22 .' . ' .. .. .. : 4 j: 1 m. i ; . .. I OF I ORNL P 1395 1 *** . . . 1 :13 . . . . 2 . : . . - SS $ - 2 Il '40 17 - 1.1.25 1.1.4 1.6 . . . i MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS - 1963 į E . e . ORNU -P- 1395 65-58 To be presented at: 5th Inter- national Congress on Acoustics, Liege, Belgium, Sept. 7-14, 1965. to be presenter 20 1965 .. 50 supor 5o CONGRES INTERNATIONAL D'ACOUSTIQUE -..... -... LIEGE 714 SEPTEMBRE 1965 ONF20509010) .. ...... - FINITE AMPLITUDE WAVI'S III LIQUIDS AND SOLIDS* Them e nten, maar M.A. Br?a::eale ...... . - . , Depertment of Physics, The University of Tennessee, and n..: Solid, State Division, Oak Ridge National Laboratory Hot T Deronia, State -..... The distortion of an initially sinusoid:ll ultrasonic wave recently observed in solids can be described in a way analogious to trat previously used for liquids, except for certain complications associinted with crystalline properties. The non- linearity parameters can be introduced in a relatiively simple way if the terms in the non-linear differential equation ar grupec properly. This results in a formalism which allows one to discuss non-linear effects in liquids and solids analogously and to point out parameters which differ significantly and thus characterize the two media. We have marie masurements by differerit techniques, using results found in one medium to suggest lir:es of investigation in the other. Reflection of Distorted Waves in Liquid:3. A It is possible to gain information abouř the behavior of a distorted ultrasonic wave on reflection from a boundary througų the use of the diffraction of light. 5 mc ultrasonic wave is generated in waser ind allowed to propagate a distance equal approximately to the discontinuity distance. At this point a reflector is placed in the water such that the distorted ultrasonic wave is reflected at an angle of approximately 15°. A beam of light from a point source passing through the water in the region of interaction of the incident and reflected waves 16 diffracted by both waves. The diffract:ion patterns resulting when one uses two extremely differený reflectors are show in Fig. 1. *Research sponsored by the U.S. Atomic Ene::gy Crimmission under contract with the Union Carbide Corporation and by the orice! :? ilaval Research under Contract No. 4289. l.. ... ........... .. .. . . . .., • :).TENT CLEARANCE OBTAINED. RELEASE U DHE PUBLIC IS APPROVED. PROCEDURES, ARE ON EILE IN THE RECEIVING SECTION, @ •••xo.••• @ : @ . . . . . . . . . . RESILIENT BOUNDARY RIGID BOUNDARY (water-air) (water-brass) P2C2 « P1C1 Pacz » 0242 Fig. 1. light diffraction resulting from interaction of distorted waves. These diffraction patterns can be understood by observing the position of the zero order and realizing that the distorted ultrasonic wave behaves in much the same way as a blaze grating. The distortion in the ultrasonic waveform leads to an a syrometrical nilfraction pattern. The diffraction pattern resulting from a single distorted ultrasonic wave progressing to The right is shown in Fig. 1(a). Diffraction patterns produced in the interference l'egion before boundaries having extreme values of acoustic impedance are shown in Fig. 1(b). These patterns can be visualized as the diffraction pattert l'esulting when each of the orders produced by the incident wave is diffractid by the reflected wave. The difference in the asynmetry of the pattern caused by the reflected wave produces a markedly different appearance of the overall pattern. There petteras can be explained by assuming that the distortions of the reflected WEL VOS in the two cases are as indicated in the diagram. (Fig. 1(c).) The wave reflected from a resilient boundary is distort. ed in a direction opposite to that reflected. Pron a rigid boundary, and actually is now in an unstable condition because the non-linear effects now cause the higher harmonics to decrease. Taese observations are consistent with that one calculates using the expression 6 - 2 tan- e n todo sin (2) for the phase shift on reflection from an interface and assuming that the phase shift of each harmonic is the same as that of an equivalent sinusoidal wave. It should be pointed out in passing that this equation predicts the unstable waveform in the reflected wave for a rigid boundary 8.8 well as the resilient boundary when the distorted wave is at grazing incider:ce. Let us consider normal incidence. If the We.Ve is normally incident the asymmetry of the diffraction patter indicates that standing waves are symmetrical for the rigid boundary and asymmetrical for the resilient boundary. This fact is very important in the study of solids, for the stress-free boundary condition obtaining when an ultrasonic wave impinges interne 13.y on the end of a solid is analogous to the resilient boundary discussed here. Solution of this problem of a standing finite amplitude wave in a solid could jresent a new means of evaluating the third order elastic constants. Such a solution is in progress. An alternative method 18 to consider the distortion of progressive finite amplitude waves. Progressive Finite Amplitude Waves in Sclids. In the study of the propagation of a firite amplitude ultrasonic wave in a solid or a liquid it is usual to consider solltiors to the non-linear differential equation satisfying the boundary condition that at x = 0, the displacement U # A Sin wit. One can formulate the problem such that the propagation of a longitudinal ultra- sonic wave in either type of medium can be described by the equation poVtt = KzlUxx + 3U_U:x) + KzUxUxx (2) where Unt is the second time derivative of the displacement, and the