AD-A133 744 FOCUSED ACOUSTICAL AND OPTICAL BACKSCA TERING FROM /SPHERES AND PROPERTIES..U) WASHINGTON STATE UNIVPULLMAN DEPT OF PHYSICS P LMARSTON 15 SEP 83 TR-3

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TECHNICAL REPORT NO. 3

FOCUSED ACOUSTICAL AND OPTICAL BACKSCATTERING

FROM SPHERES AND PROPERTIES OF OPTOACOUSTIC

SOURCES

by

Philip L. Marston

S eptember_15, 1983

ei 1 83 10 1' 43

DEPARTMENT OF PHYSICS

WASHINGTON STATE UNIVERSITY

PULLMAN, WA 99164-2814

TECHNICAL REPORT NO. 3

FOCUSED ACOUSTICAL AND OPTICAL BACKSCATTERING

FROM SPHERES AND PROPERTIES OF OPTOACOUSTIC

SOURCES

by

Philip L. Marston

September -D, 1983

Prepared for:

OFFICE OF NAVAL RESEARCH

CONTRACT NO. N00014-80-C-0838

Approved for public release; distribution unlimited

Reproduction in whole or in part is permitted for

any purpose of the United States Government

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FOCUSED ACOUSTICAL AND OPTICAL BACKSCATTERING' Interim Technical ReportFROM SPHERES AND PROPERTIES OF OPTOACOUSTIC 1 Sept 82 - 31 Aug 83SOURCES Z. PEnponem Ons. 46POmT NUuue

1. AUNWO) L C*W MMUM 100191o

Philip L. Mlars ton N00014-80j3838

9. PER11PORMING ONGAMIZAWSORIMM9 ANDiE If. a i UFI MU TAM

Department of PhysicsWashington State UniversityPullman, WA 99164-2814 _____________

I I. CONTROLLING OFPIC NAIM AftO ADDRESS II. REMORT 0ATEPhysics Division Office 15 September 1983Office of Naval Research IL. NIMDER OF PAGESArlington, VA 22217 52

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AL. SSSRIIJIIGU TA MT (6B bwM m *Am s 66~ 8W

Conain S rPe priAR s and preprints listed in block 20.

Acouatic scattering, Inverse scattering, Elastic-wave scattering,Backscattering, Sonar calibration, Physical acoustics, Underwater acoustics,Light scattering, Bubbles, Hie theory, Optical oceanography, Fog, Lightextinction, Optoacoustics, Photoacoustics, Half-order derivative

4LADTRACI ("Nm - ~ a.1do o Mwemip -d IU FAI ailM16This report consists "of publications and preprints on topics related toacoustical and optical scattering and to laser optoacoustic sources of sound.The motivation and interrelation between the topics covered are described inthe preface. The authors and titles of the papers enclosed are as follows:

1. P. L. Marston, K. L. Williams, and T. J. B. Hanson, "Observation of theacoustic glory: High-frequency backacattering from an elastic sphere,"Journal of the Acoustical Society of America 74, 605-618 (1983).

DO j~ 13 TU O I RVESIS S~mEYE UnclassifiedS/M 0 02- IF. 014. "41 gM*yy PM Ae OP WSPG U

2

Unclassified"a" C*IF"IT" 4W TW @VMN M sa n

2. P. L. Marston, "Half-order derivatiia of a sine-wave burst: Applicationsto two-dimensional radiation, photoacoustics, and focused scattering fromspheres and a torus," Journal of the Acoustical Society of America(submitted for publication).

3. P. L. MIarston, "Uniform Mie-theoretic analysis of polarized and cross-polarized optical glories," Journal of the Optical Society of America(accepted for publication).

The emphasis of Papers 1 and 2 is on the understanding of the enhanced back-scattering from spheres associated with axial focusing. The emphasis onPaper 3 is on thu modeling of certain cross-polarization and copolarizationeffects due to the scattering of laser light from small bubbles in water orfrom fog.

$/N 0102- Ipg 014- 64cUnc lassified,

3

TABLE OF CONTENTS

Page

TITLE PAGE I

REPORT DOCUMENTATION PAGE 2

PREFACE 5

PAPER NO.

1. P. L. Marston, K. L. Williams, and T. J. B. Hanson, "Observationof the acoustic glory: High-frequency backscattering from an 9elastic sphere," Journal of the Acoustical Society of America 74,605-618 (1983).

2. P. L. Marston, "Half-order derivative of a sine-wave burst:Applications to two-dimensional radiation, photoacoustics, and 24focused scattering from spheres and a torus," Journal of theAcoustical Society of America (submitted for publication).

3. P. L. Marston, "Uniform Mie-theoretic analysis of polarized andcross-polarized optical glories," Journal of the Optical Society 44of America (accepted for publication).

4

PREFACE

This technical report consists of publications and preprints which are

representative of research supported by this contract from September 1982 thru

August 1983. The purpose of this preface is to comment on the motivation of,

and background for, this research and to interrelate the topics covered.

Paper No. 1 describes the first detailed observations of, and theory

for, focused acoustical backscattering from an elastic sphere. This focusing

is referred to as "axial focusing" because the scattered pressure is locally

enhanced near the backward axis as evidenced by Fig. 9-12. This phenomenon is

called the "glory" since the enhanced optical backscattering from cloud

droplets (also known as the glory) is caused in part by axial focusing. The

first detailed model of transmitted wave glory amplitudes from spheres1

(acoustical or optical) was developed under this contract. Paper No. 1

contains several theoretical extensions of our previous (fluid sphere) model1

and it gives data which confirm those extensions. Axially-focused

backscattering is potentially significant to sonar technology. Due to the

intrinsic isotropy of spheres, there has long been an interest in the use of

fluid-filled spherical shells as sonar calibration targets. These targets may

also be useful as passive navigational beacons for which backscattering is

large for a broad range of frequencies. The theoretical considerations

described should facilitate, for the first time, a detailed understanding of

focused echoes from large spheres having ka >> 1.

Paper No. 2 describes a detailed calculation of the temporal

signatures of axially-focused backscattering from spheres. It is shown that

the time-domain response due to axial focusing is describable (under

conditions noted) by the "half-order derivative" operator. The effect of this

5

operator on a sine-wave tone burst is calculated and discussed. The quasi-

steady-state approximation used in Paper No. 1'is shown to be applicable for

I the conditions of that experiment. In addition, axially-focused

backscattering from a large torus is modeled. The frequency domain response

associated with axial focusing appears to be one of the canonical response

functions of acoustics so the temporal response is applicable to several other

problems in acoustics, some of which are noted below.

Paper No. 2 also reviews Lighthill's formulation of the pressure

radiated from a line source in terms of the half-order derivative operator. A

specific example, the transient radiation from certain optoacoustic sources,

is considered. When the optical intensity modulation is that of a half-cycle

of a sine wave, the calculation displayed in Fig. 5 should approximate the

radiated pressure. This calculation (and similar ones for other inputs)

should be useful for calibrating broad-band hydrophone systems needed in a

study (supported by this contract) of other optoacoustics sources.

Paper No. 3 describes the first uniform analysis (i.e. an analysis

which is equally valid for all scattering angles) of a novel measurement

configuration for light scattering. The measurement configuration considered

is potentially useful for the discrimination of light scattered from spheres

(small air bubbles in water or fog droplets) from that of non-spherical

particulates. It was previously introduced by us to facilitate the detection

of optical glories (i.e. axialy-focused backscattering) from air bubbles in

liquids.2 ,3 Our previous computations and unpublished visual observations

indicate that optical glories enhance the near-backward scattering from small

air bubbles freely rising in water. (Small bubbles significantly affect the

acoustical properties of water and also the air-sea interaction at the ocean's

surface.) The calculation described is applicable to the production of cross-

6

polarized light due to scattering from small bubbles (and from fog or cloud

droplets) provided the number density of scatterers is sufficiently small that

multiple scattering may be neglected. The calculation is also applicable to

non-cross-polarized (sometimes known as copolarized) scattering and to forward

as well as backward optical glories.

In other work supported by this contract we have observed the

scattering of both white light 6 ,7 and laser light from bubbles in the vicinity

of the critical scattering angle. Color photographs of the white-light

scattering are to be published.7

In preparing the papers enclosed in this report the principal

investigator (P. L. Marston) has benefited from the collaboration with, and/or

critical comments by, several students. These include: T. J. B. Hanson, D.

S. Langley, B. T. Unger, and K. L. Williams.

P. L. Marston

September 1983

References to Preface

1. P. L. Marston and D. S. Langley, "Glory- and rainbow-enhanced acousticbackscattering from fluid spheres: Models for diffracted axial focusing,"Journal of the Acoustical Society of America 73, 1464-1475 (1983).

2. D. S. Langley and P. L. Marston, "Glory in the Optical Backscattering fromAir Bubbles," Physical Review Letters 47, 913-916 (1981).

3. P. L. Marston and D. S. Langley, "Glory in backscattering: Mie and modelpredictions for bubbles and conditions on refractive index in drops,"Journal of the Optical Society of America 72, 456-459 (1982).

4. P. L. Marston, D. S. Langley, and D. L. Kingsbury, "Light scattering bybubbles in liquids: Mie theory, physical-optics approximations, andexperiments," Applied Scientific Research 38, 373-384 (1982).

7

5. P. L. Marston, "Light scattering by bubbles in liquids: comments andapplications of results to circularly polarized incident light," AppliedScientific Research 40, 3-5 (1983).

6. P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, "Scattering ofwhite light from a cylindrical bubble: Observations of colors near thecritical scattering angle," Technical Digest of the Optical Meeting onMeteorological Optics (Optical Society of America, Washington, D.C., 1983)"pp. ThA3-1 - ThA3-4.

7. P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, "Critical anglescatterind of white light from a cylindrical bubble in glass: Photographsof colors and computations," Journal of the Optical Society of America(accepted for publication).

8

Paper Number 1:

Observation of the acoustic glory:

High-frequency backscattering from an elastic sphere

Philip L. Marston, Kevin L. Williams, and Timothy J. B. Hansona)

Department of Physics-Washington State UniversityPullman, Washington 99164

Reprinted from: J. Acoust. Soc. Am. 74, 605-618 (1983).

Measurements of the scattering of 2.10 MHz ultrasonic bursts gave direct

evidence of the acoustic glory. Echo amplitudes were measured as a function

of the transverse displacement x from the backward axis; they were maximized

on the axis and they manifested side lobes. Except near minima, the amplitudes

were proportional to .30 (kbx/z n ) where b and z are the radius of, and axial

distance to, the virtual focal circles from which the echoes appeared to emanate.

This dependence on x is direct evidence of axial focusing (The principal

feature of optical glories of drops and bubbles is also axial focusing.). The

experiment was performed in water with a fused silica sphere having a radius

a=51.6 - for which Zra/X - ka-457. A theory is developed for certain of the

echo amplitudes and focal parameters. The theory typically overestimates the

amplitudes of the stronger of the echoes by 5%. Appendices describe aspects

of the theory not amenable to the experiment, including: (i) a shift of

focal parameters which should be significant when kal00, and (ii) a distortion

of transients described in part by the half-order derivative operator.

a) Present address: Defense Systems Division,Honeywell, Inc., Hopkins, NN 55343

PACS Nos: 43.20.Fn, 43.20.Px, 43.20.Dk, 42.10.Hc

9

Observation of the acoustic glory: High-frequency backseatteringfrom an elastic sphere

Philip L. Marston, Kevin L. Williams, and Timothy J. B. Hansons)Department of Physics Washington State University, Pullman, Washington 99164

(Received 28 December 1982; accepted for publication 9 May 1983)

Measurements of the scattering of 2. 10-MHz ultrasonic bursts gave direct evidence of the acousticglory. Echo amplitudes were measured as a function of the transverse displacement x from thebackward axis; they were maximized on the axis and they manifested side lobes. Except nearminima, the amplitudes were proportional to J0(kbx/z.), where b and z, are the radius of, andaxial distance to, the virtual focal circles from which the echoes appeared to emanate. Thisdependence on x is direct evidence of axial focusing. (The principal feature of optical glories ofdrops and bubbles is also axial focusing.) The experiment was performed in water with a fusedsilica sphere having a radius a -51.6 mm for which 21ra/l = ka ,457. A theory is developed forcertain of the echo amplitudes and focal parameters. The theory typically overestimates theamplitudes of the stronger of the echoes by 5%. Appendices describe aspects of the theory notamenable to the experiment, including: (i) a shift of focal parameters which should be significantwhen ka < 100, and (ii) a distortion of transients described in part by the half-order derivativeoperator.

PACS numbers: 43.20.Fn, 43.40.Ey, 43.20.Px, 42.10.Hc

INTRODUCTION ration. The principal burst (neglecting buildup and decay ofThere have been numerous investigations of acoustic the transducer's response) is four cycles of a 2. I-MHz sine

backscattering from solid elastic spheres in water. -10 (Much wave. This corresponds to a wavelength A of 0.71 mm. Theof this work is reviewed in Ref. 8.) When the sphere is very distance z, from the transducer to the near point of thelarge in comparison to an acoustic wavelength, a simple de- sphere is 171 cm which is somewhat greater than 3d 2/4,;

scription of the backscattering should be possible by ac- consequently, the sphere is in the farfield of the transducer.counting for reflection and the various longitudinal and The sphere is synthetic fused silica' 6 with a radiusshear rays internal to the sphere. 9- 2 It is insufficient to use a = 51.563 mm. It subtends a half-angle _a/z, radia , (rela-elementary ray-optical methods' ° because of a novel focus- tive to the transducer) which is less than the angle of the firsting effect.' 3 Diffraction must be included' 3 in models of this minimum (cl 1.22A /d rad) ofthe farfield radiation. The scat-"axial focusing." It is the purpose of this paper to describe tering in near backward directions was monitored with aobservations of axial focusing of backscattering and to ex- hydrophone (at the end of a bent hypodermic needle). Thetend a previous model,' 3 developed for fluid spheres, to the hydrophone could be scanned along a line transverse to thecase of the backscattering of transient signals from elastic symmetry axis. The on-axis distance z to the sphere was al-spheres. It is demonstrated that the scattering to angles ways <ca2 where k = 21r/A. Consequently, the hydrophoneslightly displaced from the exact backward direction can be was not in the farfield scattering region (in the terminologygreatly influenced by the size of the scatterer. of Ref. 13). For this experiment kac_457. Typically z- 74

It is appropriate to refer to the scattering described-here cm and displacements of the hydrophone from the axis by an

as the acoustic "glory" because the term "glory" is generally amount x of a few millimeters significantly reduced the glory

applicable to backscattering from spheres (or other spheri- scattering amplitudes.tally symmetriL; potentials) when enhauced by axial focus- Prinripal modifications of the fluid-sphere model (Ref'.

ing.,1-15 Axial focusing enhances the amplitudes due to rays 13) needed for the description of the experimental results are

backscattered with nonzero impact parameters (the "gloryrays") but it does not affect the amplitudes of the axial reflec- COAXIALtions. |3 There is no axial focusing of the backscattering from / CABLE

cylinders'" 3 and consequently the exact scattering (normal- o0ized to that of the geometric reflection from a fixed-rigid Iobject of the same size) is somewhat greater for large elastic AXIS

spheres than for cylinder.' 9 PISTON AX- - - CFigure 1 gives a conceptual description of the experi- H'DROPHONE

mert. A pistonlike transducer of diameter d = 12.7 mm is S AXISdriven electrically so as to produce a tone burst of short du- BAFFLE SCAN AXIS I

FIG. 1. Sitpled dila rm of the scatft uapubeat 7U hyd'Present addrut Defnse Systems D1ividon, Honeywell, Inc., Hopkins, may be scanned loeS aie tronverse to the symussuy saiddwd by theMN 55343. source sd the sphere e dia m t drbo to ads.

