Internal symmetry in acoustical ray theory

Wave Motion 43 (2006) 508–516

www.elsevier.com/locate/wavemoti

Internal symmetry in acoustical ray theory q

David R. Bergman *

Department of Physics, Saint Peter’s College, Jersey City, NJ 07306, USA

Received 4 July 2005; received in revised form 27 March 2006; accepted 4 April 2006

Abstract

A connection between acoustic ray theory and differential geometry suggests the existence of a local symmetry withinray theory. The local symmetry, in this case conformal symmetry, is reminiscent of that encountered in the study of elec-tromagnetic field theory, or more generally gauge theory. Acousticians can take advantage of this symmetry by choosing agauge that best suits a given problem. In this paper, the symmetry is discussed in the most general context and the trans-formations for both ray theory and paraxial ray theory are given. When applied to the paraxial equations this transfor-mation alters the stability parameters of the system. For low dimensional problems in ray theory, such as thoseincluding depth and range dependence in the local sound speed, the stability parameters become a property of the medium.Specific applications to layered media are presented and their consequences discussed.� 2006 Elsevier B.V. All rights reserved.

PACS: 43.20.+g; 42.15.Dp; 43.30.+m; 43.28.+h

Keywords: Acoustics; Ray theory; Differential geometry

1. Introduction

A complete ray theoretic description of the acoustic field in a generic fluid environment, not including caus-tic correction, may be developed using the tools of differential geometry in which the rays are null geodesics ofan effective metric space [1] and the amplitude determined by the cross sectional area of a ray tube calculatedby a set of geodesic deviation vectors [2]. When cast in the language of differential geometry the unique natureof these equations alludes to an internal symmetry common to the study of null hypersurfaces. This symmetryallows one to pick and choose from a continuum of equivalent representations of ray theory in which the formand interpretation of all equations is preserved.

A common occurrence of an internal symmetry in a field theory is the U(1) gauge symmetry, or phaseinvariance, of Fermions interacting with the electromagnetic field which is expressed by saying that the

0165-2125/$ - see front matter � 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.wavemoti.2006.04.001

q Parts of this work were completed during a 2004 summer faculty fellowship hosted at the Naval Research Laboratory in WashingtonDC and funded by the ASEE.

* Tel.: +1 6092759586; fax: +1 2019159191.E-mail address: [email protected].

mailto:[email protected]

D.R. Bergman / Wave Motion 43 (2006) 508–516 509

Dirac–Maxwell equations are unchanged by the transformation1 [3] Al! Al + olf, W! eief/⁄cW. Conse-quently this fact is used to simplify problems with an appropriate choice of gauge: Lorentz2 (olAl = 0), Cou-lomb ðr

*

�A ¼ 0Þ, radiation (A0 = 0), etc. A conformal transformation of space-time is a local scaling ofdistance and time measurements and is expressed by the transformation glm! f (x)glm, where g is the metrictensor of space-time and f is an arbitrary scalar function satisfying f(x) > 0, " x. Conformal symmetry arises inthe study of light, or photon trajectories, in curved space-times where one finds that the light cone, or nullhypersurface defined as a surface in which3 ds2 = glm dxldxm = 0, is a conformal invariant [4] (with an appro-priate change of affine parameter). Stated another way the geometry of the light cone is invariant under a con-formal transformation plus an appropriate change of affine parameter.

Although null hypersurfaces are most common in relativity they arise in general through the study of thecharacteristic solutions of a wide class of hyperbolic quasi-linear partial differential equations [5]. Acousticsfalls under this category of equations and acoustic ray theory is known to be equivalent to the theory of nullgeodesics on a pseudo-Riemannian manifold [1,2,6]. Recently, it has been illustrated that a paraxial acousticray tracing procedure of the most general type possible can be derived form the geodesic deviation equation[2]. All of the ingredients necessary for a complete description of the acoustic field in the ray approximation:ray coordinates, ray parameter, parallel propagated wavefront basis, Riemann curvature tensor, etc, trans-form in a special way under conformal transformations. In this paper, the complete set of transformationsbetween conformal equivalent representations of acoustic ray theory is presented. The procedure is explicitlyapplied to two-dimensional layered media which is common in the study of atmospheric and underwateracoustics.

