An Ultrasonic Mode Conversion Technique for Characterizing Prism-Shaped Material Samples Experimental and Numerical Results

Transcription

1 ECNDT oster 10 An Utrasonic Mode Conversion Technique for Characterizing rism-haped Materia ampes Experimenta and Numerica Resuts Abdemaek BOUHADJERA, NDT Lab, Jije University, Ageria Frank CHUBERT, Fraunhofer-IZF, Dresden Branch, Germany Abstract. An utrasonic apparatus based on a mode conversion technique is described. It invoves the measurement of the veocity of both compressiona and shear waves in prism shaped specimens with ony one transducer. The shear waves are generated through mode conversion at the interface between water and the specimen under test. The optimum couping of the transducer and the specia arrangement of the specimen enabe the evauation of not ony second order eastic constants, but third order eastic constants as we, since the specimen can be easiy subjected to both compressive and hydrostatic pressure. reiminary experimenta resuts obtained using different isotropic materias and numerica simuations of wave propagation based on the eastodynamic finite integration technique (EFIT) are presented. The resuts revea that the prism technique is a significant improvement compared to traditiona goniometer and rotating pate techniques. Introduction Measurements of sound veocities have great practica importance since they ead to the determination of eastic materia constants from we-known formuae. There are various methods based on through-transmission and puse-echo techniques [1, ]. rovided that compressiona waves ( waves) can be set up within a iquid medium, both compressiona and shear waves ( waves) can be generated in a soid specimen by a mode conversion process that takes pace upon refraction at a boundary, where an acoustic impedance discontinuity exists. Bar [3] for exampe, used this principe to measure both and wave veocities in soids. Later chneider and Burton [4] made measurements with the rotating pate technique in which the sampe is in the form of a rectanguar bock, and mounted on a turntabe that can be rotated in steps of 0.1 degrees and ocated between a pair of compression wave probes, one transmitting and the other receiving. The compete arrangement is immersed in a water tank. Reynods [5] in his technique used a cyindrica rod. Optica methods of detection are appied with accuracy, being principay imited by the precision with which the anges of the various sound beams in the immersion iquid can be measured. Rao [6] chose conditions such that the ange for incident waves was greater than the critica ange for waves within the pate. Hence, ony waves were present in the soid. Again, the ange of incidence must be accuratey known in order to determine the wave veocity. This veocity can aso be determined by directy observing the diffraction pattern produced by 1

2 shear waves within a transparent soid, as reported by Breezeae and Hiedemann [7]. Mayer [8] used the critica-ange method by a goniometer arrangement for determining the veocity of sound in a fat smooth sampe. The pronounced ampitude changes at the critica anges permit the reading of anges within 0.1 degrees. Most od and recent research in this fied is based more or ess on the same basic principes of the methods mentioned above [9-0]. The drawback with these methods is the ange of rotation, which has to be determined with high accuracy, thus increasing the compexity of the measuring system. The new method described here is an improvement to an immersion technique designed by the main author [1, ]. Both and waves are generated by mode conversion in a prism-shaped specimen. The ange arrangement aows for just one variabe to be eft in the equation that enabes the computation of the veocity of utrasonic waves [3, 4]. 1. Refection and transmission at a iquid-soid boundary If a pane sound wave in a iquid strikes a pane interface of a soid obiquey (figure 1), a refected wave in the iquid, a refracted and a mode converted wave in the soid are generated (see e.g. [1] or [5]). Refected wave Refracted wave Mode converted wave Liquid α α α α oid Incident wave ρ, c ρ, c, c Figure 1: Refection, refraction, and mode conversion of an obiquey incident wave at a iquidsoid interface. The directions of the refected and transmitted waves are determined by ne s aw, sinα sinα =, (1), c c, whereas α is the ange of incidence and simutaneousy the ange of the refected wave in the iquid, α and α are the anges of refracted and wave in the soid, respectivey, and c, c and c are the wave speeds of the wave in the iquid and of the and wave in the soid, respectivey. The wave refection and transmission coefficients with regard to intensity, R, T, and T, are given by the foowing equations (e.g. [5]): n K Z R( α ) =, () K

