I. INTRODUCTION. J. Acoust. Soc. Am. 112 (3), Pt. 1, Sep /2002/112(3)/835/8/$ Acoustical Society of America

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1 Diffraction correction for precision surface acoustic wave velocity measurements Alberto Ruiz M. and Peter B. Nagy a) Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, Ohio Received 1 December 2001; revised 15 March 2002; accepted 24 May 2002 Surface wave dispersion measurements can be used to nondestructively characterize shot-peened, laser shock-peened, burnished, and otherwise surface-treated specimens. In recent years, there have been numerous efforts to separate the contribution of surface roughness from those of near-surface material variations, such as residual stress, texture, and increased dislocation density. As the accuracy of the dispersion measurements was gradually increased using state-of-the-art laser-ultrasonic scanning and sophisticated digital signal processing methods, it was recognized that a perceivable dispersive effect, similar to the one found on rough shot-peened specimens, is exhibited by untreated smooth surfaces as well. This dispersion effect is on the order of 0.1%, that is significantly higher than the experimental error associated with the measurements and comparable to the expected velocity change produced by near-surface compressive residual stresses in metals below their yield point. This paper demonstrates that the cause of this apparent dispersion is the diffraction of the surface acoustic wave SAW as it travels over the surface of the specimen. The results suggest that a diffraction correction may be introduced to increase the accuracy of surface wave dispersion measurements. A simple diffraction correction model was developed for surface waves and this correction was subsequently validated by laser-interferometric velocity measurements on aluminum specimens Acoustical Society of America. DOI: / PACS numbers: Gp, Pt, Zc DEC I. INTRODUCTION Surface acoustic waves SAWs produce elastic displacements and stresses that are confined to the surface within a shallow depth of approximately 1 wavelength. Therefore, they are particularly sensitive to the boundary conditions at the surface and to the material properties in the near-surface region. Since the penetration depth of surface waves is frequency dependent, high-frequency SAWs are more affected by the material properties close to the surface while the low-frequency components give information on the bulk material properties of the substrate. This frequency dependence can be exploited to control the depth of the nearsurface region to be inspected. Another interesting feature of surface waves is that they are less affected by beam spreading than bulk waves, since they are confined to travel on the surface of the material; therefore, they diverge only in one dimension rather than in two like bulk modes. The use of surface acoustic waves for the characterization of surface roughness, surface residual stress, and coating thickness measurements has been studied by several authors. 1 5 The effect of surface cracks on the attenuation and dispersion of Rayleigh waves was explored by Zhang and Achenbach, 6 Warren et al., 7 and Pecorari These studies have found that the total change in the Rayleigh wave velocity is typically less than 0.3%. The challenge, of course, is to measure the Rayleigh wave velocity with sufficiently high accuracy. At this precision, the surface wave velocity a Electronic mail: pnagy@uceng.uc.edu seems to be affected by beam diffraction, and it may be necessary to use appropriate diffraction corrections in order to assess the true material-related dispersion. Of course, in many cases when the velocity exhibits a more dramatic change, e.g., in the case of coated substrates, this diffraction correction might be negligible. The theory of diffraction effects in the ultrasonic field of a piston source is straightforward and has been treated by many authors The adverse effect of diffraction in ultrasonic attenuation measurements is well known, and appropriate corrections are routinely used to increase the accuracy The application of similar diffraction corrections in ultrasonic velocity measurements is usually not necessary except for the most stringent accuracy requirements. 19,25 30 For a piston transducer radiating into a fluid medium, the complex diffraction correction can be calculated by either analytical means or numerical integration. For example, Rogers and Van Buren found a closed-form expression for the so-called Lommel diffraction correction for a circular piston radiator in the Fresnel approximation. 16 From the phase of their complex diffraction correction, the apparent local velocity of the diverging wave can be readily calculated. One finds that the apparent velocity is strongly frequency dependent even in nondispersive media and the dispersion decreases as the frequency increases, i.e., as the field asymptotically approaches a true plane wave. 19 The apparent dispersion also depends on the size of the receiver, which is important since the receiver aperture is often different from, typically much smaller than, the transmitter aperture. Using numerical integration, it can be shown that J. Acoust. Soc. Am. 112 (3), Pt. 1, Sep /2002/112(3)/835/8/$ Acoustical Society of America 835

