Three-dimensional acoustic-ray tracing in an inhomogeneous anisotropic atmosphere using Hamilton's equations

Three-dimensional acoustic-ray tracing in an inhomogeneous anisotropic atmosphere using Hamilton's equations

C. I. CHESSELL

Department of Supply, Weapons Research Establishment, Salisbury, South Australia

(Received 15 September 1971)

Three-dimensional ray-tracing equations are presented for tracing acoustic rays in an inhomogeneous anisotropic atmosphere which are based on Hamilton's equations for geometrical optics. The equations are converted to a form suitable for direct numerical integration. Simplified ray equations for the special case of horizontal stratification are obtained. Results of ray tracing in an atmosphere with both vertical and angular wind shear are outlined, and the formation of a shadow zone in the downwind direction is demonstrated. SUBJECT CLASSIFICATION: 11.7, 11.9.

INTRODUCTION

Since the pioneering work of Rayleigh • an extensive study has been made of the propagation of sound waves through the earth's atmosphere. The principal factors which cause variations in the speed of sound in a stationary atmosphere have been shown to be variations in the air temperature T and to a lesser extent humidity. The speed of sound c(m/sec) can be shown (e.g., Ref. 2) to be approximately given by

vT(l+O.275e/p)l• c=331'8L • 3'

where e is the partial pressure of water vapor and p the atmospheric pressure. In the earth's atmosphere the mode of propagation is complicated by the presence of vertical and horizontal winds. Barton s was the first to

recognize the anisotropic nature of the propagation under these conditions and to derive an approximate equation describing the path of a sound ray in the presence of temperature and wind refraction. Sub- sequently a number of authors have derived ray equations which generally apply under certain simpli- fying conditions. 4-7

The purpose of this paper is to present ray-tracing equations based on Hamilton's equations for geometrical optics. 8 These equations have been used by Haselgrove ø to derive ray-tracing equations for radio waves in the ionosphere. Since the derivation of these equations has been given in full by Haselgrove, only the results will be quoted here and their application to acoustic propagation discussed.

I. THE ACOUSTIC REFRACTIVE INDEX AND

THE RAY THEORY APPROXIMATION

In the presence of a wind of speed w the acoustic refractive index with respect to a fixed set of axes is a function of the angle between the wave normal and the wind direction, x. The magnitude of refractive index u can be written as (e.g., Ref. 10)

cosx), (2)

where c is the speed of sound with respect to the medium and Co is some reference speed.

The geometrical theory of optics is well known to be a valid approximation to the wave theory, provided changes in the refractive index are small over distances of the order of a wavelength and for durations of the order of a period of the wave motion. In acoustics the situation is analogous. Blokhintzev n has shown that the propagation of sound in a continuous medium can be treated by a geometrical theory even when the medium is in motion, provided the variations in the refractive index are sufficiently small.

II. THE HAMILTONIAN EQUATIONS

Haselgrove ø has derived the ray equations in terms of general curvilinear coordinates. Here the special case of Cartesian coordinates will first be considered.

In the Hamiltonian equations a point on a ray path in three dimensions has three position coordinates xi, x2, xa, and three direction coordinates, pi, P2, pa. At any point (xi,x•.,xa) the pi are the components of a vector p which has the same direction as the wave normal at the point and a length p=(pl•q-p•'q-pa•) « equal to the refractive index u for that direction of the

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C. I. CHESSELL

wave normal. A function G is also required, defined by

G=p/t•, (3)

which is always equal to unity by the above definition of the vector p. With these definitions the Hamiltonian equations for a ray path in Cartesian coordinates take the form

dxi OG --=-- i=1, 2,3, (4) dt

dpi OG .... i=1, 2,3, (5) dt

where the independent variable t is some function of the distance along the ray path. These six first-order differential equations are sufficient to completely determine the ray path. Equations 4 and 5 are Eqs. 29 and 28 of Ref 9. They have also been given by Landau and Lifshitz. 12 The extension to spherical polar coordinates which is required if the curvature of the earth is to be included is given in Ref. 9.