606 J. Aoust. Soo. Am. 74 (2). August 1983 0C 1-4961/3/0060O1400.80 S 1963 Acoustioi Soe" of Amelma as

description of rays and focal-cirle parameters for an elasticsphere; use of reflection and transmission coefficients for wa-ter-quartz interface; modeling of nearfield diffractive e'-fects; and discussion of the transient response. The modifica-tions are first developed for the steady-state response. Ray "* .-

acoustics is used to model amplitudes at planes near the 3 d

sphere. These amplitudes then propagate (with diffraction),,to the hydrophone. Equation numbers from Ref. 13 will becited with an F prefix. Our nomenclature extends that of 0.2 --Ref. 13. ]b

I. RAYS AND GEOMETRIC FOCAL-CIRCLE _-- --- - -- F3PARAMETERS 4- -

Backscattered ray paths through large elastic cylin- 5 I 14

ders"' 2 and spheres'0 depend on the following integer pa-rameters: n, the total number of ray traverses of the interior;m, the number of traverses as a shear wave; n-rn, the numberof traverses as a longitudinal wave; and 1, where 21-1 is the FIG. 2. Backscattered rays in a sphere representative of unmixed chordnumber of times internal chords cross the axis shown in Fig. glories. When the figure is rotated about the CC' axis, theF, trace out focal!. (The parameter n' of Ref. 13 is I - 1.) To simplify the circles which are the apparent sources of backward propagating toroidalwave fronts. These waves are axially focused along the backward axis. Thedescription of focal-circle parameters, the following ap- specular reflection appears to diverge from A,, and is unfocused.proximations will be made: (i) the incident wave will be takento be a plane wave, and (ii) the geometric condition on gloryrays will be that they leave the sphere traveling parallel to theaxis. Condition (i) is an approximation because zp/ a is finite agle of incidence 0 is again found by considering rays trav-axis.lCondiion (i) is an approxi ation because z o finit se eling parallel to the axis when they are incident on the spherewhile (ii) is an approxi:--sstion because zia is also finite. These and exiting from it. As in the unmixed case, b describes all ofapproximations facilitate the use of geometric parameters the following: the radius of the virtual source (the focal cir-from Ref. 13. As in Ref. 13 the axial rays which contribute to c) and the ray's distance from the axis at impact and whenthe scattering along the backward axis have a vanishing in- in the ane. dhetanerom nels aw and whenpact parameter while the impact parameters are finite for cthe exit plane. The generalized Snel s law and the spheri-glory rays. cal geometry give the following condition °-M2 on

When the ray traverses are either all longitudinal irj1 - 4) = (n - m)arccos(b /ML(m = 0) or all shear (m = n), they will be referred to as "un- + m arccos(b /Ms) - arccos(b). (2)mixed" chords. Geometric parameters are as described in This has been solved numerically and the glory ray's angle ofRef. 13 with the acoustic refractive index M replaced byML = Cw/CL and M, = cw/cs for the respective caseswhere cL and c, are the bulk longitudinal and shear velocitiesfor fused silica and cw is the sound velocity in water. These -

indices depend slightly on temperature (see Sec. VII); a typi-(3,31) (4,4,1)(5,)cal temperature was 22 "C which had ML = 0.2511 and

M = = 0.3957. The form of ray paths with 1 = 1 is illustratedin Fig. 2 which is drawn forM = 0.6. Figure 2 also serves todefine several angles and geometric parameters derived inRef. 13. To an external observer located near the backward - raxis, the scattering appears to emanate from a virtual ring-like source, thefocalcircle. 3 The circle has a radius b and is a

distance a behind the dashed vertical plane, the exit plane.(The subscript n of Ref. 13 will be omitted; here a and bdepend on n, m, and I.) The angle of incidence 0 which satis-fies the glory condition of exact backscattering will be de-

J noted by 0f"m." For the unmixed cases, it is given by Eq.(F 17) when I = I and n = 3 or 4 and by the numerical proce-dure described in Ref. 13 when n > 4. The focal-circle radius (30,0 (4,0j)

is b - ba where

= sin o.m., (I)a is given by Eq. (FlOb). Several unmixed chords, for the case FIG. 3. Representative unmixed-chord glory ray fora fud-silica spereof fused silica in water, are shown in Fig. 3. in water. The parameters (n. i, 1) are indicated. Dashed and solid rays rep-

When the chords are mixed, the glory condition on the resent shear and longitudinal waves, respectively.

606 J. Acouet. Soc. Am., Vol. 74, No. 2. August 1983 Marstono tal.: Bctcatterng from sphere 6a6

Consider the propagation phase t between the crow-... . ing of the dashed vertical plane (Fig. 2) of an incoming ray

(3, , 1) and the second crossing of that plane as the ray exits. For theI -exactly backscattered ray, this phase shift is

-7,,.j ---2ka [ I - cos 8..I

+ mMs cos vS + (n - M LCos VL] (4)

S- ,- in both the mixed and unmixed cases. For the mixed cases,'-" . (and the associated pulse delay) are independent of the

(4,2,1). ordering of the shear and longitudinal chords..". Section IIdemonstrates, however, the combined reflection-transmis-

-" "'"-"/ sion coefficient, needed for the description of scattering am-plitudes, depends on the ordering.

A dimensionless parameter q, describing the spreading- of the wave front (q is the limit of the ratio of the arc lengths

/- .. - - of d 'e' and de in Fig. 2 as the latter vanishes), is needed to- model the scattering.'" For unmixed chords, q is given by

Eq. (F12). For the mixed chord cases, neither a 3: q have/ been computed due to the complexities required to extend

-- - the derivation given in Appendix A of Ref. 13. Consequent-FIG. 4. Representative mixed-chord glory rays for a fused silica sphere in ly, the description of mixed-chord scattering will be limitedwater. The upper and lower groups show all sequences for the (3, 1. I) and (4, to modeling of the dependences on scattering angle and re-2, I cases, respectively. flection-transmission factors.

II. REFLECTION AND TRANSMISSION FACTORSincidence found from (I). Snell's law gives the following con-

dition on the internal angles v, and vs (relative to the normal To model the strength of the virtual ringlike source atat each vertex, see v in Fig. 2) for the longitudinal and shear each focal circle, the procedure from Ref. 13 will be fol-chords, respectively, lowed. Reflection and transmission coefficients appropriate

to an unbounded plane surface are used in conjunction withML sin vL = M, sin 3s = b. )31 geometric factors and energy arguments. Consider an inci-

There are n!/[m!(n - m)!] different ways of ordering the dent traveling plane wave carrying an energy E,. in a speci-chords.'' Figure 4 illustrates all choices for the cases (n, m, fled time interval. After a reflection from, or transmission) =(3, 1, 1) and (4, 2, 1). through, the interface let the corresponding temporal por-

RLL \

06[

Eout TWLRLL FIG. 5. Square root of the energy ratios] Eout for reflection and trnsmission-fol-

lowed-by-reflection. These were com-04 / puted for a plane water-fused silica in-~terface. The ray in the water 1which gives

rise to L and Sinternal rays) has an angle

\ of incidence 0.

02 -

.- wSRSS \,

0

8cL ecS8 (degrees)

607 J. Acoust. SOC. Am., Vol. 74, No. 2, Augut 1983 Marston etal.: Backacatterng from sphere 607

TABLE 1. Reflection and focal parameters and form function for backscattering for selected unmixed-chord glories. The sphere is fused silica in fresh waterat 22*C.

(n, m, 1) 8,,..,(deg) RTRo B a/a q b/a IG,(457,O01

3,0.1 7.77 0.27 0.14 2.iE- 2 1.038 27.3 0.135 0.0594,0,1 10.70 0.13 0.06 4.4E -4 1.022 46.3 0.185 1.3E - 3"3.3, I 12.88 0.24 0.13 1.6E - 2 1.061 17.3 0.223 0.0934,4.1 17.48 0.78 0.49 1.9E- 1 1.035 29.9 0.300 1.11251,, 19.59 0.79 0.48 1.5E - 1 1.022 46.5 0.335 0.7796.6,2 12.11 0.13 0.06 9.2E - 6 1.030 34.7 0.210 3.6E - 57.7.2 14.95 0.75 0.50 7.8E - 2 1.022 45.6 0.258 0.324

IE - I and E - 3 are factors of 10' and 10-3,

respectively, etc.

tion of the wave carry an energy Eo,,. The quantity TWS Ts, TWL = TLw, RLs L, (5)(Eot/Ei)j 2 will be denoted by the symbols Ti, and R,, R 2 + R 2 + T w= 1,where i andj denote the type of the incident and outgoing

wave, respectively. The following subscripts are used for i R 2 + R SL + w 1,andj: W for an acoustic wave in water, L for a longitudinal R' + T~5 2 T2L = ,wave in the solid and S for a shear wave in the solid with the where the ray angles 0, v, and VL are related by the general-particle motion lying in the plane formed by the ingoing and ized Snell's law. There are critical angles of incidence ,,outgoing rays. The To and R defined in this way are trans- such that Tw, = 0 for 0>0,,, wheremission and reflection coefficients which take on real valuesbetween 0 and 1; phase changes will be accounted for by a sin 0cL = ML, sin Os = Ms. (6)separate phase factor. The T,, and R. apply to an unbounded A water-fused silica interface at 22 °C has 0,L = 14.54* andplane wave so that complications related to the beam shifts Os = 23.3 1. For this case Rss, RLL, RSL, and relevant pro-near critical angles (e.g., the Schoch effect and Rayleigh ducts with TWL and Tws are shown in Fig. 5. Zeros arewaves) are not explicitly described in this approximation. evident in RLL near both 9.8* and 0,L and in Rs, near both

Expressions for the T, and R, have been derived by 11.2°and 0,L. These features in RLL and Rss are discussed inusing boundary conditions which approximate the water as detail in Ref. 18 (see, e.g., Figs. I and 7).an inviscid liquid. These expressions are well known in seis- To model the virtual source strength, the product ofmology (see, e.g., Ref. 17, Eq. 7 to 17 of Ref. 18, and the (Eo,,/Ein) |2 at each vertex is required. This may be writtenreview in Ref. 19). The FORTRAN computer program listed inRef. 19 facilitates their computation with minor modifica- B = TwiI TI, + ,w fl Rk JAk , (7)tions which make it possible to set the shear wave speed of k 7

one media (the water) to zero. Results from our version of the where: i(k ) andj(k ) are the incident and outgoing wave typesprogram were checked and found to agree with standard at the k th vertex; k = I corresponds to the first refraction ofresultss' over the full range of 0, the ray angle relative to the incident ray; and i(k) =j(k - 1). We have computed Bthe normal in the water. (Reference 20 gives Rww, RLL, and for several mixed and unmixed chords. Results for unmixedRss for an aluminum-water interface.) Energy conserva- chord cases are summarized in i able I. Of these cases, thetion," symmetry arguments, and numerical computations largest values of B are for those with both m = n andverify the following relationships -.. / > OL

TABLE 11. Reflection and focal parameters of chord sequences for selected mixed-chord glories. The sphere is fused silica in fresh water at 22 *C.

(1, m, il , ... ,Adeg) b/a B Chord sequence

3,1.1 8.88 0.154 2.IE- 2 LLS, SLL1.88 0.154 L.8E- I LSL

3,2, I 10.41 0.181 2.IE- 2 SSL, LSS10.41 0.181 l.5E- I SLS

4.2.1 12.84 0.222 .6E - 2 LLSS, SSLL12.4 0.222 9.9E - 2 LSLS, SLSL12.84 0.222 I.9E- 2 LSSL12.34 0.222 8.2E - 2 SLLS

5.3, I 14.45 0.250 2.2E - 2 LSLSS, LSSLS, SLSSL, SSLSL14.45 0.250 2.SE - 3 LLSSS, SSSLL14.45 0.250 LSE - 2 SLLSS, SSLLS14.45 0.250 1.2E- I SLSLS14.45 0.250 4.OE- 3 LSSSL

s0 J. Acouat. So. Am., Vol. 74, No. 2, August 1383 Marston etal.: Backsctteng from spwe 608

For mixed-chord glory rays having I = 1, 2 and n = 3, IV. FARFIELD SCATTERING4, and 5, B was computed for all n!l[(n - m)!m!] chord se- In this section, the amplitude in the region rka isquences. Table II gives results for some of the rays. Evident- considered where r is the distance from C' in Fig. 2 to thely B depends on the sequence."1 Sequences with mode con- observation point (not shown) to be denoted as Q. Let the leftversion at each reflection tend to have larger B than extension of the CC'axis make an angley with the line C'Q.sequences for which the L chords are grouped. For example, The Fraunhofer approximation expresses the farfield dif-the largest B values of the (4, 2, 1) glory are for the LSLS and fraction due top'(s) as a Hankel transform ofp'(s). The trans-SLSL sequences; for the (5, 3, 1) glory it is the SLSLS se- form may be evaluated using stationary-phase and angular-quence. These enhancements of B are due to the relative spectrum techniques.' 3 The farfield pressure due to a givenmagnitudes of R,,, RLL, and Rss for the relevant ray angle (n, m, 1) glory ray and chord sequence will be denoted byp,.in the water 9,,,,,. The equality of the B for certain of the The results in Sec. IIIB of Ref. 13 become for the elasticsequences for a given (n, m, 1) can be understood via Eq. (5). sphere case

III. AMPLITUDE AND PHASE AT THE EXIT PLANE p.(ry,r) = tpta/2rer +' G, (9)

Let p, exp( - iwt) denote the incident pressure in the G, (ka,u,) = (ka)' 2 ?cBJO(u,)dashed plane through C' in Fig. 2 where o and p, are the X exp[i(),... +uy - V1 -17r)], (10)frequency and amplitude of the wave. To an observer in the u. kb s y, (11)water, each resulting virtual source radiates a toroidalwave.' The amplitude in the exit plane of this outgoing wave = - ka ! - cos i,., j12)

is given by ray optics to be' . =2b (2 7ra i/q! 2a - (13)

'(s) [. k (s - b (8) where for notational convenience the grouping of factors inp' q _ exp2[ i(8)....1 + U - (S) + 2a (9) and 110) differs from that in IF281 and (F29). The signifi-

cance of G,,. u,,, T'. and /. are as follows: G. exp(i) is thewhere s is the radius in the exit plane and the parameter P farfield form function which describes the glory amplitudeand function ' s) depend on n, m, las do q, B, b, and a. The relative to the geometric reflection from a fixed-rigid spherephase shifty accounts for the phase advance of 7r/2 associat- of the same size: u,,. t he argument of the zero-order Besseled with each crossing of a focal curve prior to reaching the function J,, corresponds to the stationary phase point of theexit plane." Crossings occur at points L, and L. in Fig. 2. diffraction integral (F24) when , is small; the phase shift o' isThe total p becomes - ir/2 times the number of crossings due to a shift of the stationary-phase point from its small-yfor a given (n, m. 1) and ray sequence. The modulus factor B value: and / is a dimensionless geometric parameter whichand the spreading factor q are each evaluated for rays with includes the effects of q. b. and a on the scatter amplitude.0 = . Eq. 8) neglects their dependence on s in anticipa- The use of the stationary-phase approximation in the diffrac-tion of our use of the stationary phase approximation. tion integral motivated our approximation of B and q in Eq.

