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On the validity and improvement of the ultrasonic pulse-echo immersion technique to measure real attenuation
Ultrasonics 54 (2014) 544–550
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On the validity and improvement of the ultrasonic pulse-echo immersion technique to measure real attenuation Miguel A. Goñi, Carl-Ernst Rousseau ⇑ Mechanical Engineering, Wales Hall, Univ. of Rhode Island, Kingston, RI 02881, USA
a r t i c l e
i n f o
Article history: Received 25 April 2013 Received in revised form 7 August 2013 Accepted 8 August 2013 Available online 20 August 2013 Keywords: Attenuation Reflection coefficient Ultrasonic Immersion Alternative method
a b s t r a c t A fundamental assumption embraced in conventional use of the ultrasonic pulse-echo immersion technique to measure attenuation in solid materials is revisited. The cited assumption relies on perfect and immutable adhesion at the water to sample interface, a necessary condition that allows calculating the reflection coefficient at any interface from elastic wave propagation theory. This parameter is then used to correct the measured signal and obtain the real attenuation coefficient of the sample under scrutiny. In this paper, cases in which the perfectly cohesive interfacial condition is not satisfied are presented. It is shown also that in those cases, the repeatability of the conditions at the interface is always uncertain. This implies that the reflection coefficients are unknown, even when density is known. A new method of simultaneously measuring the reflection coefficients for both exposed interfaces that are normal to the transducer, and the attenuation coefficient of the specimen is developed and is presented here. The robustness of the new method is proven, as we demonstrate that the proper value of attenuation is achieved independently of the continuously varying interfacial conditions of these non-ideal cases. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The ultrasonic immersion technique has a very wide range of applications. It is commonly used to determine the elastic constants of homogeneous, heterogeneous, isotropic, and anisotropic composites [1,2]. It can also be used for material inspection, to detect defects within materials, lack of bonding, voids, or cracks [3,4]. Within our sphere of interest, it is also frequently used to measure wave velocities, attenuation, density, and thickness [5,6]. The present work will focus on the application of the technique to measure the attenuation coefficient of materials which, as referred in this work, is a material property, in contrast to the apparent attenuation measured by other practices such as described in ASTM Standard E664 [6]. This property is also referred to as real or true attenuation. There exist alternate techniques capable of measuring the attenuation coefficient of solid materials, which are based on the use of direct contact instead of immersion transducers. One such technique developed by Treiber et al. [7] is able to calculate the attenuation coefficient of any material even when reflection coefficients are unknown. This technique overcomes the unknown conditions at the three-media-interface formed by the transducer, the coupling agent, and the specimen by using an additional trans⇑ Corresponding author. Tel.: +1 401 874 2542; fax: +1 401 874 2355. E-mail address: [email protected] (C.-E. Rousseau). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.08.008
ducer to measure the reflection coefficient at that interface. This provides a valid measurement of the real attenuation of the material. However, based on the authors’ experience, direct contact transducers are not always as reliable as immersion transducers, sometime producing highly distorted second and third echoes that lend themselves poorly to analysis. Inconsistencies associated with the contact transducers led us to adopt immersion transducers in the measurement of attenuation coefficients of solid materials under the well-known pulse-echo immersion technique. The ultrasonic pulse-echo immersion technique uses a broadband immersion transducer positioned perpendicularly to the specimen and records the successive reflections occurring at the front and back walls of the specimen [8,9]. The spectrums of the first and second echoes are typically used to calculate the attenuation coefficient. In addition, the reflection coefficients at both faces of the sample are needed to correct the measured signals. Sometimes, the transmitted pulses are used instead of the reflected ones and the process is applied with the only difference that the transmission coefficients might be needed, depending on the setup of the test [10]. This variant of the immersion technique is known as through transmission mode. In both test setups, the reflection and transmission coefficients are calculated from elastic wave propagation theory [8–10] for the case of a plane dilatational wave impinging upon a plane interface between two media under the condition of a perfectly bonded interface [11,12]. In this paper, this assumption is analyzed by performing experiments in different
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materials for which the reflection coefficients are measured and compared to those predicted by the theory. It is shown how, for some cases, the reflection coefficients do not match the theoretical ones and therefore, care must be taken when calculating the attenuation coefficient to avoid introducing large errors. Some relatively recent versions of the classical immersion technique are able to provide the attenuation coefficient without the knowledge of the reflection or transmission coefficients [13–15]. However, these versions require equal, [14,15], or consistent, [13], reflection coefficients at both faces of the specimen in order to proceed with the calculations to obtain the attenuation coefficient. The question, indeed, is whether these assumptions do hold true during the execution of the ultrasonic immersion technique, especially for those cases in which the reflection coefficients do not match the theoretical calculation.
(ks), as well as the transducer size (a), and the distance between the transducer and the closest face of the specimen (L):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2p 2p 2p 2p cos ; DðsÞ ¼ þ sin J0 J1 s s s s where the s variables are described as s00 ¼ 2Lkw =a2 , s1 = (2hks + 2Lkw)/a2, s2 = (4hks + 2Lkw)/a2. The functions J are Bessel functions. 2.1. Attenuation coefficient calculation Dividing Eq. (2) by Eq. (3) and performing the requisite simplifications and reformulation, an expression for the attenuation coefficient a is obtained as shown below:
1 V1 Dðs2 Þ ; ln RA RB 2h Dðs1 Þ V2
2. Theory
a¼
Since this work is based on the pulse-echo mode of the Ultrasonic immersion technique, only one transducer is necessary, which works alternatively as an emitter and a receiver. Fig. 1 illustrates the supposed trajectory of both emitted and received pulses, as interactions take place between the ultrasonic pulse generated by the transducer, V0, and the specimen immersed in the liquid medium. The inclined pulse trajectory illustrated in Fig. 1 is so rendered to assist with the conceptual clarity of the phenomenon, as the actual paths would be directed perpendicularly with respect to the face of the specimen. The chosen notation defines V0 as the emitted pulse; V 00 corresponds to the front wall (A) reflection; V1 and V2 are the first and second echoes reflected off the back wall (B), respectively. The magnitudes of these reflections, as a function of that of the emitted pulse, are derived presently, as shown. In the relations presented below, the following phenomena are taken into account:
where V1 and V2 are known, since they are direct readings of the transducer. If the reflection coefficients were to be calculated following conventional methodologies, that is to say, according to elastic wave theory, then:
The pulse will partially transmit and partially reflect at the liquid to solid specimen interfaces. Attenuation will occur only within the specimen’s internal boundaries, and is neglected as the pulse traverses the liquid medium. The traveling pulses will diverge with distance. Thus,
V 00
¼ V 0 RA Dðs00 Þ;
ð1Þ
V 1 ¼ V 0 T 2A RB Dðs1 Þ expf2hag; V2 ¼
V 0 T 2A R2B RA Dðs2 Þ expf4h
ag;
ð2Þ ð3Þ
where RA, RB, TA are the reflection and transmission coefficients of faces A and B, respectively. D(s) stands for the beam spreading of the pulse, a is the attenuation coefficient of the specimen, and h is the thickness of the specimen. For a description of the beam spreading function, D(s), the reader is directed to the derivation given by Rogers and Van Buren [16]. It is a function of the wavelength of the ultrasound in liquid medium (kw) and within the specimen
RA ¼ RB ¼
Zs Zw ; Zs þ Zw
ð5Þ
where Zs is the acoustic impedance of the specimen, and Zw is the acoustic impedance of the water. In the above equation, directionality is irrelevant. 2.2. Measurement of reflection coefficient As discussed in the Introduction, the main three assumptions used by current methodologies to calculate the attenuation coefficient of a material are being tested. On the one hand, it is desired to know if the reflection coefficients match the value given by Eq. (5) for all materials. On the other hand, it is desired to know if the reflection coefficients are equal at both faces of the specimen during a particular instance of immersion and if they are consistent between consecutive immersions. In order to ascertain the veracity of these assumptions, it is necessary to measure the reflection coefficients. This measurement was performed by the method illustrated in the first two subsets of Fig. 2. The method is capable of measuring the reflection coefficients at both faces of the specimen for a unique immersion and is expounded upon in depth in Section 3.2. By means of Steps 1 and 2, the reflection coefficient at face B, RB, can be measured. Likewise, using Steps 3 and 4, the reflection coefficient at face A, RA can be measured. The mathematical fundamentals behind this measurement are relatively simple and proceed as follows: In the process pertaining to each of the previous steps, one signal is recorded. Though several options present themselves, the first echo of each signal was chosen to evaluate the reflection coefficients. Referring to Eq. (2), it is clear that the only parameter altered between the two configurations of Steps 1 and 2 is RB since the transducer and the specimen were never moved. In the configuration corresponding to Step 1, it is assumed that exposure to air results in complete reflection, i.e., RB = 1. Therefore, utilizing Eq. (2), and taking the ratio of the respective first echoes from Step 2 and Step 1, yields the water–specimen reflection coefficient at side B, RB. Alternatively, use of Steps 3 and 4 would yield RA. ð2Þ
RB ¼
Fig. 1. Interaction between an emitted pulse V0 and a specimen immersed in water.
ð4Þ
V1
; ð1Þ
V1
ð3Þ
RA ¼
V1
ð4Þ
V1
:
ð6Þ
where the subscripts stand for the first echo and the superscripts stand for the corresponding step in the experimental sequence.
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Fig. 2. Procedure for implementation of new method.
This procedure is capable of measuring the reflection coefficients that correspond to a unique immersion since each side of the tank is only filled once. Thus, repeating this procedure several times allows the experimentalist to assess the various assumptions formulated above. 3. Experiments and results Experiments were conducted on two Teflon specimens and two polycarbonate specimens. One of the polycarbonate specimens was coated with a minuscule layer of a hydrophobic substance known commercially as Hydrobead-TÒ, measuring a few microns. This coating was applied to test the validity of the assumption of perfectly bonded interfaces, a condition that is necessary for the proper implementation of Eq. (5). The polycarbonate specimen was sprayed with this material on only one of its faces, and was left to cure for 24 h. The Teflon specimens were designated as Teflon Plate 1 and Teflon Plate 2, respectively. Similarly, the Polycarbonate specimens were denoted as Polycarbonate Coated Plate and Uncoated Plate, respectively. Both Teflon plates were 3 3 in (76 76 mm) and 0.39 in (10 mm) thick, whereas both polycarbonate plates were 4 4 in (100 100 mm) and 0.476 in (12.1 mm) thick. These dimensions were carefully selected to prevent lateral wall reflections from interacting with either the first or second echoes. The density of the Teflon specimens was measured to be 2160 kg/m3, and that of polycarbonate 1210 kg/m3. The respective longitudinal wave speeds for the Teflon and Polycarbonate specimens calculated from the first two echoes are 1285 and 2208 m/s. A 1 MHz Panametrics (Olympus NDT) V303 Immersion transducer was used, together with a Panametrics 5058PR pulser/ receiver unit. The signals were displayed and recorded by a Tektronix Digital Oscilloscope with a 100 MHz bandwidth and a 1.25 GS/s sampling rate. Each signal recorded with the oscilloscope was transferred to a computer to calculate its spectrum by performing a FFT analysis using the software MATLAB. Note that each echo was individually analyzed and transposed onto the frequency domain from which all the calculations described in Section 2
proceeded. Consequently, all reflection and attenuation coefficients calculated are presented as a function of frequency. 3.1. Experimental evaluation of reflection coefficient measurements The reflection coefficients of all four specimens (Teflon and Polycarbonate), at their respective interfaces with water, were measured according to the procedure described in Section 2.2 and illustrated in Fig. 2. Each measurement was compared to the value provided by Eq. (5). Fig. 3 sequentially shows several measurements of the reflection coefficients for the Teflon Plate 1 and the Polycarbonate Coated specimen at both faces during separate instances of immersion. Subsequent measurements within those specific settings yielded experimental errors well within 1%, not appreciable enough to be displayed within the plots. Following each independent instance of immersion and corresponding measurements, each specimen was removed from the water, then reimmersed for additional measurements. This was necessary to test the consistency of the reflection coefficients. Before proceeding further, it must be underscored that the behaviors observed in Fig. 2 were consistent for both specimens during all further tests conducted. Identical experiments were performed on the other Teflon specimen, designated as Teflon Plate 2, as well as the other Polycarbonate specimen, designated as Polycarbonate Uncoated Plate. Regarding Teflon Plate 2, a similar behavior to Teflon Plate 1 was observed even though a better match was achieved most of the times for the reflection coefficient between the experiments and the theoretical value provided by Eq. (5). For the Polycarbonate Uncoated case, excellent agreement between experimental and theoretical reflection coefficients was achieved for all faces and immersion instances, identically to the uncoated face of Polycarbonate Coated Plate (Fig. 3). With respect to the layout of the Polycarbonate Coated plate, it was determined by Scott and Gordon [17] that the reflection coefficient of an interface consisting of a thin layer interposed between two substantially larger media is given by:
M.A. Goñi, C.-E. Rousseau / Ultrasonics 54 (2014) 544–550
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Fig. 3. Comparison of the reflection coefficients obtained from two separate instances of immersion for Teflon Plate 1 and two separate instances of immersion for Polycarbonate Coated Plate.
