Wave solutions in three-dimensional ocean environments

Wave solutions in three-dimensional ocean environments C. H. Harrison

BAeSEMA, Biwater House, Portsmouth Road, Esher, Survey KTlO 9SJ, England

(Received 1 April 1992; accepted for publication 2 December 1992)

A three-dimensional wave solution is developed for variable depth but isovelocity environments by transforming the wave equation into a suitable coordinate system. These solutions can be compared with existing known solutions derived from a ray point of view where the vertical modes propagate like horizontal rays. A general approach for arbitrary profiles is used to give explicit analytical solutions for various particular topographies including troughs and ridges. The behavior of these solutions is well known since they all have analogs in ordinary two-dimensional refracting media. Because this approach does not invoke the adiabatic approximation or ray invariants but evidently has similar limitations it throws some light on the limits of validity of these alternative approaches. In passing, some interesting problems in normalization are encountered, and investigations into slope and curvature restrictions reveal some general relations between average slope, curvature or higher derivatives for realistic surfaces. Numerical modelers, who currently only have the ocean wedge as a 3-D benchmark solution, now have an extended repertoire of benchmarks.

PACS numbers: 43.30.Bp

INTRODUCTION

Analytical solutions to the 3-D wave equation in an ocean context are hard to come by although they are useful as benchmarks for numerical solutions that are increas-

ingly capable of operating in these conditions. They also give useful insight into mode coupling and the validity of the adiabatic approximation in range-dependent environments. Most analytical work to date has been done on the wedge-shaped ocean, notably by Buckingham ( 1983, 1987, 1989) although a conical seamount has also been considered (Buckingham, 1986). It is well known that the vertical normal modes behave in the horizontal plane like rays [Weinberg and Burridge ( 1974); Brekhovskikh and Lysanov (1982) ]. Harrison (1977) derived sets of analytical ray trajectories for a variety of troughs, basins, and seamounts and showed that these led to horizontal shadow

zones for each mode (Harrison, 1979). One would expect the special profiles that he considered to be suitable for a wave treatment as well, and indeed this paper shows that they are. The novel approach here provides solutions that are strictly approximate although in a typical ocean environment the approximation is very good. They are, in fact, exact analytical solutions to an equation that is an approximation to the true three-dimensional Helmholtz equation. The approach may be regarded as a generalization of the "wedge mode" solution in Primack and Gilbert (1991).

Starting from the Helmholtz equation the first problem is to find suitable coordinate systems in which the coordinates are separable. A systematic approach is through conformal mapping [Morse and Feshbach ( 1953)], but it is difficult to see how this can produce any more than the well-known separable systems such as cy- lindrical, elliptical, etc. Alternatively, in the general case of isovelocity, variable depth water there is an easy way of seeing what this coordinate system looks like. In the

mapped system the boundary corresponds to one of the coordinates being constant (so this coordinate must be normalized depth •, as we will see, with •=0 at the surface and •= 1 at the bottom). With a true conformal mapping these coordinates look in real space like (and are identical to) flow lines and equipotential lines in an imaginary in- viscid fluid flowing between the two boundaries (Fig. 1). The flow lines are locally parallel to the boundaries and the equipotentials are locally at right angles. Thus, in this system normal modes are measured along a curve (an equipotential) between surface and bottom rather than a vertical line, and modal phases are measured along flow lines rather than horizontally. This is a natural system for the normal modes and there is no need for mode coupling because the wave equation separates in the new coordinates. An obvious example of this is the ocean wedge where the equipotentials are arcs of a circle and the flow lines are straight but they radiate from the wedge axis. The solution in this system is truly separable and requires no coupling. Because we can calculate some general and special solutions without invoking mode coupling or the adiabatic approximation we will be able to make some interesting comparisons with standard adiabatic solutions and well-known criteria for their validity.

I. REVIEW OF THE WEDGE

The type of geometry treated will be restricted to ones where depth varies in one coordinate only. Although it is possible to deal with circular symmetry (basins, seamounts), we will restrict this coordinate to a Cartesian one, i.e., troughs and ridges with constant cross section. Since we will adopt Buckingham's scheme for calculating the Green's function we briefly review the procedure and result for a wedge. Here, an appropriate coordinate system

1826 J. Acoust. Soc. Am. 93 (4), Pt. 1, April 1993 0001-4966/93/041826-15506.00 @ 1993 Acoustical Society of America 1826

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is polar with the polar axis (x) is along the wedge axis. Thus the Helmholtz equation with a source at (O, Os, r s) and wave number k=w/c is

2 =--- 6(r--rs)6(x)6(O--Os). (1)

A natural choice is to start by taking a sine transform in 0 since this automatically obeys the boundary conditions (e.g., pressure release or hard bottom). In the x direction there are no boundaries so one can take a Laplace transform or a Fourier transform. Finally, in the case of a wedge the sine transform in 0 results in a residual constant di-

vided by an r 2 term, which combines with the k 2 to make Bessel's equation. The natural transform to perform in r is then the Hankel transform, but clearly a different transform kernel will be necessary for other geometries. The order of carrying out the transforms is a matter of mathematical convenience, and the final solution is obtained by inverting the transforms in reverse order. In the more general case, having chosen separable coordinates, the kernel of the transform for each coordinate is always a solution of the appropriate homogeneous equation that fits the boundary conditions. Thus, in principle, a refracting medium could be accommodated, but in a wedge, for instance, it would have to be a function of 0 only, or r only, or x only.

For an isovelocity wedge of slope ]/the exact solution at (x,O,r), provided m is an integer, is

4•r • sin(mOs)sin(mO)Im (2) and

[m: Jm(Krs)Jm(Kr)K• e

p= (K2_k 2) •/2, rn=n•r/y,

dK, (3)

with n being an integer for both boundaries pressure release or an integer plus one half for mixed boundaries. Here, K is a dummy variable corresponding to a wave number in the vertical r, 0 plane. Because the integrand of I m is highly oscillatory it is only possible to make any further progress with an analytical solution by employing an approximation such as first- or second-order stationary phase (Buckingham, 1989). No attempt will be made in this paper to do this for other geometries.

II. GENERAL GEOMETRIES: NEW EQUATION

Starting with the homogeneous Helmholtz equation, Appendix A derives an approximate transformation to the curvilinear coordinate system ,/, • giving Eq. (A24). Here, ,/ is a modified horizontal distance measured along the "flow" lines (see Fig. 1) and • is a dimensionless depth normalized by the local "depth" h(,/) measured down the curve defined by ,/=const. Thus in the •/, • system the

Y

(a) dq dp

r,=0

(b) dr== dq/h(p) =dp

FIG. 1. (a) Real space showing the Cartesian coordinates y,z and the relation between them and the curvilinear coordinates p,q. (b) Mapped space showing vertical lines of constant p and horizontal lines of constant q. The final mapped system is •/,• with • being dimensionless.

seabed (•= 1) looks flat [Fig. 1 (b)]. In effect we have an approximation to the three-dimensional Helmholtz equation for which we seek exact solutions.

