Joris Degrieck

Naefeh A. H.

Nico F. Declercq

Oswald Leroy

Schoch A.

Tamir T.

English

Declercq, Nico

Degrieck, Joris

Leroy, O

Ghent University Academic Bibliography

The double-sided ultrasonic beam displacementNico F. Declercqa) and Joris DegrieckSoete Laboratory, Department of Mechanical Construction and Production, Ghent University,Sint Pietersnieuwstraat 41, 9000 Gent, BelgiumOswald LeroyInterdisciplinary Research Center, Katholieke Universiteit Leuven Campus Kortrijk, E. Sabbelaan 53, 8500Kortrijk, Belgium(Received 25 February 2004; accepted 1 September 2004)It is a well-established idea in optics as well as in ultrasonics that a bounded Gaussian beam, whenreflected from an interface, can be displaced in the forward or in the backward direction, dependingon the propagation direction of leaky waves that are generated by the incident beam. Such adisplacement is often accompanied by the so-called Schoch effect characterized by a null strip inbetween a specular and a nonspecular reflected beam, and a trailing field that is much furtherdisplaced. The current letter shows experimentally and numerically that a simultaneous forward andbackward displacement is possible accompanied by two null strips and being only the result offorward propagating Lamb waves. © 2004 American Institute of Physics.[DOI: 10.1063/1.1812363]In optics, the Goos–Hänchen theory predicts a lateraldisplacement of a light beam that is internally reflected froma dielectric interface.1 This phenomenon appears when inci-dent from an optically denser medium at an angle close tothe critical angle, resulting in the transfer of a portion of theenergy into the rarer medium by means of excitation of anelectromagnetic field that travels along the interface. Thisenergy leaks back into the denser medium and becomes partof the reflected beam exhibiting a lateral displacement thatappears as a forward beam shift. This was studied by Tamirand Bertoni.2 The early experiments of Schoch3–5 applyingthe acoustic analog of the Goos–Hänchen effect for anacoustic beam reflected from a liquid–solid interface showeda forward lateral displacement of the reflected ultrasonicbeam. It has been shown before6–8 that a backward displace-ment of an ultrasonic beam is also possible when backwardpropagating surface waves are stimulated on periodicallyrough surfaces. The expression “null strip” was mentioned inNeubauer and Dragonet.9 The effect consisting of both a for-ward and a backward displacement on a smooth interface,accompanied by two null strips, has never been observed andhas certainly never been published before. The experimentsin this letter are performed by means of a Schlieren experi-mental setup10 and the liquid/solid interfaces are smooth.There is an unwritten law that whenever leaky Rayleighwaves or leaky Lamb waves are generated by means of anincident bounded beam, the Schoch effect occurs and viceversa. As an example, a typical Schoch effect is shown inFig. 1, where sound is incident on a thin aluminum plateunder a Lamb wave angle. It is seen that there is no back-ward displacement of the reflected beam. What is visible, is aforward displaced (nonspecular) sound lobe separated fromthe main (specular) reflected sound beam by a null strip. Thisis because the nonspecular lobe is out of phase with thespecular sound beam. The null strip is the result of phasecanceling. A trailing field at larger distances is also visible.We have performed experiments on glass plates. Thelongitudinal sound velocity in glass is 5660 m/s, whereasthe transversal sound velocity is 3520 m/s. The density is2500 kg/m3. Water is characterized by a longitudinal soundvelocity of 1480 m/s and a density of 1000 kg/m3. Thewell-expected Schoch phenomenon was visible at certainangles depending on the frequency that was used and de-pending on the thickness of the plate. Nevertheless, for thecombination of a plate thickness of 1.23 mm and a frequencyof 3 MHz, a completely unexpected phenomenon appeared.The physical beam width was 1 cm. The result can be seen inFig. 2. The glass plate is indicated by means of white dashedlines. The white arrow denotes the direction of incidence.There is a forward displaced lobe and also a backward dis-placed lobe. Both lobes are separated from the specular lobeby means of a null strip. If the left lobe was not backwarddisplaced and was hence the specular lobe, the effect coulda)Electronic mail: nicof.declercq@ugent.beFIG. 