T T R ðcos h R R j sinðhÞ 12 21 23 21 23 R ¼ R12 þ ; ð1 R21 R23 cos hÞ2 þ ðR21 R23 sin hÞ2
ð7Þ
where subscript ‘2’ represents the thin layer, subscripts ‘1’ and ‘3’ are the two media on either sides of the layer, and h = 4ph/k2. The variable h is the thickness of the thin layer, and k2 is the wavelength of the pulse acting on the thin layer media. Eq. (7) has been used
widely, including in works by Vincent [18] and Rose and Meyer [19]. If we consider the condition for which h k2, then, h 0. For Z Z this case, knowing that Rij ¼ Zji þZji , it is trivial to demonstrate that 1 Eq. (7) reduces to Eq. (5), in the form R ¼ ZZ31 Z . Note that the thickþZ 3 ness of the coating was measured with an optical microscope to be approximately 25 microns. Unfortunately, the properties of the coating were not provided by the manufacturer, but considering
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its polymeric nature, the wavelength of this material corresponding to a 1 MHz frequency is estimated to fall in the range of 1.5 to 3 mm. Hence, the thickness of the coating is two orders of magnitude smaller than the wavelength. Thus, an approximation that neglects the coating would result in less than a 0.1% error and therefore Eq. (5) was used for this specimen also. The evaluation of the reflection coefficients, obtained for the purpose of this work, shows that these coefficients are seldom accurate when applied to materials exhibiting hydrophobicity. The assumptions stated in the Introduction and at the beginning of Section 2.2 can be easily violated by these materials. Clearly, these materials do not always fulfill requirements of a perfectly bonded interface condition necessary to use Eq. (5). The use of these erroneous reflection coefficients would, in turn, introduce significant errors in subsequent calculations of the attenuation coefficient. Moreover, different results are observed between opposite faces of a unique specimen during a given immersion. Therefore, the validity of the prevalent practice that universally adopts identical reflection coefficients on both faces (even if their value is unknown) is greatly weakened. Furthermore, the present experiments show that each test has intrinsic values for the reflection coefficients (they differ from test to test) providing evidence of inconsistency of the reflection coefficients during different immersions. Consequently, attenuation measurements must be captured using the exact conditions in existence during the assessment of the respective reflection coefficients, in order for the former measurements to retain their validity. The materials for which these special considerations must be held in place seem to be those that present a very low interaction, that is to say, a minimal adhesion force to water, and the inability to adequately sustain the tensile stress component of the ultrasonic pulse. Besides this, and more importantly, another consequence of this small adhesion force is the presence of air molecules at the surface roughness level (micro-scale), that further hastens serious imperfection at the interface. Hence, it is necessary to develop an alternative method capable of correctly measuring the attenuation coefficient (a) for any material. The remedial method, developed as a solution to the numerous uncertainties described above, is presented next.
the four signals recorded, where the values for RA and RB would be calculated as follows: if signal V(1) is used, RB = 1 and RA is a measured value by the new method; if signal V(2) is used, RB and RA are the measured values by the new method; if signal V(3) is used, RB and RA are the measured values by the new method; if signal V(4) is used, RA = 1 and RB is a measured value by the new method. Attenuation coefficient measurements were conducted with this new procedure for the same Teflon and Polycarbonate specimens, described above. Regarding the Teflon specimens, thirty measurements were obtained for Teflon Plate 1 and twelve measurements for Teflon Plate 2. An average value was calculated for each Teflon plate. Both averages were compared to verify the reliability of the new method. Since both plates are made of the same material, the same value for the attenuation coefficient is expected. The identical process was repeated to the Polycarbonate plates for which eight measurements were averaged for the uncoated plate and five for the coated plate. The results are plotted in Fig. 4 for both Teflon and Polycarbonate plates. Error bars, at each of the
3.2. Proposed method for measuring true attenuation As a solution to the difficulties enumerated above, a method is proposed, that will allow the researcher to measure reflection coefficients on both faces of the specimen during a unique immersion and, simultaneously, will use one of the signals involved in that process to calculate the attenuation coefficient. A configuration that fulfills this goal has already been introduced in a previous section, and is illustrated in Fig. 2. The method fragments the experiment into four distinct steps, and uses a tank divided in two by a partition, which serves to both provide complete separation between the two sides of the tank, and performs the function of a holder for the specimen. In the first step only side A of the tank is filled with water, and the transducer, immersed in the water on that side, is positioned perpendicularly to the specimen, at a distance beyond the near-field. Signal V(1) is recorded. In the second step, side B of the tank is filled with water while the transducer remains untouched, still immersed in water on side A of the tank. Signal V(2) is recorded. In the third step, the transducer is moved to side B, precisely placed such as to maintain alignment with respect to its former position within side A. Signal V(3) is recorded. In the fourth step, side A is drained of the liquid, while the transducer remains untouched, immersed in water on side B of the tank. Signal V(4) is recorded. First, the reflection coefficients for the water–specimen interfaces A and B are calculated according to Eq. (6). Next, the attenuation coefficient can be calculated according to Eq. (4) with any of
Fig. 4. Real attenuation for the Teflon specimens (top) and Polycarbonate specimens (bottom).
M.A. Goñi, C.-E. Rousseau / Ultrasonics 54 (2014) 544–550
Fig. 5. Real attenuation for the Teflon specimens (top) and Polycarbonate specimens (bottom), obtained when Assumption 1 is violated.
frequencies evaluated, are also shown in the figure, along with the linear regression equation of the attenuation coefficients as a function of frequency. It is clear that for each material, the average curves depicting the behaviors of independent specimens are visually indistinguishable, and the corresponding equations are virtually identical. Fewer tests were conducted for the Polycarbonate specimens than for the Teflon ones because of the narrower range of the standard deviation achieved by the results of the former. Possible sources of error could include: lack of perpendicularity between transducer and specimen, lack of parallelism between the specimen faces, and random noise in the signals, among the most significant. 4. Summary and conclusion Three common assumptions used to calculate the attenuation coefficient of materials through the ultrasonic immersion technique were revisited, namely: 1. The water–specimen interface fulfills the condition of a perfectly bonded interface at all instants of time during any interaction with ultrasonic pulses [8–10]. 2. The reflection coefficients are equal on both faces of the specimen during a particular immersion [14,15]. 3. The reflection coefficients are consistent and therefore are exact duplicates of each other during different immersions, though not requiring a fulfillment of Assumption 2 [13].