Adding the third dimension •=x, which defines the direction of the axis of a trough or ridge, and a source term we can write the inhomogeneous Helmholtz equation in •, ß /, • with a source at 0, •/s, •s as

1 a2•b h" a•, a:•b h'&b o•b og-- a

(4)

where a prime denotes differentiation with respect to •/. As it stands this equation is not separable, but it can be made separable under several alternative conditions. The sim- plest is h" =0, and the implications of this restriction are discussed in Appendix A. Other possibilities that are not pursued further here are either h"h=const or h"h=f(•) only.

Here we adopt the first condition that hh"•_O [i.e., the radius of curvature of the seabed is considerably bigger than the water depth, see Eqs. (A35) and (A38) in Ap- pendix A] so that we can neglect the second term. We can now take sine transforms of both sides, i.e.,

F(vn) = •b(•)sin(vn•)d •, (5a)

•b(•')--2 • F(vn)sin(vn•). (5b) n=O

Note that the functions •b and F are also implicitly functions of • and ,/. In order to fit the boundary conditions we

1827 J. Acoust. Soc. Am., Vol. 93, No. 4, Pt. 1, April 1993 C.H. Harrison: Wave solutions in 3-D environments 1827


must have the vertical (dimensionless) wave number defined by

Vn=nrr, (6)

where n is an integer for pressure release boundaries or an integer plus one half for pressure release surface and hard bottom. The sine transform of the second derivative of •p can be written in terms of F by integrating by parts and making use of the boundary conditions. Equation (4) becomes

02Fh, OFO2F( ¾2) o. 4•

= --• sin(v•s)• (n-- ns) • (•). (7) At this point in the derivation it is interesting to digress and show the connection with horizontal refraction. First

we make the substitution

W(n)=hl/2F(n) (8)

in order to get rid of the OF/O• term. If we temporarily neglect te•s of order h"/h and (h'/h)2 compared with k 2 [see Eq. (A31 ) in Appendix A] we obtain

02W 02W ( V2n• O• 2 + • + k 2-- W 4•

=-- h•sin(v•s) 6(n-n•)6(g). (9) This equation describes a wave that refracts in the (nearly) horizontal •, • plane because of the varying wave number [k2-•/h2(•)] •/2, and this behavior is well known. Re- membering that the second part of this expression is a (dimensioned) vertical wave number K•=v,h we can think of the wave as composed of rays at elevation angle 0 where sin O=K•/k=vn/kh=n•/kh. In effect there is a spatial refractive index of cos 0. Equation (9) is analogous to the usual two-dimensional Helmholtz equation in the vertical plane through a stratified medium but with the modified source strength shown in square brackets. Indeed by expressing the solution in terms of WKB normal modes

. this reduces explicitly to horizontal refracting rays (Har- rison, 1991, 1992a, 1992b).

Returning to Eq. (7) we can Fourier transform out the • dependence using the following definitions:

G(Kg) = F(•)em; • d•, (10a)

F(•) =• G(Kg)e-•X; • dKg, . (10b) and we obtain

02 G h ' OG (k 2 V2n• 4•

= --• sin(v½s)• (n-- ns). ( 11 )

We now express G as the sum of eigenfunctions U j of the homogeneous equation

2

02uj h' Ouj ( k2 ¾n ) 0,/2 +•- •+ --•--K• uj=0, (12) G= • ajuj . (13)

J

At this stage with the horizontal boundaries in r/undefined we do not know whether this sum is a finite set over some

discrete modes or an infinite set that could be expressed as an integral. We will return to this point in Appendix C with an alternative derivation assuming a continuous spectrum, but for now we assume pressure release boundaries at some given values of r/. The usual method (Morse and Feshbach, 1953) for closed boundaries is to substitute Eq. (13) in Eq. (11 ), multiply both sides by u•, and integrate over r/. Using the orthogonality and normalization of

• ujuiw( */ )d*/=gij , (14)

where •ij is the Kronecker delta, the unknown coefficients aj can be obtained. The weighting function w is often 1 but can in any case be calculated for the given uj and Eq. (12) (see Appendix C). Inverse Fourier transforming [Eq. (10b)] we then obtain F as

1 f_, 4rrw(*/s) F(•)=• •- h(*/s) o• j

sin (Vn•s) u j ( */s) u; ( */r) -- iK• X (K•--K•) e dK•.

(15)

Using standard techniques this is

F(•) =2rri • W(*/s)Sin(Vn•s)Uj(*/s)Uj(*/r) j h(*/s)Kj eiKfi (16) and inverse sine transforming [Eq. (5b)] we obtain

lp : Z b n E tl J ( */ s ) tl J ( */ r ) eiK fi, (17) n j gj

where

sin ( Vn• s) sin (Vn•r) W( bn=4rd h(*/s) ' (18)

At first sight the w( */s) /h ( */s) term appears to violate reciprocity, but it is easy to show (see Appendix C) that the orthogonality weighting term w must be related to h. It is, in fact, exactly equal to it so that w( */s) /h ( */s) = 1.

For future reference, if we had used Eq. (8) to elimi- nate the second term in Eq. (12) and neglected (h '2 - 2hh" )/4h 2 [see Eq. (A31 ) ] the denominator of b n would have been [h(*/s)h(*/r)] 1/2 instead of h(*/s). In this case the above theorem still ensures reciprocity since absence of the second term implies w= 1.

If the horizontal plane is unbounded for positive or negative */ (or both) then there is a continuous spectrum of eigenfunctions, for example the nth-order Bessel functions Jn (Kr) in the wedge. The boundary condition is now



that the eigenfunctions tend to zero as r/-. o•. In general, the eigenfunctions U(K,r/) have a different orthogonality condition (Appendix C), namely,

f f w(•7)w•:(g n) $(gn,•7) $(g•,•7)d•7 dry: 1, (19)

where U (K n, r I ) = u j ( r I ) and

u(g;,n) (20) It is shown in Appendix C that the equivalent of Eq. (17) is now

f U(Kn,•7s ) U(Kn,•7r) eit:• tp= • bnw•:(K) K dKn' (21)

where K= (k2--K2•) •/2, k=co/c, and b n is still given by Eq. (18). Although Kn is only a dummy variable in the integral it is clearly the total wave number in the •/,• plane [see Eq. (12)]. Again ifEq. (8) is used the denominator in b n is [h (•/s) h (•r) ]1/2 rather than h (•/s).

The reader may be concerned that the solution does not appear to exhibit the explicit outward going waves that we would expect for ,/> */s. This is an illusion, and it is easy to show, by using the Sommerfeld integral, that even the simple source and image formula for Lloyd's mirror (with which the horizontal "direct" and "reflected" paths have a lot in common) shows the same behavior. For a horizontal Lloyd's mirror with the source at 0, Ys, the re- ceiver at x, Yr, and the direct and reflected slant ranges being rl,r2, we have

exp(ikr l) R exp(ikr2) ;o • i tp = r• + r 2 -- • eipxLK dK, where p= (k2--K 2) •/2, R is the reflection coefficient, and

L=Jo{K(yr--Ys) )+ RJo{K(Yr+Ys) )

= ( 1 +R )So(Kys)So(Kyr)

• 2 • [ 1 • (-- 1 )nR ]Jn(Kys)Jn(Kyr). n=l

This equation not only has the same form as Eq. (21 ), but putting R =- 1, it differs from the wedge formula [Eq. (3) ] for mid-depth (where the argument of the sine terms is n•r/2), only in the Bessel function order.

