Measurement of shear viscoelasticity using dual acoustic radiation pressure induced by continuous-wave ultrasounds

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1 Japanese Journal of Applied Phsics 53, 7KF7 (4) REGULAR PAPER Measurement of shear viscoelasticit using dual acoustic radiation pressure induced b continuous-wave s Kaori Tachi, Hideuki Hasegawa,, and Hiroshi Kanai, * Graduate School of Biomedical Engineering, Tohoku Universit, Sendai 9-579, Japan Graduate School of Engineering, Tohoku Universit, Sendai 9-579, Japan kanai@ecei.tohoku.ac.jp Received November 9, 3; revised Januar 3, 4; accepted Februar 3, 4; published online June 9, 4 It is important to evaluate the viscoelasticit of muscle for assessment of its condition. However, quantitative and noninvasive diagnostic methods have not et been established. In our previous stud, we developed a method, which used ultrasonic acoustic radiation forces irradiated from two opposite horiontal directions, for measurement of the viscoelasticit. Using two continuous wave s, an object can be actuated with an ultrasonic intensit, which is far lower (.9 W/cm ) than that in the case of the conventional acoustic radiation force impulse (ARFI) method. In the present stud, in vitro experiments using phantoms made of polurethane rubber and porcine muscle tissue embedded in a gelatin block were conducted. We actuated phantoms b ultrasonic radiation force and measured the propagation velocit of the generated shear wave inside the phantoms using a diagnostic sstem. The viscoelasticities of phantoms were estimated b fitting a viscoelastic model, i.e., the Voigt model, to the frequenc characteristic of the measured shear wave propagation speed. In the mechanical tensile test, a softer polurethane phantom exhibited a lower elasticit and a higher viscosit than a polurethane phantom with a higher elasticit and a lower viscosit. The viscoelasticit measured b showed the same tendenc as that in the tensile test. Furthermore, the viscoelasticit of the phantom with porcine muscular tissue was measured in vitro, and the estimated viscoelasticit agreed well with that reported in the literature. These results show the possibilit of the proposed method for noninvasive and quantitative assessment of the viscoelasticit of biological soft tissue. 4 The Japan Societ of Applied Phsics. Introduction Viscoelastic properties of muscle tissue are closel related to the pathological state. For example, owing to pramidal tract disorders or peripheral neuropath, the elastic modulus of muscle decreases. In amloidosis, atroph and elevation of the hardness of muscle occur. Also, polmositis and moglobinuria lead to muscle weakness (decrease in muscle elasticit).,) Therefore, it would be valuable to measure the viscoelasticit of muscle for earl detection and quantitative diagnosis of muscle disorder. Over the past decade, some remote actuation methods based on acoustic radiation forces have been reported. Fatemi and coworkers proposed an imaging modalit that uses the acoustic responses of an object, which are closel related to the mechanical properties of the medium. B measuring the acoustic emission with a hdrophone, hard inclusions, such as calcified tissues in soft materials, were detected experimentall. 3,4) However, the spatial resolution was limited b the sie of the intersectional area of beams at two slightl different frequencies. Nightingale and coworkers proposed an alternative imaging method (acoustic radiation force impulse: ARFI), in which focused is emploed to appl a radiation force to soft tissue for a short duration (less than ms). The viscoelastic properties of the tissue were investigated from the magnitude of the transient response, which was measured with as the displacement of tissue. 5 7) However, in order to generate a measurable displacement b several ultrasonic pulses, high-intensit pulsed at, W/cm was required. According to safet guidelines for the use of diagnostic, it is recommended that the intensit be below 4 mw/cm (I SPTA ) for pulsed waves and W/cm for continuous waves. ) The intensit of the pulsed emploed b Nightingale and coworkers was therefore far greater than that indicated b the safet guidelines. Later man groups studied methods for actuation 7KF7- using high-intensit pulsed and the measurement of viscoelastic properties of tissue. 