UNCLASSI FIED

AD 0 i046

DEFENSE DOCUMENTATION CENTERFOR

SCIENl4FIFC* AND TfICHNICAL INFORMATION

CAME1OiH STATIOi. ALEXANDRIA, VIRGINIA

UNCLASSIFIED

NOTICE: When goverment or other drawings, speci.ficati~ons or other data are used for any purposeother than in connection with a definitely relatedgovernmt procurment operation., the U. S.

Govermaent thereby incurs no responsibility, nor anyobligation whatsoever; and the fact that the Govern-ment may have fozuilated.# furnished, or in ary waysupplied the said. drawings, specifications, or otherdata is not to be regarded by Implication or other.wise as in aW manner licensing the holder or anyother person or corporation, or conveying aW ri~itsor permission to manufacture,, use or sell anypatented invention that may in any way be relatedthereto.

"C - eta g..t --

•': WADD TR 61-42

STUDY OF ULTRASONIC TECHNIQUES FOR THENONDESTRUCTIVE MEASUREMENT OF RESIDUAL STRESS

TECHNICAL DOCUMENTARY REPORT NO. WADD TR 6142, Pt. IIImay 1963

Directorate of Mateiais and ProcewsAeronautical Systems DivisionAir Force Systems Command

Wright-Patterson Air Force Base, Ohio-~ 0L C

Project No. 7360, Task No. 73606 4i

(Prepared under Contract No. AF 33(616)-70M5by the Midwest Research Institute, Kanwsa Chy, Misouri;

Fred R. Rliawns, Jr., and Peter WUldow, authors.)

NOTICES

When Government drawings, specifications, or other data are used forany purpose other than In connection with a definitely related Governmentprocurement operation, the United States Government thereby incurs no re-sponsibility nor any obligation whatsoever; and the fact that the Govern-ment may have formulated, furnished, or in any way supplied the said draw-ings, specifications, or other data, is not to be regarded by implication orotherwise as in any manner licensing the holder or any other person or cor-poration, or conveying any rights or permission to manufacture, use, or sellany patented invention that may in any way be related thereto.

Qualified requesters may obtain copies of this report from the ArmedServices Technical Information Agency, (ASTIA), Arlington Hall Station,Arlington 12, Virginia.

This report has been released to the Office of Technical Services, U. S.Department of Commerce, Washington 25, D.C., for sale to the general public.

Copies of this report should not be returned to the Aeronautical SystemsDivision unless return is required by security considerations, contractualobligations, or notice on a specific document.

This report was prepared by Midwest Research Institute under UWA

Contract No. AF 33(616)-7058. The contract vas Initiated under Project No.7360, "The Chemistry and PkysicI of Materials," Tas No. 73606, "Nondestruc-tive Methods." The work vas administered under the Directorate of Materialsand Piceses, Deputy for Technolog, Aeronautical Systems Division, withMr. Harold Km acting as project engineer.

This report covers work conducted fro 1 February 1962 to

31 January 1963.

The vork vat performed under the direction of Mr. Fred R. Rollins,Jr. Other personnel directly involved in prosecution of the research havebeen Messrs. Paul Outshall, Iqrle Taylor, L. S. Srinath, and Peter Waldov.

The theoretical and ewermental Investiation of utrasonic beoointeraction in solid materials has been contined. Intersection of pulsedbeans (3 - 15 no/a) urLer "resonant" conditions reveals that interaction does-occur in nAW materials. A theoreticaly predicted third beum is generatedat the "point" of intrsection and has been e~erimentally observed in sam-pies of fuse silica, polyarystalline alumima, and polycrystalline magnesiu.A potential method of three-dimensional stress analysis is discussed. Anoptical system for Mtudying beam interaction in transparent solids is alsodescribed.

This technical report has been revieved and is approved.

Chief, Strength and Dynamics BranchMetals and Ceramics LaboratoryDirectorate of Materials and Processes

i1i

TAME OF COMM

PAGE

I. IINRODUCTION ..... . .. .. .. ... .. ... . ... .. 1

II. STRESS-INDUCED 'TMRINGCE .................. I

III. GRAIN BOLUDARY SCATTERIN FROM A SINGIE UIEAUSONIC MEAM4 .... 4

IV. INTERACTION OF TWO UIERASONIC WAVES .............. 6

A. G UMAL TEE.OR ...................... 6B. INTENSITY OF INTERACTION ................. 10C. EPERUIMTAL ]DETAILS AND RESUIS . . . ... 18

V. VISUALIZATION OF UI, RASONIC WAVE PACKITS .... ............ ... 27

A. INSTRtMENTATION .............. ..................... 27B. PHOTOGRAPHIC MEMUIS ............................. ... 30

VI. THREE-DIMENSIONAL STRESS ANALYSIS WITH UtIERASONICS . .... ...... 33

APPENDIX A - AMPLITUDE EXPRESSIONS FOR SCATTERED WAVES ..... ........ 35

BIBLIOGRAPHY ............. ............................. ... 39

iv

FIGOUR TIM PAM

1 DISK I== IN C IPSSIO1N POR STtIY OF BIAXIAL B . . . 5

2 SCEDIATIC DIAGRAMI OF (a) TIST K= W= TRANSIX , AND(b) SECTIO SXWINO APPAsRM PATH OF PRIUM WAVE ANDSCATT S ......... . . . . . 5

3 AMPLITtUB FACTORS P0M TIB IfFEACTICU OF TWO TRANSVERSEWAVES TO PRIX A scAT-DaRD WIE~fUDINA wAVE (c+szI-A). OT PRnm WAVS POam PERPDICU TO

fife Pium . .. .. .. .. .. .. .. .. .. .. .. . 3.4 AMPLIM FACTORS FOR THE IN'•ERACTICK OF TWO TRANSVERSEWAVES TO PRODUCE A SCATTERED T WAVE (CASEI-B). DYBM PRIMAY WAVES POLIARUIM PARALlEL TOf 1k PAHE . .. .. ..... . ...... ........ ................ 12

5 A14P1LTU FACTORS PMR THE nTAICTIO OF TWOI~0TUIAWAVES TO PRODUC A SCATTERED NSM WAVE (CASM II) . 13i

6 AMPLITLD FACTORS FOR TIM IMTERALTICH OF A IM1GITMUINALAND A WRANSVEE WAVE TO PRODU E SUA M 1M UMCYWONGITfl•IAL WAVE (CAMS In-A). THE PRIMBY TRANS-VERSE WAVE IS POIARIMZD PARALLEL TO TM Elk PLIAE.. . 14