III. EXAMPLES OF SOLUTIONS

Equations (17) or (21 ) combined with (18) are general solutions, respectively, for horizontally bounded and unbounded depth profiles. The horizontal mode functions uj(r I) or U(Kn, r I) are determined by the profile h(r/) when inserted in Eq. (12). There are many known solutions for particular cases, and general techniques are avail- able for finding others (DeSanto, 1979). Here we review and comment on several solutions with particular regard to those profiles for which ray trajectories were found by Har- rison (1977). In each case there is the possibility of solving Eq. (12) as it stands (labeled "three-term") or alterna-

tively absorbing the middle term (in the first derivative) into uj by the transformation of Eq. (8) (labeled "two- term"). Unless otherwise stated the notation is that of

Abramowitz and Stegun (1970).

A. Troughs

All the profiles below have a range of wave number for which bounded solutions are possible. Some also have a continuous spectrum. The shorthand notation K2• = k 2- K• is used throughout for the total wave number in the r/,• plane. The wave number in the • direction K• takes on discrete values Kj in the bound cases. 1. "Lorentz".' h(v)=ho/(l + •l•/t•a)•/•. ' Two-term equation

This is the trough that resulted in sinusoidal horizontal rays in Harrison (1977). The equation in r/is

d2uj 2 ¾n d•72 + Kv--h- • 1+•0 uj-0, (22)

which is the same as that for the "simple harmonic oscil- lator" with a solution in terms of Hermite polynomials Hn(x)'

tp= Z Z g,v4 exp --T ( r/s2 + r/r2) ß

n j

ix. 2Jj!rrl/2 Kj

( %2 • ,/4 ( 1 + r/r2/•) '/4 X 1+•] ho ' (23) where gn is defined here and in all subsequent equations as

gn = 4rci sin (Vn•s) sin (¾n•r) (24)

and

A = (¾n/horo) 1/2.

The discrete mode eigenvalues are defined by

1 horo( ¾2n• J+2--2Vn K2n--ho2) ' (25)

2. The ellipse: h(v)=ho(1-V2/t•o)•/2.. Three-term equation

Inserting the profile into Eq. (12) directly we obtain 2

d2uj rl duj ( ¾n ) dr/2 -- •( 1 -- r/2/•) d--• + g2•--h02 ( 1-- r/2/•) uj--O. (26)

This reduces to the standard form for Mathieu's equation (Abramowitz and Stegun, 1970, Eq. 20.1.7) on putting t = •l/r o:

(1--t2)•-•---t•-+ • K2•--h20/ (27)

The further transformation with cos v= t leads to



d2uj dv = + (a + 2q -- 4q cos = v) u j • 0,

where

-h•]'

q = r2oK•/ 4.

(28)

(29)

(30)

The eigenfunctions are "odd periodic" Mathieu functions of odd or even order sej(v,q). The usual orthogonality relation quoted for Mathieu functions is

;••r sej ( v,q )sei( v,q )dv= 7r•ij , but this is only true if the value of q is independent of v,j, or i, i.e., both the jth and the ith mode have the same q. In fact, Eq. (30) shows clearly that q is a function of eigenvalue Kv and therefore mode number j, so we expect a different relationship even though i and j are obviously constants in the integral. Substituting Eqs. (29) and (30) into Eq. (28) and isolating the K• term, Appendix C shows that the r/weighting must be sin2v and assures us that the following orthogonality relation must be true:

f o• Sej( v,qj)sei( v,qi)sin 2 v dv--ASij . (31 ) Here we show explicitly the different q values for each mode.

The normalization integral, which can be written in terms of r/ (with cos v = rl/r o) as

sej cos 1 drl=Ar o -go ;00 --70 (32)

is not exactly soluble, but for reasonably high-order horizontal modes (j>> 1) the WKB solution shows that

rr E( rr12,a ) A--

2 F(rr/2,a) '

where ct 2 1 2 2 = = --¾n/h•t, and F and E are elliptic integrals of the first and second kind, respectively.

The solution for the ellipse is then

n i 1'0 'q sej

'q Kj Ahoro (33) with gn defined as before [Eq. (24)].

The discrete mode eigenvalues are defined by the characteristic values bj(q) being equal to a. For constant a, q this would be simply a matter of reading off the value of bj for the appropriate j curve and the value of q (Abramow- itz and Stegun, Fig. 20.1 ). Since q is a function of K n we must write

2 2

bj- 2q= -- ¾nr2o/ho, (34)

which defines a straight line on the same graph, and read off the intersections with each j curve. For large q the function bj is such that the condition must reduce to Eq. (25).

It is interesting to compare Eq. (33) with the exact solution for an ellipse (having separated the Helmholtz equation in elliptic coordinates, Morse and Feshbach, 1953, p. 1421). The depth • equation, where we had, in effect,

a2f d• 2 q-const X f= 0 ( 35)

becomes the modified Mathieu equation,

d2f df (l+t a) (36)

and putting sinh v = t,

d2 f dv 2 - ( a-- 2q cosh 2v)f=O, (37)

leads to the "radial" functions Ce(v,q) and Se(v,q). These therefore replace the two sin (¾n•) terms in Eq. (24). How- ever, in an ocean context it is clear that the scale of • is very much smaller than r/ so that t and v are much less than unity in their working range in the above equations. Therefore they will both reduce to the simple form of Eq. (35).

Clearly the approximation that h' is not too large will be violated at the outer horizontal limits in this case, and the depth function h(r/) will begin to differ considerably from the shape defined in Cartesian coordinates by H(y) :ho2(1-y2/4) 1/2, which is assumed in the exact ellipse solution.

3. "œpstein':' h-e(q)= - A seche(q/ra) + B tanh(q/ra) + C: Two-term equation

This profile represents a trough with a shallow shelf as q- o• and a deep shelf as r/-•--o•, i.e.,

(h+)-2=C+B,

(h_)-2=C--B,

(ho)-2=C--A.

The equation can be written in terms of t =,//% as

d2uj dt 2 +(r20K•--•V2n(--A sech 2 t+ B tanh t+C))u•=O.

