracy of this simple correction is illustrated for some test problems in Sec. III. For the acoustic case, 3 the acoustic pressure is denoted by p, the sound speed is denoted by c, and the equivalent leading-order correction is

(pC) • 1/2pt= ( pC)• 1/2pi. (19)

As in the elastic case, the energy-conservation correction is an amplitude adjustment. The correction is of larger mag- nitude and in the opposite direction for the elastic case.

We solve the energy-conserving elastic PE using the numerical solution described in Ref. 5 with a modified

treatment of fluid-solid interfaces. A sloping fluid-solid interface is approximated by a sequence of stair steps with each of the runs placed halfway between two depth grid points, which are separated by the depth increment •z. This grid design was used in Ref. 3 to solve the energy- conserving acoustic PE. At the vertical interfaces between range-independent regions, the energy-conservation correction is implemented by adjusting the amplitude of A using Eq. (18). This contrasts with the original approach for implementing the energy-conserving acoustic PE, which involves a change of dependent variable. 3

For problems involving a sloping interface between homogeneous fluid and solid layers, only the grid points that lie on the rises of the steps are corrected using Eq.

(18). Since one of the interface conditions for the runs involves two depth derivatives (see Ref. 2), this numerical solution provides O(•z) accuracy. Numerical accuracy can be increased to O(•z) 2 by using either a difference formula for c92/c9z 2 that involves four grid points or the alternate formulation described in Appendix A. The solution of Ref. 5 involves placing each of the runs at the depth of a grid point and defining A and to in both layers at that grid point. Although this solution provides O(•z) 2 accuracy, it is not suitable for range-dependent problems.

III. EXAMPLES

In this section, we demonstrate the accuracy of the energy-conserving elastic PE using the rotated elastic PE to generate reference solutions. We also apply the energy- conserving elastic PE to illustrate mode cutoff and coupling into shear wave beams in the ocean bottom and the propagation of interface waves along sloping ocean bottoms. Each of the examples is in cylindrical geometry and involves a point source placed in a homogeneous fluid wa- ter column in which c- 1500 m/s, p= 1 g/cm 3, and/3=0.

Example A is a generalization of a range-dependent acoustic benchmark problem that involves a sloping ocean bottom. 13 A 25-Hz source is placed at z= 100 m in a homogeneous ocean in which depth decreases linearly with range from 200 m at x =0 to zero at x = 4 km. In the ocean bottom, we consider three cases for cp and Cs and take /3•=/3s=0.5, and p= 1.5 g/cm 3. A wide-angle PE starter TM is used to initialize the elastic PE at x=0. The solutions

appearing in Fig. 1 were generated using the energy- conserving elastic PE, the rotated elastic PE, and the range-independent elastic PE. The solutions appearing in Fig. 2 for the original acoustic problem were generated using the energy-conserving acoustic P E and the range- independent acoustic PE.

For the case c•= 1700 m/s and Cs=800 m/s, the energy-conserving elastic PE solution is accurate. The error in the range-independent elastic PE solution is in the opposite direction of the error in the range-independent acoustic PE solution. This behavior is consistent with the

discussion in Sec. II. For the case c•o=2400 m?s and Cs = 1200 m/s, the energy-conserving elastic PE solution has an error of a few decibels; however, the correction is in the right direction. The range-independent elastic PE solution begins to grow for x > 3 km. For the case c•o= 3400 m?s and Cs = 1700 m/s, the energy-conserving elastic PE solution has only a small error. The range-independent elastic PE solution begins to grow for x > 2 km. Since these cases involve a significant number of interactions with a sloping bottom and a relatively large range of sediment types, it appears that the energy-conserving elastic PE is accurate for a large class of problems.

