The reflection of a Rayleigh wave from the edge of a sharp elastic wedge in oblique incidence is aaalyzed. It is
shown that the nature of the angular dependence of the modulus of the reflection coeffrcient is strongly
affected by the phase shift of the symmetric field mode associated with reflection from a caustic. The results
are tested experimentally

S.V. Korolev (7122038)

Victor V. Krylov (1252356)

English

Loughborough University Institutional Repository

   This item was submitted to Loughborough’s Institutional Repository (https://dspace.lboro.ac.uk/) by the author and is made available under the following Creative Commons Licence conditions.      For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/  Reflection of a Rayleigh wave from the edge of a wedge in obliqueincidenceS. V. Korolev and V. V. KrylovPhysics Faculry, M. Y. Lomonosou State Uniuenity, Moscow{Submitted April 23, 1985)Akust. Zh. 32, 799-803 (Novembcr-Decembcr 1986)The reflection of a Rayleigh wave from the edge of a sharp elastic wedge in oblique incidence is aaalyzed. It isshown that the nature of the angular dependence of the modulus of the reflection coeffrcient is stronglyaffected by the phase shift of the symmetric field mode associated with reflection from a caustic. The resultsare tested experimentally.The interaction of a Rayleigh wave w'ith the edgeof an elastic wedge plays an i.nportant part in vari-ous branches of solid-state acoustics, includng ultraJsonic flaw detection and acoustoelectronics. r-3 Anapproximate theory has been proposed3 for the re-flection and transmission of Rayleigh waves in asharp wedge. This theory is based on the r€presen-tation of the ineident Rayleigh wave on the edge ofthe wedge by the sum of the two lowest (symmetricand antisymmetrie) Lamb (plate-wave) modes for apiate of variabie thickness. The propaga.tion ofthese modes to and from the edge is treated sepanate-ly with the application of the known solutions of thereflection problems for each of them at the free edgeof an infinitesimally thin piate. Then allowance ismade for the fact that the sSrmmetric and antisSrm-metric Lamb modes propagate with different veloei-ties in the vicinity of the edge of the wedge. As aresult, a difference in the phase shifts" which de-pends on the wedge anglee, builds up between thesemodes in propagation from the driving point to thereception point. This phase di.fference ls responsi-bie for the experimentally observedl multiple oscil-lations of the modub of the reflection coefficient lRland the transmission coefficient I tl as a function of0 for a Rayieigh wave in normal incidence on theedge. Simple geometrical considerations are also em-pioyed in Ref. 3 to anaiyze the case of oblique in-cidence of a Rayteigh wave on the edge at smallangles c. The obiique-incidence problem is reducedto the probiem of normal wave incidence on a wedgewith an equivalent wedge angle 0' smaller than 8.The possibility of oscillations of lRl and lfl as afunction of the angle of incidenee B for a fixedwedge angle a is predicted in t}.is work.In the present article we investigate the reflec-tion of a Rayleigh wave from the edge of a wedgetheoreticaliy for the case of oblique incidenee atarbitrary angles o, We also give the results of ex-perimental work on the investigated effect.As in the case of normal ineidence, we proeeedfrom the representation of the incident Rayieigh waveby the sum of a s5rmmetric (iongitudinai) and anantisymmetric (flexurai) Lamb mode propagating in aplate of variable thickness h. The Rayleigh-wavereflection and transmission coefficients can aiso beexpressed in terms of the difference in the totalphase shjft a4(a) = Sa(a) - $s(c) of the antisym-metric and symmetric modes during their propagationto the edge of the wedge and back again, and interms of the difference in the phase strifts ao(a) =o.