9,) To decrease the ultrasonic intensit for actuation of soft tissue, we chose continuous-wave, as in the work carried out b Fatemi and coworkers. The maximum intensit of W/cm for continuous waves given b the safet guidelines generates an acoustic radiation force of.7 Pa, which is ver small. Therefore, to generate a measurable displacement b acoustic actuation, a method for effective application of acoustic radiation forces should be developed. However, a single acoustic radiation force does not generate deformation in an object effectivel because it primaril produces a change in object s position. In our previous stud, we developed a method, in which two cclical radiation forces were simultaneousl applied to a phantom from two opposite horiontal directions to cclicall compress the object in the horiontal direction. Furthermore, the resultant regional displacement and strain in the tissue were measured using a different ultrasonic probe. This method enables the effective generation of deformation b acoustic actuation.,) There is another stud using two sources of shear waves, 3) in which a method for evaluating the mechanical properties of an object b making two shear waves interfere with each other is presented. However, this method requires that two shear wave sources are perfectl opposite, and that shear waves are insonified from the skin surface b an external vibrator. On the other hand, our method does not have these requirements. Also, the shear wave sources (corresponding to focal spots of two ultrasonic s) can be placed inside an object and can be localied using focused s. In the present stud, the propagation of the shear wave generated b our ultrasonic actuation method was measured also b to evaluate the viscoelastic properties of an object. 4 The Japan Societ of Applied Phsics

2 Jpn. J. Appl. Phs. 53, 7KF7 (4). Materials and methods. Acoustic radiation pressure at frequenc difference of two continuous s When the at a single frequenc propagates in a medium, a constant force is generated in the direction of propagation. This force is called the acoustic radiation force. The acoustic radiation pressure is defined as the acoustic radiation force per unit area. 4) When inserting an object (densit µ, sound speed c ) in the medium (densit µ, sound speed c ), the wave, which is incident perpendicularl on the surface of the object and progressing in the object, is given b Euler s fluid motion equation: tþ tþ þ vð; tþ ; where vð; tþ and pð; tþ are the particle velocit and the sound pressure in an object, respectivel. Equation () is modified @ v ð; tþ tþ ; tþ=@t and v ð; tþ= correspond to the acoustic radiation pressure and the kinetic energ of the, respectivel. Therefore, the acoustic radiation pressure is given b P R ð; tþ ðeð; tþþpð; ð3þ When two s with the same sound pressure, p, at slightl different frequencies, f and ( f + f ), cross each other, an acoustic radiation pressure which fluctuates at the frequenc difference, f, is generated in the intersectional space. The sound pressure, p sum (t), in the intersectional space is given b pð; tþ ¼e ½p cosðk f tþ þ p cosfk ðf þ fþtgš; ð4þ where and k are the attenuation coefficient and the wave number, respectivel. The kinetic energ eð; tþ is expressed as eð; tþ ¼ v ð; tþ ¼ pð; tþ ¼ c c p ð; tþ: ð5þ Therefore, the following equation is obtained from Eqs. (3) (5). P R ð; c e ½p cosðk f tþ þ p cosfk ðf þ fþtgš þ½p cosðk f tþ þ p cosfk ðf þ fþtgš ¼ p c e ð þ cos ftþ þ p c e ½cos ðk f tþ þ cosfk ðf þ fþtg 7KF7- þ cosfk ðf þ fþtgš þ kp c e ½sin ðk f tþ þ sin fk ðf þ fþtg þ sinðk ðf þ fþtþš p e ½ cosðk f tþ þ cosfk ðf þ fþtgþksinðk f tþ þ k sinfk ðf þ fþtgš: ðþ Since the displacement of the object generated b the highfrequenc component is negligible because it is sufficientl smaller than that generated b the low-frequenc components at DC and f, the acoustic radiation pressure P R ð; tþ is approximatel given b P R ð; tþ p c e ð þ cos ftþ:. Estimation of shear modulus from propagation velocit of shear wave using viscoelastic model of medium When acoustic radiation pressures are applied to an object, a shear wave is generated and propagates. The shear modulus G is estimated from the measurement of the shear wave propagation speed c s expressed as G ¼ c s ; ðþ where µ is the densit in the object. To determine the shear wave propagation speed c s, displacements of the object produced b acoustic radiation forces were measured also with at multiple points along the propagation path of the shear wave. From the measured displacements, the shear wave propagation speed c s is estimated as c s ¼ fl ; ð9þ where l and ª are the interval of ultrasonic beams for measurement and the phase difference between displacements of the object measured in neighboring ultrasonic beams, as illustrated in Fig., respectivel. On the basis of Eq. (), onl the shear modulus can be estimated from the shear wave propagation speed. In the present stud, the viscoelastic properties of an object were estimated as described below: The propagation of the shear wave is expressed b the wave equation as þ k U ¼ ; ðþ c s ¼! ReðkÞ ; ðþ where U, k, and ½ are the temporal Fourier transform of displacement, wave number, and angular frequenc, respectivel. When the Voigt model is used as a viscoelastic model of an object, the shear wave velocit is modeled as 7) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c s ¼ ð þ! p Þ ð þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ;! Þ ðþ where µ,, and are the densit of an object, shear elasticit, and shear viscosit, respectivel. ð7þ 4 The Japan Societ of Applied Phsics