3 n n 4Z Z T ( α) = cos α, (3) K with and Z n n n n 4Z Z T ( α) = sin α, (4) K K = Z cos α + Z sin α + Z ρ c ρ c n n =, Z =, Z =. cosα cosα cosα n n ρ c In these equations, the anges α, α, and α are inked via equation (1) and conservation of energy requires R + T + T = 1. In order to discuss the formuae, we choose an eastic haf space with ρ = 133 kg/m³, c = 3950 m/s, c = 355 m/s, according to a typica mortar specimen, and use ρ = 1000 kg/m³ and c = 1470 m/s as materia parameters for (distied) water. Figure shows the different curves cacuated by equations ()-(4). α =45 α =45 R R T T T α α 1 α α Figure : Intensity refection coefficient, R, transmission coefficient for the refracted wave, T, and transmission coefficient for the mode converted wave, T, as a function of the ange of incidence α for a water/mortar combination (ρ = 1000 kg/m³, c = 1470 m/s, ρ = 133 kg/m³, c = 3950 m/s, c = 355 m/s). At an ange of incidence α = 0 ony a refected wave in the iquid and a transmitted wave in the soid (with α = 0 ) exist. In this case, refection and transmission coefficient with regard to intensity amount to and Z Z R (0 ) = (5) Z + Z 3

4 4Z Z T (0 ) = 1 R(0 ) =, (6) ( Z + Z ) in which Z = ρ c and Z = ρ c represent the acoustic impedances of the iquid and the soid medium, respectivey. For our water/mortar exampe we obtain R (0 ) = and T (0 ) = (compare figure ). At increasing ange of incidence, α > 0, an additiona shear wave with increasing intensity is generated in the soid by mode conversion of the incident wave. imutaneousy the transmission coefficient of the refracted wave decreases. ince in most practica cases the wave speed of the iquid, c, is smaer than the wave speeds in the soid, c and c, the anges of refraction, α, α, are arger than the ange of incidence, α. Therefore, at a first critica ange of incidence, α 1, the ange of refraction of the wave in the soid reaches 90 and thus, at this point the wave disappears from the soid. This first critica ange is given by c α = 1 arcsin. (7) c In our present exampe this happens at α 1 = 1.85 (see figure ). At this critica ange of incidence the refected wave in the iquid increases significanty with a refection coefficient cose (but not identica) to one. In the soid for α > α 1 the incident wave is competey transformed into the mode converted wave. The intensity of the atter rapidy increases at the expense of the refected wave. If we further increase the ange of incidence, the ange of refraction of the wave aso reaches 90 and at this point the wave disappears, too. This second critica ange of incidence, α, is given by c α = arcsin. (8) c In our water/mortar exampe this happens at α = 38.6 (see Fig. ). For α > α the refection coefficient for the incident wave in the iquid is equa to one since a other wave modes are no onger present.. Basic principes of the prism technique The main piece of the apparatus is the transducer ce shown in figure 3. It consists of a water tank in which the specimen under test (UT) is fixed. The singe transducer that acts as both transmitter and receiver is put on a circe that turns around UT with a radius R (about 4 cm). The utrasonic beam makes an ange α that varies in a continuous manner from 0 to 90. The main face of UT is put against the diameter ine XX and has to be paced in such a way that its center coincides with the center of the transducer circe (point O in figure 3). This is done by moving UT parae to XX foowing the arrows and pointing the right ange of the prism in the direction of the positioning ine. The atter is perpendicuar to the main prism surface. 4

5 Additiona refector (optiona) a/ Figure 3: Transducer ce configuration as used for the prism technique. They key idea of the prism technique is that ony waves inside the prism with a refraction ange of 45 significanty contribute to the sensor signa. These refracted waves impinge normay on one of the side faces of the prism with ength a, are refected at the prism/water interface, and propagate back to the receiver using exacty the same trave path as before (reciprocity principe). From equation (1) we see that this happens if the ange of incidence is 1 c α = arcsin (9) c for the refracted wave, and 1 c α = arcsin (10) c for the mode converted wave. For our water/mortar exampe we obtain α = 15.6 and α = 6.19 (see dashed vertica ines in figure ). The determination of the eastic constants of the prism materia is then based on the foowing three steps..1 First step: Transit time of refected wave wave refection with α = 0 from the main face of UT occurs when the ange of incidence α is equa to zero (norma incidence aong the positioning ine). This enabes the evauation of time-of-fight t = R / c. A numerica simuation of this situation using the eastodynamic finite integration technique (EFIT [6]) is shown in figure 4. According to equation (5) approximatey 49% of acoustic energy is refected back to the transducer. The remaining portion is transferred to the wave in the prism. In this initia step the transmitted wave in the prism is not needed for evauation. 5