2 the apparent velocity dispersion is perceivably higher when a point receiver is moved along the axis of the same transmitter instead of a finite-aperture piston receiver. To the best of our knowledge, similar results are not available in the literature for surface waves. The main goals of this paper are to present a simple diffraction correction model for surface waves and to experimentally validate this model using laser-interferometric surface wave velocity measurements on smooth aluminum specimens. It will be demonstrated that a readily measurable apparent dispersion on the order of 0.1% is exhibited by untreated smooth surfaces, which is significantly higher than the inherent experimental error associated with the measurements and comparable to the expected velocity change produced by near-surface compressive residual stresses in shot-peened metals. Therefore, high-precision surface wave dispersion measurements used in nondestructive materials characterization should be corrected for diffraction effects in order to achieve maximum accuracy. II. THEORETICAL MODEL Diffraction effects begin at the face of the transmitter and continue out to a point where the transducer is so far away that it might be considered as a point source, and the diffraction effects can be easily approximated as the spreading of a divergent beam of given directivity pattern. Generally, both the amplitude and phase of the generated acoustic field deviate from the ideal case of a plane wave. In this section we will consider Rayleigh wave propagation on the free surface of an elastic half-space. For different transmitter/ receiver configurations of practical importance, we will analyze the phase of the complex diffraction correction and derive appropriate dispersion corrections for precision velocity measurements. Let us assume that the acoustic field of a given radiator is scanned by a receiver moving along the axis of the radiator. Generally, the receiver is sensitive to the weighted average of the measured field parameter displacement, velocity, pressure, etc. over the receiver aperture. The size of the receiver aperture can be either negligibly small point sensor or finite. At a given axial distance z from the radiator, the measured field parameter m can be written as m zm 0 e i kzt D zm 0 e i kzt F z e iz. 1 Here, m 0 is a constant, t denotes time, is the angular frequency, k/c is the wave number, where c is the nondispersive sound velocity, D(z) is a complex diffraction coefficient that represents the deviation of the field distribution from that of an ideal propagating plane wave, F(z) denotes the amplitude variation along the axis, and (z) is a diffraction phase correction. The apparent local phase velocity c a can be calculated from the measured field parameter using c a z Arg m z c z. 2 Let us introduce the normalized velocity diffraction correction (c a c)/c as the relative change of the apparent velocity with respect to the nondispersive real sound velocity FIG. 1. Coordinate system used to calculate the acoustic field of a line source on the free surface (y 0) of an elastic half-space. c. We will also use z/a for the normalized distance and s ak for the normalized frequency, where a is an appropriate reference dimension, e.g., the half-length of a line source. Then, the normalized velocity diffraction correction can be expressed from Eq. 2 as follows: 1 11/s / 1 1 s. In ultrasonic nondestructive materials characterization, we often use Rayleigh-type surface waves for precision velocity measurements. The coordinate system used to calculate the acoustic field of a line source on the free surface (y 0) of an elastic half-space is shown in Fig. 1. The normal surface displacement of the Rayleigh wave produced by a line source of finite length on an isotropic half-space can be obtained from Lamb s classical solution for a point source 31 u x,zu 0 e 2 i k i t a e ikr dx a r, where r 2 x 2 z 2, and u 0 is a constant determined by the load intensity and polarization of the source and the shear modulus and Poisson s ratio of the material. Equation 4 is the one-dimensional equivalent of the well-known twodimensional Rayleigh integral, 32 which is often used to calculate the complex diffraction correction for bulk waves. It