7 A1PGLITU FACTORS FOR THE INTMECTICW OF A IGITUDINALAND A TRANVERS WAVE TO PROD= A Dn, CM FRMAL-ECYWITinW)IKAL WAVE (CASE IV-A). TM PRIARY TRAMNVESESWAVE IS POLARIED PARALUEL TO TIE FIk PIANE......... 15

8 AMPLUE FACTORS 101 TB INERA.CTICW OF A IMGITMIUDIALAND A TRANSVERSE WAVE TO PRODXUE A DIFPMM FM "WCYTRANSVERSE WAVE (CASE V-A). TIM PRIA"Y TRANSVERSE WAVEIS PORI PERPMDICuMAR TO THE Elk PAM ... . . . . 16

9 AMPLIUZ FACTORS FOR Tim nmEwR oN OF •IMITMINALAND A T WAVE TO PROMME A DIFXMC IMQX3W. YSWAVE (CASE V-B). TB PRMARY." TPANS SE WAVEIS POLM:I PARALLEL TO T E112 PIE . ......... 17

10 Bt=CK DIAGRAM OF MST PICE AND ASSOCIATED*S . 19

v

LIST OF ILLUTRATIONS

FIGURE TITLE PAGE

1 DISK IDADED IN COMPRE31ON FOR STUDY OF BIAXIAL STRESS 3

2 SC=4ATIC DIAGRAMS OF (a) TEST BU)CK WITH TRANSDIER, AND(b) SECTION SHOWING APPARENT PATH OF PRIMARY WAVE ANDSCATTERED SIGNAL ........ ........................ 5

3 AMPLITUDE FACTORS FOR TBE INTERACTION OF TWO TRANSVERSEWAVEWS TO PRODUCE A SCATTED LONGITUDInAL WAVE (CASEI-A). BOTH PRIMARY WAVES POLARIZED PERPENDICULAR TOEjl 2 PLANE. ............... ........................ 1.

4 AMPLTUDE FACTORS FOR THE INTERACTION OF TWO TRANSVERSEWAVES TO PRODUCE A SCATTERED LONGITUDINAL WAVE (CASEI-B). BDTH PRIMARY WAVES POLARIZED PARALLEL TOF 1 2 PILANE ............. ........................ ... 12

5 AMPLITUDE FACTORS FOR THE INTERACTION OF TWO ONGITUDINALWAVES TO PRODUCE A SCAOTTERED TRANSVERSE WAVE (CASE II) • . 13

6 AMPLITUDE FACTORS FOR THE INTERACTION OF A LONGITUDINALAND A TRANSVERSE WAVE TO PRODUCE A S==4D FREQUENCYIDNGITUDINAL WAVE (CASE III-A). THE PRIMARY TRANS-VERSE WAVE IS POLARIZED PARALLEL TO THE FiF2 PLANE... 14

7 AMPLITUDE FACTORS FOR THE INTERACTION OF A LONGITUDINALAND A TRANSVERSE WAVE TO PRODUCE A DIFFERENCE FRQUENCYLONGITUDINAL WAVE (CASE IV-A). TBE PRIMARY TRANSVERSEWAVE IS POLARIZED PARALLEL TO THE F172 PIANE ...... .... 15

8 AMPLITUDE FACTORS FOR THE INTERACTION OF A LONGITUDINALAND A TRANSVERSE WAVE TO PRODUCE A DIFFERENCE FREQUENCYTRANSVERSE WAVE (CASE V-A). THE PRIMARY TRANSVERSE WAVEIS POIARIZED PERPENDICULAR TO THE E172 PIE .......... 16

9 AMPLITUDE FACTORS FOR THE InERACTION OF A .LONGITUDINALAND A TRANSVERSE WAVE TO PRODUCE A DIFFMOCE FRQUMCYTRANSVERSE WAVE (CASE V-B). THE PRIMARY. TRANSVERSE WAVEIS POlARIZED PARAIlZL TO TB1 E E2 PlAE ............ 17

10 BlOCK DIAGRAM OF 1TE PIECE AND ASSOCIATED EEICTRONICS : . . 19

V

ILST OF ILLUSTRATIONS (Concluded)

FIGURE TITLE PAGE

11 OSCILLOGRAM SHOWING SIGNAL PRODUEED BY SCATTERED WAVE . . . 21

12 DIAGRAM SHOWING LOCATION OF SCATTERED SIGNAL FORDIFFERENT TRANSDUCER ARRANGEMENTS ......... ............ 23

13 INTIEACTION OF LONGITUDINAL AND SHEAR WAVES IN SPECIMENOF POLYCRYSTALLINE 6061-T6 ALUMINUN ALLOY ... ....... ... 24

14 DISK SHAPED SPECIMEN FOR STUDY OF UITASONIC BEAMINTERACTION .............. ........................ ... 26

15 OPTICAL SYSTEM FOR THE OBSERVATION OF UILRASONICWAVE PACKETS. ................. ....................... 29

16 SEQUENCE OF PHOTOGRAPHS SHOWING UIWRASONIC PULSETRAVELING THROUGH GLASS SPECIMEN. . ...... ............. 31

17 INTERSECTION OF TWO 7--mc. SHEAR WAVE PACKETS IN A GLASSSPECIMEN .................... ......................... 32

18 SCHEMATIC DIAGRAM OF POTENTIAL STRESS ANALYSIS TECHNIQUEUSING REAM INTERACTION TO PRODUCE SEMER WAVES .... ...... 34

4 vi

LIST OF TABLES

TABLE TITLE PAGE

1 INRACTION CASES WHICH PRODUCE A SCATTERED WAVE ..... 7

2 INTERACTION CASES THAT CAN BE STJJDM D ON SPECIMEN SHOWNni FIGURE 14 ............. . 25

vii

I. INTRODUCTION

During the past several years Midwest Research Institute has beenengaged in a study of certain stress-dependent aspects of ultrasonic propaga-tion in solid materials. A t~horough understanding of these stress-dependentproperties should greatly facilitate the ultimate use of ultrasonics in alltypes of stress measurement problems including the nondestructive measurementof residual stress.

The propagation velocity of ultrasonic waves in solids varies withstress in a manner that depends on the third-order elastic constants. Inaddition, shear wave velocity depends upon the angle between the polarizationaxis and the stress direction. These properties can combine to produce aneffective rotation of the polarization axis as a shear wave travels througha stressed medium. Techniques for measuring these stress-dependent prop-erties have been described in previous reportsl_2/* along with experimentalresults for several different materials.