2. Conformal transformation

Given a differential manifold, M, endowed with a metric glm (xa), which measures magnitudes of vectors andangles in the tangent spaces of M, Tx (M), a conformal transformation is defined by

1 Gr2 Ein3 Co4 He5 Fo

Slmab ¼

gNewlm ¼ f ðxÞgOld

lm ; ð1Þ

in which f (x) is an arbitrary scalar function. The inverse metric transforms as gNew lm = f�1 (x)gOld lm. This inturn affects the Christoffel symbols and Riemann curvature tensor by4

Clab

New

¼ Clab

Old

þClab; ð2Þ

Rlmab

New

¼ Rlmab

Old

þSlmab: ð3Þ

The coordinates remain unchanged xNew = xOld.5 Henceforth, we refer to a manifold and all its particularsafter the application of this transformation as XM.

The combination of any two conformal transformations f (x)g (x) is also an allowed conformal transforma-tion and since f (x) 5 0 for all x and any f, the inverse transformation 1/f (x) always exists. Furthermore, byconstruction these transformations obey the associative law and commute since multiplication of real numbersis commutative. Hence, the collection {f (x)|f > 0:x, f 2 R} forms an abelian Lie group.

3. The acoustic ray equations

The basic ingredients used here consist of a four-dimensional space time coordinate system (space and timetreated equally), xl, and an affine parameter, k. Acoustic rays are parameterized curves in space-time, xl(k).

eek indices may take the values (0, 1, 2, 3) with 0 reserved for time.stein sum convention is used throughout, a double occurrence of any index, once up and once down, implies summation.ntravariant (covariant) vectors are denoted Al (Al), and are connected by the relation Al = glmA

m.re, we define f(x) = X2 (x) to simplify equations appearing later.r s impl i c i ty , fo l lowing the notat ion of Ref . [4 ] , we define Cl

ab ¼ dlboa ln Xþ dl

aob ln X� gabglmom ln X andDaCl

mb � DbClma þ Cl

akCkmb � Cl

bkCkma, in which Dl is the covariant derivative on M.

510 D.R. Bergman / Wave Motion 43 (2006) 508–516

The four dimensional tangent vector (velocity) is _xlðkÞ ¼ dxl=dk and is constrained by the equationglm _xl _xm ¼ 0.6 These ingredients arise naturally by application of the method of characteristics to the equationsof hydrodynamics. A parallel propagated ray centered basis is needed to construct the geodesic deviationequation. This basis consists of two vectors ea

I , I = 1,2, which satisfy the relations eaI eb

J gab ¼ dIJ , eaI _xbgab ¼ 0.

3.1. Geodesics

The equations of hydrodynamics for an ideal, isentropic fluid consists of Euler’s equation and the equationof continuity given by

6 Un

ot t*þ t

* � r*

t*þq�1r

*

p þr*

U ¼ 0 ð4Þ

otp þ t* � r

*

p þ qc�2r*

� t* ¼ 0; ð5Þ
in which p is the fluid pressure, t
*the fluid velocity, q the fluid density, and where the fluid obeys an equation

of state q (p), with q 0 = c�2 defining the local sound speed. The potential U is due to external forces such asgravity acting on the fluid. Eqs. (4) and (5) are a system of first-order quasi-linear partial differential equations.Applying the method of characteristics to this system leads to the characteristic equation,