should be mentioned that the numerical prefactor in front of the integral could be easily incorporated into the constant u 0, but then its physical meaning would be changed. As it is, u 0 is the asymptotic limit of the normal surface displacement amplitude of the nondiverging Rayleigh wave produced by the line source when its length approaches infinity. A. Line source and point receiver Let us assume that we are using a point receiver to scan the normal surface displacement along the axis of the transducer (x 0). Then, we can write the displacement field as u(z) u 0 e i(kzt) D a0 (z), where the complex diffraction correction can be written in a normalized form as follows: 3 4 D a0 2 i s 1 e is 2 2 d 4, where x /a, and z/a and s ak as before. Due to the above-described normalization, D a0 asymptotically approaches unit as s. Equation 5 can be solved numeri- 836 J. Acoust. Soc. Am., Vol. 112, No. 3, Pt. 1, Sep M. A. Ruiz and P. B. Nagy: Diffraction correction for velocity

3 Surface waves are routinely detected by laser interferometers that are essentially point receivers. However, in order to increase the accuracy of the measurements the detection spot is often either extended laterally to a line of substantial length using a cylindrical lens or the focus spot is optically scanned in the lateral direction, while the received signal is digitized and averaged. This is often necessary to reduce the otherwise very high sensitivity of the point receiver to spatially incoherent structural noise in inherently inhomogeneous materials such as polycrystalline metals. 33 The question arises as to how much the velocity diffraction correction changes as a result of this averaging. Assuming that the line receiver has a total length of 2b, the average normal surface displacement can be calculated from Eq. 4 as follows: u avg zu 0 e i t k 2 i 1 b a e 2b dx b ikr dx a r FIG. 2. Normalized diffraction correction for surface wave velocity measurements made by a point receiver along the axis of a line transmitter. cally, and then the velocity diffraction correction can be calculated from Eq. 3. As an example, Fig. 2 shows the calculated normalized diffraction correction for surface wave velocity measurements made by a point receiver along the axis of a line transmitter. In the far field of the line source 6. lim D a0 2s i 1 is From Eqs. 3 and 6, the far-field asymptote of the normalized velocity diffraction correction can be written as follows: lim a It should be mentioned that the complex diffraction correction D could be more concisely written if the axial distance z were normalized to the near-field/far-field distance N a 2 /, as it is customary in the literature. For example, Eq. 6 could be rewritten as follows: lim D a0 S 2 1 S is 3S i, 8 where S z/n 2/s with our previous notation. However, the normalized velocity diffraction correction requires differentiation with respect to the normalized frequency s according to Eq. 3 ; therefore, it is actually simpler when expressed in terms of than in terms of S see Eq. 7 and our results later. B. Line source and line receiver 6 u 0 e i kzt D ab z, 9 where the complex diffraction correction can be written in the previously introduced normalized form as follows: D ab s 2 i 1 1 e 2 d is 2 2 d 4, where b/a denotes the length ratio between the receiver and the transmitter. In general, the length of the receiver b can be either smaller or larger than the transmitter length a, and Eq. 10 can be solved only numerically. In the far field of both transducers, i.e., when z a 2 / and z b 2 /, r can be approximated by rz (x x ) 2 /2z in the rapidly changing exponent and simply by rz in the slowly changing denominator of Eq. 9. Furthermore, the exponential function can be approximated by the two leading terms in its Taylor expansion, so that we get D ab s i 2 d d 1 1 is 2, 2 11 that can be readily integrated to get the following approximation for the complex diffraction correction: lim D ab 2s i 1 is Finally, from Eqs. 3 and 12, the far-field asymptote of the normalized velocity diffraction correction can be written as follows: lim ab Clearly, for b 0 ab ( ) a0 ( ), while for b a the apparent excess velocity is doubled, i.e., aa ( ) 2 a0 ( ). This increase in the apparent phase velocity with respect to the case of a point receiver is due to the contribution of those oblique wave components that reach the receiver at larger angles, therefore at lower group velocity, but higher phase velocity along the axis. C. Wedge transmitter and line receiver Most angle-beam SAW transducers consist of a longitudinal transducer of circular cross section mounted on a polymer wedge. This configuration produces an apodized SAW J. Acoust. Soc. Am., Vol. 112, No. 3, Pt. 1, Sep M. A. Ruiz and P. B. Nagy: Diffraction correction for velocity 837