(38)

Following Morse and Feshbach ( 1953, p. 1652) we set

ttj=½ -at sech ø tF( t)/N 1/2 with the normalization N and the function F to be found, and a and b given by



and

2ab=v2nB4 ,

1

O• to( 2 -2--g•)l/2--ro(gS--(k2 ¾2nh72)) 1/2 = vnh + -- __ ,

•: rO ( ¾2nh :2_K•)1/2: ro(K• _ (k2_ •2nh Changing the variable to v=«(1-tanh t) we obtain the hypergeometric equation

The solution for F is then the hypergeometric function and uj is given by

e -at ( 1 1 uj=(et+e_t)bF b+•--D, b+•+ D, a+b+ 1,

( 1- tanh t) ) 2 N-1/2, (40) where

If Kv (or Kj) is such that both a and /3 are real (i.e., Kn < v2•h-2) then there are discrete modes and

d2F dF

v(l-v) •-•v2 +(a+b+ 1-2(0+ 1)v) dv

+(V•o-b(b+ 1))m=o. (39)

i 1/2 j +-• (v•4•+•) b. 2 • -- (41)

However under these conditions the hypergeometric function becomes a Jacobi polynomial p(,•,ls)(x) and the solution for the Epstein trough is

exp[ (/•-- a) (r/s+ rlr)/2ro] , •P= • •gn [(2 cosh r/s/r0) (2 cosh r/r/r 0) ] (a+t•)/2P}a'/•)(tanh rls/rø•r) ' (tanh r/r/r 0) ß

n j

2j!al•r (a + l• + j + 1 ) eiS:• • r0r(a +j + 1 ) 1 )(a q-[•)gj(h(•qs)h(•qr)) 1/2'

(42)

where F ( ) is the gamma function. This can be shown to be the same as the discrete part of the solution given by Deavenport (1966).

The orthogonality condition for the modes . •,( a fi ) [e -at sech ø tr)' (tanh t)] is not the usual one for Jacobi

polynomials. This is the same problem as was encountered for Mathieu functions and is a consequence of a and/3 being functions of the wave number Kj and hence mode number j rather than being simple constants. Applying the method in Appendix C to Eq. (38) we see that the r/ weighting in the orthogonality integral, written in terms of t, must be unity, and this assures orthogonality.

The normalization integral used in deriving Eq. (42) can be written in terms of x--tanh t, and after some ma-

nipulation and use of standard integrals (Gradshteyn and Ryzhik, 1980, Eq. 7.391/5) it can be shown that

( 1--x)'•(1 +x)l•(P3•'l•(x)) 2 dt

(/•+ a) 2•+/•F (a +j + 1 ) F(/•+j + 1 ) 2al•j!F(a +l•+j + 1 )

(43)

This differs from the usual normalization for Jacobi polynomials in that dx is replaced by dt.

There are also two more continuous spectrum contri- butions [see Morse and Feshbach (1953) and Deavenport

2 2 --2

(1966)] for the wave-number ranges V2n h-2 <K n < vnh + (horizontally "reflected" waves) and K• > V2n h • 2 (horizon_ tally "transmitted" waves). These can be regarded as branch line integrals or extensions of the j sum into a j continuum, which can be written as two wave-number in-

tegrals with respective limits (k2--v2nh• 2)1/2 to i oo and (k 2_ v2nh- 2) 1/2 to i

f exp[(l•-a)(rls+rlr)/2ro] (l+a+/• -- •n gn [(2 cosh rls/ro)(2cosh rlr/ro)](a+th/2F 2 l+a+/• 1--tanhrh/r0) (l+a+/• D, •-O, •+O, 1 +a, 2 f •--

l+a+/• •+ O, 1 +/•,

1 +tanh r/r/r0 r0F(«( 1 +a +/3)- D)r(«( 1 +a +/3) + D)e is:g 2 ) 2rriF ( 1 +a)F( 1 +[•)(h(rls)h(rlr)) 1/2 dK,

where K 2 = k 2-- K•.

1831 d. Acoust. Soc. Am., Vol. 93, No. 4, Pt. 1, April 1993 C. H. Harrison: Wave solutions in 3-D environments 1831


4. WKB: h(v) arbitrary but trough-like: Three-/two- term equation

If we start with the three-term equation, make the transformation of Eq. (8), but retain the residual terms, the distinction between three- and two-term equations becomes irrelevant.

The equation in r/is 2

• + K•--•+E uj=O, (44) where e is the residual term

1 h '2 1 h"

4h2-2 h '

The WKB solutions are

and

2 n 7I' f j(r/) =sin n-•+e dr/+•

X (g• -- ¾n/h 2 q- 6 ) - 1/4,

u(n) = f (n)lN m,

f 2 The solution for the trough is

(45)

•P= • •gnf j(r/s)f j(r/r) NKj(h(r/s)h(r/r))•/2 (46) n j

with g. given by Eq. (24). The discrete mode eigenvalues are given by the "phase integral"

2 1/2

j+ rr= K•-- • + E d v, (47) where the upper and lower integral limits are the points where the integrand is zero. For many environments one can set e to zero, but Eqs. (45) and (47) enable one to estimate how serious this approximation is for any particular case. In the case where the number of horizontal

modes is large the mode sum can be turned into a wave- number integral and the solution becomes indistinguish- able from the "ridge-like" case described later [Eq. (67)].

B. Ridges

1. The wedge: h(v)=yV a. Three-term equation. In real space 1' is the bottom

elevation angle whereas in the r/, h(r/) system it is the bottom gradient since dh/dr/=y. Inserting this into Eq. (12) directly with K n = (k 2- K 2) 1/2, having dropped the subscript j, we obtain

02U 10U ( 'V2n • (48) which is exactly the equation that gave rise to Bucking- ham's solution [Eq. (3)] in terms of Bessel functions Jm (Knr/), where from Eq. (6) m = ¾nl'= nrr/y. Inserting this in Eq. (21), remembering that the orthogonality

weighting for Bessel functions is r/K n, we obtain 4rci

•P = •n -•- sin (Vnet) sin (Vn•'r)

f e•r• X grl Jm(grlr/s)Jm(grlr/r) •-• dgrl. (49) This is identical with Eq. (3) if we note that 0•1'•, r=r/, and/p--=Ks= (k2--g•) 1/2.

The reason that the approximate approach has pro- duced an exact result is that the assumptions, namely that h" =0 and that the constant r/line is an arc of a circle, are both exactly true in this case. Consequently Eq. (4) becomes exact.

b. Two-term equation. Although a three-term solution is possible it is interesting to use the special case of the wedge as a means of investigating the magnitude of errors incurred generally by neglecting the middle term.

If we change the variable to

v = h 1/2 U = (yr/) 1/2 U (50)

and do not ignore any further terms we obtain

Or/2 + ) r/- 2 u=O (51)

and this obviously still has the solution U=Jm(Knr/) as before [see Abramowitz and Stegun (1970) Eq. 9.1.49]. However if we do ignore (h')2 terms as suggested after Eq. (8), the 1/4 disappears and the Bessel function order simply changes from m to m' where

2 1 1/2

(52)

Remembering that even for a low-order vertical mode (small n), m' is approximately nrr/y, which is a very large number for small slopes (y), we see that the difference between m and m' is always very small. A minor consequence is that the conditions under which m' is an integer no longer coincides exactly with the wedge angle being a sub-multiple of rr as one would expect from the coinci- dence of images (Buckingham, 1989). It is easy to see that despite the fact that v is now (l'r/)l/2j m, (Knr I) the change to the denominator of Eq. (18) (see the subsequent note) and the new orthogonality weighting ws: (=Kn/y) con- spire to bring back an identical solution to Eq. (23) except for substitution of m' for m.