coeficients K2 and Ka are introduced to make notatio:1 simple in describing a non-linear solid. For a cubic crystal the coefficients Kind K, are combinations of the ordinary elastic constants and the third order élastic cor.stants, respectively. These combinations depend on the direction of propiagation of the ultrasonic wave with respect to the crystal axes and are given fo:c three crystal directions in Table I. For a liquid the combination - 3K, + Kz)/K, is replaced by B/A + 2, where A and B are coefficients of the Taylor expansion of the pressure in terms of the conden- sation. For a gas this would be y + 1. In deciding how much an initially sinusoidal wave will distort in propagating through a giver medium, it is useful to consider the discontinuity distance L, the propagation distance required for an inicial.ly sinusoidal wave to develop a dis- continuity in the particle velocity. This is given by 2o. I E / 3K2 + K3 1 2262) U (3) K2 where Vo is the particle displacement sum itude at the source. As an example of. the behavior of solids compared with frids, values of L are calculated for a 30 megacycle wave in water and for varius directions in germanfun. These values are given in Table II. For an assumed source andlitude V. = 1 A, a typical value of the discontinuity distance in water vould be 15 cm; for germanium it would be between 120 and 500 cm depending on the direction of propagation in the crystal. Direction . .- :ܕc - . [100] (210) ----- C12 C21 + C12 + 2C44 2 Cu2+ 20,2 + 4C44 C111 + 301.12 + 12C766 4 (122) C212 + 60212 + 3.20244 + 24C266 + 2C423 + 16C456 Table I. Ką and K3 for (100), (110), and (211) Directions. 3 Direction K2 L(u. 18 in cm.) 5.04 x 10-6 2004 (110) 1.288 x 2012 dynes/cm2 -2.20 x 1023 dynes/cm2 (120) 1.053 x 2012 -3.93 x 1013 1.18 x 10-6 V (211) 0.9748 x 1012 -3.36 x 2073 1.2 x 10-6 cm. Table II. Discontinuity Distance in Germanium. For propagation distances small compareri rith the discontinuity distance, a perturbe ation solution of the non-linear differential equation is accurate enough to allow one to calculate the nonlinearity param:ter B/A for fluids or the third order elastic constants of solids. Compared with the discontinuity distances calculated above it is clear that the ultrasonic priti in so].id samples of ordinary sizes would be quite small so that a perturbation solution of Eq. I should be quite accurate. Therefore, when the distortion of the finite amplitude wave is large enough to measure, this should be an accurate means of determining the third order elastic constants of solids provided the attenuation can be controlled sufficientiy. A perturbation solution of Eq. 2 ingtenis 03. the discontinuity distance I is Vox U = U. Sin(kx - wt) + no Cos 2(kx - wt) + ... This solution predicts that the second harmonic in a distorted ultrasonic wave is proportional to the distance from a sinusoidal driver and proportional to the square of the fundamental amplitude. These functional relationships serve as a check on experiment and as a means of deteniining whether absorption is negligible as has been assumed. I . ; .. ..-. Measurements have been made on (121) sample:: of copper single crystals using a pulse technique. Results of measuremoniis o:: the first transmitted pulse in an annealed 9.1 cm sample of copper are given in l'ig. 2, which is a log-log plot of the amplitudes of the second harmonic and the fundamental, relative to an i arbitrarily established reference, as a function of source amplitude. It is seen that there is a considerable range of source anilitudes for which the second · harmonic curve has a slope of 2 which is expected if there is a square relation- ship between them. للبيبليبللا 2009 In Fig. 3 is given the distance dependence of the second harmonic using samples of various lengths. SECOND HARMONIC Note that the above results on liquids show that one cannot simply use successive reflections in the same sample to study patr length dcpend- ence, since on each return trip the second harmonic is generated with phase opposite to that produced by reflection, with the result that the wave "undistorts". The behavior = before and after a neutron bombard- > ment of 3.6 x 10+5 neutrons/cm2 is shown. One sees that the amplitude FUNDAMENTAL of the second harmonic increases with distance as it should. Further, the expected linear in- crease with distance is improved after bombardment. As the distance increases the second harmonic evid- · ently is approaching a maximum value; however it appears that at shorter distances the curve is reasonable linear, in agreement with theoretical predictions. The Fik. 2. Behavior of Fundamental rapid initial increase in the pre- and second harmonic in annealed bombardment second harmonic can be [111] copper sample &function attributed to the non-linear inter- of source amplitude. action of dislocations, which evid- ently reduces the effective dis- continuity distance. Neutron bomb- ardment appears to reduce this dis goo . x 3OTORE NEUTRON BOMBARDMENT QUARTZ location interaction in copper to • AFTER NEUTRON BOMBARDMEN. VOL the point that the effects of the : non-linearity of the crystal - 600 lattice become dominant. 15 20 50 100 200 500 1800 ONA o # 5 12 16 20 24 28 32 36 40mm l'iz. 3. Behavior of the second harmonic in (111) copper samples as function of clistance, showing the effect of neutron bombardment. RFERIENCES 2. A.A. Gedroits and V.A. Krasil'nikov, J. Exptl. Theoret. Phys. (USSR), 43, 1592 (1962). 2. M.A. Breazeale and 'D.0. Thompson, Appl. Phys. Lett., 3. 77 (1963). 5 3. A. Aikata, B. Chick and C. Elbawa, Appl. Phys. Lett., 3, 195 (1963). END DATE FILMED 11/ 15 /65 ** S ' memorationem