III. THE EQUATIONS IN INTEGRABLE FORM

In order to express the ray equations (Eqs. 4 and 5) directly in terms of the properties of the medium in a form suitable for numerical integration, the partial derivatives of the function G with respect to the xi and pg must be obtained. These may be readily evalu- ated from Eq. 3, resulting in the following six ray equations, written in terms of the usual Cartesian coordinates x, y, and z:

dx t• 0u •/t•:, (6) 5=( (7)

az (8) •=\ opt!

dp•_(_l•O!a, (9) dt -\u/Ox

The partial derivatives of the refractive index can be obtained from Eqs. ! and 2 and the particular meteoro- logical conditions of wind speed and direction, temperature, and humidity. Equations 6-1! are completely

general and enable rays to be traced under any desired variations of the atmospheric parameters in three dimensions, subject only to the limitations of ray theory, as outlined in Sec. I. If experimentally observed atmospheric conditions are used, then values of wind speed and direction, temperature and humidity, and their spatial derivatives between the given data points must be calculated by interpolation procedures. Several other useful quantities such as time of flight and path length may be calculated simultaneously with the ray equations. Differential equations for these variables are readily obtainable in terms of the independent variable t, which in the formulation of the ray equations presented here, represents the product of the reference speed Co and the time of flight of a wavefront along a ray.

IV. SPECIAL CASE: HORIZONTALLY STRATI-

FIED ATMOSPHEREs PROPAGATION IN A SINGLE PLANE

Consider the special case in which the wind speed and the speed of sound are functions of height only and in which the wind direction and the initial wave normal direction are coplanar. Thus t• is independent of the horizontal coordinates x and y and if the ray is propagated initially in the x-z plane, then pu is initially 0 and by Eq. 10, it remains zero. Further, if a new variable 0 is introduced, defined as the angle between the wave normal and the vertical, then

px = t• sin0, p z = t• cos& (12)

Thus t• may be written as t•(z,O) since t• depends only upon px and pz in the combinations px/p=sinO and p•/p = cos& Then

op• O0 op• u O0 and

O• O• O0 sinO O•

opz 00 opz t• 00

Since ot•/ox=O, Eq. 9 may be written as

(13)

(14)

dp• dO dt• •-=t• cosO--+sinO--=O (15) dt dt dt

(p• constant along the ray is simply Snell's law), so that

dp• dO d• • =--t• sin0---}-cos0-- dt dt dt

=--t• dO/sin& (16) dr/

Equations 13, 14, and 16 are then substituted in the first, third, and last ray equations (Eqs. 6, 8, and 11)

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RAYS IN INHOMOGENEOUS ANISOTROPIC ATMOSPHERE

and lead to

dt = sin0--cos0• , (17)

dz 1 t• cos0q-sin0•), 18) 7,=7( dO Ou / --= --sin0-- td. (19) dt Oz

These equations fully define the ray path for a horizontally stratified atmosphere for the case in which the wind and wave normal directions are coplanar. This assumption is often a reasonable approximation to the atmosphere, so that the three equations (Eqs. 17-19) may be integrated numerically rather than the full set of equations (Eqs. 6-11).

An expression for the ray curvature for this case can be readily obtained using Eqs. 17-19 and the relation

for the refractive index which can be written as

•=Co/[-c(z)-l-w(z) sin0•. (20)

Substituting this expression into Eqs. 17-19 and elimi- nating the independent variable t, the following rela- tions are obtained-

dO (tanO•dc_[ sinS0 dw (21) dz \ c /dz c cosO dz

dx w --= tan0nt-• dz c cos0

= tanqO, (22)

where qO is the angle between the ray direction and the vertical (see Fig. 1). Using Eqs. 21 and 22 and the normal expression for the curvature of a function in two dimensions, K, the following relation for the curvature can be obtained'

(dc/dz)[-sinO -- (w/c) cos20• q- (dw/dz)[-1 q- (w/c) sin303

c[- 1-]-(2w/c) sinO+(w/c)•'• • (23)

This equation illustrates the important result that the ray curvature depends upon the wind speed w, even when the wind gradient dw/dz is zero. This relation has also been obtained by Gutenberg 4 from a con- sideration of the geometry of the ray path.

V. FORMATION OF A SHADOW ZONE IN CON-

DITIONS OF VERTICAL AND ANGULAR

WIND SHEAR

In this section the ray-tracing equations (Eqs. 6-11) will be used to trace rays in an atmosphere in which the wind speed and direction are functions of height.