2. "Curved shelf".' h-2(v)=Av 4- B: Two-term equation

This profile approximates the rapid fall off at the edge of the Continental shelf and in contrast with the plane wedge might be expected to show interesting effects from the curvature of the sea bed. The equation is

d2U 2B v2nAr/) U=0. (53) dr/2 + (K•-- ¾n --

Putting



2B) ]/(A¾2n)2/3 t= [ rlA v2n -- ( K2n -- vn (54) we obtain

a2u

dt 2 -tU=O (55) with Airy function solutions Ai(t). The solution for this profile is then

;0 •P= •n gn Ai(ts)Ai(tr) e iK• 2((Ar/sq- B)(Ar/rq- B)) 1/4

X-• (.4•2) 1/3 KndK n , (56) where K•= (k 2- K2•) 1/2. The orthogonality condition and normalization are derived in Appendix C.

& "Inverse parabolic".' h=ho/(1- v2/r•o)•/2. ß Two-term solution

This profile resembles a railway carriage roof, and clearly the assumption of reasonably low slopes will be violated as r/approaches r0. The equation is

d2U 2 ¾n aw+ +h0-;0lu=0. (57)

Putting

and

t=r/(2¾n/h0r0) 1/2 ( 58 )

hørølr2n we obtain the standard form

(59)

at 2 + •--a U=0, (60) whose solution is the parabolic cylinder function W(a,t). Examination of tables of this function show that it displays all the expected behavior of modes leaking from one duct to another across a parabolic barrier; namely for large positive a the solution is periodic with large amplitude on one side (- t) and periodic with small amplitude on the other ( + t) and decays monotonically in between. For negative a the solution is oscillatory everywhere.

The solution for the profile is

•n ;0 • eiK• 2 ( hOgOIl/2 lp= gnW(a'ts) W(a'tr) K•b ,rho 2¾n ] X ( (1-- %2 / r2o ) (1-- r/ r2 / r2o ) )1/ 4 K n d K n , (61)

where b= ( 1 q-e 2va) 1/2-eva and normalization is discussed in Appendix C.

4. 'Hyperbolic'.' h = h o( l + V2/ r;o) • / 2. ß Three-term equation

Inserting this profile into Eq. (12) and putting t--r//r 0 we obtain

d2 U dU

(1 +t2) •-+t •-+ 2

,r,,-hq + e=o.

(62)

This reduces to the modified Mathieu equation on substituting sinh v = t:

d2U

do 2 q- (4q cosh 2 o-- ( a + 2q) ) U = 0,

a=•20 m ,

(63)

(64)

q= r2oK•/4. (65) The solution for this profile is in terms of radial Mathieu functions of the first kind MsS) (z,q)'

f0• ( sinh- 1 ) lp-- Z gnMSm r/s • ro

( sinh- 1 ) X Msm *Is e ifg• ro ro • •oo K. dK. . (66) -- 4v,/ho) through the The order rn is related to 2q+a( 2 2

characteristic values of a. In the limit of large r/ the asymptotic solution for Mm can be shown to be just the Bessel function Jm (Knr/), where rn = v, ro/ho= mrro/ho for n •> K•ho. In this case the complete solution reduces to the solution for a wedge of angle y= ho/ro as indeed the profile suggests.

5. WKB: h(v) arbitrary but ridge-like: Three-/two- term equation

The equation and the WKB solutions are still given by Eqs. (44) and (45), but now, as discussed in Appendix C, the K weighting is easily shown to be wf(K) = K/2rc so the final solution is

f o o e •K• lp= •n gnf ( r/s) f ( r/r) • Krt dKrt , where

f(r/) =sin

2 1/2

dv+ 2 2 2 --1/4 X (Knh --¾n+6h 2) ß

(67)

(68)

IV. CONCLUSIONS

The Helmholtz equation has been transformed into a separable equation with range •, r/and normalized depth • as the variables [Eq. (4)]. To do this we have made the approximation that locally lines of constant r/and • look, respectively, like the radials and arcs of a wedge's polar coordinate system. Using this equation but assuming low bottom curvature so that we can neglect the second term (•h"/h)O•/Oh a number of three-dimensional wave solutions have been found in trough-like or ridge-like environments. The general form of these solutions is given, respectively, by Eqs. (17) and (21 ). Of the particular solutions derived, some are exact [in the curvilinear system h (r/)]

1833 d. Acoust. Soc. Am., Vol. 93, No. 4, Pt. 1, April 1993 C.H. Harrison: Wave solutions in 3-D environments 1833


and others (labeled "two-term") have assumed low bottom curvature by neglecting terms of order (2h"h-(h')2). For the wedge it turns out that the solution is exact because h"-O and the complete equation [e.g., Eq. (4) including the second term] is exact too.

The conditions of validity of the equation itself are discussed in Appendix A. We find in Sec. 3 of Appendix A that neglect of (•'h"/h)0$/0h in this equation is the main error contributor, and it can be identified with terms that must be neglected for validity of the adiabatic approximation. Although this derivation naturally leads to the neces- sity for h"h being small (i.e., bottom radius of curvature much greater than depth) it is shown in Appendix B that for all reasonably realistic surfaces, meaning those with no overlying trend in depth, this condition is the same as (h')2 being small since the average of (h') 2 is the same as the average of (h"h). There are similar relations between higher-order derivatives.

ACKNOWLEDGMENTS

The author is indebted to Dr. Alvin Robins and Dr.

Mike Ainslie for many fruitful discussions and suggestions.

APPENDIX A: SEPARABLE HELMHOLTZ EQUATION

FOR VARIABLE DEPTH, ISOVELOCITY WATER

1. Derivation

In this Appendix we derive the equation

h2() art+kh/'=o (At)

from first principles, the objective being to find a separable coordinate system that describes a variable depth isovelocity environment. In the process it shows that ignoring the c9/c9• term is equivalent to ignoring coupling between modes in a conventional rectangular normal mode treatment. The behavior of this term therefore throws some

light on the validity of the adiabatic approximation. We start with the Helmholtz equation in the two Car-

tesian coordinates y (horizontal) and z (downwards vertical). In this system we have an arbitrary-shaped bottom profile z=H(y) as shown in Fig. 1 (a). We then choose a separate orthogonal system r/,• on to which we map the seabed. In order to reach the r/,• system it is convenient to set up one other related system p,q. Figure 1 (b) shows the mapped system r/,• in which the sea surface and seabed are both flat at •=0 and 1, respectively, and the verticals are lines of constant r/. The deformed r/,• grid can be seen in Fig. 1 (a) as it would appear superimposed on Cartesian y,z space. The relation between r/,• and y,z can be understood through the intermediate variables p and q. In Fig. 1 (a) q is an ordinary length measured along the constant r/line from surface to bottom. Similarly p is an ordinary length measured from an arbitrary origin along a line of constant •. In order that • should behave like a dimensionless depth we assume

z

(a)

•Y • dqly -dz• • +dqlz (b) dy

-dply (c)

FIG. A1. (a) A construction for determining the relation between p,q and y,z. The intersection of the lines at p,q is blown up in (b) and (c).

d•'=h(p) , (A2) r/•p, (A3)

where h(p) differs from H(y) in that it is the water depth but measured from surface to bottom along a line of constant r/, i.e.,

h(p) h(p) = dq. (A4) dO

The Helmholtz equation can be written in terms of any of the pairs of variables, and by writing out the expansions for cg/Ozly and on/Oylz we obtain