549
Under Assumption 1, expressions for the reflection and transmission coefficients can be derived theoretically, as shown in Eq. (5) for the reflection coefficient. On the other hand, the techniques based on Assumptions 2 and 3 are capable of extracting the attenuation coefficients in specimens for which neither reflection nor transmission coefficients are known. They evidently do not require the conjecture of perfectly bonded interface. Based on the present study, all three assumptions have been proved erroneous for some materials. In particular, materials that present a level of hydrophobicity similar to or higher than that of Teflon are shown to behave otherwise, with an apparently very weak adhesive interfacial condition. Moreover, the interface conditions are very difficult to control due to the random presence of air molecules at the surface roughness microscale, resulting in significant differences between the two directly opposite sides of the same specimen and between successive tests conducted on the same specimen. In other words, immersing the specimen in water several times does not guarantee same interface conditions every time. Strong evidence of the above-stated fact is provided by Fig. 3, which clearly shows high probabilities for the absence of perfectly bonded interfaces in some materials, as well as for unequal reflection coefficients at opposite sides of a single specimen during a unique immersion and also lack of consistency in the reflection coefficients between different immersions. Consequently, it became necessary to develop a new method to apply the ultrasonic immersion technique, that could simultaneously measure reflection coefficients and attenuation coefficient, in order to ensure that identical water–specimen interface conditions are preserved throughout the entire process. It was shown in Fig. 4 how several tests with very different interface conditions from each other resulted in the same attenuation coefficient for a given material, proving the conceptual and practical validity of the new method presented. In contrast, Fig. 5 shows the attenuation coefficients that would have been obtained for these specimens if Assumption 1 had been embraced without question. Discrepancies in those results are evident since attenuation cannot be different between specimens made out of the same material. This error is a consequence of using erroneous reflection coefficients in the calculation of the attenuation coefficient. In order to further corroborate the validity of the method, the magnitude of the attenuation coefficients obtained using the present methodology were compared with those found in literature. Mobley and Vo-Dinh [20] studied the attenuation of Polycarbonate (Lexan) by means of two different methods, obtaining values of 50.8 Np/m at 1 MHz for the attenuation coefficient, and 56.8 Np/m/MHz for the slope of the linear regression curve describing variation of attenuation with frequency. These values are very similar to the ones obtained in this study. In addition, Selfridge [21] and Kaye and Laby [22] provided values of approximately 267 Np/m and 240 Np/m at 5 MHz respectively for the attenuation coefficient in Polycarbonate. Considering a linear relationship between attenuation and frequency, the quoted values would correspond to 53.4 Np/m and 48 Np/m for attenuation at 1 MHz, respectively. This is done by simply dividing the values corresponding to the 5 MHz frequency by 5. Once again, these values are very similar and within the range of the attenuation measured in this study for Polycarbonate. Regarding the Teflon material, a curious situation was encountered. While Selfridge [21] measured an attenuation value of 44.9 Np/m at 5 MHz, Kaye and Laby [22] provided a value of 430 Np/m at that same frequency. Such a large difference can be explained by the perfectly bonded interface assumption violation revealed and proven in this study. It is very likely that the reflection coefficients involved in the tests performed by Selfridge were higher than those provided theoretically by Eqs. 5. Similarly, the transmission coefficients involved in Selfridge’s work were likely
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M.A. Goñi, C.-E. Rousseau / Ultrasonics 54 (2014) 544–550
lower than the theoretical ones [11,12]. Since the theoretical transmission coefficients were used to calculate the attenuation coefficient of Teflon, the measured signals were overcorrected and hence an underestimation of the attenuation coefficient was obtained. Summarizing: this unusually large range is in fact somewhat similar to the observations pertaining to Fig. 5. In order to compare values, the attenuation coefficients obtained by Selfridge, and Kaye and Laby are divided by 5, providing the values of 8.98 Np/m and 86 Np/m, respectively at a 1 MHz frequency. The tests conducted in this paper, with the method developed here, show an attenuation coefficient value of 97.5 Np/m at 1 MHz. This value is slightly higher than that provided by Kaye and Laby, which could indicate that the reflection or transmission coefficients in their tests were slightly higher or lower than the theoretical values. On the other hand, the value provided by Selfridge is significantly smaller than the one obtained in this work indicating that the reflection or transmission coefficients in his tests were respectively much higher or lower than the theoretical values. It can, thus, be concluded that the proposed method overcomes the problems found for some materials in the conventional use of the ultrasonic immersion technique to measure attenuation. It was shown in Fig. 5 that using conventional methods may lead to wide range of erroneous results that may span an entire order of magnitude, whereas using the method proposed herein provides repeatable results, as demonstrated in Fig. 4. It could also be pointed out that the reflection coefficient of the coated interface showed significant dependency with frequency. This could be due to a high concentration of air molecules at the interface, which then translates into dependency on varying wavelengths. Furthermore, the proposed method is not limited with respect to material type. Neither are there any restrictions in regards to the use of other immersion fluids besides water. Nevertheless, we do realize that the accuracy of the method is contingent upon consistent amplitudes, V0, of the emitted pulses. In the present experiment, this was not an issue, but must generally be verified by the experimentalist. As a final note, it is acknowledged that in some materials, values of the reflection coefficients could display time dependency, since the air molecules present at the interface could slowly drift with time. For these cases, the authors suggest waiting until the interface assumes stability and only then proceed with measurements. Nevertheless, the proposed method is highly recommended for obtaining the attenuation coefficient measurement of any material, due to its simplicity, its low cost (only requires one transducer), and the fact that it is not bound to questionable assumptions.