The stress-dependent rotation of shear wave polarization is verysimilar to certain photoelastic effects. It also suffers many of the samelimitations. One such limitation is that the total rotation observed is theintegrated effect of travel through a given thickness. Thus, stress varia-tions through a given thickness may produce a canceling effect and giveresults that are identical to those obtained in a stress-free medium. Thisproblem has been solved in the case of photoelasticity through the develop-ment of a three-dimensional techniqueV. which depends on certain peculiaritiesof scattering from a polarized light beam. A somewhat analogous technique isnow theoretically possible using ultrasonic waves and the results describedin this report.

II. STRESS-INDtUED BIMUFPRINGENCE

In the following sections of this report several areas of work aredescribed relative to the use of ultrasonics in a technique of three-dimen-sional stress analysis. As a prelude to this study it was felt that addi-tional information was needed about the behavior of shear waves in biaxiallystressed samples. In earlier work we had limited our birefringence experi-ments to uniaxial stress conditions. Results of the earlier workLk /

Manuscript released by the authors February 1963 for publication as an ASDTechnical Report.

*Numbers refer to references in Bibliography.

I

indicated that AV/V was linearly proportional to the uniaxial stress. Anexperiment was therefore set up to determine the relationship between AV/Vand P-Q , the difference between the principal otresses in a biaxiallystressed specimen.

The specimen shape and loading conditions are shown in Figure 1.The 8-in. diameter disk was 5/8 in. thick and made from 6061-T6 aluminumplate. Values of AV/V were determined at several different locations onthe face of the plate using the pulse-echo technique described previously.l/The compressive load was applied along a diameter of the disk either parallelor perpendicular to the rolling direction of the aluidinum stock. The aniso-tropic rolling texture of aluminum produces a bieUefringence effect withprincipal axes parallel and perpendicular to the rolling direction. With theloading parallel to one of these principal axes, superposition of the originalstate (anisotropy due to preferred orientation) and the stress state producesa rotation of the resultant principal axes at all points on the face of thedisk except along the two diameters that are parallel and perpendicular toloading. It was along these two (primary) diameters that measurements ofAV/V were first made.

The value of (P-Q) at any point (x,y) on the disk can be cal-culated using the expressionS/

p-Q = . 4rF(x2 +4y2 -r 2 )

rr[I(x2+y2-r2) 2 + 4r2x23

where r is the disk radius and F is the diametral load. The results ofmeasurements at the center of the disk and one additional point on each ofthe two primary diameters conclusively showed that AV/V does vary linearlywith (P-Q) .

In addition to measurements along the primary diameters, we madeseveral attempts to verify the stress state at other points on the face of thedisk. As mentioned before, the principal stress axes at these points rotatewhen the load is varied. Thus, the value of (P-Q) as well as the direc-tions of the principal axes varies from point to point and from one load toanother. The fact that the axes at a given location rotate as the load isvaried makes it necessary to rotate the transducer in order to evaluateAV/V . Another problem arises because the transducers cover an area overwhich considerable variation may occur in both (P-Q) and the principalaxes' directions. These factors made it difficult to obtain clean pulse-echo patterns of the type used in accurate evaluations of A/V .

2

x

Figure 1 - Disk Loaded in Compression for Study of Biaxial Stress

III. GRAIN BOUNDARY SCATTING FROM A INIGIE

UILERABQUC AMM

The "scattering" technique-/ of three-dimensional photoelasticityhas recently been refined to the point of being a very useful stress analysistool. We will not go into the details of this technique, but a cornerstoneof the whole procedure is the fact that polarized light waves produce maximumscattered radiation in directions perpendicular to the E-field vector, andminimum scattered intensity parallel to the E-field vector. It was felt thatsome of the techniques used in three-dimensional photoelasticity might beextended to the use of polarized shear waves and thus permit stress analysisin structural metals as well as transparent models. It was first necessaryto study beam scattering in polycrystalline metals and determine if a spatialdistribution exists similar to that exhibited by polarized electromagneticradiation.

Our investigation of beam scattering has been limited to ge*-.tricarrangements similar to that shown in Figure 2. Initial experiments involvedtwo 5-inc. AC-cut quartz crystals mounted on a block of aluminum as shown inFigure 2(a). The arrows on the transducers denote polarization direction.The small electrical signals produced by the receiver crystal were amplifiedand displayed on a CRT whose sweep was initiated when the transmitter waspulsed. With this arrangement we were able to determine the time delay be-tween the transmitted pulse and receipt of the scattered signals. It was notsurprising to find that a period of "quiet" followed the sweep initiation,but it was rather surprising to find that the first significant signalsoccurred at a time that corresponded to shear wave propagation along the path(de+d2 ) as shown in Figure 8(b). This fact was verified at several valuesof dl and d2 . A few very weak signals are sometimes present between toand t(dl+d2) but they are usually much weaker than those occurring at

t(dl+d2 ) . We believe the earlier signals come from beam spreading, mode

conversion, and multiple scattering which may permit a more direct path of

travel between the two transducers.

The magnitude of the first strong signal is very sensitive toslight translations of either the transmitter or receiver crystals. This isprobably due to variations in the grain structure that produces the scatter-ing. The lack of reproducibility in the received signal amplitude made itimpossible to observe any systematic variation in scattered intensity as thepolarization direction of either crystal was varied. Thus, we have beenunable to definitely pinpoint a spatial variation in the energy scatteredfrom a polarized shear wave.

4

TRANSMITTER

Il

RECEIVER-I

-- 4v

(a)

Wave~ ~ an Scated Sim

Iz ZaZld

(b)

Figu~re 2 - Schematic DIlagrszs of (a) Test Block With Transducers,and (b) Section Shoiiing Apparent Path of Primr

Wave and Scattered Si•ia

5

The study of beam scattering was next extended to include variouscambinations of X-cut and AC-cut transmitter and receiver crystals. In eachcase, the resonant frequency of both crystals was the same. The magnitude ofthe received signal was much greater when AC-cut crystals were used for bothtransmitter and receiver than any other combination of AC-cut and X-cutcrystals.

In the previous discussions we have been interested only in the in-tensity of the first signals to reach the receiver crystal. Subsequent sig-nals are numerous and ultimately run together to form an almost continuous"noise" along the CRT sweep. Multiple reflections, beam spreading, and modeconversions give rise to this relatively high intensity noise which finallydies out as the energy of the primary wave is dissipated.