ðDuÞ2 � c2r*

u � r*

u ¼ 0 ð6Þ
in which the function u ðx*; tÞ is a surface in space-time on which the initial data of the fields are specified andthe operator D � ot þ t
* � r*

is defined. Defining the four-dimensional gradient of uðx*; tÞ, pl � olu we maywrite Eq. (6) as glmplpm = 0 where the contravariant metric tensor has been introduced (g00 = �1, g0i = �ti

and gij = c2dij � titj). The covariant metric tensor, glm, is defined via the relation glmgmb ¼ db

l.Related to the characteristic equation is a family of curves called bicharacteristics [5] defined by dxl/

dk = glm om u, in which k is a suitably chosen parameter along the bicharacteristic. The change in the quantitypl � olu along these curves is determined by the equation dpl/dk = � pa pb (olgab)/2. Together the set ofequations for the variables xl (k) and pm (k) form the bicharacteristic equations for the system given by Eqs.(4) and (5). These equations may be combined to give a second order non-linear ordinary differential equationfor xl (k) which is known as the geodesic equation for a manifold endowed with the metric glm. Solutions tothis equation subject to the constraint glm _xl _xm ¼ 0 and called null geodesics, curves of zero length in four-di-mensional space-time. Eq. (6) may be multiplied through by any function f(xl) without affecting the charac-teristics. This imposes a change in glm and similarly in the definition of the bicharacteristics. A detailedtreatment of the same problem with the pressure, density and fluid velocity written as a background term plusa perturbation (associated with acoustic vibrations) p = p0 + p1, q = q0 + q1, t

* ¼ w*þ t1

*leads to the same

characteristic equation as before with the full fluid velocity t*

replaced by the background w*

.For convenience we choose to multiply Eq. (6) by an overall factor c�2. With this choice the acoustic rays in

a time-dependent moving fluid medium are identical to the null geodesics of a Pseudo-Riemannian manifoldwith a metric given by

glm ¼�c2 þ t2 �tj

�ti dij

!: ð7Þ

The Cartesian coordinates of the ray and travel time are given by the geodesic equation,

€xl þ Clab _xa _xb ¼ 0; ð8Þ

subject to the constraint glm _xl _xm ¼ 0. Applying Eq. (2) to Eq. (8) leads to

€xl þ Clab _xa _xb þ 2 _xl _xaoa ln X ¼ 0 ð8aÞ

less stated otherwise _q � dq=dk for any quantity, q.


for the geodesic coordinates. The last term can be absorbed into a redefinition of the affine parameter,dkNew = aX2dkOld (a is an arbitrary constant). Hence, Eq. (8) is invariant under the combined transformation(glm,dk)! f (x)(glm,dk).7

3.2. Parallel propagated ray centered basis

A ray centered parallel propagated basis, heaI i, is need to construct the paraxial equations. This basis obeys

the equation

7 Th

_elI þ Cl

ab _xaebI ¼ 0: ð9Þ

In general as this basis is parallel propagated along the ray in four-dimensional space-time the basis vectorswill acquire a time component. A more natural basis for the construction of the paraxial procedure, referred toas an auxiliary basis, is given in reference [2] by ~el

I � elI � e0

I dxl=dt. The auxiliary basis is purely spatial everywhere along the ray path (i.e. it never acquires a time component) and by application of Eq. (7) one can provethat h~eI ; n̂i forms an orthonormal basis along the ray in three-dimensional space, where the unit normal to thewave-front surface, n̂ is defined by cn̂þ t

* ¼ d x*=dt. This basis defines a plane along the ray in three-dimen-

sional space, which is always tangent to the wave-front. A first order differential equation for the auxiliarybasis may be derived form Eqs. (8) and (9).

d~ekI

dkþ dt

dkXki~ei

I ¼ 0; ð10Þ

in which 2Xki ¼ ejkiðr*

� t*þ s

*�n̂þ 2r*

c� n̂Þ, sk = njSjk and Sjk = ojtk + oktj.The basis defined by Eq. (9) is used in the definition of the geodesic deviation equation while projecting a

particular solution of the deviation equation into the basis defined by Eq. (10) gives a small displacement awayfrom the acoustic ray within the wave-front (more precisely within a plane tangent to the wave-front relative toa point defined by the intersection of the ray with the wave-front).

