CHARACTERIZATION OF A PERIODIC SURFACE PROFILE BY POLE-ZERO PARAMETERIZATION OF ELASTODYNAMIC PULSE REFLECTIONS * R.A. Roberts and J.D.

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1 CHARACTERZATON OF A PERODC SURFACE PROFLE BY POLE-ZERO PARAMETERZATON OF ELASTODYNAMC PULSE REFLECTONS * R.A. Rberts and J.D. Achenbach The Technlgical nstitute Nrthwestern University Evanstn, L NTRODUCTON This paper cnsiders the prblem f determining the height and perid f a surface prfile which is peridic in ne directin thrugh the analysis f reflected elastdynamic pulses. Bth thery and experiment are presented. n earlier wrk,l,2 it was nted that the peridic structure f the surface displayed a resnance at a fundamental resnant frequency, and at the subsequent harmnics. n additin, the peridic surface acts as a diffractin grating which mde cnverts the incident field int diffracted lngitudinal and transverse spectral rders. n this paper. the "quality" f this surface resnance. and the efficiency f the diffracted rder mde cnversin. are quantified by mdelling the reflected signal frequency spectrum as a ratinal plynmial ver the cmplex frequency plane. The ple and zer lci f this ratinal plynmial are btained as functins f the heip.ht and perid f the surface prfile. thereby prviding quantitative data suitable fr use in an inversin prcess. EXPERMENTAL DATA Experimental data fr scattering by several triangular peridic prfiles machined in stainless steel were btained using a pulse-ech ultrasnic technique. All prfiles had a perid D f 1.6mm. The fllwing ratis f depth h t perid D were cnsidered: hid = 0.0,.125,.25,.375 5,.625. The prfile gemetry is depicted in Fig. 1. The lngitudinal and transverse wave velcities are cl = 5900 mlsec and c T = 3230 m/sec. The steel samples were halfsubmerged in a water bath with the peridic prfile expsed t air. A.75 inch, 2.25 MHz transducer was placed belw the steel samples in a psitin fr nrmal incidence. The utput f the pulser-receiver was digitized using a digital scillscpe, and then transferred t a mainframe cmputer fr further prcessing. 145

2 146 R. A. ROBERTS AND J. D. ACHENBACH D Fig. 1. Prfile Cnfiguratin The digitized time dmain wavefrms fr three values f hd are shwn in Fig. 2. The decaying sinusidal tail seen in the wavefrms f Fig. 2 is due t the radiatin f the surface resnance int the lngitudinal field. n the previusly cited papers 1,2, it is nted that this resnance ccurs at apprximately the frequency fr which the Rayleigh wavelength equals the prfile perid D. Nte that the amplitude and decay f the sinusidal tail varies with hd. The p1ezer parameterizatin t be presented quantifies these resnant characteristics. THEORETCAL DATA Theretical frequency spectra fr the reflectin f a unit amplitude time harmnic lngitudinal plane wave under nrmal incidence were btained using a numerical slutin t the bundary integral prblem. 3 The amplitude and phase f the reflected lngitudinal field emerging nrmal t the surface are pltted in Fig. 3 fr the values f hd fund in Fig. 2. The phase spectra display a rapid 360 degree phase shift arund 1.9 MHz, indicating a surface r e s n a The n c e minima in the amplitude spectra are due t mde cnversin int the m = ±1 transverse spectral rders. Fr example, the deep minimum in the hd =.375 spectrum at 2.5 MHz implies the existence f large amplitude transverse waves prpagating away frm the surface at angles f e = 36 degrees and e = 144 degrees with respect t the Xl-axis. The spectra f Fig. 3 crrespnd in the time dmain t the reflected respnse t an incident plane wave delta-functin displacement. n principle, experimental decnvlved spectra, crrespnding t the theretical spectra f Fig. 3, can be btained by applying a Furier transfrm algrithm t the experimental pulses. t was shwn 1, hwever, that large discrepancies between the theretical and experimental spectra can arise near the resnant frequencies as a result f the electrnic time gating applied t the experimental signals. The effects f time gating diminish the usefulness f the

3 PERODC SURFACE PROFLE 147 H/D-.125 H/D-.375 H/D '".. '" d.. '" V'.. Fig. 2., 0.0 S.O S.O TME (MCRO SEC) Digitized time-dmain wavefrms fr three values f hd.