1. The typical Schoch effect of a bounded ultrasonic beam on a thinaluminum plate. Only a forward displacement is visible accompanied by atrailing field.APPLIED PHYSICS LETTERS VOLUME 85, NUMBER 18 1 NOVEMBER 20040003-6951/2004/85(18)/4234/3/$22.00 © 2004 American Institute of Physics4234have been regarded as a very strong trailing field. However itis clear that the first lobe is not a specular lobe but is indeedbackward displaced. In order to make sure that the effect wasnot caused by edge effects of the plate that was used, wehave studied different positions of incidence and we havealso studied the effect of smooth edges and rough edges. Nodifference was visible except when incidence occurred soclose to the edge that the forward displaced sound beam“touched” or even surpassed the edge.Simulations were performed by means of a decomposi-tion of the incident bounded beam into plane waves in ac-cordance with the Fourier transform. Continuity of normalstress and normal particle displacement was considered11along the water–glass interfaces of the plate. The reflectedprofile was studied along the interface water/upper side ofthe plate. The same parameters were used as in the experi-ments. The reflected beam profile as a function of the angleof incidence can be seen in Fig. 3. It is seen that at 18.29°and at 13.91° there is a regular Schoch effect. Nevertheless,at 28.35° there are two null zones separating three reflectedbeams. This is the double-sided beam deformation that cor-responds to Fig. 2. The mentioned angles correspond to thedispersion curves of Fig. 4, denoting the pole position of thereflection coefficient for pure harmonic plane waves. It isseen that the double beam deformation effect occurs at theangle that stimulates the A0 Lamb mode. The Schoch effectis not visible for the S0 stimulating angle. The calculatedprofile in amplitude and in phase corresponding to the angleof incidence of 28.35° is shown in Fig. 5, where it is seenthat the left and the right lobes are out of phase with thecentral specular reflected lobe. A backward propagatingsound field could be characterized by a negative group ve-locity. However, according to the dispersion curves in Fig. 4and according to knowledge obtained from Ref. 12, this isnever the case and hence there is no backward propagatingsound field present. Therefore the effect is the result of a(complicated) interaction between the amplitudes and phasesof the plane waves that are the building blocks of the inci-dent bounded beam and the generated Lamb waves.As a conclusion, it is shown experimentally and numeri-cally that double beam deformations are possible on smoothplates swamped in water. It is also shown that this effect, inthe example studied here, is not caused by backward propa-gating sound in the plate.Work supported by “The Flemish Institute for the Pro-motion of the Scientific and Technological Research in In-dustry (I.W.T.).”FIG. 2. The double-sided beam deformation. A backward and a forwarddisplaced beam are visible together with the central specular reflected beam.FIG. 3. Simulation (filled contour plot) of the reflected beam profile as afunction of the angle of incidence. Negative positions correspond with back-ward positions. At 28.35° the situation of Fig. 2 can be seen.FIG. 4. The dispersion curves of a glass plate. The horizontal line corre-sponds to the experiments reported here. The vertical lines correspond to theangles of incidence of 28.38°, 25.62°, 18.29°, and 13.91°.FIG. 5. Simulation of Fig. 2, i.e., incidence at 28.35°. Dotted curve: incidentbeam profile, solid curve: reflected beam profile. Note that the forward andthe backward displaced beams are out of phase with the specular centrallobe.Appl. Phys. Lett., Vol. 85, No. 18, 1 November 2004 Declercq, Degrieck, and Leroy 42351Von F. Goos and H. Hänchen, Ann. Phys. (Leipzig) 6, 333 (1947).2T. Tamir and H. L. Bertoni, J. Acoust. Soc. Am. 61, 1397 (1971).3A. Schoch, Nuovo Cimento, Suppl. 7, 302 (1950).4A. Schoch, Acustica 2, 1 (1952).5A. Schoch, Acustica 2, 18 (1952).6M. A. Breazeale and Michael Torbett, Appl. Phys. Lett. 29, 456 (1976).7N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, J. Acoust. Soc. Am.112, 2414 (2002).8N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, Appl. Phys. Lett. 82,2533 (2003).9W. G. Neubauer and L. R. Dragonet, J. Appl. Phys. 45, 618 (1974).10M. A. Breazeale, J. Opt. A, Pure Appl. Opt. 3, S1 (2001).11A. H. Naefeh, Wave Propagaton in Layered Anisotropic Media with Ap-plications to Composites (North-Holland, Amsterdam, 1995).12M. F. Werby and H. Überall, J. Acoust. Soc. Am. 111, 2686 (2002).4236 Appl. Phys. Lett., Vol. 85, No. 18, 1 November 2004 Declercq, Degrieck, and Leroy

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