4 over the transducer s aperture smoothes the phase variations, while further away it increases. The effect of apodization is twofold. First, it effectively reduces the transmitter length. Second, it smoothes the near-field oscillations. III. EXPERIMENTAL METHOD AND RESULTS FIG. 3. Diffraction corrections for velocity measurements made by a finitelength line receiver along the axis of another line transmitter for s 20 without a and with b apodization. displacement distribution because of the elliptical footprint of the circular piston on the surface. Although peripheral rays are further attenuated by leakage back to the wedge as the surface wave leaves the transmitter, a simple apodized model can be constructed by assuming a cosine SAW displacement distribution. Assuming that the receiver is not apodized, the complex diffraction correction given in Eq. 10 should be modified as follows: D ab s 2 i 1 2 d 1 1 d cos eis Such apodization slightly reduces the velocity correction by weakening the oblique rays responsible for the apparent increase in phase velocity. Figure 3 shows the calculated normalized diffraction corrections for surface wave velocity measurements made by a finite-length line receiver along the axis of a line transmitter for s 20 a without and b with apodization. As before, most of the near-field oscillations were cut off for clarity. Close to the transmitter the correction decreases with increasing receiver length as the averaging Figure 4 shows a schematic diagram of the experimental arrangement used in our surface wave dispersion measurements. A wedge transducer was mounted on the specimen and exited by a Panametrics 5270 broadband pulser. Three 12.7-mm-diameter Accuscan-S Panametrics screw-in transducers of 2.25-, 3.5-, and 5-MHz nominal center frequency were used on an ABWML-5ST 90 wedge also made by Panametrics. The propagating surface wave was detected by a LUIS 35 Fabry Perot interferometer. The laser beam was aimed at the surface of the specimen at a small angle of incidence, and an objective lens focused the diffuse reflection onto the tip of an optical fiber connected to the interferometer. The ultrasonic signal detected by the interferometer was digitized and averaged by a LeCroy 9310 oscilloscope and then sent to a computer for further processing. The specimen was mounted on a Velmex translation table and the relative position between the wedge transmitter and the laser spot was changed by a computer-accessible stepping motor controller. The scanning resolution of the translation table was m. Both the data acquisition and the scanner were controlled by a LABVIEW program. In order to assure the absolute accuracy of our velocity measurements, the temperature of the specimen was stabilized at 22 C within 0.7 C temperature variations are not expected to lead to dispersion unless the surface temperature of the specimen is significantly different from that of the interior. The specimens were scanned in two directions. Spatial averaging was used during lateral scanning normal to the wave propagation direction to reduce the incoherent scattering by i surface roughness and ii the inhomogeneous microstructure as well as iii coherent diffraction effects, especially in the near field of the transmitter. A slower stepwise scanning was per- FIG. 4. Experimental setup for laserultrasonic SAW velocity measurement. 838 J. Acoust. Soc. Am., Vol. 112, No. 3, Pt. 1, Sep M. A. Ruiz and P. B. Nagy: Diffraction correction for velocity

5 formed in the axial direction, i.e., parallel to the wave propagation, in order to map the phase of the surface vibration as a function of the propagation distance. Six 2024-T351 aluminum bars of 50.8 mm width203 mm length12.7 mm height were carefully polished with 2000-grade sandpaper parallel to the length of the specimen to minimize scattering by surface irregularities. The specimens were shot-peened over a 50.8-mm 50.8-mm square area at their center using 1.4-mm-diameter steel shots and 100% coverage. Two identical series of specimens were prepared using three different Almen intensities 4A, 8A, and 10A. r 34,35 1hThe at350 unpeened Cbefore smooth the ultrasonic side of one tests. of the specimens was used as a reference 0A. The first series of specimens was evaluated directly after shot peening and the second set was heat treated fo The objective of the heat treatment was to remove nearsurface material variations caused primarily by the shot peening, but, to a lesser degree, potentially present in the original cold-rolled bar stock material as well. These variations are associated with subtle near-surface effects such as the presence of residual stresses, increased hardness, dislocation density, and anisotropic texture. With the exception of surface roughness-induced