+ + 2

+ on

a 2 an +2aav + ' The terms in square brackets can be determined geometri- cally as shown in Fig. At. In the blowup (b) we see that, in the infinitessimal right angle triangle (where y and z are orthogonal )

I • •=sin a, (A6)

I • =cos a, (A7) y

so that

+ = ]. (AS) Similarly in (c) we see that



'1 •zz = -sin/5, (A10) y

so that

2 0a2 (All)

Now, by definition, we require p and q to be orthogonal so that a=•5, which leads to

aq ap aq ap •y •y+•zz •zz=0. (A12)

From Eq. (A2), remembering that • is independent of p, we have

ag' ag' aq 1 aq ay - aq Oy -- h (p ) Oy

(A13)

and the similar equation for 0/0z converts Eq. (A8) to

'O•') 2 (0•) 2 I 1 ,•yy + •zz -h2(p)-h2(rl) ' (A14) In addition Eq. (A 12) becomes

an an 3y 3y +•zz •zz =0 (A15)

and Eq. (A11) becomes

-+- •zz =1. (A16)

Equations (A 14), (A 15), and (A 16) now determine the contents of the first, fifth, and third square brackets in Eq. (A5).

Referring to Fig. A1 (a) the second derivatives in the second and fourth square brackets are related to the respective curvatures of the œ and q lines, or the rate of change of a and/3. For instance,

3y •yy =•yy (+sina) Oct Oct

= + cos a •yy = + cos a cos/• OR (A 17) and

O(Oq) O Oa •z •zz = •zz (cøs a) = -- sin a •zz = -3- sin a sin •5 (A18)

Combining these we have

a2q O2q Oot 1 Oy2-[- • • -[--• •--pp (AI9)

where pp is the curvature along the p (or •/) direction. The curvature at the seabed (corresponding to negative Oa/Op) is h" for practical cases, and for intermediate depths q [referring to Fig. 1 (a)] it is •h" to a good approximation. Using Eq. (A2), Eq. (A 19) becomes

' ' gh" ay2+ =

Similarly it is easy to show that

02p 02p 0/3 1 Oy,+ Oz = +-. Pq

(A20)

(A21)

From Fig. A1 we see that in practical cases a line of constant p is virtually an arc of a circle. Since p is assumed orthogonal to q it is easy to show that the curvature is h'/h (corresponding to negative OB/Oq) so that

029/ 029/ h' 0y 2 +•= +•-. (A22)

Both Eqs. (A20) and (A22) are exact if the lines of constant p are truly arcs of circles, as is the case for a wedge.

Substituting Eqs. (A14)-(A16) and (A20) and (A22) into Eq. (AS) we have

0 2 0 2 1 0 2 •'h"O 0 2 0y2+0z2--h2(•/) 0•2-h(•/) 0•-5-•2 -5-••

h' 0

h(•l) On (A23)

and the Helmholtz equation is

1 O 2•, gh "b O 2•, h'&b h2 a•2- • O•+•+T•+k2•-O. (A24)

Although the approximate values for pp and pq [Eqs. (A20) and (A22) ] on which this equation relies are intu- itively reasonable it is difficult to write a general evaluation of what precisely has been rejected. This is because the degree of approximation depends on the detailed transformed coordinate system. As a poorman's alternative we can compare the new equation with some known separable systems, where they exist.

First we take the wedge, and inserting h(•)=y• into Eq. (A24) we obtain

1 O 2• O 2• 1 O• •n 2 0•2 +•+• •+k2•--0. (A25)

This can be recognized as the exact equation in polar coordinates r,O with r• and 0• y•. This is reassuring, but no sunrise since our approximate curvatures, pe and pe happen to be exactly correct in this case.

Second, we take the ellipse, and inserting h(•)•h0( 1 _•2/•)•/2 with, for shorthand, t•/•o and s•h•o we obtain

0 2• s • •2•

The Helmholtz equation for elliptic cylinder coordinates (Abramowitz and Stegun, 1970, p. 722, with the substitution t•cos • and s•sinh u) can be written as

-+-A [$2-3-( 1- t 2) ] •b=0, (A27)



where A is a constant. Again we see that the t (or r/) derivatives in the equation are correct and the errors in the coefficients of the s (or •) derivatives are correct to first order in s and t (i.e., small but not necessarily negligible slope). However, note that the second term is already smaller than the first by a factor of order bottom slope. Note also that the discrepancy here is not entirely due to any problems with the accuracy of Eq. (A24); in addition there is the fact that the exact solution is for the Cartesian, truly vertical depth H(y)=Ho(1--y2/yo2) 1/2 rather than h (,/), which is measured down the curved constant-v/lines.

Similar analysis with similar results is possible for the hyperbolic ridge. The exact solution is the same as Eq. (A27) but now with s and t interchanged. Insertion of h=h0(1-•-T]2/T]02) 1/2 into Eq. (A24) results in

a2•p s a•p a2•p a•p as2-1+••ss+ (1+ t2) •-•+t •-+k2•/02(1 +t2)•P =0.

(A28)

2. Separable solutions

Writing Eq. (A24) as

a21p •h.halp h2021p . 01p 0;2- •-•+ •--•2+n'h •-•+h2k2fi=0 (A29) it is clear that this is separable if the second term is negligible. The condition for this to be true will be discussed shortly. When this is so we can write

•p=Z(•) Y(r/) (A30)

and we have

a2z

a• 2 + v2z=o (A31 ) with the solution

Z=A sin v•, (A32)

where A and v are constants, and we have assumed pressure release boundary conditions implying that v is the dimensionless vertical wave number given by

v=nrc. (A33)

In addition we have

a2 y aY

h 2 •--•2+h'h •--4+ (h2k2-v 2) Y=0 (A34) and writing this as

c92y h'cgY (2 v2) 0,/2+•-•-•+ k --• Y=0 (A35) it is clear that we have, in effect, a position-dependent wave number (k2-¾2/h2(•l))l/2 and it is this that controls horizontal refraction. There is a minor effect from the middle

term, but this can be eliminated by making the substitution it= Yh 1/2 so that

Or/2+ --•+ •7--2 •- u=0. (A36)

(a)

(b)

FIG. A2. (a) Zig-zag rays obeying a ray invariant in isovelocity water in real space. In the mapped system (b) the effect is as if there were a horizontal variation of sound speed. Thus both upgoing and downgoing rays bend towards the direction of increasing water depth.

The residual in square brackets is now extremely small being of order bottom slope squared or curvature as compared with the square of vertical wave number. It is, in fact, proportional to the curvature of h 1/2 rather than the curvature of h.