(c) - os(c) assoeiated with the reflection of eachmode separately from the edge.3 We calcuiate thephase.shifts 4"(a) and Ss(a) in the geometrieal-optics (geometrieal-acoustics) approximation, whichgeneralizes the WKB approximation used in Ref. 3 tothe case of oblique incidence. These calculations,of eourse, are more complicated than in the case ofnormal incidence.Fi.rst o-f all, it must be borne in mind that thesSmmetric and antisSrmmetric modes are both sub-jected to refraetion as they approaeh the edge, wherethe nature of theb refraetion differs as a result ofthe difference in the iaws governing the variation oftheir veiocities near the edge. The velocity of theantisynmetric (flexural) mode decreases from theRayleigh-wave velocity cR to zero (as h+0), so thatttris mode approaches the edge practicallyin the nor-maldirection (Fig. 1). The veloeity of the symmetric(Iongitudinal) mode, converseiy, increases in the di-rection of decreasing h from the veiocity cp to theso-ealled rrplateil velocity cn = 2ct(1 - c{l e L2yl 2,where eu,1 denotes the velocities of longitudnal andshear bulk waves. As a result, the symmetric modeis incident on the edge at a more oblique angie. Sec-ond, it must be taken into consideration that the raypath of the s1'rnmetric mode has a turning point at acertain angle of i:rcidence c,o, i.e., the symmetriemodeonlonger reaches the edge (Fig. 1). Conse-quently, rays with angies of incidenee a 2 oo from asimple caustic, at which the symmetric mode acquiresan additional (caustic) phase shift =T/2inreflectionfrom it.qUsing the well-kno$'n geometricai-optics (geo-metrical-acoustics) relation for the phase of a wavepropagating in a medium that is inhomogeneous inonedirection (see, e.g., Refs' 4 and 5)' we canwrite an equation for the difference in the phaseshifts of the symmetric and antisymmetric modes ex-cited and received at a point infiniteiy far from theedge:te@)--2k"{if53-,io,o ]"d, (1)FIG, 1. Oblique inci.dence of a Rayleigh wave on the edge of awedge. 1) Antisymetric nodei ?) slmelric mode.497 Sov. phys. Acoust. 32(6), Nov.-Dec. 1986 OO38-562V86/O6 O497-03 S03.40 @ 1987 American lnstitute oi Physics- i tqg-sin,z ]'r,)/i "'Here k","(x).denotes the wave numbers of the sym-metric and antisymmetric modes as a function of thecoordinate x, whieh is directed away from the edgeof the wedge; kg is the Rayleigh wave number in ahalf-space; and xt(c) is the coordinate of the turn-ing point. Making use of tbe fact that h(x, 8) =2xtan€/2), we can rewrite Eq. (1) in the convenientformAq (a) :-26 (a) lts(912) ,6("):+{jt{p-,id,ol* ar- [ [ r'"(r'l -.io.o ]naa ]. e)n"L,L k^'The functions k"."(h) do not have analytical expres-sions, and so it is often necessary in calculations touse suitable approxi.mations for the conespondi.ngdispersion curves calcuiated by numericai methods(see, e.g., Refs. 6 and ?). We use the approxima-tions proposed in Ref. 3. The'value of the iocalwedge ttrickness h1( a) characterizing the turning pointof the symmetric mode is determined from the equa-tionk,2(h")1k"2-sin'a:0. . (3)The value of the critieal angle ao,' at which thispoint emerges, corresponds to the repl,aeement ofks(ht) in Eq. (3) by ks(0) = kn = urlep. It foilowsfrom this result, in particular, that go = 32o foraiuminum.We now discuss the behavior of the phase shiftsQ",a(o) of the symmetric and antisymmetric riodes inthe reflection of each from t.Le edge of the wedge forthe case of oblique incidence. Inasmuch as the anti-symmetric mode is incident on the edge practically ata right angle due to refraction, its phase shift inreflection does not depend on the angle of incidenceo and coincides with the phse shift in the case ofnormal incidenceona wedge with vertex angle 0,i.e.,Qu = (n+8)lz (Ref' 3)' The phase shift for thesymmetric mode is readiiy determined from the solu-tion of the problem of oblique incidenee of the sym-metric mode on the free edge of a thin plate Thisprobiem is exactly anaiogous to the problem of obliqueincidence of a plane Iongitudinal wave on the bound-ary of an elastic half-spaee. It follows from thesolution that the vaiue of 6=(a,) is identically zeroat o < co (we recall thAt onl.y under tlris conditiondoes the symmetric mode reach the edge), as in thecase of normal incidence. T The modulus of the sym-metric-mode reflection coefficient, however, differsfrom unity for 0 < o < oo, because part of