6 Transducer Water Incident wave Mortar prism t = 6 µs t = 4µs Refected wave Transmitted wave t = 33µs t = 4µs Figure 4: -D EFIT simuation of norma incidence case for the water/mortar exampe (size of the mode = cm, side ength of the prism = 5 cm, center frequency of the input puse = 1 MHz). The wave front snapshots represent the absoute vaue of partice veocity using a inear grey scae. They were taken at t 6, 4, 33, and 4 µs after start of excitation. Driving puse Refected wave Figure 5: Time-domain signa of norma partice veocity according to figure 4, cacuated by integrating over the transducer aperture. From the sensor signa the time-of-fight of refected wave, t, can be determined (step 1 of the prism technique). 6

7 . econd step: Transit time of refracted wave By increasing the ange of incidence, the refected wave echo disappears, and a second echo appears, which is reated to refracted wave within UT (due to the finite transducer aperture the utrasonic beam carries some divergent portions so that refraction anges of 45 aways appear even when the main beam is not refracted in the 45 direction). In this region the measured time-of-fight of the refracted wave echo is independent (!) on the ange of incidence, α, but certainy its ampitude varies with varying α. The maximum echo is obtained if the ange of refraction is 45, i.e. α = α according to equation (9). In our water/mortar exampe this happens at α = It is worth mentioning once again, that for measurement of the echo transit time it is not necessary to exacty adjust α to α. It is sufficient to manuay tune the echo ampitude to obtain a good signa-to-noise ratio. This fact represents a significant improvement over goniometer arrangements where the ange of incidence has to be measured exacty. A numerica EFIT simuation using an incidence ange of α = α is shown in the foowing figure 6. Transducer Water Incident wave Mortar prism t = 6 µs t = 4 µs Refected wave Mode converted wave t = 30 µs Refracted wave t = 33 µs 7

8 wave echo wave t = 36 µs transmission t = 39 µs wave echo to be detected t = 4 µs t = 51 µs Figure 6: -D EFIT simuation for a wave refraction ange of 45 for the water/mortar exampe (size of the mode = cm, side ength of the prism = 5 cm, center frequency of the input puse = 1 MHz). The wave front snapshots represent the absoute vaue of partice veocity using a inear grey scae. They were taken at t 6, 4, 30, 33, 36, 39, 4, and 51 µs after start of excitation. Driving puse Echo of refracted wave Figure 7: Time-domain signa of norma partice veocity according to figure 6, cacuated by integrating over the transducer aperture. From the sensor signa the time-of-fight of the refracted wave, T, can be determined (step of the prism technique). 8

9 From the wave front snapshots in figure 6 it can be seen that not ony a refracted wave but aso a mode converted shear wave is generated in the prism. However, since the refraction ange of this wave is significanty smaer than 45 no significant contribution appears in the cacuated time-domain signa of partice veocity as shown in figure. 7. The refracted wave however is party refected by the eft side face of the prism (norma incidence!), runs back to the main face where it is once again refected, mode converted, and refracted. Ony the refracted part (refraction ange = origina incidence ange α!) in the iquid propagates back to the transducer where it produces a we-defined echo signa (see figure 7). From this sensor signa the time-of-fight of the refracted wave, T, can be easiy determined..3 Third step: Transit time of mode converted wave A further increase of the ange of incidence α makes the wave echo disappear, and after that a new echo wi appear, which is reated to the mode converted shear wave in the prism. The maximum echo is obtained if the ange of refraction is 45, i.e. α = α according to equation (9). In our water/mortar exampe this happens at α = An EFIT simuation of this case is shown in the foowing figure 8. Transducer Water Incident wave Mortar prism t = 6 µs t = 1 µs Refected wave Mode converted t = 30 µs wave t = 36 µs 9