Example B involves mode cutoff as ocean depth decreases with range. A 25-Hz source is placed at z--25 m in an ocean in which depth decreases linearly with range from 150 m at x=0 to zero at x= 10 km. In the ocean sediment,

we take c•,=3400 m/s, Cs = 1700 m/s, /3•=0.25, /•=0.5, and p= 1.5 g/½m 3. A wide-angle PE starter 14 is used to initialize the elastic PE at x=0. We also consider the re-



40; 5O

60-

70-

80-

90

0

(a)

I I I

1 2 3

Range (kin)

40

60-

70-

80-

90

0 1 2 3 4

Range (kin)

40

50-

60-

70-

80-

90

0

Range (kin)

50 • ! •', _

• 70

8O

9O

(c)

I I I

o 1 2 3 4

Range (kin)

FIG. 1. Transmission loss at z= 30 m for example A, which involves a sloping elastic ocean bottom and three cases for the wave speeds: (a) %= 1700 m/s and Cs=800 m/s; (b) %=2400 m/s and cs= 1200 m/s; (c) c•= 3400 m/s and cs= 1700 m/s. The solid curves were generated with the energy-conserving elastic PE. The dashed curves are the reference solutions, which were generated with the rotated elastic PE. The broken curves were generated with the range-independent elastic PE.

lated acoustic problem for which the sediment parameters are c= 1700 m/s,/3=0.5, and p= 1.5 g/½m 3. The energy- conserving elastic PE solution appears in Fig. 3. The energy-conserving acoustic PE solution appears in Fig. 4 for the related acoustic problem. The shear potential was generated from the elastic PE solution using an approach that is described in Ref. 2. Two trapped modes are excited by the source for both problems. For the acoustic case,

FIG. 2. Transmission loss at z= 30 m for the benchmark problem of Ref. 13, which is related to example A. The solid curve was generated with the energy-conserving acoustic PE. The broken curve was generated with the range-independent acoustic PE.

which is similar to the problem of Ref. 15, the modes are cut off as the ocean depth decreases and couple into beams in the ocean sediment. For the elastic case, the modes are cut off as the ocean depth decreases and disappear from the compressional potential (i.e., the dilatation). This energy reappears in shear wave beams in the ocean sediment that resemble the beams of the acoustic problem. We also solved this problem with the rotated elastic PE and obtained identical results.

Example C consists of two problems involving an interface wave propagating along a sloping ocean bottom. Both problems involve a 5-Hz source placed 15 m above the bottom of the ocean. In the sediment, we take c•,= 2400 m/s, Cs = 1200 m/s,/3p=0.1,/3s=0.2, and p= 1.5 g/cm 3. The self-starter, which is accurate for problems involving interface waves, 16 is used to generate an initial condition at x=0. For the upslope case, the ocean is 150 rn deep for x < 5 km, 10 rn deep for x > 10 kin, and linearly decreasing in depth for 5 km < x < 10 km. For the downslope case, the ocean is 100 rn deep for x < 5 km, 250 rn deep for x > 10 km, and linearly increasing in depth for 5 km < x < 10 km. The energy-conserving elastic PE solutions appear in Fig. 5. For the upslope case, the interface wave propagates up the slope and interferes with a trapped mode. The mode is cut off as ocean depth decreases and couples into a compressional wave beam in the sediment. The interface wave propagates all the way up the slope into the shallow region. The interface wave also remains attached to the interface

for the downslope case. Since cutoff does not occur, however, there is no coupling into a beam in the sediment.

IV. COMPLETE ENERGY-CONSERVATION CORRECTIONS

In this section, we present a complete energy- conservation correction for the acoustic PE that is a generalization of the leading-order correction of Ref. 3. The complete correction is of limited practical importance for ocean acoustics problems because the leading-order correction is accurate for most problems. Since the leading-order



2OO

400-

600 -

200

400

600 -

800

(a) o 1

Range (km) lO

800

0 2 4 6 8 10

Range (krn)

200

.C 400-

600

800 '

(b) 0 2 4 6 8 10 Range (kin)

FIG. 3. The energy-conserving elastic PE solution for example B, which involves the cutoff of trapped modes and coupling into shear wave beams. This problem is related to the mode cutoff problem of Ref. 15. The compressional potential (a) and shear potential (b) appear on a decibel scale with red corresponding to high intensity and black corresponding to low intensity.

correction is not as accurate for the elastic case (as the examples in Sec. III indicate), however, a complete correction for the elastic case would be of practical use. Since the complete correction is related to the self-starter, 16 which has been generalized to the elastic case, it is likely that it can be generalized to the elastic case.