3 Jpn. J. Appl. Phs. 53, 7KF7 (4) probe focal point...,, shear wave propagation x distance Δl x+ Δx Δθ, Δt Fig.. (Color online) Schematic diagram of the method of estimating shear wave propagation speed. Δf t t diameter 9 mm porcine muscle 5. mm beam focal point mm gelatin measured region porcine muscle gelatin height 3 mm (c) width [mm] gelatin porcine muscle 33 snchroniation x Fig. 3. (Color online) Illustration of porcine muscle phantom. signal generater power AMP signal generater power AMP amplitude [μm] oil water probe shear wave Focal point tissue block Low-Pass Filter Diagnostic sstem amplitude [μm] Fig.. (Color online) Schematic diagram of the experimental setup..3 Experimental methods Figure shows a schematic diagram of the experimental sstem. To measure the displacement distribution at a high spatial resolution, we emploed ultrasonic diagnostic equipment (Hitachi-Aloka ProSound F75) with a MH linear probe (Hitachi-Aloka UST-545). The frame rate and interval between ultrasonic beams were set at 4 H and.75 mm, respectivel. The number of beams used in the measurement was 3 (: ). As shown in Fig., the -th beam was located at the center of the two ultrasonic focal points for actuation. To improve the spatial resolution in the measurement of the response of an object to the acoustic radiation force, an correlation-based method, the phased-tracking method,,9) was used to measure the distribution of minute displacements. The accurac of the displacement measurement b the phased-tracking method has alread been evaluated to be. µm b basic experiments using a rubber plate ) and, also, has alread been applied to measurements of vibrations and viscoelasticities of the arterial wall,) and heart wall. 3,4) We used two phantoms made of two different polurethane rubbers that simulate soft biological tissues and another phantom containing porcine muscle tissue to measure the 7KF7-3 Fig. 4. (Color online) Spatial distributions of amplitudes of displacements at f of 5 H estimated b the Fourier transform. The results were obtained for phantoms of ASKER C hardnesses with and 5. propagation of the shear wave and estimate the viscoelasticit. The dimensions of the two polurethane rubber phantoms were 9 mm in diameter and mm in height, and their hardnesses were determined to be and 5 using the ASKER C-tpe durometer. The porcine muscle phantom used in the present stud is illustrated in Fig. 3. As illustrated in Fig. 3, the phantom was obtained b embedding a porcine muscle tissue in gelatin to prevent the deterioration of the porcine muscle b immersing in water. The position of the measured region is indicated b the blue dashed line. 3. Experimental results and discussion 3. Basic experiments using polurethane rubber phantoms Figures 4 and 4 show the spatial distributions of displacements in polurethane rubber phantoms with ASKER C hardnesses of and 5, respectivel, estimated at f = 5H using the Fourier transform. Focal points of ultrasonic beams for actuation were set at a depth of approximatel 5 mm. The intensit of each for actuation was.9 W/cm. As can be seen in Fig. 4, the shear wave is attenuated in accordance with its distance from the focal point. 4 The Japan Societ of Applied Phsics

4 Jpn. J. Appl. Phs. 53, 7KF7 (4) displacement [μm] displacement [μm] time [s] time [s] (distance from focal point) 4 ( mm) 5 (3.75 mm) (.5 mm) 7 (9.5 mm) ( mm) (distance from focal point) 4 ( mm) 5 (3.75 mm) (.5 mm) 7 (9.5 mm) ( mm) Fig. 5. (Color online) Displacements of polurethane phantoms with ASKER C hardnesses of and 5 at f of 5 H. Displacements were measured at 5.5 and 5. mm in depth. Fig.. (Color online) Spatial distributions of phases of displacements at f of 5 H obtained for phantoms with ASKER C hardnesses of and 5. Figure 5 shows the displacement waveforms measured in the respective polurethane rubber phantoms located 5.5 to mm awa from the focal point. The depths, where the displacements were measured, were 5.5 and 5. mm. It can be seen that the phase lag occurs between the waveforms in Fig. 5, which is