scattering, annealing above the recrystallization temperature of the material effectively eliminates all other near-surface variations, which might otherwise contribute to the observed surface wave dispersion. To some degree, annealing also changes the microstructure and can lead to perceivable grain coarsening when it is overdone. As a result, the surface wave velocity might also change slightly, but, since the effect is essentially the same throughout the whole volume of the specimen, it does not necessarily lead to surface wave dispersion, though it might cause some microstructural dispersion via grain scattering in both bulk and surface wave propagation. In our measurements we first placed the front of the wedge transducer at a 30-mm distance from the laser spot and then scanned the beam along the propagation direction over a total distance of 50 mm, i.e., up 80 mm from the wedge, in steps of 80 m. At each of the 625 axial positions, we averaged 1000 pulses to increase the signal-to-noise ratio. In addition, during averaging we also used lateral scanning normal to the propagation direction to further improve the accuracy of the measurement. The lateral scanning dimension was 2.5 mm, i.e., significantly smaller than the beam width the diameter of the piezoelectric transducer mounted on the wedge was 12.7 mm. In this way, both temporal and spatial averaging was achieved at the same time and the laser interferometer effectively acted as a 5-mm-long line receiver. The LABVIEW program gradually changed the delay time of the LeCroy digital oscilloscope with respect to the trigger signal by using the programmable digital delay and highly accurate quartz master clock of the oscilloscope. At each step the new delay time was calculated from the scanning position using a nominal tracking velocity. In this way, the recorded signal moved only very slightly either forward or backward within the data window depending on whether the carefully chosen tracking velocity c n was higher or lower, respectively, than the actual surface wave velocity. In an ideal case, the tracking velocity is very close to the actual FIG. 5. Example of the measured normalized phase versus propagation distance curves on an Almen 4 shot-peened specimen at 16 different frequencies. one, and the broadband pulse does not move appreciably at all, though its shape slightly changes as a result of dispersion. The recorded data were spectrum analyzed by a discrete Fourier transform algorithm that determined the phase of the signal at different frequencies depending on the bandwidth of the transmitter the bandwidth of the interferometer is essentially flat from 2 to 100 MHz. As an example, Fig. 5 shows the normalized phase as a function of the propagation distance obtained from a 4Aintensity shot-peened specimen using a 3.5-MHz transducer. In order to achieve the highest possible phase accuracy, we have to limit the overall range over which the phase might change. To a large degree, this is achieved by carefully selecting the nominal tracking velocity that is used to control the delay of the digitizer. However, some time shift will inevitably occur, which in turn will produce large changes in phase that are linearly proportional to frequency. These changes might hide smaller frequency-dependent changes caused by dispersive wave propagation, and therefore must be further suppressed. This was achieved by normalizing the actual phase measured at any given frequency f to the center frequency f n of the transducer according to n f n / f. This normalization of the phase allowed us to observe the change in the slope of the phase due to dispersion, as well as to avoid folding back of the phase at 360 intervals. The velocity of the SAW was calculated from the slope of the normalized phase using the following formula: c n c m 1c n / / z 1c n / n n / z, 15 where c m is the measured surface wave velocity, c n is the tracking velocity chosen to minimize the observed total phase variation over the fairly long propagation distance. Equation 15 is the same as Eq. 2 with the measured surface wave velocity c m taking the place of the apparent local phase velocity c a, and the constant tracking velocity c n taking the place of the nondispersive sound velocity c. On the left side of Eq. 15 we introduced the angular frequency c n J. Acoust. Soc. Am., Vol. 112, No. 3, Pt. 1, Sep M. A. Ruiz and P. B. Nagy: Diffraction correction for velocity 839