In a range-independent region of a fluid waveguide, the acoustic pressure satisfies

x2-PzzP- zz+k2P =0, (20)

where k=o•/C•,. Factoring Eq. (20) and selecting the outgoing solution, we obtain

Ox-iQp, (21) where

FIG. 4. The energy-conserving acoustic PE solution for the acoustic problem that is related to example B and is similar to the mode cutoff problem of Ref. 15. The acoustic field appears on a decibel scale with red corresponding to high intensity and black corresponding to low intensity

0•

200

400

600

800

1000

(a) o 4 8 12 16 20

Range (kin)

O-

200

400

600

800

.

1000 I • (b) 0 4 8 12 16 20

Range (km)

FIG. 5. The energy-conserving elastic PE solutions for example C, which involves an interface wave propagating along sloping ocean bottoms: (a) upslope propagation; (b) downslope propagation. The compressional potential appears on a decibel scale with red corresponding to high intensity and black corresponding to low intensity.

979 d. Acoust. Soc. Am., Vol. 94, No. 2, Pt. 1, August 1993 Michael D. Collins: Elastic energy conserving parabolic equation 979


Q= . The first condition in Eq. (9) is equivalent to

•Pt •Pi p•lpt •xx=p•-lpi sx . (23)

This nonlinear boundary condition is difficult to apply. In the limit of arbitrarily small propagation angles, Q reduces to k and Eq. (23) reduces to Eq. (19). The complete energy-conservation correction is given by the linear boundary condition,

- l/2Q1B/2pt l/2Q1A/2pi Pa =PX , (24)

where

a -1 a k2 ) 1/4 Q1/2= p•p •q_ . (25) To show that Eq. (24) is a mode-space equivalent of

Eq. (23), we apply the normal-mode representation, 17'18

p= • aj(x)c•j(z), (26)

where each range-independent region has its own coeffi- cients a j, normal modes •bj, and eigenvalues kj. The normal modes and eigenvalues satisfy

f p-l•,4j dz=•ij , (27) Qrcf j=k]r. cf j, (28)

where 1' is an arbitrary constant. Applying the depth operator appearing in Eq. (24) to Eq. (26), we obtain

p 1/2Q1/2p p 1/2 Z-,_1/2. - = - c•jnj q•j. (29)

Squaring both sides of Eq. (29) and integrating over depth using Eqs. (21 ), (27), and (28), we obtain

(p 1/2Q1/2p)2dz = p 1/2 Z otjkj q•j dz

= f-'pQpdz

From •s. (24) and (30), we obtain

; ; Although Eq. (31) may appear to be a weaker condition than the pointwise condition given by Eq. (23), it is actu- ally just as strong because it is required to hold for an arbitrary incident field.

The depth operator Q•/: of the complete energy- conservation correction is also involved in the boundary-

value problem for the self-starter. 16 If Eq. (24) is multi- plied by p•2 and discretized in depth, it becomes identical to the problem for the self-starter with an array of weighted and phased point sources located at the grid points, which are separated by the grid spacing Az. As the range from this array becomes arbitrarily small, the energy flux through a depth interval of length Az centered at the depth of one of the sources is entirely due to that source (this is not true for a line source). Based on this observa- tion, it would not be surprising if there exists a complete energy-conservation correction for the elastic PE that is linear and related to the self-starter for a point source in an elastic waveguide.

We have implemented the complete correction for the acoustic PE using a Pad• approximation for the operator fourth root. However, we were not able to find a problem for which the leading-order correction has a significant error. This surprisingly robust performance is due to the cancellation of terms beyond leading order. To show this, we break a vertical interface between two range- independent regions into two vertical interfaces, with changes in p occurring across the first interface and changes in k occurring across the second interface. We show in Appendix B that, in the limit of small propagation angles, eigenvalues are more sensitive to changes in wave number than to changes in density. We have tested this asymptotic result numerically and found it to be robust. Since Eq. (24) is linear, it suffices to assume that only one mode is excited. We also assume that mode coupling 17 is a higher-order effect. Under these assumptions, we apply the results of Appendix B to Eq. (24) for the first interface and obtain

-- 1/2--1/2 1/2--1/2 P B Kn Pt=P• Kn (32)

Since Eq. (32) reduces to Eq. (19), the leading-order energy-conservation correction is correct to beyond leading order for the first interface.