IV. ITERACTIC OF TWO UlaRASONIC WAVES

A. General Theory

The details of the theoretical analysis of elastic wave interactionin nonlinear solids were given in a previous report2_ and therefore will notbe repeated here. Let it suffice to say that an interaction between twowaves is predicted and a "scattered" wave of combination (sum or difference)frequency theoretically originates at the region of intersection. The in-tensity of the scattered wave is expected to be a maximum when so-calledresonant concitions exist. Table 1 summarizes the various types of inter-action predicted by theory and gives expressions for (cos rp) where • isthe angle between the prinary beams at resonance.

in reference 2, it was pointed out that from a quantum mechanicalstandpoint the phonon-phonon interaction should give rise only to phononswith the sum of the frequencies of the primary phonons due to conservation ofwave vector (3 - 1i+12) and energy (w3 - wI+w2 ) . It will be shown in

the following discussion that the classical treatment which predicts a scat-tered wave with the difference frequency (wl-w2 ) also has its quantum

mechanical analogy. The argument is based on a paper by Slonimskii5J onsound absorption in solids.

It seems obvious that, if it is possible for two phonons to combine

to give a third one of frequency w5 - wl+w2 , the reverse process can occur

as well, i.e., one phonon of frequency "5 may split into two under the

condition: w3 - cl + (us3 -. 1 ) . If (w3 -w1 ) is set equal to w2 , the

6

C4

ri rI r4cu c

0 i

LJL..J +

Ht

u uu

00

4204+

020 04-

iH H HiH0

.7u

equivalence of the two relations is clear. If the wavelengths or the phononsinvolved are sufficiently greater than the lattice spacing so that dispersionand tMklapp processes need not be considered, it is easily seen from the con-servation laws that only longitudinal phonons may split into two phonons.The resultant phonons ray both be transverse -r one may be longitudinal andthe other transverse.

The elastic waves are represented as a sum of harmonic waves

with A.O5 it (a.,e +arxe )()

where A is a unit vector in the direction of polarization, a. is theamplitude, and . the propagation vector. If N,, represents the number ofphonons in a particular mode a , the matrix elements of the displacementoperator, according to the theory of the harmonic oscillator, are:

<Na-ljadj'Nd> Ul -W2

(2)

<N5It4IN,-l1> e _

where m is the mass of' the volume of interest. We are interested in thetransition probability from a state with number of phonons N, , N2 , N3 toa state with number of phonons Nl-l , N2 +1 , N3 +1 . The part of the elasticenergy expansion containing cub c strain terms is taken as the perturbingHamiltonian H. It is shown• that H. can be evaluated to give:

41) -V'#.() , 3)r3 (3)

for the decay of a longitudinal phonon into one longitudinal and one trans-verse phonon, and

8

for tne decay of a longitudinal phonon into two transverse phonons. A andB are third-order elastic constants and a , b , c , d , f are expressionscontaining the wave vectors and polarizations, i.e., a = (1 1 ,1 2 ,k 3 ,elA 2 ,t 3 ).The matrix elements are then

< NI,N2,N N1ý(1) NI-I,N2+I, N3+I>

- i[2B(a+b)+2Ac]L ) (("+ / e(i[N'+l') t (5)

for the cab.- (3) where K = kl--k2-_kZ3 , and f Wl-w2-w3 . Integrating overthe volume V and calculating the transition probability per unit timeaccording to perturbation theory one obtains:

W(l) =2ft NI(N2+I)(N3+I) 2ei' d~r]2 6

( 2i) N [2B(a+b)+2Ac] 2 6(0)[ / d r (6)(~3 w1w2w3 V

The volume integral goes rapidly to zero for K ý 0 if V ls large comparedto the wavelengths involved. When K = 0 , we see that W( is propor-tional to V2 . Using expression (4) a similar expression is obtained forthe transition probability of a longitudinal phonon into two transversephonons.

It is now apparent how the generation of an elastic wave of fre-quency (wl-w2 ) , as predicted classically, can also be predicted from aquantum mechanical standpoint. Furthermore, the quantum mechanical treatmentallows an easy calculation of the intensities. Equation (6) shows that thetransition probability is proportional to the number of phonons involved inthe process. Now, consider the state 1NlN 2,N,> , from which the transi-tion to the state INI-l,N2 +1,N3 +÷> will be made. Note that the transitionprobability is enhanced by a large N2 just as it is by a large N1 . Inother words, if two elastic waves with frequencies (wI > w2) intersectunder the right conditions as outlined above, the interaction results in areduction of the intensity of the wave o1 . The amount is partly added to

9

the intensity of the wave w2 and partU. gives rise to a third wave of fre-quency (il-o 2 ) . One might say that the second wave stimulates the split-ting of the phonons of the first wave. This "moser"-lile interaction mayhave some further significance.

The correspondence between the classical and quantum mechanicaltreatment of two interacting elastic waves in a solid now appears morereasonable. Both treatments lead to the same conditions for the angle ofintersection of the two waves to produce a third one of either the sum ordifference frequency, and also give the right polarizations of the threewaves relative to each other. It is expected further that, for the sameintensities of the two primary waves, the intensity of a scattered wave withfrequency (wl-w2) is less than the intensity of a scattered wave with fre-quency (wl+w2 ) , since in the latter case two phonons combine to give athird one corresponding to the transition from a state INI,N 2 ,N 3> to astate .Nl-lN2-lN3+l>

B. Intensity of Interaction

Using the techniques described in a previous report, expressionshave been derived for the amplitude of scattered waves from all of the pos-sible interaction cases. These expressions are given in Appendix A. Themost favorable conditions for detecting beam interaction should occur whenthe amplitude of the scattered wave is maximum. Obviously the maximum shouldoccur under conditions of resonance, but it should be remembered that reso-nance can occur over a finite range of (a) values. The amplitude expressionsshotn in Appendix A were therefore programmed for an IBM 1620 computer andnumerical results were obtained for various values of the frequency ratio,(a) , and for most of the materials for which third-order elastic constantsare available._,7/

Figures 3 - 9 are plots of amplitude factors for four different

materials in each of the seven cases that yields a nonzero amplitude expres-sion. The amplitude is directly proportional to the factor y throug~h theexpression

amplitude ( f x 1kf) (7)

where X1 and X2 are the amplitudes of the primary waves, fl is the fre-quency in mc/sec, V is the volume of intersection, and R is the distancebetween the "point" of intersection and the point of observation.

10

104 1

0..0ý .0 .=0 _0 __0 _0_ 0 _0..0 --00 0 "0opolyst '

103 3. 0•' Copper

Ora x..x Pyrex

o/ x~..~x

I', 102 - .A..A Iron

102 i1 1"3 /4O

II /!