10 wave t = 39 µs echo t = 4 µs wave in iquid to be detected t = 51 µs t = 57 µs Figure 8: -D EFIT simuation for wave refraction ange of 45 for the water/mortar exampe (size of the mode = cm, side ength of the prism = 5 cm, center frequency of the input puse = 1 MHz). The wave front snapshots represent the absoute vaue of partice veocity using a inear grey scae. They were taken at t 6, 1, 30, 36, 39, 4, 51, and 57 µs after start of excitation. Driving puse Echo of mode converted wave Figure 9: Time-domain signa of norma partice veocity according to figure 8, cacuated by integrating over the transducer aperture. From the sensor signa the time-of-fight of the mode converted wave, T, can be determined (step 3 of the prism technique). 10

11 The wave front snapshots in figure 8 revea that inside the prism the incident wave is neary competey transformed to the mode converted wave since in this case the incidence ange α is greater than the first critica ange of the wave, α 1 (compare figure ). Thus, at east for ideaized pane waves no wave in the prism shoud exist. Nevertheess, in the third snapshot of figure 8, a weak refracted wave in the prism can sti be observed. This wave is caused by the divergent parts of the incident beam yieding to sighty differing incidence and refraction anges. A comparison between the signas in figures 7 and 9 shows that the ampitude of the detected wave echo is significanty arger than that of the wave echo. Two main facts seem to be responsibe for that. First of a, according to figure at α = α more than 60% of incident acoustic energy is transferred to the refracted wave. In contrast to that at α = α ony about 3% is transferred to the refracted wave. econdy, the refection of both waves at the side face of the prism (i.e. the mortar/water interface) is significanty different. Whie the refracted wave is party transferred to the surrounding water and thus, oses energy, the wave is totay refected and maintains its initia energy. For a detaied quantitative study of the ampitudes of the different wave modes, aso the third refection/transmission process at the main face of the prism must be taken into account. Moreover, in addition to the refection and transmission coefficients with regard to intensity as given by equations ()-(4), aso their counterparts with regard to wave ampitude (ike partice veocity or acoustic pressure) must be considered. Unfortunatey, this is beyond the scope of the present paper. From the time-domain signa given in figure 9 the time-of-fight of the mode converted wave, T, can easiy be determined. Together with the transit times for the refracted wave, T (from step ), and that of the refected wave, t (from step 1), one can deduce a simpe formua to determine shear and pressure wave speeds of the prism materia by using the specific geometry of the prim (see figure 3). For the transit times T, we obtain R a a T, = + = t +, (11) c c c and finay for the wave speeds,, c, a a = =, (1) T t T R c,, where a is the side ength of the prism. If one is interested in eiminating the dependence on the prism size one coud introduce an additiona refector as shown in figure 3, remove the prism and perform another measurement of the transit time t R of the refected wave in water (norma refection from the refector). ince the height of the prism is given by h = a -1/ we have h a t R = t + = t + (13) c c and together with equation (1) we obtain c, = c ( t t ) R ( T t ), = R t ( t t ) R ( T t ),. (14) 11

12 If we aways use a standardized prism size a, distied water at the same temperature and thus, a constant c, as we as a constant R, then the transit times t and t R in equations (13) and (14) can aso be seen as characteristic constants of the apparatus. In this case the phase speeds c, to be determined are a function of the transit times T, ony and the formua takes the simpe form K c, =, (15) T k where K and k are characteristic constants of the measurement system.,. The Apparatus 3.1 ystem description The schematic diagram of the utrasonic testing apparatus is shown in figure 10. It consists of an Utrasonic user/receiver (anametrics 5077R), various Immersion Transducers (anametrics, 1-.5 MHz), a Longitudina Wave Contact Transducer (anametrics V106RB), a Digita Oscioscope (Tektronix TD 100), a Laptop Computer with Wavetar software for data-acquisition, and the Transducer Ce. Figure 10: chematic diagram (eft hand side) and photograph (right hand side) of the utrasonic apparatus. 3. ystem operation At the start of the experiment, the incident waves from the immersion transducer are made to impinge normay on the argest face (main face) of the prism. The reated echo gives the time-of-fight of the wave within the water. Next, the ange of incidence is increased sowy unti the disappearance of the first echo and the emergence of a second echo reated to refracted waves. After that, the ange is increased further, a third echo appears, and this time it is reated to shear waves within the prism. The accuracy of measurements depends ony on one parameter, namey the time-offight of utrasonic waves. It coud be determined with high precision using we-known signa processing techniques, such as the overap, cross-correation or cepstrum agorithms. However, since the echoes are quite strong, no compex signa conditioning is required in 1