6 FIG. 6. Surface wave velocity measurements on four different aluminum specimens using two different transmitters of 3.5- and 5-MHz nominal center frequencies. FIG. 7. Surface wave velocity measurements on unpeened smooth aluminum specimens before and after annealing HT using three transmitters of 2.25-, 3.5-, and 5-MHz nominal center frequencies. used for normalization n 2 f n, that is typically chosen to be either the nominal or actual center frequency of the transmitter. The average slope of the normalized phase n / z was calculated at each frequency by finding the best-fitting linear approximation to all data points using the least-meansquare method. In Fig. 5, the low-frequency slopes are slightly negative, which, according to Eq. 15, means that the low-frequency components propagated at a velocity slightly higher than the tracking velocity. In comparison, the high-frequency slopes are positive, which indicates that they propagated slower than the tracking velocity, i.e., the velocity decreased with increasing frequency over the bandwidth of the transmitter. Figure 6 shows the results of our surface wave velocity measurements on four different aluminum specimens using two different transmitters of 3.5- and 5-MHz nominal center frequencies. On the three shot-peened specimens we can observe that the dispersion increases with peening intensity. This effect is mainly caused by surface roughness-induced scattering, though the other previously mentioned effects of shot peening also contribute. For our immediate purposes, the most important observation is that, although the shotpeened specimens obviously exhibit much stronger dispersion, some dispersion, that is well above the experimental uncertainty of the measurement 0.02%, is also exhibited by the smooth, untreated specimen. Figure 7 shows the results of our surface wave velocity measurements on unpeened smooth aluminum specimens before and after annealing using three transmitters of different nominal center frequencies. As was mentioned above, the observed dispersion even on an unpeened smooth surface suggests that the apparent velocity is affected not only by the surface roughness and other material variations residual stress, dislocation density, local texture, and hardening produced by shot peening, but also by the intrinsic diffraction of the surface acoustic wave. Because of the inevitable microstructural change caused by annealing, the absolute surface wave velocity is 0.2% higher in the heat-treated specimen. Actually, the apparent dispersion is also somewhat higher in the heat-treated sample than in the original one, which indicates that some contribution from the microstructure is also present in the dispersion. It should be mentioned that at even higher frequencies the true material dispersion caused by grain scattering in the polycrystalline aluminum is expected to dominate, assuming that no imperfections are present. 36 Figure 8 shows the comparison between the experimental dispersion curve measured on a smooth aluminum specimen and the theoretical predictions of our diffraction model. The normalized surface wave velocity was calculated from the apodized diffraction model using the experimental parameters including a 22-mm added propagation length within the wedge from the transducer to the surface of the specimen and the best-fitting c mm/ s surface wave velocity. The measured velocity is more affected at low frequencies, while it becomes more frequency independent at high frequencies. When we plot the experimental normalized velocity against the theoretical prediction, there is a fairly good qualitative agreement between them. These results suggest that the measured velocity is perceivably affected by the previously discussed diffraction effects, and the apparent dispersion should be corrected for diffraction in order to realize the full potential of high-precision dispersion measurements in ultrasonic materials characterization. These experimental results demonstrate that a measurable dispersive effect is exhibited by well-polished smooth FIG. 8. Comparison between the experimental dispersion curve measured on a smooth aluminum specimen and theoretical predictions of the diffraction model (c mm/ s. 840 J. Acoust. Soc. Am., Vol. 112, No. 3, Pt. 1, Sep M. A. Ruiz and P. B. Nagy: Diffraction correction for velocity

7 surfaces, which cannot be attributed to surface roughness scattering. This behavior can be explained by our theoretical results that demonstrate that diffraction effects do cause an apparent dispersion that has very similar features magnitude and frequency dependence to those of the experimentally observed results see Fig. 8. Of course, the measured dispersion could also be caused, at least in part, by the intrinsic attenuation in the solid, e.g., grain scattering. As we mentioned above, the perceptible increase of dispersion in annealed aluminum see Fig. 7 could be attributed to excess material dispersion, especially at higher frequencies. In order to quantitatively estimate the degree of material dispersion caused by grain-scattering effects on SAW propagation, we could rely on existing, though extremely complicated, theoretical predictions. 