Al

II'* I

II

100 II

10"1 I

0 0.2 0.4 0.6 0.8 1.0

(a)Figure 3 - Amplitude Factors for the Interaction of Two Transverse Waves

to Produce a Scattered Longitudinal Wave (Case I-A). BothPrimary Waves Polarized Perpendlciilar to 1Ri2 Plane.

11

104 I I I

, oPolystyrene

~ /101 0b • Cpper

,O Iron0 ).,.A , X .•X,,,,.X( Py•r ex

102 A 100 • l

100

ZO ZI I I I I0 0.2 0.4 0.6 0.8 1.0

(a)Figure 4 - Amplitude Factors for the Interaction of Two Transverse Waves

to Produce a Scattered Longitudinal Wave (Case I-B). BothPrimary Waves Polarized Parallel to klk2 Plane.

12

106Polystyrene

//105 o/ Copper

// A 1 Iron

Pyrex

lOAl

0 I

'I

iI/,

xif I

101 I I I I I0 0.2 0.4 0.6 0.8 1.0

(a)

Figure 5 .- Amplitude Factors for the Interaction of Two LongitudinalWaves to Produce a Scattered Transverse Wave (Case II)

13

1041O/010•Ool~ystyrene

0'"/0

0 U . -. Copper &-A" Pyrex.

0~ Ow- \~ o-ft xy ao " 0 I

a' • , I Il/ ooIv 0I0 &J-INi,4 1

xY

II'

'I

101 II

10 0

I I I I0 0.5 1.0 1.5 2.0 2.5 3.0

(a)Figure 6 - Armlitudo Factors for the Interaction of a Longitudinal and a

Transverse Wave to Produce a Ammed Frequency LongitudinalWave (Case II-A). The Primary _Transverse Wave 1i Polarized

Parallel to the k1 k2 Plane*.

U4

o04

103

0/ 0''0 Polystyrene

I cfv0 Copper10 x--X Pyrex

xoý

A " , A Iron

101 %

A-"* II I I

0.2 0.4 0.8 0.8 1.0

Figure 7 - Amplitude Factors for the Interaction of a Longitudinal

and a Transverse Wave to Produce a Difference Frequency

Longitudinal Wave (Case IV-A). The Primary Transverse

Wave is Polarized Parallel to the -1ý. Plane.

15

10 p

o.-O-0 Polystyrene

l

/

A

Iron

io3

)

/

AI

102x

I

101

X Ix I

1Ix

III

100

I

I

.

..

I

0 0.2 0.4 0.6 0.8 1.0

(a)Figure 8 - Amiitude Factors for the Interaction of a Longitudinal

and a Transverse Wave to Produce a Difference FrequencyTrimisverse Wave (Case V-A). The Primary TransverseWave is Polarized Perpendicular to the k1k 2 Plane.

16

0o 0-0-" ON• 0 Polystyrene

103 0 Copper

0

13 X Pyrex

o /o. x A Iron

AA xX,.

l °2 Ax.x.-

IY

101

1o0

I 1 1 I .I0 0.2 0.4 0.6 0.8 1.0(a)

Figure 9 - Amplitude Factors for the Interaction of a Longitudinaland a Transverse Wave to Produce a Difference Frequency

Transverse Wave (Case V-B). The Prima~ry TransverseWave is Polarized Parallel to the k lk2 Plane.

17

y has the dimensions of sec 3 /cm 3 in the c.g.s. system used here. We seethat y is free of all of the experimental variables except the frequencyratio (a) and the physical constants of the material under investigation. Ithas been mentioned previously2_/ that the amplitude expressions were derivedfor the following conditions:

R >>V1/3 >>

where Xs is the wavelength of the scattered wave. These conditions, ofcourse, prohibit R from going to zero in (7) above.

In Figures 3 - 9, we have plotted the absolute values of y . The

real value of y may be negative as well as positive, but the intensity ofthe scattered wave is proportional to y2 and therefore always positive. Inplotting the data we have used dashed lines near those areas where the realvalue of Y changes sign. The absolute value of y always shows a minimumnear such areas as it actually goes to zero. The exact locations of thesezero intensity points have not been accurately determined because we are more

interested in the intensity maxima.

The information contained in the (y versus a) plots can be used inoptimizing the experimental variables. The curves are particularly helpfulin locating relatively sharp maxima such as those shown in Figure 6. Ingeneral, we see that the intensity of the interaction should be larger inpolystyrene than any of the other materials examined here. One might haveguessed this on intuitive grounds and we expect that other "plastics" wouldprobably compare favorably with polystyrene. It should be remembered, how-ever, that the calculations consider all materials as dissipationless. Ex-pression (7) indicates the scattered amplitude has a very strong dependenceon frequency as well as amplitude of the primary v,.aves. It is possible (andindeed probable) that a material with a high y will be particularly absorp-tive and therefore attenuate both the primary and scattered -wves much morerapidly than materials with a low y

C. Experimental Details and Results

Figure 10 shows a block diagram of a typical setup for observationof scattered waves produced by ultrasonic beam interaction. The triangular

specimen is cut so that the intersection angle y satisfies the resonantcondition for interaction. Transducers (A) and (B) are driven by separate rfpulse generators and produce the two primary waves. An Arenberg pulsed oscil-

lator model PG-650C is used to drive one transducer and the pulsed oscillator

18

IiJ0-0-C jzU

I-QjI

0 Z 0.-

44 u

41 1--I

0.

cr1

40 41

w 00

_j 0

LLI,C*z4

19

portion of a Sperry ultrasonic attenuation comparator is used to drive thesecond transducer. Transducer (C) is used as a receiver for the scatteredwave. At time to a signal from the blocking oscillator triggers PulseGenerator No. 1 and the oscilloscope sweep. A variable delay is used totrigger Pulse Generator No. 2 at (to+td) where td is the delay. Thetunable amplifier has a bandwidth of approximately 1.5 mc/sec at the 3 dbpoints, and operating into the scope it produces a maximum deflection ofapproximately 5 mv/in at 5 mc/sec. Quartz transducers have been used ex-clusively; therefore, impedance matching circuits have been used between thetransducers and the generators or amplifier.

Isolation and identification of the scattered signals are accom-plished in a number of ways. First, the time of arrival of the scatteredsignal at transdvcer (C) should correspond to acoustic propagation alongpaths dl and d2 . It should be recalled that the velocity of propagationalong leg dl may be different than that along d2 . For example, thetriangular specimen illustrated in Figure 1 was made up for the interactionof two transverse waves of frequencies wl and ui2 to produce a longi-tudinal wave of frequency )3 - w÷ . Thus, the scattered signal shouldarrive at (C) at time (to+dl/ct+d2/cl) . The time can easily be determinedwith the calibrated sweep of the oscilloscope. Transducer (C) is, of course,selected by frequency and mode to match the expected scattered wave. Ampli-fier tuning reveals the frequency of the detected signal. In addition, thesetting of td must be selected so that both primary wave packets passthrough the volume of interaction at approximately the same time.