caused b the wave propagation. For example, the peak of the waveform at 4 (distance from the focal point is mm) is formed slightl before that at 5 (distance from focal point is 3.75 mm). Figures and show spatial distributions of phases of displacements measured in the phantoms with ASKER C hardnesses of and 5, respectivel, estimated at f = 5H using the Fourier transform. As shown in Fig., it is confirmed that the phase increases with the distance, which is shown b the red arrow in the figure. Means and standard deviations of phases at f evaluated for three measurements are shown in Fig. 7. The depths, where the phases in Figs. and were estimated, were 5.5 and 5. mm, respectivel. The phases were obtained as the values relative to the phase measured at 4 (distance from focal point: mm). From Fig. 7 and Eq. (9), it can be seen that the shear wave propagates in the direction awa from the focal point. Shear wave velocities of polurethane rubber phantoms with hardnesses of and 5 were estimated to be. and.7 m/s, respectivel, using Eq. (9) with the slopes corresponding to ª/ l in Eq. (9) of the regression lines (green lines in Fig. 7) determined b the least-squares method. To discuss the accurac of the estimated values, we measured the shear wave velocit of the polurethane rubber phantom with an ASKER C hardness of for three times. The mean and standard deviation were.7 and. m/s (at f of 5 H), respectivel. Therefore, in the present paper, the number of digits after the decimal point was set at one. There is a relativel large variation in the estimated phase. The reason for the variation is not perfectl clear, but the generated displacement is ver small and might be affected b an undesirable vibration from the environment. 7KF distance from focal point at axis [mm] distance from focal point at axis [mm] Fig. 7. (Color online) Averaged phases of displacements at f of 5 H plotted as functions of distance obtained for phantoms with ASKER C hardnesses of and 5. The green lines show the regression lines determined b the least-squares method. We need to clarif the reason for the variance and to develop a method for separating the displacement induced b acoustic radiation force from the environmental vibration because we aim to actuate an object within the maximal acoustic output suggested b the safet guideline. The propagation velocities of shear waves in the phantoms with hardnesses of and 5 are plotted as a function of f in Fig.. The measured frequenc characteristics of shear wave propagation velocities fit well to those obtained using the Voigt model shown b Eq. (). The estimated elastic moduli and viscosit constants of the phantoms with hardnesses of and 5 were ( ¼ : kpa, ¼ 4: Pas) and (5 ¼ 4 The Japan Societ of Applied Phsics

5 Jpn. J. Appl. Phs. 53, 7KF7 (4) velocit [m/s] 5 hardness : hardness : frequenc [H] Viscosit (universal tester) [kpa s] Elasticit [kpa] Viscosit () [Pa s] viscoelasticit measured using universal tester hardness : hardness : 5 viscoelasticit measured b hardness : hardness : 5 Fig.. (Color online) Shear wave propagation speeds in phantoms plotted as functions of f. The solid lines show the frequenc characteristics obtained using the Voigt model. Fig.. (Color online) Comparison of viscoelastic constants of polurethane phantoms measured using precision tensile tester with those measured b. Elasticit E [kpa] 4 Elasticit Viscosit 4 3 Viscosit η[kpas] amplitude [μm] 5 hardness of phantom determined using ASKER C-tpe durometer Fig. 9. (Color online) Viscoelastic constants of polurethane phantoms measured using precision tensile tester. 5: kpa, 5 ¼ : Pas), respectivel. The root mean square difference (RMSE) between the measured values and the values estimated using the model shown in Fig. was also evaluated. RMSEs with respect to phantoms with hardnesses of and 5 were 7 and 3%, respectivel. Figure 9 shows the elastic moduli and viscosit constants of the respective phantoms measured using a precision universal tester (Shimadu AG-X kn). In Fig. 9, plots and vertical bars show means and standard deviations for measurements, respectivel. As shown in Fig. 9, viscoelastic properties of the polurethane rubber phantoms estimated using the precision universal tester were ( ^ ¼ 5:9 :4 kpa, ^ ¼ 33:4 5: kpas) and ( ^ 5 ¼ 3:3 : kpa, ^5 ¼ 4:54 :5 kpas). There were significant differences between the absolute values obtained b and those obtained using the precision universal tester. In the measurement using the universal tester, the actuation frequenc was about H, and the maximum strain in the object was about % due to limitations of the specifications of the tester. In general, the viscoelastic properties depend on the magnitude of strain, and the different levels of deformation ma be one of the reasons for the differences in the absolute values of the viscoelastic constants. However, as shown in Fig., similar tendencies of the viscoelastic constants (higher elastic modulus of the phantom with the hardness of 5 and higher viscosit constant 7KF7-5 Fig.. (Color online) Spatial distribution of amplitudes of displacements at f of 5 H measured in porcine muscle phantom. of the phantom with the hardness of ) were found in both measurements with and the universal tester. 