Upon M! XM the basis eaI transforms as ea

I ! X�1eaI þ X�2fI _xa where ea

I is the basis on M and fI obeys theequation _f I ¼ �ea

I oaX along the null geodesic. The wave-front basis ~eaI obeys a much simpler transformation

rule, ~eaI ! X�1~ea

I .

3.3. Paraxial rays, geodesic deviation

The acoustic intensity is approximated in ray theory by the geodesic deviation vector, which represents afirst order perturbation of the ray coordinates, Yl = dxl, constrained by Y l _xmglm. The geodesic deviation equa-tion is

€Y I þ KIJ Y J ¼ 0; ð11Þ
in which KIJ � Rlamb _xa _xbel
I emJ is the sectional curvature and Y I ¼ ea

I gabY b. The area of a cross section of the raytube is obtained from the triple product ðY � �Y Þ � t̂, in which Y and �Y are two deviation vectors with indepen-dent sets of initial conditions and t̂ is a unit tangent vector along the ray in three-dimensional space. The inten-sity is then proportional to the inverse area along the ray [7,8].

The sectional curvature determines the stability of a neighboring ray about a suitably chosen fiduciary ray.The conformal transformation alters the components of K in such a way that rays that appear stable in onerepresentation may be unstable in another and vice versa. Since Yl = dxl, it will not change after the confor-mal transformation. The field YI however will transform into XYI. The additional term contributing to ea

I dueto a conformal transformation has no affect on the behavior of YI. Furthermore the sectional curvature maybe rewritten as KIJ � RlambK

laI Kmb

J , in which 2KlaI � _xlea

I � _xaelI . It is easy to show that Kla

I ! X�3KlaI under the

action of a conformal transformation. The invariance of Eq. (11) may be demonstrated by inserting all thechanges in k, ea

I , Rlamb, KlaI , _xa and YI into it and some algebra.

is invariance only occurs for null geodesics.

Table 1A complete list of all relevant quantities used in a ray theoretic construction of the acoustic field

Manifold M XM

Coordinates xa xa

Affine parameter k X2kTangent vector _xa X�2 _xa

Ray centered basis eaI X�1ea

I þ X�2fI _xa

Wave-front basis ~eaI X�1~ea

IDeviation Ya, YI Ya, XYI

Metric glm X2glm

Connection Clab Cl

ab þ Clab

Sectional curvature KIJ X�4(KIJ + LIJ)*

From left to right the first column gives the name of each quantity, the second column the symbol used to define that quantity on M andthe third column gives an expression for the transformed quantity appearing on XM in terms of the associated quantity on M (i.e. columnthree provides a formula for computing all XM quantities given their values on M and the transformation X).

* LIJ ¼ Slamb _xl _xmeaI eb

J .


3.4. Summary

A summary of the above results is presented for completeness and convenience in Table 1 bellow. All quan-tities belonging to XM are expressed in terms of those on M.

4. Applications

The collection of transformations outlined in Sections 3.1–3.3 take the geometry of the acoustic field in onerepresentation, M, into the appropriate geometry in a second representation, XM. In this section the confor-mal transformation is applied to a couple of well known systems and the results interpreted.

4.1. Moving layered media

In the study of oceanic (atmospheric) sound one encounters problems where the local sound speed and cur-rent (wind) can be assumed to be functions only of depth (height), given by c (z) and wðzÞ̂i respectively. Addi-tionally there exists a subset of rays which remain in a two-dimensional plane throughout their evolution,namely those for which _y ¼ 0. In this case, the acoustic propagation may be viewed as a purely two-dimen-sional problem. When w (z) = 0 rays are torsion free and their osculating plane contains the z-axis, hencethe entire acoustic ray skeleton for a point source in three dimensions can be constructed by finding the raysin a single vertical plane and rotating this picture about the z-axis. For the general case described in the firstparagraph of this section this simplifying trick is not possible and only the x–z plane contains torsion free rays.In the remainder of the section we shall consider only this two-dimensional subset of rays as a system. Due tothe existence of cyclic variables the equations for the ray coordinates and travel time may be reduced to firstorder. This form of the ray equation has appeared numerous times in the literature in a variety of forms [9,10].The resulting first order equations for travel time, range and depth8 presented here were derived using anothergeometric symmetry principle, isometry [11].