4 148 R. A. ROBERTS AND J. D. ACHENBACH AMPLTUDE SPECTRA PHASE SPECTRA g H/D ; ; ;!::ci! l.. 0 :E::ij : 0 > i... lq a::- V 0:: 8 ::t: a::.,; l.., fre:ouency [MHZ) H/D R 1..1 V... g... 0::0 _0 cg'- '" l.. 0 :E::ij :fl 0 ::t: l..ij ci H/D R > i... lq a::- V 0:: 8 a:: V ", ::;)... 0::0 t::ci cg'- 1..1'" 0 l.. :E::ij :fl > i... lq a::-..j V 0:: a:: 8 ::t:.,; l.. - i i Fig. 3. Amplitude and phase spectra fr three values f hd.

5 PERODC SURFACE PROFLE 149 experimental spectra fr btaining quantitative infrmatin. The ple-zer parameterizatin technique discussed in the next sectin perates n the experimental data in the time dmain, thereby aviding the prblems assciated with time gating. t was previusly nted 2 that the resnant decay characteristics f the reflected time dmain signals display a dependence f the width f the incident beam used t insnify the reflecting surfaces. Hwever. numerical results btained using a theretical mdel which accunted fr finite beam width2 revealed that, fr the beam widths cnsidered here, the errrs intrduced by assuming plane wave incidence are negligible. POLE-ZERO PARAMETERZATON The pulse-ech experiment can be thught f as a series f cascaded electrnic filters, ne f which is the peridic reflecting surface. The input t this filter, x(t), is defined as the pulse-ech system respnse t reflectin frm a plane surface, while the filter utput, y(t), is defined as the pulse-ech system respnse t reflectin frm the peridic prfile. n the cmplex frequency (s) dmain, the filter respnse functin, h(s), is defined as A A A h(s) = y(s)/x(s) (1) A A where x(s). y(s) are the Laplace transfrms f x(t). y(t). respectively. n the ple-zer parameterizatin technique. the respnse functin A h(s) is apprximated as a ratinal plynmial. The ples, and zers f the apprximating plynmial are parameters which depend n the gemetry f the surface prfile. and thus yield infrmatin useful in the inverse prblem. The ple-zer data is btained frm the experimental data by fitting a discrete recursive filter functin t the digitized time dmain signals via an errr minimizatin t e c h n i qthe u e representatin. f this recursive filter functin in the cmplex frequency dmain takes the frm f a ratinal plynmial. frm which the ples and zers are extracted. The ple-zer data is btained frm the theretical spectra by fitting a ratinal plynmial directly t the cmplex frequency spectra using an errr minimizatin technique. 5 When apprximating the scatterer respnse functin as a ratinal plynmial. a chice must be made regarding the rders Nand M f the numeratr and denminatr plynmials. Assume that the number f actual ples cntained in the data t be analyzed is L. Then chsing M > L will yield M ples; L f which crrespnd t actual ples in the data. and M-L f which are "extraneus" ples. lying in psitins which minimize the defined errr criterin ti A similar statement applies t the zers f the ratinal plynmial. t is pssible t determine which f the M ples and N zers crrespnd t actual ples and zers in the data by nting the ples and zers which repeatedly ccur as Nand M are increased. Anther technique fr recgnizing actual ples and zers in the data is t repeat the parameterizatin using different weighting functins in the errr minimizatin rutine fr fixed Nand