Pha-se changes due to transmission and reflection at (81 as constants evaluated at the stationary value of s which iseach vertex are combined to give '. When 0...., < ,' is 0 s = b.or 1r and is independent of s. For unmixed shear glories hay- In addition to the glory contributions, axial rays alsoing 0,.,,, > 0, the phase shift at each vertex differs from 0 contribute to the farfield scattering. The amplitudes haveor r; it depends on 0 in the corresponding plane interface the following form according to ray acoustics"problem. 9 Consequently P depends on s according to the PA = P, (a/2r)exp(ikr + i_, /,, where n = 0.2,4... is the num-original angle of incidence 0 for that ray which crosses the ber of internal chords and thef, are the axial form-functionexit plane at s. This dependence is approximated in Appen- moduli. The strongest amplitude is due to the first axial re-dix A with the following results when ka> 1. (i) P (s) may be flection (n = 0) which has f, = Rw and bo given by Eq.replaced by 'P (b) in Eq. (8) with b and a replaced by an effec- (F32). Here R,, is the reflection coefficient evaluated withtive focal radius b and distance a,. (ii) The magnitudes of 0 = Y/2. For small values of y, axial ravs due to shear chords(b - b, )Ia and (a - a,)/a are roughly (ka)- '. (iii) The cor- are weak and only the longitudinal-chordf,, n = 2, 4,..., arerections to b and a are so small in our experiment that they significant. Geometric approximations for these f, and anwill be omitted. In the remainder of this paper (with the approximation for , follow from results in Sec. IIC andexception of Appendix A) ' will denote Oi (b). Appendix C of Ref. 13.

For the mixed-chord glories, Eq. (8) describes the am- In the geometric approximation for the axial form func-plitude due to each sequence. The total amplitude for a given tion moduli, thef, are independent of ka. Consequently, if*1 (n, m, 1) requires that p' be summer over all chord sequences. attenuation is negligible, as assumed in the derivation of(10),As noted in Sec. I, the a and q have not been modeled for glory amplitudes are enhanced by a factor proportional tomixed-chord glories. (ka)"2 relative to those due to axial rays. When y = 0", Rw

The definition of B as the product of (E,/nE,.) at is given by the usual result for waves at normal incidence,each interface is such that p' is specified only for the same Rww = (ZL - ZW)/(ZL + Z ), where the impedancesmedium (water) in which p, is given. Modeling of stresses ZL = pCL, Zw = poCw, and p, and po are the densities ofwithin the sphere requires the use of material dependent con- fused silica and water, respectively. At 22 "C, the resultingfoversion factors and it need not be described here. is 0.796. Comparison with the IG.(457,0) listed in Table I

so J. Acoust. Soc. Am.. Vol. 74, No. 2, August 1983 Marston ea. : B1cksC0t"rkng fr phee SOiw

.-

shows that the glory amplitudes exceed that due to the stron- of Q. The integral can now be simplified by introducing thegest axial ray. following polar coordinates: x' = s cos *, y' = s sin b,

x = h cosg, andy - h sing. Equation (14) becomesV. NEARFIELD SCATTERING D,, eibt+H ) r2w o

In this section we extend our approximation so as to P e = z. djf s ds

include points with rna which did not satisfy the previous ik (x'x +yy - s212)condition r~oka2. It is again assumed that ka>i 1. The coordi- x ex - b), (15)nate system to be used is shown in Fig. 6. The location of the z /

origin C. and the longitudinal coordinate z. of the observa- where H = h 2/2'.. This expression reduces totion point Q depends on the ray parameters (n, m, 1); C. is the p.(xz.) = (kbD./iz.)Jo(kbh /z.)e'sz- + ff K) (16)center of the focal circle of the (n, m, 1) ray. Figure 2 showsone ofthese centers. Withz defined as in Fig. l,z. = a + z. where K = b 2/2z2 and h/z. =tan y.. For points Q not farScalar diffraction theory22" 3 gives the following diffraction from the axis, z. (I + H) r, the distance from C. to Q. Tointegral for the pressurep. at Q when r.' > evaluate D. we require that as K--O, p.(0,

D.e' , - b)z. 0,Z )-p. ( y( = 0, r = z. - a) from Eq. (9). This evaluation

P. n(x =Y "Lb dx'dy', (14) when combined with (10) yields the following approximationU rx d)dy' , (- 1r for the pressure in the near- and farfields

where s = (x'2 + y,2)112,r.' is the distance from a point on the p. (xyz.) = (pja/2z.)exp [ ik (r. - a + z.K)]focal-circle's plane to Q, and D. describes the strength of X G. (ka,kbh /z.), (17)virtual source. Three approximations simplify the evalua- where G. is given by (10) evaluated with u. = kbh /z.. It cantion of (14). In the exponent take be shown that when y is not large and rkb 2 that

r. =.. + [(X - X')2 + (V -y') 2

]/2Z.. k (r. - a + z.K -kr + q). Consequently (17) reproduces

This is the Fresnel approximation23 which is applicable pro- the previous farfield result both on and off of the axis.

vided22 To approximate the pressure due to the axial reflectionin the near field, we use the geometric result' 3 that for h /

m[ + ..I. z< 1, the reflection appears to come from a virtual pointlike

where x' 2 + y2 = b 2. Let h denote the radial coordinate of source located a distance a/2 behind C'. This source is at A.Q' in Fig. 6. The condition becomes z3 . - '(h + b )4 which in Fig. 2 and Ref. 13, Fig. 4. The reflected pressure at Qfor on-axis locations of the hydrophone reduces to becomesz. >b (b/A )13. The following approximations are less re- P0 = (Pja/2ro)Rwwekoi a-o (18)strictive and require that z, >b and z. >h. Replace r, in the where r is the distance from A0 to Q and R is evaluated atdenominator by z., a constant. Let the focal circle be uni-form in strength2 4 so that D. does not depend on the location the geometric angle of incidence. From Weston's asymptotic

results for electromagnetic nearfield backscattering,25 it isexpected that the leading correction to (18) will be of magni-tude (ka) '(or smaller) relative to unity provided ro is some-what larger than a/2.

VI. TRANSIENT PULSES: DELAY AND DISTORTIONIn this section the pressure of the incident plane wave is

changed from plexp( - it ) to a transient function of time( ,y~z) p,(t) where the reference plane is again the dashed plane in

r Fig. 2. To simplify our description ofthe nearfield scattering,hh Q is now restricted to lie on the backward axis so that

x = y = 0. After the incident wave crosses the dashed plane,(x., the propagation time delay r required to reach Q is deter-

mined by the combined k-dependent phase terms in (10) and

r =o + ( (19)

c Cw"..-= 2a I- cos 0..-., + mM s cos vS

+(n -m)ML cosvL +b 2 [2z +a)l'. (20)

C - 1"o = z/cw is the propagation delay for the first axial refiec-tion to reach Q. For the (n, m, I)th glory wave, rL,, is the

" zdelay relative to that of the axial refiection. The presaure ofthe first axial reflection is

FIG. 6. Coordinate system which facilitates the description of nearfeld po(t) = aRww (2z + a)- 'p,(t - ro).atterin The x'y plane contains the local circle aproiate to the gloUwe

wave uder consieration and C. is the center of the circle. Unlike the axial wave, the glory waves are ditotW as

610 J. Acoust. Soc. Am.. Vol. 74, No.2. August 1 U3 Marston *t/: Bacscttslng from sphere 610

well as shifted in time. To describe the distortion, it is con- cilitate a test of the model developed in Secs. V and VI. Fig-venient to define ure 7 shows the scattering chamber (an 8 2 f 2 It water

f-j/ - II, (21) aquarium), the transducers, and the sphere. The sphere waswhere i and 1' are phase shifts due to internal caustics and supported by three plastic-coated aluminum rods. (The usu-

al practice" of hanging the target from a wire wasl not usedreflection-transmission factors, respectively (Sec. III). Since du trisks of dagent the epie sphre The ose < 0, the phase is advanced and this k-independent shift dis- due to risks of damage to the expensive sphere.) The rods

torts transients3 216 .27 Further distortion is predicted as a were positioned so as to avoid paths of glory rays. Rod ech-

consequence of thek "2 exp(- i4)factor in (0). Appendix oes were identified and eliminated using strips of anechoicB models the combined distortions; combined with (1 7) and rubber (not shown in Fig. 7). Echoes from the tank's rear wall

were similarly eliminated. The water was filtered and deion-(19) it yields the following for the pressure of the (, m, l)th ized. The x value of the hydrophone probe (Fig. 1) was con-glory wave:

atrolled by the stepper motor which is visible in Fig. 7. Rails,a 3/2 WB()/ 2pJt = -2 )t p, (t - r), (22) which extended the length of the tank, supported the driven

2 W = transducer and the hydrophone positioner. The rails facili-

where (d/dt)"12 is the half-order deriative operator [Eq. tated the location of the backward axis (the truex =O,y = 0,(B5)], p, is given by (B3) evaluated at (t - r). In the evalua- location) to within I mm. Distances zP and z were deter-tion of (B3), pH (t - r) is the Hilbert transform26 ofp, (t - r) mined sonically by measuring tone-burst propagation delayswhich follows from (B4). with the aid of a digital delayed-pulse generator.

The phase shift by e of p,(t - r) yields p,(t - r). This The hydrophone's piezoelectric element was sufficient-description of the distortion is only approximate since the ly small (0.6 mm diam) that its directivity was insignificantshift c is only applicable to spectral components having for the range of x encountered in the experiment. The bentka> 1. Application of this result to p,(t) which turn on ab- hypodermic needle which supported the hydrophone wasruptly at t = 0 predicts precursor in p, which extends un- sufficiently small in diameter (1.3 mm) that its perturbing ofphysically2" to t = - oc. Due to the asymptotic limitations the incident wave was insignificant in the measurements ofof (17), the resulting temporal expressions (B3) and (22) in- the glory scattering.correctly describe the early time behavior of the precursor. The source transducer (Panametrics model V309, ac-In the experiment to be described the incident pressure is tive element diameter of d = 12.7 mm) was excited at fre-approximated by quencies below its 5-MHz resonance. It was driven by a low

P(t)= P sin(wt), 0<t< 21rN/ w, (23) impedance source consisting of two matched linear 40-Wpower amplifiers (Amplifier Research model 40A 12) operat-

with p,(t) = 0 elsewhere; N is an integer number of cycles, ed with their outputs in parallel. The voltage V(t ) across theeach of amplitude p,. Cron and Nuttal 26 have graphed the transducer was closely approximated by the functionresulting p, (I) for N = I and 2 and various c (their Figs. 2 V = P sin( t (for 0< < 2irN / and V = Oall other t, whereand 4). Their analysis indicates that for integer N>2, the N = 4. This excitation was repeated at c- 10-ms intervals;peak-to-peak amplitude ofp, in the central cycles of the tone this interval was sufficiently long for the decay of spuriousburst is close to the undistorted value, 2,. Furthermore for reflections. The tranducer's radiation pattern with wthese N, both the after-signal and the precursor are weak. 2r = 2.1 MHz was measured near the z = 0 plane as a func-

To estimate the effect of the half-order derivative on the tion of the impact parameter b. (This was facilitated by re-peak-to-peak amplitude of the scattering we have considered versing the direction of the hydrophone from that shown in(didt)'2p, for thep, given by (23). When N>4, the peak-to- Figs. 1 and 7.) Measurements of the incident wave's ampli-peak amplitude of the central cycles of the resulting function tudep, as a function of b (over the range of relevant b ) agreedis within 0.2% of the steady-state value 21,l "Ip which fol-lows from (B6). This steady-state approximation applies tothe scattered waveform when sin e = 0; however, it is to beexpected that even when sin E:(#0, it approximates ihe peak-to-peak (didt) " 2p, due to the aforementioned smallness ofthe effect of e on the central peak-to-peak amplitude. Theeffect of the (didt) 2 operator on certain other functions hasbeen tabulated.28' 2 q

In addition to the distortions due to c and the (didt)"2

operator, broadbanded signals in water are distorted due tcthe preferential attenuation of the high w components. Thisdistortion is estimated to be negligible in the experiment toI ~be described.