the ener-gy of the symmetric mode is converted into the ener-gy of shear SH waves, which ar€ generated in re:flection and are poiari.zed in the plane of the wedge.The difference does not exeeed -10% for the majorityof materiais in the investigated range of angles, andthis is consistent with the error limits of the approxi-matd theory used here. We shall therefore ignore theindicated difference from now on. When c 3 co,i.e.,when a turning point is present, SH waves are notgenerated. In this case, as mentioned, the reflectedsymmetric node acquires a caustic phase shift equallo - nl2.The expression for the modulus of the coefficienlof reflection of a Rayleigh wave from the edge of awedge in oblique incidence can be written as followsin light of the foregoing considerations:1n1: lsinti#-+.+ll. (4).where Agk is the caustie phase sbift; which is equal!o ^zglo for a < oo and is equal to - t-l2 for cr ; c,o.It foilows from Eq. (4) that the function R@, a) Gsa eomplex osciliating behavior with respect to both0 and c. When a= 0, we obtain the equation forthe reflection coefficient in normnt incidence3 fromEq. (4):r"r: i''[#,a -+]1. (5)where -,6-6 (Q) - ('|/:) J t[. tll -lr. tl,l ]an.In the case of small angles a (smaller than ao),the expression for 6(a) from Eqs. (Z) can be "i*pU-fied by replacing the sines with their arguments andexpanding-the square roots in a power series, whiehare then restricted to the first two terms. Hereao)= 1{ i,o",o,-0., h)tdh++(6)' k^' ,, Ix J Itr.(D) -/c, (h)l --------::- a61 .; h.(h)k.(h) )Aecording to the theorem of the mean, which is wellknown in anaiysis, the second integral in Eq. (6)can be represented in the form l(h,) IW"O>-f,tolloo,where f(ho; = kRz/kso (h)ks(iro) t afr" value of thefunction f(h) at some point ho beionging to the do-main of integration. A numerical calculation usingapproximative functions k",r(h) shows that f(h):;0.75. Makinguse of the fact Urat (1/2) J lk"(h)-k,(h)ldh:6(0), we can now write Eq. (6) in tfr'. torrn5(") i 6rc)U + o.rs*'f 2) " (T)We note that the function (7) differs somewhatfrom the result obtained3 for small cr, by reductionof the problem of obiique incidence of a Rayleighv/ave on a wedge at small angies o to the probiem ofnormal incidence on an equivaient lvedge with angle8t = Ztan-t ltan(g/Z)cos o]. But the coefficient infront of 6(0)a,/2in Ref. 3 is equal to unity. Thisimplies that the influence of refraction is eonsider-ably smaller in the first approximation with respectto c2.The approximation (?) beeomes invalid for angleso i co, and it is necessary to carry out a completenumerical calculation of the function 6(a,) and the'quantity lR l. We have performed such a calcuia-tion on a computer for an aluminum wedge for twovertex angles I = 30 and 600. The calculated func-tion 6(a) was multiplied by a cotection factor of0.915, which was determined fmm the condition ofexact agr€ement of the theor.eticai value of lnl innormal incidence on a wedge with angle 0 = 30o andSov. Phys. Acousr. .32(6), liov.-Dec. 1 986 S. v. Koroiov and V. V. Krylov 498tFlI0,6tRlI0,60,6tftl0,6l0 30 50 d, dezFIc. Z. Iheoretical and experi-@ntal graphs of the oodulus of thereflection coefficient I Rl vs angle of ircidence c . a) I =?O"i b) 600; c) 90o.the experimental value for" this case. I Ttris corre-sponded to replacement of the value 6(0)= 2.75 *t-culated in Ref. 3 aeeording to the approximate func-tions by the quantity 6(0) = 2.52. Such a correla-tion was necessitated by the establishment of theinitial correspondence in a comparison of the calcula-tions with the above-described experiments on Ray-leigh-wave reflection from the edge of a wedge inoblique incidence. The comparison would be mean-ingless if the theoretical values were alr€ady in dis-agTeement with the experimentai at a = 0.An aiuminum prism with a height of 15 cm and abase in the shape of a right triangie with 30, 60,and 90o angles was used for the experjmental deter-mination of the moduius of the reflection coefficientlRl as a function of the angie of incidence a. ThisoUject enabled us to measurL the functions ln<"llfor three values of the angie 0 on