Figure 11 is an oscillogram showing a typical scattered wave pulse(indicated by arrow) and illustrates the signal to noise relationship. Thesignal amplitude will drop severely if (i) the amplitude of either primarywave is reduced, (2) the frequency of either primary wave is changed, (3) theamplifier is tuned off the expected w3 frequency, or (4) td is variedfrom the calculated value.

During the past few months we have experimentally confirmed each ofthe five theoretical interaction cases enumerated in Table 1. The successfuldetection of a "scattered" beam was first attained in a triangular specimenof fused silica using Case I, i.e., interaction of two transverse waves toproduce a sum frequency longitudinal wave. For short-hand purposes let usrefer to such an interaction as follows: T(wI)+T(w2 ) = L(wi+w2 ) . The actualcase studied in the fused silica specimen was T(5)+T(5) - L(10). Theory pre-dicts (as shown in Appendix A) that a scattered wave should be generated whenthe two transverse waves are both polarized in the (Fl,i 2 ) plane or whenboth are polarized perpendicular to the ( 1Fl,1E2 ) plane. It further predictsthat the signal amplitude should be zero when one primary wave is polarizedin the (El,k 2 ) plane and the other is polarized perpendicular to the plane.

20

Figure 11 - Oscillogram Showing Signal Produced by Scattered Wave

21

We have experimentally confirmed all these predictions using the fused silicaspecimen. Using the same specimen we have replaced the two 5-mc/sec crystalswith one 3- and one 7-mc/sec crystal. Although the sum frequency is still

10 mc/sec we have been unable to detect a scattered signal. These negativeresults agree with theory, of course, because the specimen was not cut toproduce resonant interaction between beams of 3 and 7 mc/sec.

We have also performed several experiments to determine the degreeof collimation exhibited by the scattered wave. One such experiment isillustrated in Figure 12. With the primary transducers arranged symmetricallyas shown in Figure 12a, the maximum scattered signal was detected with thereceiving transducer centered at position 0 . When the receiving transducerwas moved 1 cm. either way to position X or Y , the amplitude of the scat-tered signal fell to less than one-tenth the value cbtained at 0 . If theprimary transducers are rearranged as in Figure 12b (and td is adjusted tothe appropriate delay) we now find the maximum scattered signal at position0' .

The interaction case, L(5)+T(5) = L(1O) , was studied in a specimenof 6061-T6 aluminum similar to that illustrated in Figure 13. A 10-mc/seclongitudinal wave was detected approximately 18 p sec after to . The cal-culated travel time along dl and d3 is ehown to be 17.8 I.sec. Further-more, the scattered signal was observed only when the delay time td wsapproximately 3 1sec. This agrees well with the calculated delay of 3.5Isec. In the above case, it was found that the magnitude of the scattered

signal was maximum when the polarization of the transverse wave was in the(!Flk2) plane and zero when rotated to the perpendicular position. Thisresult also agrees with theory (see Appendix A).

In the experiments just described the two primary waves were alwaysthe same frequency. This leads to the production of a sum frequency equalto 2wo , a rather unfortunate case since the second harmonic of the primary

frequency is involved. In subsequent experiments different primary frequen-cies were often used. Many of these experiments have been performed on mag-nesium due to the relatively large interaction effects exhibited by thatmaterial. The following cases have all been investigated in magnesium.

T(5)+T(5) = L(lO)

L(lO)+T(5) - T(5)

L(7)+T(3) - L(lO)

L(io)+T(3) = L(7)

22

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00

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44

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00

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23

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0

00) 0 0

toW

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z0 .

24

L(lS)+L(lO) - T(5)

L(J5)+T(5) - L(l0)

L(lO)+T(5) - L(15)

L(5)+T(3) - L(8)

L(5)+L(8) - T(3)

L(8)+T(3) , L(5)

Most of the above cases were studied using disk-like specimens similar tothat shown in Figure 14. Small flats are machined at selected points on the

disk circumference to permit the application of transducers. The flats areperpendicular to the disk radii; thus, interaction between the primary beamsalways occurs at the center of the disk. If each flat has a parallel flat

on the opposite side of the disk, as represented by positions A' , B' andC' , then several interaction cases can be studied. For example, considera 5-mc. longitudinal wave, Rl , intersecting a 3-mc. shear wave, k2 , toproduce an 8-mc. longitudinal wave, 7-3 . The angle 91 must satisfy theresonant condition for this interaction and the scattered wave travels in the73 direction as defined by (F 1 +2 2 ) . By close examination it becomes ob-vious that all of the interaction cases listed in Table 2 can be studied.

TABLE 2

INTERACTION CASES THAT CAN BE STUDIEDON SPECIMEN SHOWN IN FIGURE 14

Direction ofResonance Direction of Scattered

Interaction Case Angle Primary Waves Wave

L(5)+T(3) = L(8) Yl K- and '2 F3

L(8)+T(3) = L(5) cP2 U- and R2 Fl

L(8)+L(5) = T(3) IP3 K-3 and k1

25

10!

/

/

/ / /"•

///

Figure 14 - Disk Shaped Specimen for Study of Ultrasonic Beam Interaction

26

Another experiment was performed on a magnesium disk to furtherexplore the collimation characteristics of the scattered wave. An additionalflat was prepared on either side of position C such that their nornals were

t12e from that represented by IE3 in Figure 14. Although a relatively strongsignal was observed at C (on a 6-in. diameter disk), no signals at all weredetected at X or Y. These results definitely rule out the possibility thatthc scattered signals we observe are due to independent scattering of the twoprimary beams from grain boundaries with subsequent "mixing" of these signalsat the receiving transducer. Such scattering would not be particularly di-rective and produce such a high degree of collimation. The relatively sharpcollimation also suggests that the scattered wave is nearly plane and probablysuffers less than the R-1 fall-off assumed in calculating the amplitude fac-tors of Appendix A.

The relative intensities of the primary and scattered waves havebeen examined but current results are only qualitative. Generally, thevoltages measured on the receiving transducers are 40 - 60 dbs below thoseproduced by primary wave echos returning to the transmitting crystals.