VII. EXPERIMENTAL CONSIDERATIONS AND v :

HYDROPHONE CALIBRATIONFIG. 7. Photograph of the scattering facility. The source transducer is on

This section supplements the experiment's description the lef, the bent hydrophone needle is in the center, md the sphere is on thegiven in the Introduction. We note considerations which fa- right. Adjacent to the base of the aquarium is a rule marked in centimeters

611 J. Acoust. Soc. Am., Vol. 74, No. 2, August 1983 Marston otl : Backscatterng from sphere 611

approximately with the elementary diffraction model of the is denoted as KPo(x = 0) where K is an undetermined constantcentral radiation pattern of a piston: P, (b) and P,, denotes the pressure amplitude. Then, for a specific=P _.(b = 0)(1 - b 2v) where'" v = (kd /2zp )2/8. Glory rays in glory echo (which is identified from its r.,,.,), the peak-to-our model 3 have b<bm.. = Msasr2.0 cm. The resulting peak voltage KP(x) is measured for the central cycles of thecorrection to Eq. (17) is small yet significant: b 2 z,,' 0.05; it echo. The measurements are made from the centers of theis incorporated into Eq. (24). noise broadened oscilloscope traces to facilitate a partial

Figure 8 shows oscilloscope records of the first axial cancellation of noise. The result of Sec. VI is that the steady-echo and several glory echoes. Figure 8(c) snows the axial state model is applicable to the peak-to-peak central portionecho, and after delays T,,,. > 40/us, several distinct glory of the glory echo's amplitude. Consequently, the model pre-echoes. Inspection of the time interval between the axial and diction for the voltage (and amplitude) ratio isglory echoes indicates that the noise amplitude is l 10% of P(x) (z + ja)IG. (ka,kbx/z.)I (I - b 2v)the typical echo's amplitude. Tests indicate that the princi- = , (24)pal noise source was the wideband preamplifier which was 2VX i0

matched to the hydrophone. The preamplifier's self-noise where the aforementioned source factor (I - b 2v) is includedwas equivalent to an input " 2/uV rms; however, the hydro- and Rww is evaluated at normal incidence.phone output voltage was small due to the combined require- A technique for calibrating hydrophones which refer-ments of high spatial resolution (small size of piezoelectric ences elastic response to the amplitude of the axial reflectionelement) and good temporal resolution (heavily damped ele- is described by Dragonette et al.' Our technique uses geo-ment for wide bandwidth). Not having signal averaging ca- metric and reflection factors whch differ from those of Ref.pabilities limits the extent of these experiments to measure- 31 in ways evident from Eqs. (18) and (24). (i) The spreadingments of echo times and (as a function of x) amplitudes but factor in the backscattered axial echo's amplitude isnot their distortions. (z + a)- ',not (z + a)' as it would be ifthe wave appeared

Our procedure of testing Eq. (17) is to first measure the to spread from the sphere's center. (ii) The reflected ampli-peak-to-peak preamplifier output voltage for the central cy- tude is less than that for reflection from a rigid sphere due to

cles of the axial echo. This voltage is measured when x = 0; it the coefficient R,,,. That the amplitude of the specular re-flections is altered by a target's impedance has been con-firmed previously with exact calculations of the backscatter-ing of tone bursts, e.g., those of Davis et al.2 for variousrubber cylinders with ka = 12.

(a) Some comments on attenuation due to bulk absorptionof the 2.1 MHz bursts are in order. The bulk absorptions of

sshear and longitudinal waves within the fused silica are cal-S /sculated to be negligible from published attenuation data

even when the internal path length was long, as is the case forthe (7, 7. 21 path shown in Fig. 3. [The reasons for choosingfused silica were (il low attenuation even at high frequencies,(ii) isotropy of elastic properties, and (iii) dimensional stabil-ity.] Absorption losses within the water over the propagationdistances z, and z are not negligible if absolute amplitude is

(b) the quantity of interest. The experiments, however, yieldedthe ratio of axial and glory amplitudes. The path lengths in

2 us water of the axial and glory echoes differed by < 1 cm. Con-sequently these losses do not significantly alter (Eq. (24)], themodel prediction.

The water temperature T varied with the room tem-perature. Associated with periods of rapid change in T, theglory echo amplitudes sometimes contained irregularities intheir dependence on x. These irregularities were apparently

(c due to the influence of gradients in Ton the acoustic propa-gation. Data collection was limited to periods for which the

U ments of T(j) (via a vertical array of thermalcouples) gave

vertically averaged gradients (dT/dy) having moduli-1 "C/m though local gradients were as large as I "C/n.

FIG. 8. Photographs of oscilloscope traces of the preamplifier output for During the series of experiments, the mean water tem-the hydrophone on axis and z_36 cm. (a) Shows the first axial reflection. (b) perature (T) ranged from 2-25 C. Details of the 5 .,, andBegins 67 ps after the axial reflection and shows several glory echoes. (c) P (x)/Po data were only weakly dependent on (T); however,Begins a few ps before the reflection and it shows both the reflection and theprincipal glory echoes. The strongest echo in (b( and (c is the (4,4. 1 ) glory comparisons with the theory are always made for materialecho. parameters appropriate to the T of the measurement. The

612 J. Acoust. Soc. Am., Vol. 74, No. 2, August 1983 Marston eta/: Backacattering from sphere 612

fresh water data of Ref. 33 for c w were used; e.g., at 22 "C, cw tion is the first echo in Fig. 8(b). It is followed by large distin-= 1.488 km/s. Use was made of Fraser's measurements34 of guishable echoes in Fig. 8(b) which are the (4, 4, 1) and (5, 5,

c, and c, for synthetic fused silicas having the normal OH 1) echoes. These are followed by a broadened echo which is aion concentration of 10 ppm by weight. The measurements superposition of echoes for rays having n = m > 5 and I = 1.were adjusted for Tof 22" to 25 'C by using the temperature Other than the axial reflection, Fig. 8(a), only one othercoefficients3": dcs/dT-0.28 m/s "C and dcL /dT,-0.68 m/ axial-ray echo could be identified. This was the (2, 0, 1) echos *C. At 22 *C, this procedure gives cs = 3.761 km/s and CL listed in Table III which has a predicted delay= 5.928 km/s. The density of synthetic fused silica is 2.201 r2,.. = 4a/cL. The signal-to-noise ratio was noticeablyg/cm3 ; at 22 "C, the density of water is 0.9978 g/cm 3. smaller for z = 74.4 cm than for 36.4 cm; consequently this

echo was not clearly detected in the large z case.VIII. EXPERIMENTAL RESULTS AND DISCUSSION For the measurements with z = 36.4 cm, the contribu-

tion to ".,,., of the second term on the right of Eq. (20) was asA. On-axle echo amplitude, and delays large as 0.24/zs. Evidently this term is significant since its

We now compare measurements taken from oscillo- omission decreases agreement with the data. This term has ascope traces, similar to those of Fig. 8, with the model pre- simple physical interpretation. It approximates the time dif-dictions of Eqs. (20) and (24). The hydrophone was placed ference for propagation along a straight path in water fromwithin 1 mm of the true backward axis. (The proximity of the focal circle to the hydrophone with that from the on-axisthis axis is evident from the symmetry of off-axis amplitude point C, to the hydrophone. When computing this term indata described later.) Time delays were measured from the the mixed chord cases the exact expression for a is notcentral cycle of the axial reflection to the corresponding cy- known. We take a = a since elementary considerations indi-cle in the glory echo. The latter time could only be deter- cate that the C, lie close in the sphere's center. The values ofmined with an uncertainty which slightly exceeded ± I cy- a/a tabulated in Table I give evidence of proximity in thecle ( ± 0.24 /s). The sphere's radius was chosen to be case of unmixed chords.sufficiently large that most principal glory echoes would not Let 9, and M, denote the ratio of peak-to-peak ampli-overlap as illustrated by Fig. 8. Representative delay and tudes P(x = O)/Po(x = 0) as measured and as predicted byamplitude measurements for distinguishable echoes are giv- Eq. (24), respectively. Table III compares R, and 99, withen in Table III together with predictions. The echoes are two values ofz at the indicated temperatures. For most of theordered according to their delay, echoes, 9P, was measured on more than one day at the indi-

The difference between the observed and modeled delay cated conditions and the value is an average. With apparent-in each case was <0.5/us. When making this comparison it ly similar water condition individual measurements variedshould be remembered that Eq. (20) does not include the by as much as ± 0.04 from the indicated jP,.. This uncer-effects of the k-independent phase shift c on the temporal tainty is also representative of comparisons among measure-displacement of the cycles near the center of each echo,26 nor ments taken at other temperatures. The deviations often ex-does it include displacements due to the (d /dt )/2operator in ceeded + 0.04 in measurements having z = 74.4 cm sinceEq. (22). We estimate the magnitude of these apparent time the signal voltage was smaller relative to the amplifier'sshifts to be $0.25 ps. The agreement of the model with these noise. Inspection of Fig. 8(a) reveals a weak signal whichmeasurements facilitates the identification of these echoes. appears _ 2/us after the first axial reflection. Tests indicate

One prominant echo appeared to be the superposition that this was due to the hydrophone. This signal may intro-of the (5, 3, 1) and (4, 3, 1) mixed chord echoes; both are duce a nonrandom source of error in the measurement of ,predicted to have r,..,, -- 70.0us. The resulting superposi- when the echo follows an earlier echo by < 2 /is as is the case

TABLE III. Predicted and measured backscattering delays and amplitudes relative to those of the specular echo. The upper group was obtained with z = 36.4cm and T = 24.9 *C. The lower group had z = 74.4 cm and 7 = 22.0 'C. The incident wave had W/21r = 2. 10 MHz.

Echo Delay, .. , s Ratios of peak-to-peak amplitudes(n, m. II Eq. 120) Measured .J, o,?. /.4,

2,0, 1 34.8 34.8 0.049 0.04 0.83.0, 1 44.6 44.3 0.069 0.08 1.163.2, 1 62.0 62.1 ... 0.54 ...

4.2, 1 63.4 63.5 ... 0.70' ...

4,4. 1 74.8 74.7 1.271 1.23 0.97

5,5, 1 77.0 77.0 0.883 0.90 1.027.7,2 147.9 148.1 0.373 0.27 0.72

3,0, 1 44.6 44.5 0.074 ......

3.1,1 53.5 53.5 0.874,4,1 74.7 74.7 1.036 1.19 0.915, 5. I 77.0 76.5 0.907 0.87 0.967, 7, 2 147.9 148.0 0.384 0.35 0.91

' The (4, 2, I) amplitude may contain interference from the (5, 2, 1) echo; rs. 2. 1 from Eq. (20) is 63.8 js.

613 J. AcOtwt. Soc. Am.. Vol. 74, No. 2. August 1983 Marston tal.: BackWotte ngfrom sphere 618

for (5, 5, 1). echo. The reflected wave is not axially focused. For both z ofFor unmixed chord glory echoes, Eq. (24) was used to 36.4 and 74.4 cm, there was no measurable dependence of

compute 9P, and the ratios R9P,, were computed. The KPoonxfordisplacements lxi 2cm. In sharp contrast withtheory overestimates the amplitudes of the glory echoes with this behavior, the peak-to-peak glory echo voltages KP werethe uncertain exceptions of(3, 0, 1) and (5, 5, 1) echoes. Note, strongly dependent on x.however, that the spread of values for .,/ , is much Figure 9 displays the measured ratio P (x)/PoSx = 0) as-smaller than the spread of the 9P,. Evidently the theory cor- sociated with the (4. 4, 1) glory whenz = 36.4cm. The proce-rectly describes the relative ordering of glory echo ampli- dure which was followed was to record KP after each incre-tudes. The largest echo is the (4, 4. 1); its amplitude on the ment in x. Starting with x = 0, the scan was completed inbackward axis is almost as large as that for a perfectly re- one direction; x was reset to 0 and the scan completed in theflecting sphere of the same size. In terms of P0 and Rww of other direction. The increments were uniform; in most scansthe fused silica sphere, the perfect reflector would have P / they were 250jum. Division of the data by KPo(x = 0) yieldedPo = (Rw)- - 1.256. This value is only slightly larger the plotted points. With increasing x, the echo's delay in-than the . of the (4, 4, 1) echo. creases due to the x dependence of r, in Eq. (17).

The relative values of 0,,,., and 0 L significantly influ- To find if the dependences on x is described by theence the ., for those echoes with n = m. The (4, 4, 1), (5, 3, steady-state model, the following quantity was graphed to-1), and (7, 7, 21 glory rays all have 0,..,. > 0,. For compari- gether with each data set:son, the (3, 3, 1) echo is predicted to be small, havingM, = 0.11. Equation (20) gives -. = 69.8 jus which is x =. *)Jo(kbx/Z,)I, (25)close to the delays for the (5, 3, 1) and (4, 3, 1) echoes. Conse- where R. is the measured P (0)/Po(O). Thex dependent factorquently the (3, 3, 1) echo could not be discriminated from the is that of Eq. (24). Normalization to 9 , was used since in(5, 5,1) and (4, 3, 1) echoes. For aluminum spheres in water, plotted cases R , <.,f , where .W, is given by (24) with x = 0.03.3., > 0,L and the (3, 3, 1) echo is predicted to be the stron- It is evident from Fig. 9 that (25) approximates the data ex-gest unmixed-chord glory echo.9 cept near the minima.

Consider now the -', for the distinguishable mixed- Most of the scans were taken with z = 74.4 cm wherechord glory echoes (3, 1, 1), (3, 2, 1), and (4, 2, 1). For each of the maxima were sufficiently broad that the finite width ofthese cases there are several chord sequences as illustrated in the hydrophone's active element (0.6 mm) was insignificant.Fig. 4. Table II indicates that the B factor for certain of these Figures 10 and 11 give results for unmixed-chord and mixed-sequences are similar in magnitude to the B of the stronger chord echoes, respectively. It is again evident that the echoesglories having n = m. It is consistent with our model that are localized or "focused" along the backward axis. Fluctu-these three echoes were easy to detect, though their param- ations of the data are larger than in Fig. 9 due to the lowereters a and q are not known. amplitude of the voltages. When graphing (25) in Fig. 11, it

TableIIIincludes R, and.R, forthe(2,0, l)axialecho. was necessary to assume a value for a to be able to evaluateHere R, = (z + ) f21(Z2Rww) 'where extension of results z, = z + a. Here we take a =a because of the expectedin Appendix C of Ref. 13 to the elastic sphere case with 0 = 0 proximity of the C, to the sphere's center. The graphs ingive the following: I2M = Rww (I - R2, -M L/ Figs. 9 and 10 use the modeled a given by Eq. (Fl0b). The(2- ML), z2=z+ a 2, and a2/a = I + iM. The resulting values of R, for these data sets are similar to those of Table9, agrees with the measured Wf, within experimental error. III.