3. Condition for separability

The curvature term (•h"/h)d•p/cg• in Eq. (A24) pre- vents the equation from being separable in general (never- theless see the comments on the exact solution for an el-

lipse) so the condition for its being negligible is of interest. Consider rays that obey an invariant. In real space [Fig. A2(a) ] they zig zag between boundaries becoming steeper in shallow water. In the mapped space [Fig. A2(b)] the same rays must still hit the boundaries but they are allowed to curve [because of the effective horizontal velocity profile (k2--v2/h2(•])) 1/2] in order to shorten their cycle distance in shallow water. As the ray angle is reduced [Fig. A3 (a)] it will be possible for rays to graze the humps and miss troughs altogether; at this point ray invariance fails. The equivalent of this in the mapped system [Fig. A3(b)] is that rays refract upwards and turn without hitting the boundary. This refraction is caused by the term (•h"/

To demonstrate this imagine a number of horizontal rays at different depths in real space, Fig. A4(a). In mapped space, Fig. A4(b), they have increasing curvature with depth that tends to focus rays near the surface. This behavior is exactly the same as that given by a semi- parabolic refracting duct with profile

k2(z) =k•(1-a2z 2) (A37) and a2=h"/h. The wave number at the surface is kl and at the seabed is

•=k•(1-h"h). (A38) The condition we seek is equivalent to the condition that we can ignore refraction and treat the parabolic duct as an



(a)

(b)

FIG. A3. (a) An example of a low angle ray in real space that does not obey an invariant. In the mapped space (b), remembering that the vertical coordinate is normalized depth, it is clear that there must be some residual refraction from a vertical gradient in addition to the effects shown in Fig. A2. This refraction is identified with the term (•'h"/h)cg•p/

isovelocity one. We propose both a loose and a stringent criterion.

(1) Loose criterion: Rays must be steep enough to reflect. Mathematically this condition is that

k• cos 0 < k 2 , (A39)

which using Eq. (A38) reduces to

sin 0> (h"h) •/•. (A40)

(2) Stringent criterion' All modes must be in the reflecting region. The eigenvalue of the first mode in a semi- parabolic duct (actually the second mode in a full parabolic duct) is

(a)

(b)

FIG. A4. A demonstration that parallel horizontal rays in isovelocity water in the real system (a) behave in the mapped system (b) like the focused rays in a semi-parabolic refracting duct.

K•---k•--3ak]=k•(1-3(h"/h) ]/2/k]). (A41) So the condition is'that

K] < k2. (A42)

Combining (A41) and (A38) we have

( h"h ) •/2hk• < 3.

Using the rough relation hk• sin O=nrr this becomes

sin O/n> (h"h)•/2(rr/3). (A43)

These two conditions are directly analogous to the condition for validity of ray invariants (loose) and the Milder criterion for the validity of the adiabatic approximation (stringent) (Milder, 1969; Harrison and Ainslie, 1991 ). The stringency appears in the additional n in Eq. (A43) [cf. Eq. (A40)]. The resemblance is reinforced by noting that there is a rough relation between bottom curvature and bottom slope for all well-behaved surfaces, namely (see Appendix B),

h"h•(h') 2. (A44)

Inserting this in Eqs. (A40) and (A43) makes them identical with the criteria in Harrison and Ainslie (1991 ) bar factors of order unity. Another way of stating the stringent criterion [Eq. (A43)] is that the radius of curvature of the seabed must exceed h(h/2•)2.

The conclusion is that the second term in Eq. (A1) can be dropped under conditions virtually identical to those that satisfy the adiabatic approximation. This leaves an equation that describes multiple surface and bottom reflections in water of variable depth by treating them as a horizontal refraction phenomenon.

APPENDIX B: A RELATION BETWEEN DERIVATIVES FOR WELL-BEHAVED SURFACES

Criteria for the breakdown of ray invariance, the adiabatic approximation and the separability considered in this paper have been quoted in Appendix A as depending on various derivatives of the reflecting surface H(y), e.g., H', H", H'". This Appendix shows that there is a hint of "red herring" here since the average values are related.

Example 1: Curvature and slope

Write the spatial mean of HH" as

(B1)

with a, b constant. Integrating by parts we have

a-- (H') 2 dy. (B2)

Thus, provided H' (or H) tends to zero at the ends of the average (a,b) we have the relation

(HH"} = -- ((H')2}. (B3)

More generally we can use exactly the same procedure starting with the product of higher-order derivatives. The



result is a rule of invariance of the total number of primes N on each side of the equation provided the highest-order- but-one derivative in question is zero at the ends,

(HmH N-m) = (-- 1 )n--m (HnHN-n), (B4)

for all n, m <N. The alternation of signs can be understood most easily in the first example. It is caused by the XH bias that weights negative curvature highly but discounts positive curvature.

APPENDIX C: NORMALIZATION FOR DISCRETE AND CONTINUOUS SPECTRA

1. Weighting in •/

Equation (12) in the main text (a "three-term equation") can be written in the form of the Sturm-Liouville equation as

drl p(rl) • + [q(r/) +A,r(rl) ]4=0, (C1) i.e.,

2

dr/ h(v)• + -h(v•+ (V) u= Clearly p=r•h, q=-v•/h, and X=K•(=k2--K}). It is well known (see Morse and Feshbach, 1953, p. 720) that the weighting factor in •, labeled w(•) in the main text, can be identified as r(•) in Eq. (C1). In the case of Eq. (12) this is h(•). This is easily seen by writing Eq. (C2) for Un, multiplying by urn, subtracting the same with n, m swapped, and integrating in •. If we make use of the transformation equation (8) in the main text and its associated approximations to obtain a "two-term" equation

d2v ( V2n • the equivalent of Eq. (C2) is now

2

d• 1X + -h2(•) So the weighting is w= 1.

2. Almost continuous spectra: Weighting in K

If the spectrum is dense but still strictly discrete we still have

• q•( Kn,•l ) q•( Km,•l ) W( •l )d•l=Zl•nm . (C5) Therefore we can write a sum over rn as

• f q•(gn'rl)q•(gm'rl)w(rl)drl m

=A= f [4(Kn, r/)]2w(r/)dr/. (C6) For very large m we write Em--} f dm and Kn=K,

ff dm qb ( K, rl ) qb ( K', rl ) w ( rl ) dr I •-• dK= A, and this can be written as

(C7)

f f 4(K, rl)4(K',rl)w(rl)ws:(K)drl dK= 1, (C8) where w•: is the desired K weighting that is clearly given by

dm dm(f )-1 w•:(K) :•A-1 :• [•b(K,r/) ] 2w(r/)dr/ . (C9)

In some cases this is a viable technique for calculating w•:(K). For instance, for the WKB method it is easy to show that w•:(K)= 2K/rr. It is often possible to calculate w•: from the asymptotic form of special functions. An even simpler technique for continuous spectra is to "calibrate" the special function's asymptotic solution against the WKB solution for the same differential equation. Surprisingly, this is an exact method because the integral in r/extends to infinity when the spectrum is continuous so that in the limit the asymptotic solution occupies a higher and higher proportion of the entire range of r/and is, in effect, exact everywhere. This method was used for the modified Mathieu functions and the parabolic cylinder functions. It can also be used for Bessel functions and Airy functions, but for these there are alternatives.