8 Thintegra

dtdk¼ k0c�2ð1� awÞ ð12Þ

dxdk¼ k0ðaþ c�2wð1� awÞÞ ð13Þ

e ray parameter a = cosh0/(c(z0) + w (z0)cosh0), in which h0 is the initial angle between the wavefront normal and the x axis, and thetion constant, k0, is chosen such that (dt/dk)0 = 1.

9 A rchoose10 He


dzdk

� �2

¼ k20c�2ðð1� waÞ2 þ a2c2Þ ð14Þ

respectively. Combining Eqs. (13) and (14) yields an expression for _s, where s is the arc-length if the ray inthree-dimensional space.

dsdk

� �2

¼ k20c�2ð1þ 2a2c2 þ w2c�2ð1� waÞ2 � w2a2Þ: ð15Þ

One notices immediately that each of Eqs. (12)–(15) represent a possible conformal transformation. Using Eq.(12) transforms the problem to a representation in which travel time is the natural choice of affine parameter9

while Eq. (15) can be used to turn three-dimensional arc-length into an affine parameter (time and arc-lengtheach being good choices as they are strictly increasing along the ray). Eqs. (13) or (14) can be used to turnrange or depth respectively into the affine parameter (although it should be noted that the transformationis only valid when _x 6¼ 0 or _z 6¼ 0, hence when rays with horizontal or vertical turning points exist the trans-formation can only be done on patches of the acoustic ray skeleton). For the simplified two-dimensional prob-lem presented in this section there is only one basis vector and the corresponding deviation vector only has onedegree of freedom. The curvature matrix KIJ has one non-trivial component,

K ¼ ðac�2w00ð1� awÞ � a2c�2ðw0Þ2 � ac�3c0w0ð1� awÞ þ a2c�1c00Þk20; ð16Þ

in which a prime denotes differentiation with respect to z. The solution to Eq. (11) in this case is

Y ¼ k1 _zcZ

_z�3c�2dzþ k2 _zc; ð17Þ

in which k1 and k2 are constants of integration.

4.2. Stationary two-dimensional problems

Problems in underwater acoustics frequently assume that rays travel in a two-dimensional vertical planeand that there is no bulk motion of the fluid, or that the effects of these currents on the acoustic field are neg-ligible. Typically, it suffices to model the environment by a depth-dependent sound speed, c, but in many casesperiodic range dependence is included. When range dependence is turned on rays are no longer torsion freeand the ray system is no longer truly two-dimensional except for one plane of rays which contains the soundspeed gradient. We assume the same conventions in this section as were discussed in Section 4.1.

The metric in Eq. (7) may be written in the compact form glm = �c2 � dij, and the space-time ray velocity is_xl ¼ ðc�2; c�1n̂Þ, in which the purely spatial part, c�1n̂, is the slowness. The ray and deviation equations are

€t þ _t_x* � r

*

ln c2 ¼ 0;€x*þ _t2cr

*

c ¼ 0 ð18Þ€Y þ c�3cggY ¼ 0; ð19Þ

respectively, in which cgg ¼ c2ð _x2czz þ _z2cxx � 2_z _xcxzÞ. In practice, this representation can be difficult to imple-ment as one typically does not know c(g) a priori. Thus full solutions, usually numeric, to Eq. (18) are requiredto make use of Eq. (19). Since the sectional curvature, K, is a quadratic form one can assess whether or notK > 0 or K < 0 by evaluating the conditions czzcxx � c2

xz > 0 and czz > 0 or czz < 0 along the ray. However, whenczzcxx � c2

xz 6 0 no decision can be made.Alternatively, choosing X = c�1 transforms the system into one in which time is an affine parameter, here

we set k0 = 1 for simplicity. In this representation the metric is �1 � c�2dij and the space-time ray velocity is_xl ¼ ð1 cn̂Þ. The rays and deviation are given by10

elation between t and k like that expressed in Eq. (11) is necessary for choosing time as an ‘‘affine parameter’’. One can alwaystime as a ray parameter (including time dependent problems) at the cost of losing the identification of the rays with null geodesics.

re, an over dot refers to differentiation with respect to time.