Each data set is the result of a single scan and each isplotted without smoothing. Figures 9 and 10 are representa-

B. Off-axis echo amplitudes tive of II scans of(4, 4, 1) or (5, 5, 1) echoes obtained atThe dependences of the peak-to-peak glory amplitudes various water temperatures. Some comments on the symme-

on x (the displacement from the backward axis) give direct try of the data in Figs. 10 and II are merited. To symmetrizeevidence of axial focusing. First consider measurements of these sets, the assumed origin was shifted by 0.5 mm for allthe peak-to-peak voltage KPo for the ;imply reflected (n 0) sets having z = 74.4 cm. This shift is consistent with the

A 1.2

]0 (n, A.) . (4. 4.I1a .36. . FIG. 9. The peak-to-peak voltage for

1.0 the(4,4, 1) echo plotted as a ncton

0.s of x. The voltages are ratioed with- the on-axis value of the frt axial

echo. Measurements were obtained0.6 when the water temperature was

24.9 'T. The solid curve, Eq. (251, isthe modeled dependence on x nor-malized so as to fit the measurenmt

0.2 -at x = 0. The peak at x =f 0 and the

side lobes are direct evidence of axial0.0 tocusing.

-2 -I 0 2TRANSVERSE DISPLACEMENT A OF HYOROPMONE (c)

614 J. Acoust. Soc. Am., Vol. 74, No. 2, August 1983 Marstn : acL:iKtttrlngomlWre 614

1.2(a) (n. m.1) - 4.4. 1

.0 z= 74.4 cm

_ 0.6

0.2

0.0 9

-3 -2 -i 0 2 3TRANSVERSE DISPLACEMENT x OF HYDROPHONE (cm) FIG. 10. Data and models (as in Fig.

9) for the (4, 4, 1)and (5,5, 1) echoesa 0.8 ( in (a) and (b). respectively. Here

() z~ )(4.4 cm iz = 74.4 cm and the temperature(b) ,z.74.4 c, was 22C.

" 0.6

UA0.4 -

-2 -2 0 2 I

TRANSVERSE DISPLACEMENT x OF HYDROPHONE (cm)

aforementioned alignment imprecision. Since Figs. 10 and dent wave is described by Eq. (23) with N 4. This assump-11 were plotted with a common x - 0, y = 0 location, the tion is suspect for x near the zeros of Jo(kbx/z.) where thesymmetry of each data set is experimental confirmation that dependence on w is most significant. Careful observations ofthe different echoes are focused along a common axis. The the waveform near the temporal center of the echo indicatepredicted dependences on displacement do not depend on an apparent phase shift by ir as the hydrophone is scannedthe azimuthal angle as a consequence of the sphere's isotropy through each minima. This is in agreement with steady-stateand placement at the center of source's radiation pattern, theory. Unfortunately, the peak-to-peak amplitudes are dif-The isotropy of the sphere was confirmed by comparing (4,4, ficult to measure near these minima due to a small signal-to-1) measurements, Fig. 12, obtained before and after rotation noise ratio.of the sphere by an angle c- !130'.

Figures 9-11 show minima which are not as deep aspredicted by the steady-state theory. Recall the assumption IX. CONCLUSIONS AND RELEVANCE OF PARTIAL-ofx = 0 andy = 0 in the derivation of Eq. (22) and the ensu- WAVE THEORYing discussion of the applicability of steady-state theory. Foroff-axis points, the Bessel function in Eq. (10) contains a de- We observe that the backscattering of sound from elas-pendence on w not included in the derivation of (22). It is tic spheres manifests a localized enhancement similar tothatassumed in Eqs. (24) and (25) that steady-state peak -to-peak present in the optical glories of dropsI4.1 and bubbles. 1 Theamplitudes are applicable to off-axis pointsprovided the inci- scattering associated with particular internal rays is ob-

(a. m.9) - (3. 1. 1)

0 a * a.74.4 ct

05 FIG. i. Dots and model (na Fig.

9) for the (3,1. I, mixed.chord echo.0.4 - * Here z - 74.4 cm an the tempera-

* *lure was 24C.

5 0.2 - .0a a

00

0.0 L-S -4 -9 0 2 4 6

TRANSVERSE DISPLACEMENT OF HYOROPONE (Us)

G15 J. Acoust. Soc. Am., Vol. 74, No. 2, August 1983 MMs Waet: a leaksoltflrnomm mo 615

1.4 F Here r and r are the distance and backscattering angle rel-Itive to the sphere's center, 9jAcos y) is the Legendre polyno-

0 1.2 n = mial, and U. is a complex-valued function of kaj, and mate-n = rialparameters. 1-3

ao 1.0 * Before rotation When ka is large, the axially focused backscatteringX After rotation amplitude due to the (n, m, I)th glory ray can be attributed to

X" 0.8 -only some of the .Consider the asymptotic expansion of0 the Legendre polynomal3 for the case ofj> I and y I:DC 0,.(cos y) = (y/sin y)/2Jo[(j-)] + O( 3 2). (28)

0.4 The dependence on r will be similar to that of Eq. (10) pro-vided

".2 ' jkb -. (29)0.0 A _ ._1.. . -

-3 -2 - I 0 , 2 3 The recognition that partial waves can be attributed to raysTRANSVERSE DISPLACEMENT x OF HYDROPHONE (cm) according to the value of the ray's impact parameter is

FIG. 12. Data and model (as in Fig. 9) but forz = 74.4 cm and a temperature known as the localization principle of optical scattering the-of 24.7 C. Data was obtained before and after rotation of the sphere ory.14.15,36 In the "physical-optics" approximations devel-through an angle - 130'. The rotation axis was inclined to the backward oped here for elastic spheres and in Ref. 13 for fluid spheres,axis; consequently the rays traversed regions of the sphere which differed the dominant ray-optical contributions are for those raysbefore and after the rotation. having impact parameters equal to the radius b of a focal

circle. For values ofj satisfying (29), the phases of the U in

served to be axially focused. Inspection of Figs. 9-12 reveal (27) should be stationary; consequently, there will be a signif-angular widths A y of a few degrees for the central peaks; for icant contribution to /proportional to Jo(kby). For values of

example in 10(a), Ay'c_ 2 .8 cm/z,_-38 mr2.2. The agree- j not lying close to kb - I (for the b of any strong glory ray)ment with the modeled focusing pattern and the form of Eqs. the phases of the U should be such that neighboring, tend(10) and (11) suggest that 4y,-uA /nrb radians where u,-2.4 to cancel each other. A complete derivation of Eqs. (9) andis the first root of J,(u) = 0. This approximation should be (10) directly from the partial-wave series (26) would be coin-applicable when the hydrophone's distance z and sphere's plicated by the following caveats. (i) The /in (27) are phaseradius a are such that z>a>,A. Inspection of Figs. 10 and 11 referenced to the center of the sphere while Eqs. (9) and (10)together with Tables I and II demonstrate that the observed are referenced to C' in Fig. 1. The two are related by thewidth of the central peak is smallest for echoes having the unimodular y-dependent phase factor exp(ig) where , islargest focal radii b. Inspection of Table III shows that the given by Eq. (F37).discrepancy between the observed and predicted central am- (ii) Equation (9) includes an additional y-dependentplitude is approximately 5% of the predicted value. The phase factor exp(iq' ). They dependence of the net phase shift,cause of this discrepancy is not known; however, experimen- c + q = - k [2a + (a - a)(1 - cos y)], must be pre-

tal difficulties due to preamplifier noise and peculiarities of served in any asymptotic evaluation of the partial-wave se-

the hydrophone's temporal response cannot be ruled out. nes.The model could be refined by computing the reflection- Asymptotic expressions for scattering due to rays trans-transmission factor B along a path which accounts for the mitted thru solid elastic cylinders were obtained by Brill andfiniteness of the hydrophone and source distances, z and zP. Uberall. " Their method used saddle-point integration to

Interpretation of the experiments was simplified be- evaluate the Watson transformation of the partial wave se-cause the short duration of the incident tone burst made ries for cylinders. Unlike our results for spheres, theirs con-possiblc the temporal separation of echoes with different pa- tain no (ku)'!2 factor since backscattering from cylinders is

rameters (n, m, 1) in most cases. In true steady-state scatter- not axially focused.

ing the axial and glory echoes overlap in time. The mutual Our model predicts that backscattered transient pulses

interference of these echoes would be highly dependent on will be distorted in the manner described by Eqs. (22) andka when ka> i. The interference of glory and axial rays due (B3HB6). Detection of this distortion was beyond the scopeto several different parameters is the plausible cause of the ka of the present experiment.dependent structures which are present in Flax's exact com-putations of backscattering from large aluminum spheres.' ACKNOWLEDGMENTSThe exact farfield scattered pressure may be written in the Preliminary observations of the acoustic glory were de-following form'- 3 p( y, r) = (p, a/2r)/where the total form scribed in Ref. 38. Portions of this paper were presented atfunction /is a sum of the partial-wave amplitudes 1, the 104th meeting of the Acoustical Society of America (No-

4ka, y)= 'l', (26) vember 1982). This work was supported by the Office of Na-s-0 oval Research. One of us (P.L.M.) acknowledges the support

= (2/ka)U(ka),j(cos y). (27) of an Alfred P. Sloan Research Fellowship.

616 J. Acoust. Soc. Am., Vol. 74, No. 2, August 1983 Marstool Wt Back9at from apt8r 616

V1

APPENDIX A: EFFECTS OF THE REFLECTION- The aforementioned procedure gives the following values forTRANSMISSION PHASE SHIFTS ON FOCAL the (4, 4, 1) ray: I!' ' - 0.85 and A -.0.87 tad. Conse-PARAMETERS quently, the shift of the radius is negligible when ica is large,

When a ray's angle of incidence in the water 0 exceeds e.g., (b, - b )/ac- - 0.0019 for ka = 457. These results are0,L, the internal rays are all shearchords. The phase changes also representative of those for rays having 0, closer to 0,,due to transmission or reflection at each vertex vary continu- since the 0' are smaller but the I VI'1 are larger.ously with 9. Consideration of Fig. I for a case in which The skewed wave appears to originate from a focal cir-

0> ,c will convince the reader that outgoing wavelet d 'e' cle which is a distance a, = k [d 2(q - 1')/ds2]- 'behind thewill be skewed so that it is no longer described by the geomet- exit plane. Application of (A4) givesric focal parameters b and a. Instead, it is described by effec-tive parameter be and a,. The purpose of this Appendix is toillustrate how fb - b, I <a and ja - a, I <a when ka is large. For the (4, 4, 1) ray, the shift is negligible when ka - 457The skewing ,s only present for glory rays having n = m. We since (a, - a)/a - - 0.0020. The effective parameters in-simplify the notation by letting a subscript n denote the eva- fluence the scattering by replacing a and b in Eqs. (IOH 13),luation of angles or derivatives at the geometric glory condi- (17), (20), (24), and in z. = z + a. For fused-silica spheres,tion 0= 0,,.,; henceforth, we let 0, denote n.mj. the changes are negligible unlessca < 100.

Recall that s denotes the distance from C'at which a raycrosses the exit plane. Figure 2 of Ref. 13 gives an expanded APPENDIX B: FILTER MODEL OF PULSE DISTORTIONview of the crossing. We are interested in rays having 0--, Inspection of the steady-state scattering amplitude,so it is convenient to use the expansion Eqs. (10) and (17) reduces the problem of the on-axis pulse

- I. d(s' 1 - b )2 d2O (1 distortion to answering the following question: What is the- )( ). + -2! (A 1 ) general temporal effect of a filter whose response to the func-

and to define the following dimensionless quantities tion exp( - iat ) is S0S, exp( - iw.t )? Here S. and S, are the

0' = a(dO Ids), and 0" = a 2(d 20 Id).. Evaluation of Eq. following frequency responses:

(AS) of Ref. 13 at the glory condition 9=8= 0,, gives S,(w) = exp[icsgn(w)], (BI)0' = (q cos 0,) - ' where q = a/(a - a) is the spreading fac- S, (w) = IwI "2exp[ - ir sgn(w)], (B2)tor. Inspection of Table I shows that 9'< I rad. Numericalevaluation of 0" shows that it is also small; e.g., the (4, 4, 1) where the function sgn(w) = I c*l/w and c is given by (21). In

ray has '_-0.035 rad and 9" = - 0.007 rad. (B 1) and (B2), the domain has been extended to include nega-

Recall that W denotes the total phase shift due to each tive values of w by requiring that phase advancements for

vertex. It is convenient to use the expansion positive values of w are also advancements when w becomesnegative. The scattering process is modeled as a filter with a

F' - IP,,2(9 - 9,JIP" + (9 - 0,,)2IF/", (A2) response S, followed by one with a response S,. Let p,(t)

where V,, = Is = b ) in the notation of Sec. II, denote the input to the first filter; the outputp,(t ) is26" 23*P'= (dI/d),, and IP" = (d 2 P/dO2),. The constituent p,(1) =p,(t)cos c +pH(t)sin c, (B3)

derivatives may be evaluated at each vertex: for example, where PH is the Hilber transform of p,

V -'s) + (n - l)( .- s- + -j 'pt) 1 p('dO dO dO PI')PL 'f d' (B4)

(A3) " _ t -where l'ss is the shift due to reflection of a shear wave and and P denotes the Cauchy principal-value integral.'Pws and P'sw are transmission (mode-conversion) phase It is convenient to rewrite the response of the second

shifts. These shifts were computed from the complex reflec- filter as S, = ( - iw)1/ 2 where the negative real w axis is cho-tion and transmission coeffcients of a plane surface39 via the sen to be the branch cut in the evaluation of the radical. Toalgorithm listed in Ref. 19. The ' and W " were obtained obtain the temporal behavior of this filter we define the fol-numerically, lowing half-order derivative operator2 '

Let 17 denote the propagation phase delay between the d,/2entrance and exit planes, see Eq. (7). Inspection of() gives t t 2 dt B)11 - 17, .k (s - b )2/2a The following s dependence for thetotal phase at the exit plane is found by combining this result where(t') = dp(t')dt'. Bydirctevaluation of(BS)it may bewith (A l) and (A2). demonstrated that

(17- I)-( 7, -+ IX -J2, (A4) 1/2e- - S.e (B6)

where = (s - b)/a, X' -', X" = ka-' + A,& = a/a, and A = -9 "' - 0'2#"'. The effective focal- Application of(B6) to each ofthe spectral components ofthecircle radius b, for the skewed wave is the value ofs for which input p(t) to the second filter, gives the following output:

d (17 - 'P )/dj =0; this procedure gives p.(t) 1 - p(t). (B7)b, - b - = "e'p' It

a Cz &A + ka Takingp to bep, letsp. manifest the combined effect of both

617 J. ACuot. Soc. Am., Vol. 74, No. 2, Augut 1983 MarntonaL:Wt : sm attwn from shwro 617

proceaes. Cal applications," Phil. Meg. (London) 48, "4-97.567-569 I1399).