13 most cases. The cursors of the digita oscioscope were positioned on the main haf-cyce of the echo to give a fairy accurate measurement of the transit times. 3.3 rism sampes pecia care must be undertaken whie preparing the materias sampes. The starting point shoud be from a perfecty made cube, which has a side ength of about 50 mm. It is then divided through its diagona into two equa prisms. The prism coud aso be made from a cyindricay shaped specimen. The critica point here is to ensure the right ange between the two side faces. Unike cyindricay shaped specimens, the prism-shaped ones coud be tested in three different directions, which woud give a fairy good idea about the anisotropy of the materia. There is aso the possibiity of crosschecking the veocity of waves using a contact transducer. Therefore, the drawback reated to the difficuty of preparing prismshaped specimens compared to cyindricay shaped ones is more than compensated by the weath of information yieded from the former. 3.4 ystem caibration A reference prism is made from auminum. The vaues of both wave veocities are engraved on the bock. It is aso used as a refector for measuring the transit-times t R and t. The Radius R is measured with great accuracy from knowing the veocity of utrasound in distied water at a certain temperature and t. A second contact transducer that has the same center frequency as the immersion one is used to cross check the wave veocity before the start of the experiment. The system being temperature compensated is then ready for making the rest of measurement. 3. Experimenta resuts The waveforms shown in figure 11 are obtained by simpy rotating the transducer around the specimen as expained in chapter. In this case the specimen is made of Auminum, and cut from a cube that has a side ength of 56 mm. As can be seen from the two ower pictures in figure 11, the echo reated to the refracted wave in the prism is significanty smaer than the echo reated to the refracted wave. In the atter case even mutipe echoes can be identified. The observation of a strong wave echo agrees we with the energyreated considerations done in section.3. From the oscioscope dispay the transit times of the different waveforms are determined and the characteristic wave speeds of the prism materia can be cacuated using equations (13) or (14). From the wave speeds the two Lamé constants of an isotropic medium, λ and μ, as we as Young s moduus E, buk moduus K, and oisson s ratio ν 0, can be easiy derived by using we-known formuae given in any serious text book. Tabe 1 shows second-order eastic constants for various materias as obtained by the prism technique. 13

14 Driving puse Mutipe wave echoes from main surface wave echo from optiona refector Echo reated to refracted wave in the prism Mutipe echoes reated to mode converted wave in the prism Figure 11: Oscioscope dispays obtained for an Auminum sampe showing mutipe echoes from the main face of the specimen (top eft), an echo from the optiona refector (top right), an echo reated to the refracted wave in the prism (down eft), and mutipe echoes reated to the mode converted wave in the prism (down right). Tabe 1: econd-order eastic constants as obtained by the prism technique Materia ρ (kg/m³) c (m/s) c (m/s) λ (Ga) μ (Ga) E (Ga) K (Ga) ν 0 (Ga) Auminum 017A Brass Mortar (w/c=0.55) (s/c =1) Mortar (w/c=0.35) (s/c=1)

15 4. Discussions and outook This experiment shows that it is possibe to measure both and wave veocities of the prism materia with reative ease. Materia sampes are easy to prepare and ony one variabe is needed for computing the eastic constants, namey the transit-time of the utrasonic puses. The atter are measured with the digita oscioscope using the vertica cursors. The accuracy is directy reated to the samping rate of 1 Gsampes/second, which gives a definition of one nanosecond. This is more than enough for the measurement of veocities, which are needed for the determination of second-order eastic constants. However, this wi be crucia when the same apparatus is used for detecting sma changes in wave speed due to noninearity induced by externa stress, which coud be generated by appying uniaxia and hydrostatic pressures on the specimen. This means that the present method coud be easiy extended to the determination of third-order eastic constants. The main advantages of this method over other more known utrasonic techniques and especiay the rotating pate technique (which shares some of the features) are as foows: There is just one transducer used for measuring the wave speed of both compressiona and shear waves. Easy and uniform couping between the transducer and the sampe, which characterizes immersion-testing methods The measured veocities are independent of the exact incidence ange, which represents a huge advantage over goniometer methods. Measurements are temperature compensated, which means that the wave speed remains constant for sma variations around the ambient temperature. However, the iquid interface coud be used to transfer heat to the sampe for taking measurements at high temperatures. The same configuration coud be easiy adapted to test sampes under high stresses (uniaxia compression, hydrostatic pressure, etc.). Due to the strong echoes no compex signa processing is required in genera. Finay we woud ike to mention that the proposed prism technique coud aso be used to measure attenuation characteristics of the prism materia in principe. This damping can be caused by viscoeastic dissipation, scattering by aggregates and pores as we as by noninear effects. For this kind of measurement a detaied quantitative anaysis of expected ampitudes incuding compensation of system-reated attenuation (geometrica spreading and damping in water) is needed. This topic is the subject of ongoing research and wi be pubished esewhere. References [1] J Krautkrämer and H Krautkrämer, Utrasonic Testing of Materias, Berin (pringer-verag), pp , [] J Bitz and G impson, Utrasonic Methods of Non-Destructive Testing, London, (Chapman&Ha), pp , [3] R Bar, Hev hy Acta, Vo 13, p 61, [4] WC chneider and CJ Burton, Determination of the Eastic Constants by Utrasonic Methods, J App hys, Vo 0, pp 48-58, [5] M B Reynods, The Determination of the Eastic Constants by the Utrasonic use Technique, Trans Am oc Metas, Vo 45, pp , [6] B R Rao and V Rao, Nature, Vo 174, p 835, [7] M A Breezeae and EA Hiedemann, J Acoust oc Am, Vo 7, p 10, [8] W Mayer, Energy artition of Utrasonic Waves at oid-liquid Boundaries, Utrasonics, Vo 3, 15