The possibility of the formation of a shadow zone in the presence of a wind shear is well known. The case of a horizontally stratified atmosphere with a positive wind gradient and with the wind in the plane of incidence has been studied by a number of authors, e.g., Ref. 5. Solutions for the case in which the wind direction is also a function of height can be readily obtained using the method of ray tracing outlined here. The presence of angular wind shear is often observed in the boundary layer of the atmosphere and has been theoretically described by the so-called Ekman spiral. 2 If this angular shear is sufficiently severe it will give rise to significant modifications of the shadow zone.

Consider the case in which the wind is horizontal and

its speed is given as a function of height by

w (z) = w0[ 1 - exp ( - k •z) •, (24)

where w0 and k• are constants. Further, if qO(z) is the angle the wind direction makes with the x axis and if it is initially in the positive x direction at ground level,

then it is further assumed that this rotation is given by

qO (z) = qO0[- 1 - exp ( -- k2z) •, (25)

where qO0 and k•. are constants. For a ray initially launched in the x-z plane with the wave normal at an angle 00 to the z axis the expression for the refractive index (Eq. 2) becomes

la= 1/[-l+(w/co) sin0 cost-I, (26)

if the speed of sound is assumed constant at Co throughout the atmosphere (i.e., temperature and humidity are constant).

The occurrence of a new shadow zone for this case is

readily seen by examination of the Poeverlein construction, •3 shown diagrammatically in Fig. 2 for the conditions assumed here. The cross sections of the

Z I WAVE NORMA L RAY

Fro. 1. Diagram showing the wave normal ray and wind direction in the x-y plane and defining the angles used.

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C. I. CHESSELL

Z2

• S2•

FIG. 2. Diagram of the refractive index surfaces at various levels and the Poeverlein construction for two initial angles of incidence; sin00--S• and the critical case, sin00=S.,.. The speed of sound is constant and the wind speed and direction vary with height as given by Eqs. 24 and 25.

refractive index surfaces in the x-z plane for a series of increasing heights z•-z5 are shown superimposed upon each other with a common origin and oriented correctly with respect to the axes. The height z• corresponds to ground level where w=0 so the refractive index is a circle of unit radius. The surfaces for higher levels are

600

I III//

o -2-5 0 2.5 5.0 7.5 I0.0 10.5

X (Kms)

Fro. 3. Projection of the ray paths on the x-z plane for an atmosphere with a constant speed of sound and with the wind speed and direction as given by Eqs. 24 and 25. The initial wave normal direction is indicated on each ray path.

ellipses with the origin as one focus. Lines joining the origin to the refractive index surfaces, e.g., OA, represent possible directions of the wave normal, the ray direction in each case being given by the normal to the refractive index surface at the point of intersection, e.g., at A. It can be shown that points of intersection of a vertical line (e.g., AB) with the refractive index surface represent possible directions of the wave normal for a wave with an initial incidence angle of 00, where

sin00=S, (27)

and S is the perpendicular distance of the line from the origin (S•, for the line AB in Fig. 2). The existence of a reflection level where the ray becomes horizontal can thus be readily determined. For example, a shadow zone would be expected for rays propagated against the wind (in the negative x direction) since no refractive index surface cuts the axis inside the unit circle corre-

sponding to ground level in Fig. 2. For rays propagated with the wind, those with initial

take-off angles (00) near -}•r will be refracted back to the ground. For example, in Fig. 2 the vertical line AB represents the case of sin00=S•, the ray being reflected at the level z'-z2. Note that in the case considered here, although the wave normals lie in the x-z plane, the ray directions do not. As 00 is reduced the condition repre- sented by the vertical line CD is reached; for angles less than sin00=S2, the ray will not be refracted back to the ground but will proceed upwards indefinitely. Thus a shadow zone will be formed in the initial ground-

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RAYS IN INHOMOGENEOUS ANISOTROPIC ATMOSPHERE

level wind direction. The critical height zc, rotation angle 4•c, and the take-off angle 0• corresponding to the line CD in Fig. 2 are given' by

tan•=kl/{k•.•o[exp(klZ•)- 1-][-exp(-k•.z•)-]}, (28)