a single sample.The measurement were carried out in the pulsed re-gime at a frequency of 1 MHz; the puise durationwas -10 us. Rayleigh waves were generated and re-ceived by means of two Plexiglas wedge transdueers,which were moved along the surface of the sampleand were bonded to it acoustically through an epoxynesin layer. The diameters of the longitudinallyvibrating piezoceramic wafers on the wedge trans-ducers were equal to 1 cm. Ali the measurements'were repeated several times and were processed sta-tisticaiiy.The experimental curves of lR(s)l for g= 30 and600 are shown in Fig. 2 together with the corre-sponding theoretical curves cslculated according toEqs. (2)-(a). We see that the curves undergo oscil-lations as a is varied, and the ampiitude of the oscil-lations is smaller for 0 = 600 than for I = 30o.Figure 2 also shows the experimental curve of lR(")lfor I = 90o, which is practically nonoscillating. CaI-culations were not carried out for this case, becauseit exceeds the limits of validity of the given theory.A comparison of the experimental data with theresultg of the caiculations for I = 30o indicates thatthey arein satisfactory qualitative ag?eement, aswitnessed by the good correspondenee between thepositions of the naxima and rninima of the comparedeurves. The theoreticFl curve for the case 0=600only very remoteiy resembles the experimental curve.This is whst we should expect insofar as the givenvalue of e corresponds to the limit of validity of thegiven theory, which holds in the interval 0 5 0 5600,i.e., for sufficienily shar? wedges.We call attention tb the fact that the occurenceof the causdc phase shift for the symmetric mode(at c 2 32o) has a strong influence on the behaviorof the theoretical curves of lR(a) l; in particulair,it induces an abrupt variation of I R( a) | at c - 32o. Itis readily discerned from Fig. 2 th,at a similar anomalyis clearly observed in the experiment for the case0 = 30o. In contrast with the theory, however, theexperiment yields a smoother decay of the reflectioncoefficient at 30< a < 35o. This result can be ex-plained in part by the fact that the theory involvesplane surface waves, whereas all the investigatedprocesses in the experimental work took place foreylindrical waves, from wtrich the receiving trans-ducer extraeted a na*ow bundie of rays wj.th anangular width of -3o. This tended to average theexperimental curves over the indicated intervals. Itis important to note that the eaustic phase shift doesnot occur in experiments in the usual praeticalsituations, e.g., in problems of ocean acoustics, andso it does not have any practical significanee. Theabove-considered case of Rayleigh-wave reflectionfrom the edge of an elastic wedge, where ttr-is striftaffects the modulus of the reflection coeffiiient, isunique in this respect by virtue of the presence ofa reference wave (the antislrmmetric mode) that is notreflected from the caustic.1L A. viktorov, Acoustic Surface.ilaves in Solids [in Russian],Nauka, Hoscow (1981).2L. Knopoff, I'Elastic vave propagation in a wedgerrr in: itavePropagation ir Solids, J. Miklowitz (ed.), nSlE, New York (1959),3v. v. Krylov and v.c. Mozhaev, Akust, Zh. 31, 751 (1985) [sov.Phys. Acoust. g, -457 ( 19e5 ) I .\u. V, Rrar"tsov md Yu. t. Orlov, Geooetric OPtics in Inhomgen-eous Hedia [in Russim]' Nauka, Moscov (1980).sL. M. Brekhovskikh, tlaves j-n l€yered Media, znd Ed., AcadesicPress, New York (1980).5f. R. Heeker ad A. H. Meitzler, 'rGuided wave ProPagation inelongated cylinders and Platesr" in: Physical Acoustics' vo1'tA, l{. P' }lason (ed.), Aedmic Press' Nes York (1964), p' 112'TE.Slatdrzyk, Sirple and Cotsplex vibratory Systens' Penn' stateIrniv. Press' University Park-London (1968).Transiated by J. S. Wood0,4{*- . .----rrts--'t-*----._\499 Sov. Phys. Acoust. 32(6), Nov.-Dec. 1 986 S. V. Korolev End V' V. Krylov 499

Reflection of a Rayleigh wave from the edge of a wedge in oblique incidence

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Reflection of a Rayleigh wave from the edge of a wedge in oblique incidence

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