We have also been interested in measuring changes in the primarybeam intensities caused by interaction effects. As discussed in Section IV-A,the intensity of both priniary beans might be ex-pected to decrease in thoseinteraction cases where the scattered wave is a sun frequency. In the caseof difference frequencies, however, the intensity of the "splitting" v.avewould be expected to decrease while the other would increase. Experimentalefforts to observe these relatively small changes in primary beam intensitieshave thus far yielded negative results.

Although the intensity curves show that beam interaction should berelatively great in polystyrene, vm have been unable to observe a scatteredwave in this material. These results can probably be explained by the largeultrasonic attenuation exhibited by polystyrene. A high attenuation willseriously reduce the intensity of the primary beams at the point of inter-section as well as quickly attenuate the scattered wave after generation.

V. VISUALIZATION OF ULTRASONIC WAVE PACKETS

A. Instrumentation

The early attempts to detect scattered waves created by beam inter-actions were confined to the use of transducers. These transducers are quitesensitive to weak signals, but at high frequencies they are highly directive.It was felt that the negative results which came first were possibly due to

27

slight errors in the calculated propagation direction of the scattered wave.The transducer directivity might therefore prevent detection of the scatteredwave. The search for a less directive detector finally led us into variousoptical techniques. Although these techniques are limited to transparentsolids, they do have certain advantages for studying beam interaction.Several waves traveling in different directions can be visualized at thesame time and yet the sensitivity of detecting one wave in relation toanother may be adjusted.

A simple schematic of our optical detection system is shown inFigure 15. It is basically a standard schlieren system except that modifi-cations have been included to permit the use of polarization techniques.When operated as a simple schlieren system, the polaroids, P and A , willbe removed. The xenon flashtube, F , produces a rather fuzzy line sourcethat is imaged on an adjustable slit, S . The light passing through theslit is collected by the lens, L1 , and ionverted into a parallel beam. Thebeam passes through the test block, M , and then into lens, I, , whichproduces a real image of the slit source S at S' . A "stop" is normallypositioned at S' to intercept the image of the slit. However, ultrasonicwaves in the test block may produce localized gradients in the index ofrefraction. These gradients, when normal to the optic axis, cause some ofthe light passing through M to be scattered out of the parallel beam.These rays pass by the stop as illustrated by the dashed lines and are focusedonto a ground glass plate or film at G . The regions in the test block whererefractive index gradients exist will therefore appear as bright areas on adark field. When a slit source is used, maximum sensitivity of detectingwaves is achieved when the wave front is parallel to the slit. This factorcan be used in interaction experiments to accentuate the image of the scat-tered wave over that of the primary waves.

When the polarizer, P , and analyzer, A , are used as shown inFigure .5, the stop S' is not necesoary. If P and A are "crossed," theimage at G will generally be a uniform dark field. An ultrasonic wave inthe test block M , however, may "depolarize" the parallel beam and thuspermit some light to pass through the analyzer. The location of waves willappear as bright areas on a dark field. Weeks§/ has shown that the emergentlight intensity is greatest for a longitudinal mode when the plane of polari-zation of the incident light is inclined at 45* to the direction of sonicpropagation. The modulation vanishes when the plane of polarization is paral-lel or perpendicular to the direction of sonic propagation. Just the oppositeis true for transverse uaves.

In an experiment in which we wish Lo detect a relatively weak wavegenerated at the intersection of two stronger waves, the use of continuouslygenerated -aves is prohibited. It would be very difficult indeed to absorb

28

LA-

ww

0.0

0.0

00.

f4

I LiL

IlW 2I_

M-'Ir

299

all of the energy in the primary waves before it is repeatedly reflected atsurface boundaries. The reflected waves would ultimately produce index gradi-ents over the entire specimen that would probably be larger than those pro-duced by the scattered wave. This difficulty can be largely overcome byusing pulsed vltrasonic waves and a pulsed light source for the optical sys-tem.

The blocking oscillator controls the repetition rate of both therf pulse generator and the pulsing of the light source. However, a variabledelay between the blocking oscillator and the flashtube power supply makesit possible to observe the wave packet after it travels a given distance intothe test specimen. With a given delay between the pulsing of the ultrasonicgenerator and the flashtube, one can "stop" the traveling packets at a givenposition and visually observe or photograph the resultant patterns. AGeneral Radio "Strobotac" was used as the flashtube power supply. The FX-31xenon flashtube (Edgerton, Germeshausen, and Grier, Inc.) has an opticalwindow which facilitates imaging of the rather fuzzy line source, and it canbe flashed at rather high repetition rates.

B. Photographic Results

Figure 16 is a sequence of photographs showing a 7-mc. shear wavepacket at various stages of travel in a glass specimen. Standing waves areclear in (c) due to superposition of the incident and reflected travelingwaves. Side lobes have developed in (d) as the packet travels back towardthe transducer. Figure 17 shows two 7-mc. shear beams intersecting at ap-

proximately 900. Interference between the two beams is easily seen even inareas outside the region of main lobe overlap. The duration of the lightpulse in both figures was 0.8 jIsec corresponding to a shear wave traveldistance of approximately 2 mm.

During the past year, beam interactions were studied simultaneouslyusing both transducers and the optical system. Successful detection of thesought-after beams using transducer techniques lessened our immediate interestin optical techniques. Further experimentation was therefore delayed untilmore inforration on the interaction process could be collected. The prelimi-nary results obtained with transducers now suggest that under ideal conditionsone should be able to detect the scattered waves with an optical system.Furtner work along this line is warranted because of the versatility of theoptical system in studying beam spreading and structure variations as thescattered waves propagate away from the point of origin.

30

(b)

(c)

(a)

Figure 16 - Sequence of Photographs Shorwing Ultrasonic Pulse TravelingThrough Glass Specimen

31

Figure 17 -Intersection of~ Two 7-inc. Shear Wave Packets in a Glass Specimen

32

VI. THLEE-DMIMSINAL STRESS ANAISIS WITH UIWRASONICS

The successful generation and detection of scattered beams ofreasonable intensity provides us with a theoretically possible technique ofperforming three-dimensional stress measurements. Note in Table 1 that atransverse scattered wave can be generated in either case II or case III.For simplicity, let us consider only case II where the interaction of twolongitudinal waves may produce a transverse wave of difference frequency. Itcan be shown theoretically (and confirmed experimentally) that the polariza-tion of the transverse wave lies in the plane defined by the propagation vec-tors of the two primary waves. We therefore have a potential technique ofgenerating transverse waves of. kn•wn polarization at any subsurface volumeelement.