References [1] P.W.A. Stijnman, Determination of the elastic constants of some composites by using ultrasonic velocity measurements, Composites 26 (1995) 597–604. [2] J.H. Lee, Y.C. Park, Nondestructive characterization of metal-matrix composite by ultrasonic measurement, Composites Engineering 5 (1995) 1423–1431. [3] C. Kotval, Ultrasonic evaluation of aluminum master cylinder die castings for porosity, Materials Evaluation 38 (1980) 23–27. [4] G.J. Jung, Y.S. Ahn, Y.S. Cho, S.K. Park, D.S. Gil, W.H. Lee, Implementation of ultrasonic immersion technique for babbitt metal debonding in turbine bearing, Key Engineering Materials 270–273 (2004) 271–276. [5] V.K. Kinra, V.R. Iyer, Ultrasonic measurement of the thickness, phase-velocity, density or attenuation of a thin-viscoelastic plate. 1. The forward problem, Ultrasonics 33 (2) (1995) 95–109. [6] Standard practice for the measurement of the apparent attenuation of longitudinal ultrasonic waves by immersion method. ASTM E664/ E664M – 10, 2005. [7] M. Treiber, J.-Y. Kim, L.J. Jacobs, J. Qu, Correction for partial reflection in ultrasonic attenuation measurements using contact transducers, Journal of the Acoustical Society of America 125 (2009) 2946–2953. [8] A.W. Nolle, S.C. Mowry, Measurement of ultrasonic bulk-wave propagation in high polymers, Journal of the Acoustical Society of America 20 (1948) 432– 439. [9] R.A. Kline, Measurement of attenuation and dispersion using an ultrasonic spectroscopy technique, Journal of the Acoustical Society of America 76 (1984) 498–504. [10] H.J. McSkimin, Physical acoustic, principles and methods, Part 1, vol. 1, Academic Press – WP Mason, 1964. [11] H. Kolsky, Stress Waves in Solids, Dover Publications, Inc., 1963. [12] K.F. Graff, Wave motion in elastic solids, Dover Publications, Inc., 1991. [13] S. Umchid, Frequency dependent ultrasonic attenuation coefficient measurement, in: The 3rd International Symposium on Biomedical Engineering, 2008, pp. 234–238. [14] M.F. Youssef, N.K. Gobran, Modified treatment of ultrasonic pulse-echo immersion technique, Ultrasonics 39 (2002) 473–477. [15] P. He, J. Zheng, Acoustic dispersion and attenuation measurement using both transmitted and reflected pulses, Ultrasonics 39 (2001) 27–32. [16] P.H. Rogers, A.L. Van Buren, An exact expression for the Lommel diffraction correction integral, Journal of the Acoustical Society of America 55 (1974) 724–728. [17] R.W. Scott, P.F. Gordon, Ultrasonic spectrum analysis for nondestructive testing of layered composite materials, Journal of the Acoustical Society of America 62 (1977) 108–116. [18] A. Vincent, Influence of wearplate and coupling layer thickness on ultrasonic velocity measurement, Ultrasonics 25 (1987) 237–243. [19] J.L. Rose, P.A. Meyer, Ultrasonic signal-processing concepts for measuring the thickness of thin layers, Material Evaluation 32 (1974) 249–258. [20] J. Mobley, T. Vo-Dinh, Photoacoustic method for the simultaneous acquisition of optical and ultrasonic spectra, Journal of the Acoustical Society of America 4 (2003) 89–94. [21] A.R. Selfridge, Approximate material properties in isotropic materials, IEEE Transactions on Sonic and Ultrasonics SU-32 (1985) 381–394. [22] G.W.C. Kaye, T.H. Laby, Tables of Physical and Chemical constants, Longman, 1995.
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On the validity and improvement of the ultrasonic pulse-echo immersion technique to measure real attenuation Miguel A. Goñi, Carl-Ernst Rousseau ⇑ Mechanical Engineering, Wales Hall, Univ. of Rhode Island, Kingston, RI 02881, USA
a r t i c l e
i n f o
Article history: Received 25 April 2013 Received in revised form 7 August 2013 Accepted 8 August 2013 Available online 20 August 2013 Keywords: Attenuation Reflection coefficient Ultrasonic Immersion Alternative method
a b s t r a c t A fundamental assumption embraced in conventional use of the ultrasonic pulse-echo immersion technique to measure attenuation in solid materials is revisited. The cited assumption relies on perfect and immutable adhesion at the water to sample interface, a necessary condition that allows calculating the reflection coefficient at any interface from elastic wave propagation theory. This parameter is then used to correct the measured signal and obtain the real attenuation coefficient of the sample under scrutiny. In this paper, cases in which the perfectly cohesive interfacial condition is not satisfied are presented. It is shown also that in those cases, the repeatability of the conditions at the interface is always uncertain. This implies that the reflection coefficients are unknown, even when density is known. A new method of simultaneously measuring the reflection coefficients for both exposed interfaces that are normal to the transducer, and the attenuation coefficient of the specimen is developed and is presented here. The robustness of the new method is proven, as we demonstrate that the proper value of attenuation is achieved independently of the continuously varying interfacial conditions of these non-ideal cases. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The ultrasonic immersion technique has a very wide range of applications. It is commonly used to determine the elastic constants of homogeneous, heterogeneous, isotropic, and anisotropic composites [1,2]. It can also be used for material inspection, to detect defects within materials, lack of bonding, voids, or cracks [3,4]. Within our sphere of interest, it is also frequently used to measure wave velocities, attenuation, density, and thickness [5,6]. The present work will focus on the application of the technique to measure the attenuation coefficient of materials which, as referred in this work, is a material property, in contrast to the apparent attenuation measured by other practices such as described in ASTM Standard E664 [6]. This property is also referred to as real or true attenuation. There exist alternate techniques capable of measuring the attenuation coefficient of solid materials, which are based on the use of direct contact instead of immersion transducers. One such technique developed by Treiber et al. [7] is able to calculate the attenuation coefficient of any material even when reflection coefficients are unknown. This technique overcomes the unknown conditions at the three-media-interface formed by the transducer, the coupling agent, and the specimen by using an additional trans⇑ Corresponding author. Tel.: +1 401 874 2542; fax: +1 401 874 2355. E-mail address: [email protected] (C.-E. Rousseau). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.08.008
ducer to measure the reflection coefficient at that interface. This provides a valid measurement of the real attenuation of the material. However, based on the authors’ experience, direct contact transducers are not always as reliable as immersion transducers, sometime producing highly distorted second and third echoes that lend themselves poorly to analysis. Inconsistencies associated with the contact transducers led us to adopt immersion transducers in the measurement of attenuation coefficients of solid materials under the well-known pulse-echo immersion technique. The ultrasonic pulse-echo immersion technique uses a broadband immersion transducer positioned perpendicularly to the specimen and records the successive reflections occurring at the front and back walls of the specimen [8,9]. The spectrums of the first and second echoes are typically used to calculate the attenuation coefficient. In addition, the reflection coefficients at both faces of the sample are needed to correct the measured signals. Sometimes, the transmitted pulses are used instead of the reflected ones and the process is applied with the only difference that the transmission coefficients might be needed, depending on the setup of the test [10]. This variant of the immersion technique is known as through transmission mode. In both test setups, the reflection and transmission coefficients are calculated from elastic wave propagation theory [8–10] for the case of a plane dilatational wave impinging upon a plane interface between two media under the condition of a perfectly bonded interface [11,12]. In this paper, this assumption is analyzed by performing experiments in different
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materials for which the reflection coefficients are measured and compared to those predicted by the theory. It is shown how, for some cases, the reflection coefficients do not match the theoretical ones and therefore, care must be taken when calculating the attenuation coefficient to avoid introducing large errors. Some relatively recent versions of the classical immersion technique are able to provide the attenuation coefficient without the knowledge of the reflection or transmission coefficients [13–15]. However, these versions require equal, [14,15], or consistent, [13], reflection coefficients at both faces of the specimen in order to proceed with the calculations to obtain the attenuation coefficient. The question, indeed, is whether these assumptions do hold true during the execution of the ultrasonic immersion technique, especially for those cases in which the reflection coefficients do not match the theoretical calculation.