6 150 R. A. ROBERTS AND J. D. ACHENBACH M, and nting the recurring ples and zers. Bth techniques were used in btaining the ple-zer data presented here. Tw sets f cmplex-cnjugate ples and tw crrespnding sets f cmplex-cnjugate zers were bserved cnsistently in the parameterizatin f the experimental data. Crrespnding ple-zer pairs were als fund in the theretical data. The crdinates in the cmplex frequency plane f these ple-zer pairs will be referred t as R. R. aj + iw j,r. = alb, j = p,z, where the superscripts "a" and "b" are used t distinguish between the tw ple-zer pairs. By referring t the theretical frequency spectra. it was nted that the ple-zer a "a" had an imaginary crdinate w j ' j.. P.Z. near the resnant frequency f peridic surface, while the imaginary crdinate f the ple-zer pair "b" appeared near the minima due t the mde cnversin int the m = ±1 transverse spectral rders. The imaginary crdinates f these ple-zer pairs shwed little variatin fr N = M > 16. while the real crdinates tended t scillate slightly abut mean values as N = M increased. NVERSON OF SCATTERNG DATA USNG POLE-ZERO PARAMETERZATON A simple means t determine the prfile perid frm the ple-zer a data. is t nte that w j ' j = p,z, lies near the surface prfile resnance, hence The relatin f Eq.(2) can be imprved by carefully nting the shift a in w j j.. P,z. as hd increases. and. in the inverse prcess, allwing fr this shift nce hd has been determined. There are ptentially several different functins which can be btained frm the ple-zer data which will display a parametric dependence n prfile depth h. The functin which is presented here is the difference dr.. R. alb, between the real crdinates f the plezer pairs dr. - R. R. n b = az - apt " a (3) Plts shwing the variatin in dr., R. a,b with hid are shwn in Fig. 4. The agreement between thery and experiment 1s generally gd, except fr the hd.375 sample. t was nted that the prfile f the hd.375 stainless steel sample was nt truly triangular, due t imprper machining. Further experiments are needed. hwever. t verify that this is the cause f the discrepancy. The data f Fig. 4 prvides a means fr quantitative inversin f the reflected signals. Nte, hwever, that it is implicitly assumed that the surface prfile i. peridic in ne dimensin, and ba. a triangular. symmetric shape, which suggests tpics fr future wrk. Future studies need t cnsider 1) the dependence f ple-zer lci (2)

7 RESPONSE OF ELASTC PLATE 151 '" W.J.... a.. 0 n c i N _N a:!ci w a : : THEORY POLE-ZERO PAR "A" \, c l f ) W.. a.. 00 w'" N _0...N POLE-ZERO PAR "S" EXPERMENT "--(- a::: W a:: 0 ci "! r ' r ' HD Fig. 4. ) / THEORY ' HD Difference d as defined by Eq.(3) fr tw ple-zer pairs, versus hid. n the prfile shape (e.g. sinusidal, square, ramp, etc.), 2) the effects f deviatin frm true peridicity, including peridic surfaces f finite extent, and 3) the effects f peridicity in tw dimensins. The results shwn have encuraged the future refinement f ple-zer parameterizatin as a tl fr scattering data inversin. ACKNOWLEDGMENT This wrk was supprted by the Natinal Science Fundatin under Grant MEA REFERENCES 1. A. Jungman, L. Adler. J.D. Achenbach, and R.A. Rberts, Reflectin Frm a Bundary with Peridic Rugness. J. Acust. Sc. Am. 74(3): 1025 (1983). 2. R.A. Rberts, J.D. Achenbach. R.K. L. Adler. A. Jungman. and G. Quentin, Reflectin f a Beam f Elastic Waves by a Peridic Surface Prfile, WAVE MOTON 6:1, (1984). 3. J.T. Fkkema and P.M. van den Berg, E1astdynamic Diffractin by a Peridic Rugh Surface (Stress-Free Bundary), J. Acust. Sc. Am. 62:1095. (1977). 4. D.G. Dudley. Parametric Mdelling f Transient Electrmagnetic Systems. Radi Science 14: 387 (1979). 5. E.C. Levy, Cmplex Curve-Fitting. RE Transactins n Autmatic Cntrl: 37 (1959). 6. R.A. Rberts. E1astdynamic Scattering by Peridic Surface ndentatins, Ph.D. Thesis, Nrthwestern (1984).