Incorporation of beam interaction techniques into a possible methodof three-dimensional stress analysis is illustrated in Figure 18. Primarybeams _k, and k2 are first crossed at position (a) to produce a transversewave i 3 . With a detector at position (c), the total angular rotation aof the polarization axis, can be determined for propagation between (a) and(c). The primary waves can then be crossed at (b) and the rotation 0 deter-mined for path. (bc). The difference between the secondary principal stresses(in a plane perpendicular to k3 ) can then be approximated for a point mid-way between (a) and (b). The approximation should take the form

(p-q) K(-)

where p and q are the secondary principal stresses, K is the stresc-rotation constant for the material in question and L is the distance between(a) and (b). Since the direction k3 and its polarization can be variedthrough control of il and k-2 , it should be possible to determine the di-rections of p and q and the value (p-q) for several different IE di- Irections passing through a common point.

There are many practical problems to be solved before the above pro-

cedure will yield much useful information. On certain geometric shapes theseproblems will be m-ch more severe than others. The difficulty of varying the Idirection of Ri and R and accurately controlling the intersection point

is indeed great, especially in irregularly shaped specimens. However, liquidcoupling techniques may prove satisfactory in many cases and thus increaseversatility over contact bonding. At any rate, the potential applications ofbeam scattering techniques in the measurement of stress and other subsurfaceproperties are numerous and deserve considerable attention. Additional workshould be directed toward the evaluation of all influencing factors as well asthe development of techniques that will facilitate the application topractical problems.

33

k2 (a)

Figure 18 - Schsatic Diagram of Potential Stress Analysis Technique Using

Beom Interactioh to Produce Shear Waves

34

APPENDIX A

AMPLITJE EPRE&SSIONS FOR SCATTERED WAVES

In the amplitude expressions, the terms are defined as follows:

X1, X2 - amplitudes of primary waves

V - volume of intersect!4on

w - angular frequency

p - density

R - distance between point of intersection and point of observation

- velocity of shear wave

cL - velocity of longitudinal wave

c - ratio of velocities ctCA

l - shear modulus

K - bulk modulus

a - ratio of primary frequenciesWi

A , B - third order elastic constants

0- angle between primary waves

kl k2 - propagation vectors for primary waves

35

Case I - Two Transverse Waves Interacting to Produce a Scattered LongitudinalWave of Frequency (01+2)

(A) Both transverse waves polarized perpendicular to the ;EA2

plane.

a Xr1XikV4 cAk ) [(c2_1)(a3+l)+(a2+a)(c2+1)]16 TpR ct CA

- (K+ 1 + A +B) [c2(3c2_l)(a2+a)+c2(c2_j(a3+l)}

(B) Both transverse waves polarized in the k-k 2 plane.

am -X1X2Vw4 F2(l+a) A + +Bý + cos2 0 1 1+K28 wpR c4 cl 1 2 /B (3

(C) One transverse wave pooArized in •IE pia-±e and one perpen-

dicular to iE2 plane.

amp = 0

Case II - Two Longitudinal Waves Interacting to Produce a Scattered Transverse

Wave of Frequency (W2 -W1 )

-up XV'o (73 p+,•K÷2) cos 0 i--(•.P)coos 0(2a32a)÷oa2,-,,)2+(,a2,)28 irpR ct ci ~~A o ~?JU

where

P 2a 3 +a 4 c 2 +2a 2 c 2 -a 4 -6a 2 +2a+c2 -1

2(a-1)2

36

Case III - One Transverse Wave and One Longitudinal Wave Interacting toProduce A Scattered Longitudinal Wave of Frequency (wl+w2 )

(A) If the transverse wave is polarized in the Elk plane.

av -TXlX 2Vw [a(1..cos2 34)+Ba1 KK+ P- + A +2 c2(1+a)2 cos 08 wpR1 ct c +

1+ I + ý A+2B4.K) C2 Cos 0 + (~+A(2a2+4ac cos 0) cos 01

(B) If the transverse wave is polarized perpendicular to the kjk2plane.

amp m 0

Case IV - One Transverse Wave and One Longitudinal Wave Interacting to ProduceA Scattered Longitbdinal Wave of Frequency (w1 -w2 )

(A) If the transverse wave is polarized in the klR2 plane.

/ - 2 mf-X1 X2VWI \ja(l-cos2 ) 31 1 A/ B)a

"P 8~ ý,nR c2 c } l-a 2~~Jk ~-)a

+ +K ~ 2B~2(l.a)2os -( O +A+K)c2 Cos 0+(A+ A) 4ac cos 0-2a2)cos 0

(B) If the transverse wave is polarized perpendicular to the 1 •k2plane.

amp 7 0

37

Case V - One Transverse Wave and One Longitudinal Wave IM teractinG to ProduceA Scattered Transverse Wave of Frequency (wl-w2 )

(A) If the primar7 transverse wave is polarized perpend•cular tothe klk2 plane.

X1X2VU,,! 1. ( ± 01. A +B) (2ac2+c4 _e2)

- A ( :W. )(4a2+c4-2c 2+1+4ac2 -4a)}

(B) If the primary transverse wave is polarized in the klc plane.

1 + a2 1½+ 0am p = ±t - • cos4 rpR ct c, C2C

vhere

M {DI-(D2-DI)aC cos O+*c2Dla 2D2 }(-,a)2

= {D2+(D2 -D1 )ac cos 0+c2D_-a 2D2 j(1-a)2

2. l 4 K[ao ,2(2 c 0-1)-ac Cos 3 (+ _.,.K.a cos

2 ct(l-cos2 y')T 0 4/ K

D2 2iý 1-2( A4)a Cos 3 o+ K+ A + A +2B) c cos 20(-~2c c•(-cos 2 0)

Always use the positive root of (1-cos 2 0)1

38

BIBLIOGRAPHY

1. F. R. Rollins, "Ultrasonic Methods for Nondestructive Measurement ofResidual Stress," WADD TR 61-42, Part 1 (1961).

2. F. H. Rollins, "Ultrasonic Methods for Nondestructive Measurement ofResidual Stress," WADD TR 61-42, Part II (1962).

3. M. M. Frocht and L. S. Srinath, Proc. Third U. S. National Congress ofApplied Mechanics, 329 (1958).

4. E. G. Coker,and L. N. G. Filon, "A Treatise on Photoelasticity," secondedition, p. 410 (Cambridge University Press, 19b7).

5. G. L. Slonimskii, J.E.T.P. (U.S.S.R.) 12, 1457 (1937).

6. D. S. Hughes and J. L. Kelly, Phys. Rev., 92, 1145 (1953).

7. A. Seeger and 0. Buck, Z. Naturforschung, 15a, 1056 (1960).

8. R. F. Weeks, J.A.S.A., 33, 741 (1961).

3

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