(ks), as well as the transducer size (a), and the distance between the transducer and the closest face of the specimen (L):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2p 2p 2p 2p cos ; DðsÞ ¼ þ sin J0 J1 s s s s where the s variables are described as s00 ¼ 2Lkw =a2 , s1 = (2hks + 2Lkw)/a2, s2 = (4hks + 2Lkw)/a2. The functions J are Bessel functions. 2.1. Attenuation coefficient calculation Dividing Eq. (2) by Eq. (3) and performing the requisite simplifications and reformulation, an expression for the attenuation coefficient a is obtained as shown below:
1 V1 Dðs2 Þ ; ln RA RB 2h Dðs1 Þ V2
2. Theory
a¼
Since this work is based on the pulse-echo mode of the Ultrasonic immersion technique, only one transducer is necessary, which works alternatively as an emitter and a receiver. Fig. 1 illustrates the supposed trajectory of both emitted and received pulses, as interactions take place between the ultrasonic pulse generated by the transducer, V0, and the specimen immersed in the liquid medium. The inclined pulse trajectory illustrated in Fig. 1 is so rendered to assist with the conceptual clarity of the phenomenon, as the actual paths would be directed perpendicularly with respect to the face of the specimen. The chosen notation defines V0 as the emitted pulse; V 00 corresponds to the front wall (A) reflection; V1 and V2 are the first and second echoes reflected off the back wall (B), respectively. The magnitudes of these reflections, as a function of that of the emitted pulse, are derived presently, as shown. In the relations presented below, the following phenomena are taken into account:
where V1 and V2 are known, since they are direct readings of the transducer. If the reflection coefficients were to be calculated following conventional methodologies, that is to say, according to elastic wave theory, then:
The pulse will partially transmit and partially reflect at the liquid to solid specimen interfaces. Attenuation will occur only within the specimen’s internal boundaries, and is neglected as the pulse traverses the liquid medium. The traveling pulses will diverge with distance. Thus,
V 00
¼ V 0 RA Dðs00 Þ;
ð1Þ
V 1 ¼ V 0 T 2A RB Dðs1 Þ expf2hag; V2 ¼
V 0 T 2A R2B RA Dðs2 Þ expf4h
ag;
ð2Þ ð3Þ
where RA, RB, TA are the reflection and transmission coefficients of faces A and B, respectively. D(s) stands for the beam spreading of the pulse, a is the attenuation coefficient of the specimen, and h is the thickness of the specimen. For a description of the beam spreading function, D(s), the reader is directed to the derivation given by Rogers and Van Buren [16]. It is a function of the wavelength of the ultrasound in liquid medium (kw) and within the specimen
RA ¼ RB ¼
Zs Zw ; Zs þ Zw
ð5Þ
where Zs is the acoustic impedance of the specimen, and Zw is the acoustic impedance of the water. In the above equation, directionality is irrelevant. 2.2. Measurement of reflection coefficient As discussed in the Introduction, the main three assumptions used by current methodologies to calculate the attenuation coefficient of a material are being tested. On the one hand, it is desired to know if the reflection coefficients match the value given by Eq. (5) for all materials. On the other hand, it is desired to know if the reflection coefficients are equal at both faces of the specimen during a particular instance of immersion and if they are consistent between consecutive immersions. In order to ascertain the veracity of these assumptions, it is necessary to measure the reflection coefficients. This measurement was performed by the method illustrated in the first two subsets of Fig. 2. The method is capable of measuring the reflection coefficients at both faces of the specimen for a unique immersion and is expounded upon in depth in Section 3.2. By means of Steps 1 and 2, the reflection coefficient at face B, RB, can be measured. Likewise, using Steps 3 and 4, the reflection coefficient at face A, RA can be measured. The mathematical fundamentals behind this measurement are relatively simple and proceed as follows: In the process pertaining to each of the previous steps, one signal is recorded. Though several options present themselves, the first echo of each signal was chosen to evaluate the reflection coefficients. Referring to Eq. (2), it is clear that the only parameter altered between the two configurations of Steps 1 and 2 is RB since the transducer and the specimen were never moved. In the configuration corresponding to Step 1, it is assumed that exposure to air results in complete reflection, i.e., RB = 1. Therefore, utilizing Eq. (2), and taking the ratio of the respective first echoes from Step 2 and Step 1, yields the water–specimen reflection coefficient at side B, RB. Alternatively, use of Steps 3 and 4 would yield RA. ð2Þ
RB ¼
Fig. 1. Interaction between an emitted pulse V0 and a specimen immersed in water.
ð4Þ
V1
; ð1Þ
V1
ð3Þ
RA ¼
V1
ð4Þ
V1
:
ð6Þ
where the subscripts stand for the first echo and the superscripts stand for the corresponding step in the experimental sequence.
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Fig. 2. Procedure for implementation of new method.
This procedure is capable of measuring the reflection coefficients that correspond to a unique immersion since each side of the tank is only filled once. Thus, repeating this procedure several times allows the experimentalist to assess the various assumptions formulated above. 3. Experiments and results Experiments were conducted on two Teflon specimens and two polycarbonate specimens. One of the polycarbonate specimens was coated with a minuscule layer of a hydrophobic substance known commercially as Hydrobead-TÒ, measuring a few microns. This coating was applied to test the validity of the assumption of perfectly bonded interfaces, a condition that is necessary for the proper implementation of Eq. (5). The polycarbonate specimen was sprayed with this material on only one of its faces, and was left to cure for 24 h. The Teflon specimens were designated as Teflon Plate 1 and Teflon Plate 2, respectively. Similarly, the Polycarbonate specimens were denoted as Polycarbonate Coated Plate and Uncoated Plate, respectively. Both Teflon plates were 3 3 in (76 76 mm) and 0.39 in (10 mm) thick, whereas both polycarbonate plates were 4 4 in (100 100 mm) and 0.476 in (12.1 mm) thick. These dimensions were carefully selected to prevent lateral wall reflections from interacting with either the first or second echoes. The density of the Teflon specimens was measured to be 2160 kg/m3, and that of polycarbonate 1210 kg/m3. The respective longitudinal wave speeds for the Teflon and Polycarbonate specimens calculated from the first two echoes are 1285 and 2208 m/s. A 1 MHz Panametrics (Olympus NDT) V303 Immersion transducer was used, together with a Panametrics 5058PR pulser/ receiver unit. The signals were displayed and recorded by a Tektronix Digital Oscilloscope with a 100 MHz bandwidth and a 1.25 GS/s sampling rate. Each signal recorded with the oscilloscope was transferred to a computer to calculate its spectrum by performing a FFT analysis using the software MATLAB. Note that each echo was individually analyzed and transposed onto the frequency domain from which all the calculations described in Section 2
proceeded. Consequently, all reflection and attenuation coefficients calculated are presented as a function of frequency. 3.1. Experimental evaluation of reflection coefficient measurements The reflection coefficients of all four specimens (Teflon and Polycarbonate), at their respective interfaces with water, were measured according to the procedure described in Section 2.2 and illustrated in Fig. 2. Each measurement was compared to the value provided by Eq. (5). Fig. 3 sequentially shows several measurements of the reflection coefficients for the Teflon Plate 1 and the Polycarbonate Coated specimen at both faces during separate instances of immersion. Subsequent measurements within those specific settings yielded experimental errors well within 1%, not appreciable enough to be displayed within the plots. Following each independent instance of immersion and corresponding measurements, each specimen was removed from the water, then reimmersed for additional measurements. This was necessary to test the consistency of the reflection coefficients. Before proceeding further, it must be underscored that the behaviors observed in Fig. 2 were consistent for both specimens during all further tests conducted. Identical experiments were performed on the other Teflon specimen, designated as Teflon Plate 2, as well as the other Polycarbonate specimen, designated as Polycarbonate Uncoated Plate. Regarding Teflon Plate 2, a similar behavior to Teflon Plate 1 was observed even though a better match was achieved most of the times for the reflection coefficient between the experiments and the theoretical value provided by Eq. (5). For the Polycarbonate Uncoated case, excellent agreement between experimental and theoretical reflection coefficients was achieved for all faces and immersion instances, identically to the uncoated face of Polycarbonate Coated Plate (Fig. 3). With respect to the layout of the Polycarbonate Coated plate, it was determined by Scott and Gordon [17] that the reflection coefficient of an interface consisting of a thin layer interposed between two substantially larger media is given by:
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Fig. 3. Comparison of the reflection coefficients obtained from two separate instances of immersion for Teflon Plate 1 and two separate instances of immersion for Polycarbonate Coated Plate.