11 Th12 A g


€x* ¼ �cr

*

cþ _x* _

x* � r

*

ln c2 ð20Þ

€Y þ ðcr2c�r*

c � r*

cÞY ¼ 0: ð21Þ
The sectional curvature appearing in Eq. (21) is independent of
_x*

in contrast to the situation described by Eqs.(18) and (19). Hence ray instability is determined by whether or not the ray passes through regions of themedium where K 6 0 and one may conclude that a ray that remains in a region where K > 0 will be stableand that caustic formation is imminent. Regions of stability or instability can now be thought of as part ofthe environment itself.

4.3. Examples

The transformation is applied to specific problems involving stationary media described by a positiondependent sound speed, c (x, z), and the results interpreted. First consider an environment with a depth-depen-dent sound speed. In the affine representation the sectional curvature is11Kk = c (z0)4a2c�1c00, in which z0 is thesource depth. Stability is governed by the concavity of the sound speed and the ray parameter, a = c (z0)�1

cosh0, in which h0 is the initial launch angle of the ray. Moving to the time parameterization of the same prob-lem leads to Kt = cc00 � c 02. A significant difference in these two results occurs for rays that are launched ver-tically, a = 0. In the first case Kk = 0 indicating divergent behavior in k, while this special feature of verticalrays is absent in the time representation. For a purely depth dependent sound speed Eq. (16) implies range, x,is also a good choice of affine parameter along any non vertical ray resulting in the sectional curvatureKx = c�1c00. The quantity K is equivalent to the Gaussian curvature of a two-dimensional manifold. Wecan interpret the effect of the environment on the ray as being due to an equivalent curvature of the back-ground space. A well known result form the study of differential geometry is that the curvature of a two-di-mensional sphere of radius R is K = R�2 while the curvature of a plane is zero. A space with constant negativecurvature K = �R�2 is called a pseudo-sphere.

For c = c0 cosh(z/L), Kk = c(z0)2 cos2h0L�2, Kt ¼ c20L�2 and Kx = L�2. In all three representations the

geometry along each ray is spherical. The form of Kk indicates a radius of curvature, sec h0L/c(z0), whichdepends on launch angle while Kt (Kx) gives a constant radius of curvature, L/c0 (L), for all rays. This lastresult can be used to make the definite prediction that caustics will form along each and every ray from a com-mon point of origin at exactly the same time and horizontal displacement. This ideal point-like focusing is awell known feature of this sound speed profile [12]. However, the result for Kt also implies that vertical raysencounter conjugate points in the same amount of time whereas the result for Kk suggests that this neveroccurs. The reason for this discrepancy, and the resolution of the paradox, is due to the fact that in the timeparameterized representation these rays are incomplete.12 Using the known solution for the travel time showsthat vertical rays reach infinity in a finite value of time which is less than the period of conjugate point for-mation predicted above.

For a quadratic sound speed, c = c0(1 + (z/L)2) the curvatures are Kk = c (z0)2 cos2h02L�2(1 + (z/L)2)�1

and Kt ¼ c20L�2ð1� 2ðz=LÞ2Þ. Kk > 0 for all z, for all a 5 0 while Kt > 0 only for jzj < L=

ffiffiffi2p

and Kt 6 0 forjzjP L=

ffiffiffi2p

. In the time representation ducted rays may periodically encounter regions of divergence makingit difficult to easily predict the onset of caustics by inspection. The affine representation suggests that causticformation is certain for all rays with a 5 0 since Kk > 0 for all z.