Where the solution U(K,r/) can be written as U(f(K, rl)), wheie f is a simple function, (e.g., Kr/ for Bessel functions or AK 2-- Bz for Airy functions) it is possible, by change of variables, to relate the integral in r/,

f 6(K--K') U(K'rl)U(K"rl)w(rl)drl= w•:(K') ' (C10) to the integral in K,

f U(K, rl) U(K, rl')Ws:(K)drl-- (C11 ) w(v) '

and thereby calculate ws:(K). As an example we take the Airy equation

d2U

dt 2 -tU=O, (C12) which we may regard as a dimensionless version of either the equation for a linear velocity profile

k2(z) =ko2(1-az) (C13)

leading to

d2U

dz 2 + (ko2-K2r-ko2az) U=0 (C14) with

t= (zako 2- ( ko2- K• 2 ) )/ ( ak02 ) 2/3 (C15)

or alternatively the "two-term" equation for a "curved wedge"

h-2(rl)=Arl+ B (C16)



leading to

dT]2 nt-(g•-v2nB-v•T])$=O (C17) with

and

t--{flAy2, -- (K2•-- v2, B) )/(Av2,) 2/3. (C18) For clarity we will stick to the r/ interpretation but give results for both at the end. Firstly the weight in r/is w= 1. The normalization integral in r/and K [cf. Eq. (C8)] can then be written in terms of the variables t [Eq. (C18)] and e where

,2 /(Av2n)2/3, e=t'--t: (K•--K 7) at=(Av2n) •/3 dr I, (C19)

de = 2Kn (A v2, ) - 2/3 dKn . The r/part of this integral [cf. Eq. (C10)] is

_' Ai(t)Ai(t+e)dt=M•5(e), (C20) where it is clear that M must be just a number independent of e.

Therefore the complete normalization integral is

f' f_' Ai(t)Ai(t+e)dtde=M, (C21) but changing variables to t'= t+e this becomes

Ai (t) Ai ( t' ) dt dt' = Ai (t) dt 2

(C22)

This integral is well known and the solution is M= 1. Changing variables back to r/and Kv we find

wK( K n) = 2Kv/ (Av2•) •/3. (C23) In the velocity profile case the solution would have been

wg( Kr) = -- 2Kr/ ( akg ) •/3. (C24) Morse and Feshbach ( 1953, p. 764) imply that by Fourier transform- type symmetry generally one can assume that the dimensionless forms of w(r/) and wit(K) must be the same function. The examples in this paper, and in particular the Airy function, for which w(,/)= 1 and wit(K) cr K demonstrate that this is usually not true. Even if we take the eigenvalue to be K 2 rather than K the Bessel function, which would otherwise have w(r/)=r/ and wir(K)=K, would disobey the rule.

3. Continuous spectra ,

The derivation in the main text after Eq. (11) and before Eq. (15) assumed an expansion of the function G in eigenfunctions. However, this was not a necessary step. An alternative is to set up a transform with an, as yet unknown, kernel U(K•,•q),

A(K)-- f U(K, rl)G(rl)w(rl)drl. (C25) Equation (11 ) can be written as

2

han + a 4rr

Multiply both sides by U(K, rl)w(r I) and integrate in r/ [remembering that w(r/) =h (r/)]. The rhs is --4rrsin(Vn•s)U(K, rls) and the lbs after integrating by parts and assuming that [h(G dU/dr I- U dG/drl) ] evalu- ated at the integral limits (the boundaries) is zero, is

f G •--• K•--h2jUh]dr I. Now if the quantity in square brackets is zero

2

drt h + •-• Uh=O or alternatively

drt2 + •- •-•-• + K•- h2 ] U=0, (C26) then the lbs can be written in terms of the transform A (K) [Eq. (C25) 1, and

4rr sin (v•) U(K•,r h) A (K•) = -- ( 2 2 ß

If the U(K,r/) functions are normalized by

we can then return to G by multiplying both sides by U(K•,r/') X w•:(K•) and integrating in K•:

f sin(v•0 U(K•,rh) U(K•,rl)wg(K•)dK • • ( r I ) -- --4rr ( k 2 2 2 ß (c29)

Finally taking an inverse Fourier transform defined by Eq. (10b) in the main text we arrive at an equation similar to Eq. ( 16):

F(•) =2rri

X f sin(v•s) U(Kv,rls) U(Kv,rl)w•:(Kv)ei•:• dKv (C30)

where K•=k2--K•. Inverting the sine transform then results in Eq. (21) of the main text.

Abramowitz, M., and Stegun, I. A. (1970). Handbook of Mathematical Functions (Dover, New York).

Brekhovskikh, L., and Lysanov, Yu. (1982). Fundamentals of Ocean Acoustics (Springer-Verlag, New York).



Buckingham, M. J. (1983). "Acoustic propagation in a wedge-shaped ocean," in Acoustics and the Sea-Be& edited by N. G. Pace (Proc. Inst. of Acoust., Bath U.P., Bath, UK).

Buckingham, M. J. (1987). "Theory of three-dimensional acoustic propagation in a wedgelike ocean with a penetrable bottom," J. Acoust. Soc. Am. 82, 198-210.

Buckingham, M. J. (1986). "Theory of acoustic propagation around a conical seamount," J. Acoust. Soc. Am. 80, 265-277.

Buckingham, M. J. (1989). "Theory of acoustic radiation in corners with homogeneous and mixed perfectly reflecting boundaries," J. Acoust. Soc. Am. 86, 2273-2291.

Deavenport, R. L. (1966). "A normal mode theory of an underwater acoustic duct by means of Green's function," Radio Sci. I (6), 709-724.

De Santo, J. A. (1979). Ocean Acoustics (Springer-Verlag, New York). Gradshteyn, I. S., and Ryzhik, I. M. (1980). Table of Integrals, Series,

and Products (Academic, New York). Harrison, C. H. (1977). "Three-dimensional ray paths in basins, troughs,

and near seamounts by use of ray invariants," J. Acoust. Soc. Am. 62, 1382-1388.

Harrison, C. H. (1979). "Acoustic shadow zones in the horizontal plane," J. Acoust. Soc. Am. 65, 56-61.

Harrison, C. H. (1991). "An analytical formula for intensity in a wedge- shaped ocean," YARD Technical Report No. M540/TR/11 (May 1991).

Harrison, C. H., and Ainslie, M. A. (1991). "Range-dependent eriviron- ments in propagation modelling," Proc. Inst. Acoust. 13, Pt. 3, 66-73.

Harrison, C. H. (1993a). "An analytical intensity in a wedge-shaped ocean," in Proceedings of the 3rd (1991) IMACS Symposium on Com- putational Acoustics, edited by D. Lee (Elsevier, New York).

Harrison, C. H. (1993b). "Intensity from 3-D rays," to be submitted to J. Acoust. Soc. Am.

Milder, D. M. (1969). "Ray and wave invariants for SOFAR channel propagation", J. Acoust. Soc. Am. 46, 1259-1263.

Morse, P.M., and Feshbach, H. (1953). Methods of Theoretical Physics (McGraw-Hill, New York).

Primack, H., and Gilbert, K. E. (1991). "A two-dimensional downslope propagation model based on coupled wedge modes," J. Acoust. Soc. Am. 90, 3254-3262.

Weinberg, H., and Burridge, R. (1974). "Horizontal ray theory for ocean acoustics," J. Acoust. Soc. Am. 55, 63-79.



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Wave solutions in three-dimensional ocean environments