T T R ðcos h R R j sinðhÞ 12 21 23 21 23 R ¼ R12 þ ; ð1 R21 R23 cos hÞ2 þ ðR21 R23 sin hÞ2
ð7Þ
where subscript ‘2’ represents the thin layer, subscripts ‘1’ and ‘3’ are the two media on either sides of the layer, and h = 4ph/k2. The variable h is the thickness of the thin layer, and k2 is the wavelength of the pulse acting on the thin layer media. Eq. (7) has been used
widely, including in works by Vincent [18] and Rose and Meyer [19]. If we consider the condition for which h k2, then, h 0. For Z Z this case, knowing that Rij ¼ Zji þZji , it is trivial to demonstrate that 1 Eq. (7) reduces to Eq. (5), in the form R ¼ ZZ31 Z . Note that the thickþZ 3 ness of the coating was measured with an optical microscope to be approximately 25 microns. Unfortunately, the properties of the coating were not provided by the manufacturer, but considering
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its polymeric nature, the wavelength of this material corresponding to a 1 MHz frequency is estimated to fall in the range of 1.5 to 3 mm. Hence, the thickness of the coating is two orders of magnitude smaller than the wavelength. Thus, an approximation that neglects the coating would result in less than a 0.1% error and therefore Eq. (5) was used for this specimen also. The evaluation of the reflection coefficients, obtained for the purpose of this work, shows that these coefficients are seldom accurate when applied to materials exhibiting hydrophobicity. The assumptions stated in the Introduction and at the beginning of Section 2.2 can be easily violated by these materials. Clearly, these materials do not always fulfill requirements of a perfectly bonded interface condition necessary to use Eq. (5). The use of these erroneous reflection coefficients would, in turn, introduce significant errors in subsequent calculations of the attenuation coefficient. Moreover, different results are observed between opposite faces of a unique specimen during a given immersion. Therefore, the validity of the prevalent practice that universally adopts identical reflection coefficients on both faces (even if their value is unknown) is greatly weakened. Furthermore, the present experiments show that each test has intrinsic values for the reflection coefficients (they differ from test to test) providing evidence of inconsistency of the reflection coefficients during different immersions. Consequently, attenuation measurements must be captured using the exact conditions in existence during the assessment of the respective reflection coefficients, in order for the former measurements to retain their validity. The materials for which these special considerations must be held in place seem to be those that present a very low interaction, that is to say, a minimal adhesion force to water, and the inability to adequately sustain the tensile stress component of the ultrasonic pulse. Besides this, and more importantly, another consequence of this small adhesion force is the presence of air molecules at the surface roughness level (micro-scale), that further hastens serious imperfection at the interface. Hence, it is necessary to develop an alternative method capable of correctly measuring the attenuation coefficient (a) for any material. The remedial method, developed as a solution to the numerous uncertainties described above, is presented next.
the four signals recorded, where the values for RA and RB would be calculated as follows: if signal V(1) is used, RB = 1 and RA is a measured value by the new method; if signal V(2) is used, RB and RA are the measured values by the new method; if signal V(3) is used, RB and RA are the measured values by the new method; if signal V(4) is used, RA = 1 and RB is a measured value by the new method. Attenuation coefficient measurements were conducted with this new procedure for the same Teflon and Polycarbonate specimens, described above. Regarding the Teflon specimens, thirty measurements were obtained for Teflon Plate 1 and twelve measurements for Teflon Plate 2. An average value was calculated for each Teflon plate. Both averages were compared to verify the reliability of the new method. Since both plates are made of the same material, the same value for the attenuation coefficient is expected. The identical process was repeated to the Polycarbonate plates for which eight measurements were averaged for the uncoated plate and five for the coated plate. The results are plotted in Fig. 4 for both Teflon and Polycarbonate plates. Error bars, at each of the
3.2. Proposed method for measuring true attenuation As a solution to the difficulties enumerated above, a method is proposed, that will allow the researcher to measure reflection coefficients on both faces of the specimen during a unique immersion and, simultaneously, will use one of the signals involved in that process to calculate the attenuation coefficient. A configuration that fulfills this goal has already been introduced in a previous section, and is illustrated in Fig. 2. The method fragments the experiment into four distinct steps, and uses a tank divided in two by a partition, which serves to both provide complete separation between the two sides of the tank, and performs the function of a holder for the specimen. In the first step only side A of the tank is filled with water, and the transducer, immersed in the water on that side, is positioned perpendicularly to the specimen, at a distance beyond the near-field. Signal V(1) is recorded. In the second step, side B of the tank is filled with water while the transducer remains untouched, still immersed in water on side A of the tank. Signal V(2) is recorded. In the third step, the transducer is moved to side B, precisely placed such as to maintain alignment with respect to its former position within side A. Signal V(3) is recorded. In the fourth step, side A is drained of the liquid, while the transducer remains untouched, immersed in water on side B of the tank. Signal V(4) is recorded. First, the reflection coefficients for the water–specimen interfaces A and B are calculated according to Eq. (6). Next, the attenuation coefficient can be calculated according to Eq. (4) with any of
Fig. 4. Real attenuation for the Teflon specimens (top) and Polycarbonate specimens (bottom).