Another common example is illustrated by c = c0(1 + z/L). Here, Kk = 0, Kt ¼ �c20L�2 and Kx = 0. From

this we see that rays diverge linearly in k, and x, and exponentially in t. These few examples illustrate thechange in physical and geometrical interpretation that accompanies the transformation from M to XM.

In the discussion at the end of Section 4.2 it was pointed out that in the time parameterized representationthe curvature is independent of the ray velocity. Consider a simplified model of an oceanic environment withrange- and depth-dependent sound speed of the form c (x,z) � cr(x)cd (z), the Gaussian curvature can be writ-

e subscript on K serves to remind one of which representation is being used.eodesic is said to be complete if it exists for all k 2 (�1,1).


ten as a sum of sectional curvatures due to each conformal factor according to the rule c�2K ¼ c�2r Kr þ c�2

d Kd ,with KI � cIo

2I cI � ðoI cIÞ2, I = r, d.

As an example a periodic range-dependence of the form cr = 1 + e sin2 (kx), with e < 1, and a depth depen-dent sound speed with the property Kd > 0 for all points in the plane.13 For simplicity let cd (z) = c0 cosh(z/L),which produces a space with constant positive curvature. The range-dependent curvature isKr = 2k2e(1 � (2 + e)sin2 (kx)). When the two are combined in a product form the resulting expression forK is derived

13 Th14 On

movin

K ¼ c20cosh2ðz=LÞ2k2eð1� ð2þ eÞ sin2ðkxÞÞ þ c2

0L�2ð1þ e sin2ðkxÞÞ2: ð22Þ
Setting K (x,z) = 0 defines a boundary curve z0 (x) separating space into two regions defined by either K > 0 orK < 0, called convergence and divergence zones for short. The boundary curve z0 (x) depends on the param-eters that define the sound speed (such as c0, L, k, e, etc).
From Eq. (20) the critical points of z0 (x) are found to be at kxc = np/2 for n = integer and cosh2 (z0 (xc)/L) = L�2k�2(1 + e�1)/2 for n = odd, n = even leading to the unphysical condition cosh2(z0) < 0. (Anotherset of critical points may be found when the condition L2k2

6 (1 + e)/(2 + e) is valid, for the purposes of thisdiscussion we assume that this condition is not valid.) The curve z0(x) will have overlapping turning pointswhen kL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ e�1Þ=2

p� R, for kL < R there will always be a horizontal band of positive curvature,

whereas for kL > R this horizontal region disappears leaving in its wake vertical bands of alternating curva-ture. As k is increased up to the value k = L�1R there are fewer stable rays in the system, and once k growsbigger than this value it becomes impossible for rays that continue to travel horizontally to remain stable asthey will begin to diverge from their neighboring rays as soon as they enter the first region with K < 0. Itshould be noted however that in such a situation rays have horizontal turning points as well as vertical turningpoints, hence rays fired up or down could be trapped in a vertical wave guide just as rays fired horizontally inthe deep ocean are trapped in a horizontal wave guide, most research done on ray stability focuses on hori-zontal rays and their behavior under the influence of range dependence. These divergence/convergence zonesmapped in space may be used to judge the stability of any ray, not just those with small launch angles. Theabove analysis would not be possible in the affine parameterized representation.

5. Discussion and conclusions

When solving problems in acoustics, as with any problem, the right choice of coordinates as well as rayparameter can lead to a much simpler problem. Many texts have preferred representations of ray theory whichlead the reader to a comparison with other commonly studied physical systems.14 Applied to the paraxial rayequations these choices lead to the introduction of first order derivatives to the equation. The virtue of theconformal transformation is that all the equations maintain their form and the interpretation of all quantitiesis the same in any representations.