For the second interface, Eq. (24) becomes

Ut= pB • PB •'•t-kB pB • PB •-•t-k2• u i. (33)

Expanding the operator fourth roots one term beyond leading order (i.e., applying a linear Taylor series), we obtain

Ut= 1- •-•o2 P a • P a • -- •o 2 ( k•- ko 2 )

1 ( a _la ) 1 X 1-F•o ø pa • pa • -F•o2 (k•-•) U i ß

(34)

Expanding the product of operators and neglecting the (higher-order) cross terms, we obtain

1 1 (35)



Since the terms involving depth derivatives cancel, the leading-order energy-conservation correction is also correct to beyond leading order for the second interface.

V. CONCLUSION

We have derived an energy-conserving elastic PE. The accuracy of this solution was confirmed for a problem involving a sloping ocean bottom for three ranges of elastic wave speeds. The energy-conserving elastic PE was applied to solve a problem involving an interface wave propagating along a sloping interface and a problem involving mode cutoff and coupling into shear wave beams in the sediment. In contrast to the rotated elastic PE, the energy-conserving elastic PE is valid for a large class of range-dependent problems. It would be worthwhile to derive an improved energy-conserving elastic PE.

APPENDIX A: AN ALTERNATE FORMULATION

In this Appendix, we describe an alternate formulation for the equations of elasticity that is suitable for the PE method. This formulation has the advantage that the interface conditions involve only single z derivatives. Inter- face conditions that involve second z derivatives are more

difficult to implement, especially for range-dependent problems. For simplicity, we assume that each layer is homogeneous (it is easy to generalize to depth-dependent layers) so that Eqs. (2) and (4) become

p, •-•X2 +• -•-q- p(.02Wq- ()• q-p,) -• =0, (A1)

02A 02A

(A+2/.t) •-•x2+ (•+2•) •+pto2A=0. (A2)

We define the new dependent variable,

21.t Ow c=A+-•-•z. (A3)

Substituting Eq. (A3) into Eqs. (A 1 ) and (A2), we obtain

O2W ]d, ( A + 21.t ) O2w Oo' • ax 2 - • az 2 -3- pw2W q- (• q-/2 ) • = 0, (A4)

(A+2p) •-x2x2+ (A+2 p) •q-p(.o20 '--• 2•pw 2 OW

A 0z

0% 03w -- )• 0X2& -- A 0z 3 --0. (A5)

Differentiating Eq. (A4) with respect to z, we obtain

03w • ( ,• + 2• ) 03w Ow 020' • ax 2 az- zl, 023 Jr-pCo2 •z Jr- (•,•/.t) •=0.

(A6)

Using Eq. (A6) to eliminate the term involving three z derivatives from Eq. (A5), we obtain

020 ' 020 ' ( •)OW 4•(2+•) O3w

- )• 0x20z= 0. (A7) Since the system consisting of Eqs. (A4) and (A7) does not have a term involving O/Ox, it factors into outgoing and incoming solutions.

APPENDIX B: EIGENVALUE PERTURBATIONS

In this Appendix, we apply a generalization of the perturbation analysis in the Appendix of Ref. 19 to show that eigenvalues are more sensitive to changes in wave number than to changes in density. Similar perturbation methods for the eigenvalue problem appear elsewhere, such as in Ref. 17. We perturb the operator F=Q 2 with AF and obtain the perturbed eigenvalue problem,

(F+ AF) (•bn+ •bn, 1 + ' ' ' )

--(kn-[-kn, 1 -[-'" )2(•bn-[-•bn, 1 -[-.'' )=0, (B1)

where kn, j and •kn,j are corrections to the unperturbed eigenvalue and eigenvector. We apply Eq. (28) to Eq. (B 1 ) and neglect higher-order terms to obtain