Computations of the half-order derivative of a tone 15K. Ergin, "Energ ratio of the seismic waves reflected and refracte atmarock-water boundary," Bull. Seismol. Soc. Am. 42, 349-372 (1952).bunt, (did: ) 12p, with thep, from (23), show that the princi- 19G B. Young and L. W. Bra"s "A computer program for the application

pal distortion is to the leading and trailing half-cycles. of Zoeppritz's amplitude equations and Ksnott's energy eqmsatioua" Dull.Seismol. Soc. Am. 66. 1881-lUS (1976).

20J. Krautkrimer and H. Krautkrirner Eoltrasonic Testing of Materials'J. 3. Faran, Jr., "Sound scattering by solid cylinders and spheres," J. (Springer-Verlag. Berlin, 1977), pp. 603-609.

-* Acoust Soc. Am. 23.405-418 (195 1); "Sound scattering by solid cylinders "1This dependence on sequence is manifested more Clearly in the prsetand spheres" ONR Tech. Memo. No. 22. Harvard University, Cambridge MOdel than in Refs. 10-12.(1951). 2j. W. Goodman, Introduction to Fourier Opt ic; (McGraw-Hill, San Frnm-

'R. Hickling, "Analysis of echoes from a solid elastic sphere in water." J. Cisco, 1968).Acoust. Soc. Am. 34 1582-1592 (1962). 23A. D. Pierce, Acoustics, An Introduction to its Phyui cal Pnincples and Ap-

'Ut 0. Neubauer, R. H. Yogi, and L It. Dragonette, "Acoustic reflection 24plicationt (McGraw-Hill, New York. 1968).from elatc spheres. I. Steady-state signals," J. Acoust. Soc. Am. 55, "The approximation that the virtual-source strength is independent ofQ1123-1129 (1974). requires that the combined reflection-transmission and spreading factors

4L. Rt. Dragonette, R. H. Vogt, and W. 0. Neubauer, "Acoustic reflection are approximately independent of 6 near 0,_,. It is a questionable apfrom elastic spheres and rigid spheres and spheroids. 11. Transient analy- proximation when t9 is near 8,.L and 49ssis," J. Acoust. Soc. Am. 53, 1130-1137 (1974). 'H. V. Weston. "Near-zone backacattering from large spheres," AppI. Sci.

'R. H. Vogt and W. G. Neubauer. "Relationship between acoustic reftec- Res. 09, 107-116 (1962).tion and vibrational modes of elastic spheres." J. Acoust. Soc. Am. 60,15- 16B. F. Cron and A. H. Nuttal. "Phase distortion of a pulse caused by hot-22(1976). tom reflection," J. Acoust. Soc. Am. 37,486-92 (1%5).

'L. Flax, "High ka scattering of elastic cylinders and spheres," 3. Acoust. "D. A. Sacha and S. Silbiger, "Focusing and refraction of harmonic soundSoc. Am. 62, 1502-1503 (1977). and transient pulses in stratified media," J1. Acoust. Soc. Am. 49,824-440

1L. Flax, L. It. Dragonetts and H. Uberall, "Theory of elastic resonance (1971).excitation by sound scattering," J. Acoust. Soc. Am. 63, 723-731 (1978). 28M. J. Lighthill, Waves in Fluids (Cambridge U. P., Cambridge, 1978), pp.

'S. K. Numrich, L. R. Dragonette, and L. Flax, "Classification of Sub- 20-21, 86.merged Targets by Acoustic Means." in Elastic Wave Scattering and Prop- 29K B. Oldham and J. Spanier. Thse Fractional Calculus (Academic, Newagation edited by V. K. Varadan and V. V. Varadan (Ann Arbor Science, York, 1974), Chap. 7.Ann Arbor, 1982), pp. 149-175. 'OA. L. Fetter and 3. D. Walecka, Theoretical Mechanics of Particles and

"P. L. Marston and L. Flax. "Glory contributions to the backscatter from Continua (McGraw-Hill, New York, 1980), p. 323.large elastic spheres," J. Acoust. Soc. Am. Suppl. 168, S8 1(1980). "L. Rt. Dragonette, S. K. Numrich, and L. LFrank, "Calibration technique

'"G. J. Quentin. M. deflilly, and A. Hayman, "Comparison of backscatter- for acoustic scattering measurements,' J. Acoust. Soc. Am. 69, 1186-ing of short pulses by solid spheres and cylinders at large ka," J. Acoust. 1189 (1981).Soc. Am. 70. 870-878(198 1). "C. M. Davis, L. R. Dragonette. and L. It. Flax, "Acoustic scattering from

"D. Drill and H. Uberafl, "Acoustic waves transmitted through solid elas- silicone rubber cylinders and spheres." J. Acoust. Soc. Am. 63,1694-1698tic cylinders," J. Acoust. Soc. Am. 50,.921-939 (1971). (1978).

"P. J. Welton, M. die Billy, A. Hayman, and G. Quentin, "'Backscattering of "3V. A. DelGrosso and C. Wt. Mader, "Speed of sound in pure water," J.short ultrasonic pulses by solid elastic cylinders at large ka." J. Acoust. Acoust. Soc. Am. 52, 1442-1446 (1972).Soc. Am. 67,470-476 (1980). 14D. B. Fraser, "Factors influencing the acoustic properties of vitreous sili-

"P L. Marston and D. S. Langley, "Glory and rainbow enhanced acoustic ca,." .1. App). Phys. 39, 5868-5878(1968).hackscattening from fluid spheres7 Models for diffracted axial focusing," "H. J1. McSkimin, "Measurements of elastic constants at low temperaturesJ. Acoust. Soc. Am. 73. 1464-1475 ( 1983). by means of ultrasonic waves. data for silicon and germanium single crys-

"V. Khare and H. M. Nussenzveig. "The theory of the glory." in Statistical tals, and for fused silica," J. AppI. Phys. 24.988-997 ( 1953).Mechanics and Statistical Methods in Theory and Application, edited by U. ~'H. C. Van de Hulst. "A theory of the anti-coronae," J. Opt. Soc. Am. 37,Landiman (Plenum, New York, 1977), pp. 723-764; V. Khare. "Surface 16-22(19471.waves and rainbow effects in the optical glory," in Electromagnetic Sur- "G. Szego, Orthogonal Polynomials (American Mathematical Society, Newface Modes. edited by A. D. Boardman (Wiley, Chichester, 1982), pp. 4 17- York, 1939), Theorem 8.21.6.464. '8T. J. B. Hanson, "An acoustical backscattering experiment," M. S. Pro-

"3D. S. Langley and P. L. Marston, "Glory in optical backscattering from ject Paper, Washington State University. 1982 (unpublished).air bubbles." Phys. Rev. Lett. 47, 913-916 (198 1). " V. Cerveny and R. Ravindrs. Theory of Seismic Head Waves (Univ. of

"Suprmsil grade 2 manufactured by Heracus-Amersil Inc. (Sayreville, NJ). Toronto Press, Toronto. 1971) Sec. 2.4: W. M. Ewing, W. S. Jardetzky,T'he sphere, which has a maximum deviation from sphiericity of 2.5 tim, and F. Press Elastic Waves in Layered Media (McGraw-Hill, New York.was fabricated by Speedring Division ofSchiller Industries Inc. (Cullman. 1957). p. 79.AL). 4'Equation ) B3) is equivalent to (7a) of Ref. 26. Our use of the -it conven-

"C. 0 Knctt, "Refleation and refraction of elastic waves, with seismologi- lion det,~rmines the sign on the sii, c term.

ale J. Acous. Soc. Am., Vol. 74, No. 2, August1960 Marston o tal.: Secksoattsrlng from sphore 618

Paper Number 2:

Submitted (August 1983) to J. Acoust. Soc. Am.

Half-order derivative of a sine-wave burst: Applications.to

two-dimensional radiation, photoacoustics, and focused

scattering from spheres and a torus.

Philip L. Marston

Department of Physics

Washington State University

Pullman, WA 99164I-2814

The half-order derivative (d/dt) 3(t) is calculated for 3(t) given

by a burst of sine waves. The burst is N cycles in length where N is an

integer or a half odd-integer. The result contains Fresnel integrals; it may

be written in a compact for. by using the auxiliary Fresnel-integral function

usually denoted by g. Some novel properties of g were derived to

facilitate a discussion of the result. The calculation is applicable to

several d1ssipationless acoustical models for which the response to an

exp(-Iwt) input is proportional to w* exp(-iwt - i r/4). Some examples

include the pressure radiated by two-dimensional sources such as the

optoacoustic radiation from a thin modulated laser beam. Another example is

the pressure from virtual ring-like sources such as those present in focused

backacattering from large spheres and a torus. The scattering from a large

torus Is modeled for the case of an incident plane wave which propagates

parallel to the syaetry axis.

PAC Ms. 43.20.Px, 43.35.Sx, 43.20.Fn, 43.20.B1

24

2

I. Introduction

This paper gives a calculation of the temporal response of certain

acoustical systems to inputs consisting of bursts of sine waves. The

approximate frequency responses of the systems considered, Eq. (3) below, are

characterized by a 45 deg. phase advance together with a modulus which

increases with the angular frequency w in proportion to J. To describe

the temporal behavior of such systems, Lighthill used the following half-

order derivative operator on the time domain function 3(t)

( d/atA 3(t) B ()~-)r *dT (1)

where i(T) x ds/dt (evaluated at t T). By direct evaluation of the

integral, the steady-state frequency response he associated with this

operator is1'2

(d/dt)* e i = h e Wt , (2)

h W (-iw)* I* -ei*7sgn(w ) (3)

where the signum function is sgn(w > 0) s 1, sgn(w < 0) = -1, and sgn(O) = 0.

The sine-wave bursts to be considered are N cycles in length where N is an

integer or half odd-integer so the 9(t) are of the form

N t) H(wt) H(27i - At) sinwt (4)

where the Heaviside unit-step function H(y < 0) z 0, H(y > 0) a 1, and

H(y 2 0) 2 * Throughout this paper we take W > 0.

Section II reviews several acoustical systems for which h is an

approximate transfer function for a range of w. This response is associated1

with the propagation from two-dimensional sources and with the effects of

25

3

2diffraction in axially-focused scattering. All systems considered are taken

to be linear so that signal distortions due to finite amplitude effects are

omitted. The effects on signal shape of the absorption of propagating sound

are also omitted. The examples considered here constitute one class of

intrinsic distortion. Other examples of linear problems for which distortions

have been analyzed include those due to frequency independent phase shifts3

(caused e.g. by simple caustics4 or bottom reflections at grazing angles of

56incidence ) and those due to the focusing in stratified media.6

For the transients given by (4), the integral in (1) is evaluated in

Sec. III. The integral also converges for other differentiable s for which

(-t) -s(t) * 0 as t . -; however, the general distorting effects of this

operator will not be considered here. Though inputs proportional to (4) are

not exactly realizable in physical systems, the distortions calculated should

be useful in understanding properties of real systems. It is found here that

the distortion is most significant near the leading and trailing edges of

(d/dt)*3 N . The operator defined in (1) is a special case of the

"semiderivative operator" which has been used for solving the diffusion

equation7 and modeling electrochemical experiments.8 Values of (d/dt) 1 for

some other 3(t) can be inferred from tables in Sec. 7.6 of Ref. 7 and in

Ref. 8. Notice that when s = exp(-Lt), (d/dt)k (d/dt)*s z ds/dt so the

terminology half-order derivative is Justified.

II. Some Acoustical Applications of (d/dt)* 3(t)

A. Pressure radiated from two-dimensional sources

Lighthill 1 oonsidered the acoustic pressure p(r,t) generated by an

infinite-line thin uniform source at a perpendicular distance r from the

line. The mass outflow per length of source is 3(t), which has units

26

4

-1 -1g cm sec . Integrating the radiation from each line element gives the well

known result

24p(r,t) = (2i)" 1 ( 2

- r2 )-* (t - ;/c)d; (5)r

where c is the speed of sound in the medium. This may be simplified when s

is a transient of duration at. In the far-zone where r >> cAt, (F2 _ r )

may be replaced by 2r(F- r) so that (5) becomes1

p(r,t) = (/8r) * (d/dt)* s(t - r/) . (6)

The condition on r may be written as r >> X where X is the wavelength of

the principal spectral component of 3(t).

Landau and LIfshitz9 calculate the radiation from a radially vibrating

infinite cylinder which has a cross-sectional area W(t). The unperturbed

density of the surrounding fluid is P0 Their result, Eq. (73.15), is

equivalent to (5) with 3(t) 2 p i and their far-zone result, Eq. (73.17),

reduces to (6). For (6) to be applicable it is necessary that r >> I >> W .

The continuity equation for density may be used to show that the effective

mass outflow per length is indeed Oo W.

B. Photoacoustic generation of pulsed sound in a weak optical absorber

10In a review of pulsed optoaOoustic speCtrosCopy, Patel and Tam apply

the Landau and Lifshitz result equivalent to (5), to the calculation of

radiated pressure. The meahanism by which the pressure is generated is the

heating of fluid due to the absorption of a pulsed thin light beam of radius

R. The electrostrictive mechanism is neglected in their (and In the

following) prediction because it is usually small. Consider cylindrical

coordinates for which the light beam propagates along the z axis. Let the

27

optical beam power at z be

U(t,z) = U(t,O)e~a z 2w I(t,zr)rdr

00where il(t,z,;) is the optical Poynting vector at a radius from the

center of the assumed symmetric beam; a is the optical absorption coefficient

of the fluid. In the notation of Eqs. (5) and (6), the Patel and Tam source

strength prescription becomes, from Eq. (33) and (35) of Ref. 10,

3(t) a CLS& (tO)C p "1 (7)

where B is the thermal coefficient of volume expansion and C is theP

specific heat at constant pressure. For this prescription to be applicable it

10,11is necessary that beam radius be such that

R D z (DAt) < << cAt = (8)

where RD is the thermal diffusion length for the pulse duration At of the

beam and D is the thermal diffusivity. In addition to (8), it is necessary that

the pressure field at r be essentially two-dimensional so that a can't be larg(-

-1 -1r <<a and z <<a '; furthermore, At is assumed to be sufficiently long

that the optical transit time across the source volume is << At.

From the considerations giving (6) as the far-zone approximation to

(5), it is evident that p(r,t) is proportional to (d/dt) (t - r/c, 0)

where the beam power & is evaluated at the shifted time which is r/c

earlier than t. Since & is intrinsically positive, the calculation of

(d/dt) 3N is only applicable to photoacouatios when N a 1/2.

Readers not interested in focused soatterina may omit Sea. IIC and IID

without loss of continuity.