The similarities between acoustic ray theory in a generic fluid medium and the optical ray theory on acurved space-time background are striking and served as an inspiration for this and other work. The differ-ences are just as striking and important to point out. In the limit of weak and slowly varying sound speedand currents the full scalar acoustic field equation is approximated by that of a scalar field propagating ona curved space-time background with an effective metric [2] dependent on c (xl), w

*ðxlÞ and the backgroundfluid density q(xl). The pressure variations then obey DlDlp = 0, where Dl is the covariant derivative onM [4]. It is well known that this equation is not invariant under a conformal transformation of M, unlessdim(M) = 2. On the other hand the full electromagnetic field equations (i.e. Maxwell’s equations) are confor-mally invariant when, and only when dim(M) = 4, see Ref. [4] for details. Therefore, while it seems that Max-well’s equations have additional geometric symmetry in general this is only an approximate symmetry of theacoustic field, or any ray approximation to a hyperbolic quasi-linear PDE.

ese choices are purely for illustration and are not based on physical grounds.e example of this is the Newton parameter, named so because the ray equation takes the form of Newton’s second law for a particleg in the presence of a conservative force.


An historical review of the development of ray theory illustrates the wide variety of approaches taken. Infact the presence of an ignorable factor in the ray system was noticed by R.J. Thompson [9]. It is clear in lightof the results presented here that the author was playing with a conformal symmetry. Lastly, the mapping ofconvergence/divergence zones from the Gaussian curvature provides an intuitive way of judging whether raysin a given environment are likely to be stable or unstable. The author has used this technique extensivelytowards accurately predicting acoustic intensity and caustic formation in ray traces but it is also clear thatsuch information could prove useful in the study of ray chaos [13,14].

Acknowledgements

The author would like to acknowledge the Naval Research Laboratory in Washington DC and the Amer-ican Society for Engineering Education for hosting a summer faculty fellowship in 2004 during which timeparts of this research were completed. A special thanks to Dr. Roger Gauss (NRL) for proof reading themanuscript.

References

[1] R.W. White, Acoustic ray tracing in moving inhomogeneous fluids, J. Acoust. Soc. Am. 53 (6) (1973) 1700–1704.[2] D.R. Bergman, Generalized space-time paraxial acoustic ray tracing, Waves Random Complex Media 15 (4) (2005) 417–435.[3] F. Mandl, G. Shaw, Quantum Field Theory, Revised Edition, John Wiley and Sons, New York, 1984.[4] R.M. Wald, General Relativity, The University of Chicago Press, 1984.[5] R. Courant, D. Hilbert, Methods of Mathematical Physics Volume II, John Wiley and Sons, New York, 1962.[6] W.G. Unruh, Experimental black hole evaporation?, Phys Rev. Lett. 46 (1981) 1351–1353.[7] E.S. Eby, L.T. Einstein, General Spreading – Loss Equation, Letter to the Editor, J. Acoust. Soc. Am. (1965).[8] D. Blokhintzev, The Propagation of Sound in an Inhomogeneous and Moving Medium I, J. Acoust. Soc. Am. 18 (2) (1946) 322–328.[9] R.J. Thompson, Ray theory for an inhomogeneous moving medium, J. Acoust. Soc. Am. 51 (5) (1972), Part 2.

[10] P. Rothwell, Calculation of sound rays in the atmosphere, J. Acoust. Soc. Am. 19 (1) (1945) 205–221.[11] D.R. Bergman, Symmetry and Snell’s law, J. Acoust. Soc. Am. 118 (3) (2005) 1278–1282.[12] I. Tolstoy, C.S. Clay, Ocean Acoustics Theory and Experiment in Underwater Sound, McGraw-Hill, New York, 1966.[13] K.B. Smith, M.G. Brown, F.D. Tappert, Acoustic ray chaos induced by mesoscale ocean structure, J. Acoust. Soc. Am. 91 (4 Pt. 1)

(1991) 1950–1959.[14] F.J. Beron-Vera, F.D. Tappert, Ray stability in weakly range-dependent sound channels, J. Acoust. Soc. Am. 114 (1) (2003) 123–130.

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Internal symmetry in acoustical ray theory