( r - k2n) (kn, 1 = ( -- AF -[- 2knkn, 1 )(kn' (B2) As in Ref. 19, we apply the solvability condition,

p--l•bn( -- AF + 2knkn, 1 )(kn dz=O, (a3) to Eq. (B2) and obtain

kn, l =•n n p- •knAl"•kn dz. (B4) Substituting for AF, we obtain

kn, l=•n n P•- 1 (k•-- k•) •b2n dZ

( -•nn P•-lqbn PB • PB dz

d _•d•n) dz. Applying integration by parts to the second term on the right side of Eq. (BS), we obtain

if kn, l=• P; 1 (•--•)•2n d2

•• log • • p; l•n • d•. (B6) The first term on the fight side of Eq. (B6) dominates the second term in the small-angle limit:

981 d. Acoust. Soc. Am., Vol. 94, No. 2, Pt. 1, August 1993 Michael D. Collins: Elastic energy conserving parabolic equation 981


IF. D. Tappert, "The Parabolic Approximation Method," in Wave Prop- agation and Underwater Acoustics, edited by J. B. Keller and J. S. Papadakis, Lecture Notes in Physics, Vol. 70 (Springer-Verlag, New York, 1977).

2M. D. Collins, "A two-way parabolic equation for elastic media," J. Acoust. Soc. Am. 93, 1815-1825 (1993). M.D. Collins and E. K. Westwood, "A higher-order energy-conserving parabolic equation for range-dependent ocean depth, sound speed, and density," J. Acoust. Soc. Am. 89, 1068-1075 (1991).

4M. D. Collins, "A split-step Pad6 solution for parabolic equation method," J. Acoust. Soc. Am. 93, 1736-1742 (1993).

5M. D. Collins, "Higher-order parabolic approximations for accurate and stable elastic parabolic equations with application to interface wave propagation," J. Acoust. Soc. Am. 89, 1050-1057 (1991). H. Kolsky, Stress Waves in Solids (Dover, New York, 1963), p. 11. R. R. Greene, "A high-angle one-way wave equation for seismic wave propagation along rough and sloping interfaces," J. Acoust. Soc. Am. 77, 1991-1998 (1985). A. Bamberger, B. Engquist, L. Halpern, and P. Joly, "Higher order paraxial wave equation approximations in heterogeneous media," SIAM J. Appl. Math. 48, 129-154 (1988).

9B. T. R. Wetton and G. H. Brooke, "One-way wave equations for seismoacoustic propagation in elastic waveguides," J. Acoust. Soc. Am. 87, 624-632 (1990).

]øM. B. Porter, F. B. Jensen, and C. M. Ferla, "The problem of energy conservation in one-way models," J. Acaust. Sac. Am. 89, 1058-1067 (1991).

]•A. Ben-Menahem and S. J. Singh, Seismic Waves and Sources (Springer-Verlag, New York, 1981 ), p. 26.

]2M. D. Collins, "The rotated parabolic equation and sloping ocean bottoms," J. Acoust. Soc. Am. 87, 1035-1037 (1990). F. B. Jensen and C. M. Ferla, "Numerical solutions of range-dependent benchmark problems," J. Acoust. Soc. Am. 87, 1499-1510 (1990). R. R. Greene, "The rational approximation to the acoustic wave equation with bottom interaction," J. Acaust. Sac. Am. 76, 1764-1773 (1984).

•SF. B. Jensen and W. A. Kuperman, "Sound propagation in a wedge- shaped ocean with a penetrable bottom," J. Acoust. Soc. Am. 67, 1564- 1566 (1980).

]6M. D. Collins, "A self-starter for the parabolic equation method," J. Acoust. Soc. Am. 92, 1357-1368 (1992).

•7R. B. Evans, "A coupled mode solution for acoustic propagation in a waveguide with stepwise depth variations of a penetrable bottom," J. Acoust. Soc. Am. 74, 188-195 (1983). M. B. Porter and E. L. Reiss, "A numerical method for ocean acoustic modes," J. Acoust. Soc. Am. 76, 244-252 (1984).

•9M. D. Collins, "A nearfield asymptotic analysis for underwater acoustics," J. Acoust. Soc. Am. 85, 1107-1114 (1989).



Documents

An energy-conserving parabolic equation for elastic media