28

6

C. Axially-focused backscattering from spheres

Asymptotic approximations to waves backscattered from large elastic2

and fluid spheres 12 have recently been formulated. The sumary given here2

reviews the connection with the half-order derivative previously noted. This

discussion also facilitates the analysis of the backscattering from a torus

given in Sec. IID. Amplitudes due to glory rays are enhanced relative to

those of other reflections because of a weak axial focusing. Glory rays are

defined to be backscattered rays which have non-zero impact parameters; Fig. 1

illustrates several such rays. The parameter n denotes the number of

internal chords. Figure 1 displays as dashed lines, rays which lie on either

side of the n = 3 glory ray. The wavefront associated with this glory ray

corresponds to the curved wavelet d'e'. This wavefront is toroidal and it

appears to emanate from a focal circle formed by rotating the point F3 about

the CC' axis. The wave is characterized by the focal parameters b, the radius

of the circle, and a, the distance of the circle from the dashed vertical

plane.

Let p1 exp(-iwt) denote the incident wave's pressure in the dashed

plane through C'. The scattered pressure due to each glory ray for points on

the backward axis (the left extension of the CC' axis) is computed in Ref. 2

for spheres having ka >> 1 where a is the sphere's radius, k z 2r/X,

X 2fc/w, and c is the sound speed in the surrounding fluid medium. It is

necessary that the distance z of the observation point from C' be such

that z3 x 1 b' . The scattered pressure of the nth glory wave is

p n a (pI a/2z) expik(z + jb2 z')]Gn (9)

Gn S (ka)i E B exp~(n e - *7)] (10)

E a 2b(2ffovq)* a"3 / 2 (11)

29

7

which correspond to Eqs. (17), (10), and (13) of Ref. 2, respectively;

z n z + a, B is the product of plane-surface reflection and transmission

factors for each vertex [Eq. (7) of Ref. 2), and the wavelet spreading factor

q is the limit of the ratio of the arc lengths of d'e' and de as the

latter vanishes. The phase n is the propagation delay along the glory ray.

The phase C = u - I is independent of ka (assuming ka >> 1); y is the

combined phase of the vertex reflection an transmission coefficients. (T is

usually 0 or r, the exception being elastic-sphere glories for the case in

which all the chords are shear wave and the angle of incidence exceeds the

longitudinal critical angle. 2 ) The phase U is -ir/2 times the number of

internal foci crossed. Three foci are crossed for the n s 3 ray and these are

labeled L and L in Fig. 1.

As a consequence of the (ka) factor in (10), Gn and pn diverge as

ka * =. This divergence is due to the axial focusing. The geometrical cause

of this focusing is discussed in Ref. 12-14.

The phases in (9) and n in (10) are = k; they give rise to a time

delay T for propagation. Lot pl(t) be the pressure of a transient wave

incident on the dashed vertical plane. When lei x 0 or w, the frequency

response in (9) and (10) is like that of (3) so the transient response is2

+a 3 / 2 EBPU(= W d_(t p I(t - T) (12)

2z cyn

where .(-) corresponds to Ie of 0(w). For intermediate values

of e there is an additional distortion and the pressure includes a term

given by the Hilbert transform2 '3 of this Pn (t).

The salient feature of axially-focused backscattering is the a3 / 2

proportionality in (12) whereas for simple specular reflection the

30

8

pressure = a. With appropriate focal parameters, (9) - (12) are applicable to

liquid-filled spherical shells as well as elastic and liquid spheres. The

internal absorption is assumed to be negligible. The cause of large target

strengths observed or computed for certain liquid-filled shells 15,16 is axial

focusing. This is evident since reported temporal signatures 15 ,16 show

enhancements (relative to the specular echo) at delays of glory rays. For

example, Fig. 7 and 8 of Ref. 15 display measured signatures and paths for a

liquid-filled shell having an acoustical refractive index M a 1.81. The

dominant echo is from an n 2 2 glory ray which has 2 internal foci. (There is

one of type L1 and one of type L2; see e.g. Fig. 1 of Ref. 13.)

Consequently 4 z -ir and (from the impedances) T 2 0 so that e x - r. The

(d/dt)* N given in Sec. III should describe the backscattering of sine-wave

bursts, however the data in Ref. 15 do not faciliate comparison.

Liquid-filled shells have potential uses as sonar calibration targets

and, due to their large target strengths, as passive navigational beacons.

For certain N values, G may increase even more rapidly than (ka) sol n

(12) is not applicable. One such case is the rainbow-enhanced glory

122/predicted to occur when M =1.18; in that case G (3c )2"3. Another

example is the backward focusing of the rear axial reflection which should

occur when M x 2. In that case, elementary diffraction and lens theory

predicts G2 a ka when ka >> 1.

D. Axially-focused backscattering from a rigid torus

Backscattered pressures proportional to (d/dt)* p, are not unique to

spheres. They should also exist for other shapes. In this section I extend

the treatment in See. IIC to a new case: backscattering from a large rigid

imovable torus.

31

9

Figure 2 shows the problem under consideration. The surface of the

torus is generated by rotating the center of a circle CT about the point C.

The radius a of the circle is assumed to be less than the distance b which

separates CT from C. In the analysis which follows it is assumed that

ka )> 1 and that kb >> 1. It is assumed that the direction of propagation

of the incident plane wave is -z.

The reflection of the incident wave from the rigid surface gives rise

to a virtual backward-facing toroidal wavefront shown as a dashed curve in

Fig. 2. The focal circle for this wave (which is at F ) lies a distance

m = a/2 behind the dashed vertical plane. This follows from geometrical

considerations similar to those 2,12 which locate the virtual point-like source

of the first reflection (from a sphere) at A0 in Fig. 1. The radius of the

focal circle is b and the spreading factor q is unity. The reflection-

transmission factor B is unity for the rigid torus; for real materials it is

the plane-surface reflection coefficient for normal incidence.

In this analysis, multiply-reflected glory rays, such as the dotted

ray in Fig. 2, will be neglected. Their contribution to the total

backscattering has been estimated, Eq. (4.7) of Ref. 17, and it is small for

the large torus considered here and especially for a << b.

Let the polar coordinates (relative to C') of the observation point

be r and y . A polar angle of Y = 00 corresponds to backsoattering.

Consider the far zone in which r >> kb 2 . The diffraction integral for this

single bounce glory wave is identical to one evaluated in Sec. III of Ref. 12.

When the pressure incident on the dashed vertical plane is P, eXp(-iut), the

scattered pressure is

pO a (p, b/r)Grs)1 JO(U)ei(kr e" ) (13)

32

10

where the argument Of the Bessel function J0 is u0 X kb sin y and

: - ka(1 - cosy)/2. The backiscttering cross section resulting from (13) is

equivalent to that for a "thick-wire loop" in the corresponding

electromagnetic problem. 17,18 The frequency response of (13) with Y x 0 is

like that of C3).

A more general result which includesthe near zone (z3 >> X-1b4) is

similar in form to Eqs. (9) - (12). The transient response is given by (12)

with the upper sign and with the aforementioned focal parameters. These2b /a' b2z -1 eez s

give: B = 1, E z 2bw1 /a, zn X z # *a, and e - z+ib 2 Here z is

the distance of the on-axis observation point from C'.

III. Half-order Derivative of a Sine-Wave Burst

The half-order derivative of Eq. (4) may be obtained from

L(t) = (d/dt s,(t) where s3(t) a Ht) sinwt is a burst of infinite

duration which comences at t = 0. To evaluate (1), notice that the second

term ofs9 i = H(wt) w coawt + siruat(d/dt) H(wt) vanishes since

(d/dt)H(wt) z w(wt) where 6 (wt) is a delta function; consequently

I(t) a wt *J(t-T)4 owr (14&)

for t > 0 and I(t<O) z 0. Changing the integration variable to

y z (2W/) 1 (t-T)* gives

I (t) a (2w)* H(wt)EC(u) cosut + S(u) sinwt] (15)

JooJ

where u x (2wt/) and C and S are Fresnel integrals. Define the

following auxiliary Fresnel integral function20

33

11

(u) a, - c)100 ,2) U 2 S(U)1n(! u2 ) (17)22

and Eq. (15) becoms

I(t) H((A)w*Esin(wt + ) - 2t g(u)J • (18)

It is well known that as u - C(u) , 5(u) - so that g(u) 0. It is

evident from (18) that for large wt, I. approaches a sine wave which is

Phase advanced by 45 deg and altered in magnitude by w These steady-state

properties are evident directly from (3). The expressions in (18) and (15)

are in agreement with tabulated semiderivatives listed in Ref. 7 and 8

respectively.

Equation ('4) may be written as s Ct) N M (t) ;3(t -'1 2irN). Here

and below, for the upper (lower) sign, N is integer (half odd-integer). From

the linearity of the half-order derivative operator, it follows that

(d/dt)*sNtt) = zCt) ; I=(t -w-127M). (19)

IV. Discussion

For discussing the implications of (19), it is convenient to introduce

the notations IN(t) (d/dt) sN (t) and J,(t) w - IN(t). Figures 3-5

display calculated JN for N a 4,1, and r, respectively. The calculations

used the fast rational approximation to g(u) given In Ref. 20 as Eq.

(7.3.33). The errors due to this approximation are estimated to be negligible

on the scale of these plots. When N is large, J. closely resembles

sin(wt +*w) except near its leadingj and trailing edges; this is especially

evident on plots (not shown here) comparing J. with sin(t + in). The

leading maxima and minima of J4 are 2 0.852, -1.032, 0.984, and -1.010.

From the form of (19), these apply to other '1i for normalized times

34

12

T z wt/2w < N. From series expansions 20 of the Fresnel integrals in (15), it

can be shown that as T - 0, J cc T r so that JN (t) vanishes with an

infinite slope at t a 0. For a viscous fluid, the leading edge of the

pressure must be rounded. For one-dimensional radiation into a viscous fluid,

21rounding of the leading edge has been modeled for the case of a mass

outflow - 3 (t).

We now consider some implications for the acoustical systems discussed

in Sec. 1I. For scattering responses describable by (12), the peak-to-peak

values of the response will closely approximate the steady-state value after

the first cycle. For example, the peak-to-peak values of j4 for the four

complete cycles are = 1.885, 1.994, 1.998, and 1.999 which are to be compared

to the.steady-state value of 2. Figures in Ref. 3 show that the central peak-

to-peak values of Hilbert transformed multicycle sine-wave bursts lie close to

2. These combined observations justify the use of measured peak-to-peak

pressures to confirm quasi-steady-state amplitude predictions for the acoustic

glory of elastic spheres.2

Figure 5 applies to the photoacoustic pressure radiated from a thin

light beam for which the power 4(t,O) - 3 (t) and the other conditions noted

in See. IIB hold. The maximum and minimum of J (t) are a 0.852 and -0.75,

respectively.

The frequency response (3) vanishes at zero frequency so the following

integral

AN " JN(t)dt (20)

must yield A a 0 for all w in (4). Numerical Integration of the

computations for Fig. 3-5 gave only an approximate verification of this

due to the nature of the algorithm used for S. It was desirable to check

35

13

Eq. (19) by analytic evaluation of AN. For integer N, the proof that A.=

0 is trivial because the upper sign is used in (19). For half-integer values

of N the proof requires an integration of g. For brevity, we consider only

the Case N = j because the proof here can be easily generalized to N a 3/2,

512 . . .; integration of the sine terms in (18) and (19) give

w2 -" A =1 - 2wO~g(u)dt (21)

where u = (2wt/w)*. The integral in (21) may be evaluated by first using

properties20 of the Fresnel integrals to put (17) in the following form

g(u) = Of e sin(y2)dy-. (22)

Inserting (22) into (21) and changing the order of the integrations yields

A s 0 where use is made of tabulated integrals.22 Equation (22) and the

requirement that the integral in (21) yield Aj x 0 are properties of the

auxiliary Fresnel-integral function g which appear to have not previously

been noted.

The part of j for T > N may be referred to as the "tail" or

"wake" of the response. The result AN = 0 requires the tail to be more

9prominent for N x 1/2, 3/2, 5/2 . . . than for integer N. It is well known

that pressure transients from two-dimensional sources vanish slowly as t .

Consider now the trailing edge for some related radiation problems. For

finite-length cylindrical sources with mass outflows of finite duration, the

radiated pressure will be of finite duration for a dissipationless fluid.23

Radiation from a Diston driven with a velocity 3 s1(t) has been modeled by

Beaver2 4 with the result that the pressure is of finite duration. The long

tail on j should be suppressed in real axially-focused scattering because

(12) is only applicable for spectral components having ka >> 1.

36

14W

A summary of this research25 was presented at the 105th meeting of the

Acoustical Society of America. This research was supported by the Office of

Naval Research and by an Alfred P. Sloan Fellowship held by P. Marston.

III

L 37

15

REFERENCES

1. J. Lighthill, Waves in Fluids (University Press, Cambridge, 1978), pp. 21,

86.

2. P. L. Marston, K. L. Williams, and T. J. B. Hanson, "Observation of the

acoustic glory: High-frequency backscattering from an elastic sphere,"

J. Acoust. Soc. Am. 73, 605-618 (1983).

3. B. F. Cron and A. R. Nuttal, "Phase distortion of a pulse caused by bottom

reflection," J. Acoust. Soc. Am. 37, 486-492 (1965).

4. A. D. Pierce, Acoustics. An Introduction to its Physical Principles and

Applications (McGraw Hill, New York, 1981); D. P. Hill, "Phase shifts and

pulse distortion in body waves due to internal caustics," Bull. Seism.

Soc. Am. 64, 1733-1742 (1974).

5. A. B. Arons and D. 1. Tennie, "Phase distortion of acoustic pulses obliquely

reflected from a mdium of higher sound velocity," J. Acoust. Soc. Am. 22,

231-237 (1950).

6. D. A. Sachs and A. Silbiger, "Fotusing and refraction of harmonic sound and

transient pulses in stratified madia," J. Acoust. Soc. Am. 49, 824-840

(1971).I 7. K. B. Oldham and J. Spaniar, The Fractional Calculus (Academic, New York,

1974), Chap. 7.

8. K. B. Oldham and J. Spanier, "The replacement of Fick's law by a formation

involving saomdifferentiation," J. Electroanaly. Cham. 26, 331-341 (1970).

9. L. D. Landau and E. M. Lifshitz, Fluid Hechanics (Pergamon, London, 1959)

Sacs. 70, 73.

10. C. K. N. Patel and A. C. Tam, "Pulsed optoacoustic spectroscopy of condensed

matter," Rev. Mod. Phys. 53, 517-550 (1981).

38

16

11. H. M. Lai and K. Young, "Theory of pulsed optoacoustic technique," J. Acoust.

Soc. Am. 72, 2000-2007 (1982).