36 Alternatively, much simpler order-ofmagnitude type of predictions can be made based on the observed frequency-dependent attenuation using the wellknown Kramers Kronig relationship. After Viktorov, we are going to estimate the dispersion of the SAW from the behavior of the shear wave, since it can be measured much easier and much more accurately. 37 Figure 9 a shows the experimentally determined attenuation coefficient versus frequency curves for the shear wave in our aluminum specimen before and after annealing. In order to verify the similarity between the attenuation of the shear and Rayleigh waves, we also included in Fig. 9 a the corresponding SAW results for the more attenuating annealed case. One of the difficulties in applying the global Kramers Kronig relationship is the necessity of knowing the frequency dependence of the attenuation coefficient at every frequency to calculate the dispersion at a given frequency, 38,39 although approximate methods are also available in the literature to calculate the dispersion from the nearly local behavior of the attenuation. 40,41 Since in the case of grain scattering in polycrystalline materials the behavior of the attenuation coefficient is known over the whole frequency range, we used the global Kramers Kronig relationship. First, we estimated the grain size by matching our experimental data with Stanke and Kino s unified model, 42 then used this model to estimate the grain-scattering-induced dispersion. It should be mentioned that similar results also can be obtained directly from Stanke and Kino s unified model for dispersion. Figure 9 a shows the theoretically estimated material dispersion based on the global Kramers Kronig relationship the scale was kept the same as in Fig. 8 for easier comparison. These results indicate that the intrinsic dispersion of the specimen, as estimated from the measured frequency-dependent attenuation, is at least one order of magnitude lower than the measured dispersion and provides additional evidence that the measured apparent dispersion is indeed caused by the diffraction effect. IV. CONCLUSIONS Surface wave dispersion measurements are often used in ultrasonic, nondestructive testing to characterize near-surface material variations that might lead to significant dispersion. It was found, however, that a perceivable dispersion is exhibited by untreated smooth surfaces as well. Although this dispersion effect is only on the order of 0.1%, i.e., relatively FIG. 9. Experimentally determined attenuation coefficient versus frequency curves a and the theoretically estimated material dispersion based on the global Kramers Kronig relationship b. small with respect to the dispersion caused by typical nearsurface material variations, it is clearly higher than the experimental error associated with the measurements themselves. This paper demonstrated that the cause of this apparent dispersion, at least in part, is the diffraction of the SAW as it travels over the surface of the specimen. This conclusion suggests that a diffraction correction may be used to increase the accuracy of precision surface wave dispersion measurements. A simple diffraction correction model was developed for surface waves and this correction was subsequently validated by laser-interferometric velocity measurements on smooth aluminum specimens. Further work is needed to develop practical ways of separating diffractionrelated dispersion from both surface roughness and microstructural scattering and from additional dispersive effects due to the presence of near-surface residual stresses and texture. J. Acoust. Soc. Am., Vol. 112, No. 3, Pt. 1, Sep M. A. Ruiz and P. B. Nagy: Diffraction correction for velocity 841

8 ACKNOWLEDGMENTS The authors wish to thank the Universidad Michoacana de San Nicolás De Hidalgo and CONACYT-MEXICO for their support of Alberto Ruiz Marines during his Ph.D. studies. This work was supported by the Metals, Ceramics, and NDE Division of the Air Force Research Laboratory under Contract No. 01-S C1. 1 A. I. Lavrentyev and W. A. Veronesi, Ultrasonic measurement of residual stress in shot peened aluminum alloy, in Rev. Progr. Quant. Nondestr. Eval. AIP, Melville, 2001, Vol. 20, pp C. Glorieux and W. Gao, Surface acoustic wave depth profiling of elastically inhomogeneous materials, J. Appl. Phys. 88, M. Duquennoy, M. Ouaftouh, M. L. Qian, F. Jenot, and M. Ourak, Ultrasonic characterization of residual stresses in steel rods using a laser line source and piezoelectric transducers, NDT & E Int. 34, V. V. Krylov and Z. A. Smirnova, Experimental study of the dispersion of a Rayleigh wave on a rough surface, Sov. Phys. Acoust. 36, F. Lakestani, J. F. Coste, and R. 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