4•c = 4•0[-1 - exp (- k2z•)-], (29)

sin0• = { 1 q-(Wo/Co)[1-exp(-klz•)• cos•,} -1. (30)

The ray-tracing equations (Eqs. 6-11) have been solved for this case by a numerical integration procedure for values of the constants in Eqs. 25 and 26 of w0--20 m/sec, 4•0 = l,r rad and kl= 2k•.=0.018 m -1. For these constants the wind direction is within 1% of the geo- strophic wind direction at a height of 500 m. The resulting ray paths are shown in Figs. 3 and 4, where the projections of the ray paths on the x-z and y-z planes are illustrated. The formation of a shadow zone in the downwind direction is clearly seen. For rays near the edge of the shadow zone the ground range depends critically upon the initial elevation angle, indicating that the position of the boundary of the shadow zone would vary markedly with small variations in meteoro- logical conditions. The calculations were performed on an IBM 7090 computer using an Adams-Moulton integration technique. The average time required to trace a ray was about 10 sec. It is useful to note that

600

72 5 ø

400

Z 730 tin} 3oo

ooo

o

ioo

o -so o so •oo •so •oo •so

Y(m)

Fro. 4. Projection of the ray paths on the y-z plane for the atmosphere of Fig. 3. The initial wave normal direction is indicated on each ray path.

Haselgrove ø has pointed out a convenient check on the integration procedure. At the end of each step the accuracy may be estimated by comparing the two quantities t• and p=(px•'+p•2+p•') •, which should agree to the level of accuracy of the method of integration. In calculating the ray paths illustrated in Figs. 3 and 4, agreement of these two quantities to within one part in 107 was maintained throughout.

VI. CONCLUSION

Ray-tracing equations have been presented, for acoustic waves in an inhomogeneous anisotropic atmosphere, which are based on Hamilton's equations for geometrical optics. The equations have been converted to a form suitable for direct numerical inte-

gration. Simplified versions of the equations are ob- obtained for the case of a horizontally stratified atmosphere and propagation in a single plane. Finally, results of ray tracing in the case of an atmosphere with both vertical and angular wind shear are presented and the formation of a shadow zone in the downwind

direction is demonstrated. The ray-tracing equations presented here may be readily adapted to describe the propagation of acoustic rays under water provided the appropriate expression for the refractive index (Eq. 2) is known for this case.

ACKNOWLEDGMENT

The author wishes to thank Dr. P. W. Baker for

helpful discussions during the preparation of this work.

• J. W. Strutt Lord Rayleigh, The Theory of Sound (Dover, New York, 1945) (lst ed., 1878), Vol. II, Chap. 14.

2 p. N. Tverskoi, Physics of the Atmosphere (Israel Program of Scientific Translations, Jerusalem, 1965).

a E. H. Barton, Phil. Mag. 1, 159 (1901). 4 B. Gutenberg, J. Acoust. Soc. Amer. 14, 151-155 (1942). 5 E. T. Kornhauser, J. Acoust. Soc. Amer. 25, 945-949 (1953). 0 R. P. Lee, "A Model for Acoustic Ray Tracing," Proceedings

of Second Conference on Atmospheric Acoustic Propagation, Fort Bliss, Texas, 1964.

7 p. Ugin•:ius, J. Acoust. Soc. Amer. 37, 476-479 (1965). a W. R. Hamilton, "Geometrical Optics," in Mathematical

Papers (Cambridge U. P., Cambridge, 1931), Vol. I. 9 j. Haselgrove, "Ray Theory and a New Method for Ray

Tracing," in Report of the Conference on the Physics of the Ionosphere (London Physical Society, London, 1954), p. 355.

•0 p.M. Morse and K. U. Ingard, in Handbuch Der Physik, S. Flugge, Ed. (Springer, Berlin, 1961), Vol. 11/I, p. 80.

n D. Blokhintzev, J. Acoust. Soc. Amer. 18, 322-328 (1946). • L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon,

New York, 1959), p. 257. •a H. Poeverlein, Z. Angew Phys. 1, 517 (1949).

The Journal of the Acoustical Society of America 87


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Three-dimensional acoustic-ray tracing in an inhomogeneous anisotropic atmosphere using Hamilton's equations