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Fig. 5. Real attenuation for the Teflon specimens (top) and Polycarbonate specimens (bottom), obtained when Assumption 1 is violated.
frequencies evaluated, are also shown in the figure, along with the linear regression equation of the attenuation coefficients as a function of frequency. It is clear that for each material, the average curves depicting the behaviors of independent specimens are visually indistinguishable, and the corresponding equations are virtually identical. Fewer tests were conducted for the Polycarbonate specimens than for the Teflon ones because of the narrower range of the standard deviation achieved by the results of the former. Possible sources of error could include: lack of perpendicularity between transducer and specimen, lack of parallelism between the specimen faces, and random noise in the signals, among the most significant. 4. Summary and conclusion Three common assumptions used to calculate the attenuation coefficient of materials through the ultrasonic immersion technique were revisited, namely: 1. The water–specimen interface fulfills the condition of a perfectly bonded interface at all instants of time during any interaction with ultrasonic pulses [8–10]. 2. The reflection coefficients are equal on both faces of the specimen during a particular immersion [14,15]. 3. The reflection coefficients are consistent and therefore are exact duplicates of each other during different immersions, though not requiring a fulfillment of Assumption 2 [13].
549
Under Assumption 1, expressions for the reflection and transmission coefficients can be derived theoretically, as shown in Eq. (5) for the reflection coefficient. On the other hand, the techniques based on Assumptions 2 and 3 are capable of extracting the attenuation coefficients in specimens for which neither reflection nor transmission coefficients are known. They evidently do not require the conjecture of perfectly bonded interface. Based on the present study, all three assumptions have been proved erroneous for some materials. In particular, materials that present a level of hydrophobicity similar to or higher than that of Teflon are shown to behave otherwise, with an apparently very weak adhesive interfacial condition. Moreover, the interface conditions are very difficult to control due to the random presence of air molecules at the surface roughness microscale, resulting in significant differences between the two directly opposite sides of the same specimen and between successive tests conducted on the same specimen. In other words, immersing the specimen in water several times does not guarantee same interface conditions every time. Strong evidence of the above-stated fact is provided by Fig. 3, which clearly shows high probabilities for the absence of perfectly bonded interfaces in some materials, as well as for unequal reflection coefficients at opposite sides of a single specimen during a unique immersion and also lack of consistency in the reflection coefficients between different immersions. Consequently, it became necessary to develop a new method to apply the ultrasonic immersion technique, that could simultaneously measure reflection coefficients and attenuation coefficient, in order to ensure that identical water–specimen interface conditions are preserved throughout the entire process. It was shown in Fig. 4 how several tests with very different interface conditions from each other resulted in the same attenuation coefficient for a given material, proving the conceptual and practical validity of the new method presented. In contrast, Fig. 5 shows the attenuation coefficients that would have been obtained for these specimens if Assumption 1 had been embraced without question. Discrepancies in those results are evident since attenuation cannot be different between specimens made out of the same material. This error is a consequence of using erroneous reflection coefficients in the calculation of the attenuation coefficient. In order to further corroborate the validity of the method, the magnitude of the attenuation coefficients obtained using the present methodology were compared with those found in literature. Mobley and Vo-Dinh [20] studied the attenuation of Polycarbonate (Lexan) by means of two different methods, obtaining values of 50.8 Np/m at 1 MHz for the attenuation coefficient, and 56.8 Np/m/MHz for the slope of the linear regression curve describing variation of attenuation with frequency. These values are very similar to the ones obtained in this study. In addition, Selfridge [21] and Kaye and Laby [22] provided values of approximately 267 Np/m and 240 Np/m at 5 MHz respectively for the attenuation coefficient in Polycarbonate. Considering a linear relationship between attenuation and frequency, the quoted values would correspond to 53.4 Np/m and 48 Np/m for attenuation at 1 MHz, respectively. This is done by simply dividing the values corresponding to the 5 MHz frequency by 5. Once again, these values are very similar and within the range of the attenuation measured in this study for Polycarbonate. Regarding the Teflon material, a curious situation was encountered. While Selfridge [21] measured an attenuation value of 44.9 Np/m at 5 MHz, Kaye and Laby [22] provided a value of 430 Np/m at that same frequency. Such a large difference can be explained by the perfectly bonded interface assumption violation revealed and proven in this study. It is very likely that the reflection coefficients involved in the tests performed by Selfridge were higher than those provided theoretically by Eqs. 5. Similarly, the transmission coefficients involved in Selfridge’s work were likely
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lower than the theoretical ones [11,12]. Since the theoretical transmission coefficients were used to calculate the attenuation coefficient of Teflon, the measured signals were overcorrected and hence an underestimation of the attenuation coefficient was obtained. Summarizing: this unusually large range is in fact somewhat similar to the observations pertaining to Fig. 5. In order to compare values, the attenuation coefficients obtained by Selfridge, and Kaye and Laby are divided by 5, providing the values of 8.98 Np/m and 86 Np/m, respectively at a 1 MHz frequency. The tests conducted in this paper, with the method developed here, show an attenuation coefficient value of 97.5 Np/m at 1 MHz. This value is slightly higher than that provided by Kaye and Laby, which could indicate that the reflection or transmission coefficients in their tests were slightly higher or lower than the theoretical values. On the other hand, the value provided by Selfridge is significantly smaller than the one obtained in this work indicating that the reflection or transmission coefficients in his tests were respectively much higher or lower than the theoretical values. It can, thus, be concluded that the proposed method overcomes the problems found for some materials in the conventional use of the ultrasonic immersion technique to measure attenuation. It was shown in Fig. 5 that using conventional methods may lead to wide range of erroneous results that may span an entire order of magnitude, whereas using the method proposed herein provides repeatable results, as demonstrated in Fig. 4. It could also be pointed out that the reflection coefficient of the coated interface showed significant dependency with frequency. This could be due to a high concentration of air molecules at the interface, which then translates into dependency on varying wavelengths. Furthermore, the proposed method is not limited with respect to material type. Neither are there any restrictions in regards to the use of other immersion fluids besides water. Nevertheless, we do realize that the accuracy of the method is contingent upon consistent amplitudes, V0, of the emitted pulses. In the present experiment, this was not an issue, but must generally be verified by the experimentalist. As a final note, it is acknowledged that in some materials, values of the reflection coefficients could display time dependency, since the air molecules present at the interface could slowly drift with time. For these cases, the authors suggest waiting until the interface assumes stability and only then proceed with measurements. Nevertheless, the proposed method is highly recommended for obtaining the attenuation coefficient measurement of any material, due to its simplicity, its low cost (only requires one transducer), and the fact that it is not bound to questionable assumptions.
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