12. P. L. Marston and D. S. Langley, "Glory and rainbow-enhanced acoustic

backscattering from fluid spheres: Models for diffracted axial focusing,"

J. Acoust. Soc. Am. 73, 1464-1475 (1983).

13. H. C. van de Hulst, "A theory of the anti-coronae," J. Opt. Soc. Am. 37,

16-22 (1947).

14. M. V. Berry, "Uniform approximation: a new concept in wave theory," Sci. Prog.

(Oxford) 57, 43-64 (1969).

15. B. M. Marks and E. E. Mikeska, "Reflections from focused liquid-filled

spherical reflectors," J. Acoust. Soc. Am. 59, 813-817 (1976); the reader

is cautioned that in Ref. 15, the term "focused reflectors" refers to

focusing of the "incident sound waves on the near back surface of the

sphere" but not to the axial focusing in sense of my discussion and those

in Refs. 2, 12-14.

16. D. L. Folds and C. D. Loggins, "Target strength of liquid-filled reflectors,"

J. Acoust. Soc. Am. 73, U47-1151 (1983).

* 17. J. B. Keller and D. S. Ahluvalia, "Diffraction by a curved wire," SIAM

J. Appl. Math. 20, 390-405 (1971); the reader is cautioned that here (and

in Ref. 18) the definitior of a and b are reversed from those in my Fig. 2.

18. J. W. Crispin and A. L. Maffett, "Radar cross-section estimation for

simple shapes," Proc. IEE. 53, 833-848 (1965); Eq. (29).

19. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions

(Univeristy Press, Cabride, 1958), p. 31.

39

17

20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions

(Dover, New York, 1965), pp. 300-304.

21. D. T. Blackstock, "Transient solution for sound radiated into a viscous

fluid," J. Acoust. Soc. Am. 41, 1312-1319 (1967).

22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and

Products (Academic, New York, 1980); Entry Number 3.8521.

23. W. J. Remillard, "Pressure disturbances from a finite cylindrical

source," J. Acoust. Soc. Am. 59, 744-748 (1976).

24. W. L. Beaver, "Sonic nearfields of a pulsed piston radiator," J. Acoust.

Soc. Am. 56, 1043-1048 (1974).

25." P. L. Marston, "Half-order derivative of a sine-wave burst: Applications

to scattering, two-dinensional radiation, and photoacoustics," J. Acouat.

Soc. Am. Suppl. 73 S99 (1983).

4

40

18

Figure Captions

Fig. 1. Backscattered rays from a sphere for which the acoustical refractive

index N = 0.6. For the 3 chord glory ray, the associated back-facing

wavefront d'e' is shown together with the locations (L1 and L2 ) of internal

foci. This wave is axially focused along the left extension of the CC' axis.

Fig. 2. Cross-seCtional view of the torus considered in Sec. lID. The figure

may be rotated about the CC' axis. The reflected wave is axially focused

along the left extension of the CC' axis when the incident wave propagates in

the -z direction. Scattering due to the dotted ray is omitted.

Fig. 3. The solid curve is the calculated jN = W 4 (d/dt) sN for the 4 cycle

burst given by the dashed curve. The normalized time is T = wt/2r.

Fig. 4. Like Fig. 3 but for N z 1.

Fig. 5. Like Fig. 3 but for N 2 1/2. For real photoacoustic sources, the

pressure transitions ac T - 0 and 0.5 will be smoother than j due to the

convolution over the finite optical beam area discussed in Ref. 11.

1

' • . . ... .. I. : 4 1

3

I 3

424

1.5

0.5

-ob.0 15 20 2...0. 40 4 ./ NORMAIZED TIME T

-'.5

Fig. 4

0.5

o10 1.5 2L0 2.5 3.0 3.5 4.0 4.5 5.0

0.5 NORMALIZED TIME T

-10

-1.0

111.543

Paper Number 3:

Accepted for publication in the Journal of the Optical Society of America

Submitted July, 1983

Uniform MIe-theoretlo analysis of polarized

and cross-polarized optical glories

Philip L. Marston

Washington State University

Department of Physics

Pullmn, Washington 9916J4-2814

Abstract

Expressions are derived from Nie theory for the cross-polarized and non-cross-

polarized measurement configurations useful for the study of the optical

glories of transparent spheres.

II

44

Optical backscattering from cloud droplets, 1,2 bubbles in liquids,3- 5

and other transparent spheres 6 is enhanced due to a weak focusing along the

backward axis. This enhanced scattering is referred to as the glory. Axial

focusing should also enhance the forward scattering from drops and from

5bubbles. The purpose of this letter is to relate certain observable

intensities with Mie theory1 '18 for the case of plane-polarized incident light.

A diagram of the type of experiment under consideration is shown in Fig. 1. A

sphere is illuminated with a beam propagating in the z direction having the

incident E field along the x axis. The near forward or backward scattering is

viewed via ideal polarizers which may be oriented to either block or pass I-

polarized fields. The polarizers lie in planes which are normal to the z axis

in the far-scattering zone. Previous descriptions of the transmitted

"intensities based on Mie theory (e.g. Eq. 6 of Ref.. 4) were nonuniform in that

they were applicable only near the forward or backward directions. The

description presented here is not restricted by choice of angle.

It is necessary to review certain results of Mie theory. The coordinate

system shown in Fig. 2(a) is equivalent to the one used in Ref. 1. At an

observation point Q the orthogonal unit vectors eree , and e are directed

according to the displacement of Q with infinitesimal increases in r, a,

and Cf, respectively. Let a denote the radius of the sphere and

k x 21/Xo, where X 0 denotes the wavelength in the outer media (which may be a

liquid in the case of scattering from bubbles). In the far zone (r >> ka ,

without a polarizer the scattered field is E5 Ede E4 where

Ke 3 fS 2 coas, E[ u - f S1 sinf, (1)

f a (-LE /kr) exp(lwt - ikr), and 3 and S 2 are the standard Mle amplitude

functions 1'9 which are to be computed for the appropriate scattering angle 8,

45

2

size parameter ka, and relative refractive index M of the sphere, and E is

the incident wave's amplitude.

The polarizers (e.g. sheets of idealized Polaroid) do not redirect the

scattered waves; consequently the local transmitted field Ej ej is orthogonal

t.o e . The values for the parameter j are defined as follows. Consider the

effect of polarizer 1 on a z-directed wave. If it is oriented to perfectly

transmit E fields along e while perfectly absorbing those along ey, then-X -

j z 5. If it transmits E fields along aey while absorbing those along e,

then j a 6. (This convention for j is justified below.) These definitions

apply also to the effects of polarizer 2 on a wave directed along the -z axis.

The unit vector a5 my be constructed by removing from that part of

which lies along er It is

e h hla e (e Xe) h). -3~ew5 5 r x .r Sbx- mor-x-x

S h (3 snecoS0 er), (2)5 WX -

where h5 a |eX X e|-1 • El - (sine cosr)23 1/ 2 and the right side of (2)

follow from expressions 10 for ex in terms of the e , jfe Likewise, w6 may

be constructed by removing from e that part of lying along er . It is

. h e x (e Xe,) h(0 - % 3le (3)P6 6 wr wy to r h6 jy siws1r)

where h a le x E1 l - (sinesnf) " " 2 . n Fig. (2a),W6 IS shown.

h6 a y air *(-I ig sso

Notethat for 8 ,0 or 8 z that 25 "me and em"e

For each case the propagating transmitted field becomes

E"I ",'e " It Is convenient to define amplitude functions 3 such that

m f S31 0 a 5,6, (4)

where the choice of values for j were selected to avoid confusion with the

unrelated amplitude functions S3 and 3 (which vanish for spheres). It is

46

3

convenient to express the 3 in terms of the standard amplitude functions S1

and S2 via Eqs. (1) - (3). This procedure gives

S5 a h(S 2 oso 2 + S1 sin ) (5)

$6 2 h 6 snvcosMP (S2 cose - SI ) . (6)

In agreement with elementary considerations, CP: zt/2 gives 55 2 S3;

also f = 0 or w give S5 x S2 in the forward hemisphere and 5 :- S2 in

the backward hemisphere.

The amplitudes S6 and 35 are associated with 'cross-polarized" and

"non-cross-polarized" scattering measurements, respectively.3 (This definition

of cross-polarization is not universally held.) When describing their

associated intensities, it is convenient to use a normalization factor I xr

I a2/ Ar 2 where I is the incident intensity and Ir is the intensity at a

radius r >> a for a perfectly reflecting sphere of radius a (as predicted by

naive ray optics). The intensity transmitted by a J-oriented polarizer is

I I r41S 12 (ka) "2. (7)

For observations made In the backward hemisphere with the apparatus illustrated

in Fig. 1, the (partial transmission) losses due to the beam splitter reduce the

fields from those Liven by (1). The final result (7) is still applicable if Ir

is interpreted to be the perfect reflector result with losses and the

polarization dependence of the losses are negligible. A similar interpretation

facilitates the application of (7) to scatterers in liquids viewed via windows

with detectors (e.g. cameras) placed in air. In the latter case, the 6 In (5)

and (6) becomes the refracted scattering angle while the S1 and S. are

evaluated for a scattering angle within the liquid.

47

14

Equations (5) - (7) are. the principal results of this analysis. The

discussions which follow concern their application to backward and forward

scattering from large transparent spheres.

To facilitate discussion of the backscattering it is convenient to use

the angles illustrated in Fig. 2(b); S1 "and S2 are evaluated at

e = r -y. Equations (5) and (6) may be written as

S5 = h s" - S* cos21f +C( - cosy)2S2 cos , (8)

S6 - jh6 sin2cp ES* - (1 - cosY)S2 ], (9)

9where S 2 S S appear naturally in the computation Of His scattering,1 2

h5 = E1 - (siny cos) 2 1 /2,and h6 x E1 - (siny sn) 2 "1/ 2. The previous

'4approximations for polarized and cross-polarized Hie scattering are given by

replacing the h3 by unity and omitting the terms proportional to (1 - cosy).

Backward axial focusing is only significant for quite small values of Y. The

angles of interest have y < 5" for which (1 - cosy) < 0.004 and the

approximations introduce negligible errors. Evidently the errors should be

largest for the cross-polarized case as y - 0, since for spheres S * 0

though 52i 0. (Representative plots of I6 for bubbles (in which S2 was

omitted from (9)) are given in Ref. 4 and 5; the first maximum of IS+Iwas found

to be near y of X /a rad. which is << 5e in cases of strong backscattering.

In those papers the azimuthal angle ( is denoted by 9.3 The exact and

approximate 16 vanish for y a 0 and for 4 x 0, ±W/2, and 7r, these

symmetries are evident in photographs of cross-polarized glory from bubbles;3

1 6 mMifests the glory4 because that part of e due to unfocused reflections

is negligible when Y .

For consideration of forward scattering It is convenient to rewite (5)

and (6) as

48

35 h5 S* - S" co2qP- (1 - cos S2 cosp, (10)

S - h6 31n24? 3"+ (1 - cose)S2 , (11)

which show that the dependencies on S* and S" are reversed from those in (8)

and (9). When 8 a 0, 16 vanishes since 1 a 3 2 and S z 0. For spheres

large enough for axial focusing to be significant (ka > 200), 15 and 16 are

of similar magnitude in near backscattering but not in near forward scattering.

The diffraction contribution to near forward scattering is (Ref. 1, p. 210)

(S1)d a (S2)d Z *(S*)d a (ka) a " 1(a)

where a x ka8. For comparison, the leading contribution to S- [which from

(11) is evidently the intrinsic cross-polarized scatteringJ must increase less

rapidly with increasing ka; for example, transmitted toroidal wavefronts1 '3

contribute to S" (and to 3 ) in proportion to (ka)3/2 . Due to the relatively

large magnitude of ($2)d' errors due to omitting the term proportional to 32

should be more significant in (11) than in (9). Forward-directed toroidal

wavefronts should manifest themselves in measured 16; due to the non-ideal

aspects of polarizers and sources, care should be taken to remove (via a spatial

filter) the strong unsCattered beam.

This work was supported by the Office of Naval Research and by an Alfred

P. Sloan Foundation Fellowship.

I

49

REFERENCES 6

1. H. C. van do Hulat, Light Scattering by Small Particles (Wiley, New York,

1957).

2. V. Khare and H. N. Nussenzveig, "Theory of the glory," Phys. Rev. Lett.

38, 1279-1282 (1977).

3. D. S. Langley and P. L. Marston, "Glory in optical backscattering from air

bubbles," Phys. Rev. Lett. 47, 913-916 (1981).

4. P. L. Marston and D. S. Langley, "Glory in backscattering: Mie and model

predictions for bubbles and conditions on refractive index in drops,"

J. Opt. Soc. Am. 72, 456-459 (1982); In the final section, the equation

pT - -1 should be pT - 1.

5. P. L. Marston, D. S. Langley, and D. L. Kingsbury, "Light scattering by

bubbles in liquids: Mie theory, physical-optics approximations, and

experiments," Appl. Sci. Res. 38, 373-383 (1982); P. L. Harston, "Light

scattering by bubbles in liquids: comments and application of results to

circularly polarized incident light," Appl. Sd.. Res. 40, 3-5 (1983).

6. J. J. Stephens, P. S. Ray, and T. W. Kitterman, "Far-field impulse

response verification of selected high-frequency optics backscattering

analogs," Appl. Opt. 14, 2169-2176 (1975).

7. H. H. Nussenzveig and W. J. Wiscombe, "Forward optical glory," Opt.

Lett. 5, 1279-1282 (1980).

8. G. Hie, "Beitrage zur Optik trUber Medien, speziell kolloidaler

Metallsungn," Ann. Phys. (Leipzig) 25, 377-445 (1908). [English

Translation No. 79-21946 (National Translation Center, Chicago, 1979)].

9. W. J. Wiscombe, "Improved Mi. scattering algorithms," Appl. Opt. 19,

1505-1509 (1980).

10. G. Arfken, Matheatical Mathods for Physicists, 2nd ed. (Academic, New

York, 1970), p. 84.

50

---'"--POLARIZER I

"4SPHERE

BEAM \ LASERSPLITTER BEAM

-- -POLARIZER 2

Fig. 1. ethod for observing cross-plarized and non-ofs-polaritzed

scattering sitlar to thate used in Rea. 3 for the backward ilory. The

, - diameter of" the incident beam is assumed to be much largier than the sphere'ts

diameter. The cameras are to be focused on nensty d that the photographs

record the far-zone scattern. her detectors my be used in place of

cameras.

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X 0

(a) Ei r

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point Q In the forward hemisphere (a) and backarad hemisphere (b). The

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z axes such that they Interrupt the lines OQ when r is largie.

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