16 pp 6-68, [9] K Ergin, Energy Ratio of the eismic Waves Refected and Refracted from Rock-Water Boundary, Bu eism oc Am, Vo 4, pp , 195. [10] L Brekhovskikh, Waves in Layered Media, Academic ress, NewYork, [11] F Roins, Critica Utrasonic Refectivity a Negected Too for Materia Evauation, Materia Evauation, Vo 4, pp , [1] L Fountain, Experimenta Evauation of the Tota Refection Method of Determining Utrasonic Veocity, J Acoust oc Am, Vo 4, pp 4-47, [13] F R Roins, Utrasonic Examination of Liquid-oid Boundaries Using a Right-Ange Refector Technique, J Acoust oc Am, Vo 44, pp , [14] A R Gregory et a, Dua-Mode Utrasonic Apparatus for Measuring Compressiona and hear Wave Veocities of Rock ampes, IEEE Trans onics and Utrasonics, Vo 17, pp 77-85, [15] R A Kine, Measurement of Attenuation and Dispersion Using an Utrasonic pectroscopy Technique, J Acoust oc Am, Vo 76, pp , [16] J Wu, Determination of Veocity and Attenuation of hear Waves Using Utrasonic pectroscopy, J Acoust oc Am, Vo 99, pp , [17] R J Freemante and R E Chais, Combined Compression and hear Wave Utrasonic Measurements on Curing Adhesive, Meas ci Techno, Vo 9, pp 1-1, [18] D E Bray, Current Directions of Utrasonic tress Measurement Techniques, 15 th WCNDT, Rome, 000. [19] I Anto et a, An utrasonic Method for Determination of Eastic Modui, Density, Attenuation and Thickness of a oymer Coating on a tiff ate, Utrasonics, Vo 39, pp 11-1, 001. [0] I Imamura, Measuring Method for the Mechanica Anisotropy of oid by Water Immersion Utrasonic ing-around Method, 3 rd Word Congress Utrasonics, aris, 003. [1] A Bouhadjera, A Universa Utrasonic Apparatus for Measuring Eastic Constants of Materias, hd Thesis, Nottingham Univ, Engand, [] A Bouhadjera, A Mode Conversion Method for Evauating Eastic roperties of Materias, Insight, Vo 38, pp , [3] A Bouhadjera, An Improved Design of an Utrasonic Apparatus for Characterizing Materia ampes, Insight, Vo 46 N 9, pp , 004. [4] A Bouhadjera and C. Bouzrira, High-Frequency Utrasonic Testing of Young Cement-based Materias Using the rism Technique, NDT&E Internationa, Vo 38, pp , 005. [5] V A utiov, hysik des Utraschas, Akademie-Verag Berin, 1984 (in German). [6] F chubert, Numerica Time-Domain Modeing of inear and Noninear Utrasonic Wave ropagation using Finite Integration Techniques Theory and Appications, Utrasonics, Vo 4, pp 1-9,