3. In vitro experiment using porcine muscle tissue In the present stud, an in vitro experiment using a gelatin phantom containing porcine abdominal muscle tissue was conducted to investigate whether the proposed method could generate and measure the presumed shear wave. Figure shows the spatial distribution of amplitudes of displacements at f = 5 H estimated b appling the Fourier transform to displacement waveforms measured b. The focal points of ultrasonic beams for actuation were set at a depth of approximatel 5.7 mm. Figure shows the displacement waveforms measured at a depth of 5.7 mm in the porcine muscle at f = 5 H. The spatial distribution of phases of the measured displacement waveforms is shown in Fig., and the averaged phase at f of 5 H is shown in Fig. (c), where plots and vertical bars show means and standard deviations, respectivel. In Fig., a frequenc component of about H is clearl seen. The amplitude of this component increases when the acoustic radiation force increases (corresponding to the downward displacement of the object). Therefore, this component might be the high-frequenc component of the acoustic radiation force, which was neglected in the present stud. Although the high-frequenc component of the radiation force is nearl MH, the sampling frequenc of the displacement waveform is 4 H (corresponding to the frame rate). Thus, an aliasing effect might be one of the reasons that the component is seen as a H component. As shown in Fig., the displacement caused b ultrasonic actuation is attenuated during propagation. Also, in 4 The Japan Societ of Applied Phsics

6 Jpn. J. Appl. Phs. 53, 7KF7 (4) displacement [μm] time [s].7 (distance from focal point) 4 ( mm) 5 (3.75 mm) (.5 mm) 7 (9.5 mm) ( mm) velocit [m/s] frequenc [H] (c) distance from focal point at axis [mm] Fig.. (Color online) Displacement waveforms, spatial distribution of phases, and (c) average phases of displacements at f of 5 H obtained for porcine muscle phantom. Figs. and (c), phase delas due to the propagation of the shear wave are observed. The position of the analed region is indicated b the green square in Fig. 3. B determining the regression lines, as shown b the green line in Fig. (c), the shear wave propagation speed at f of 5 H was estimated using the slope of the regression line. In Fig. 3, the shear wave propagation speeds are plotted as a function of the frequenc difference f. Bfitting the frequenc characteristic of the shear wave propagation speed obtained using the Voigt model to the frequenc characteristic measured b, the viscoelastic constants of the porcine muscle tissue were estimated to be ( p ¼ : kpa, p ¼ :9 Pas), which were similar to the viscoelastic constants of cow muscle ( = 3 4 kpa, = 3 Pas), 5) beef muscle ( = kpa, = 3 Pas), ) and human muscle ( =.5 kpa, = 5 Pas) 7) reported in the literature. From these results, the proposed method could generate and measure the shear wave propagation and was also applicable to the estimation of the viscoelasticit of biological tissue. 4. Conclusions In the present stud, we actuated phantoms simulating soft biological tissue using dual acoustic radiation pressure and measured the propagation velocit of the generated shear wave as a function of the actuation frequenc f. In polurethane phantoms, the viscoelastic properties measured b tend to be similar to those measured using a precision universal tester, though there were differences in their absolute values. Furthermore, an in vitro experiment using porcine abdominal muscle was conducted and the viscoelastic properties were also estimated as in the measurements of polurethane phantoms. These results show the.3. 7KF7- Fig. 3. (Color online) Shear wave propagation speeds in porcine muscle phantom plotted as a function of f. The solid line shows the frequenc characteristic obtained using the Voigt model. possibilit of the proposed method for the noninvasive and quantitative assessment of the viscoelasticit of biological soft tissue. ) S. Shoji, Kin Shikkan no Shindan to Chiro (Nagai Shoten, Osaka, 9) p. 7 [in Japanese]. ) S. Sato and T. Inoue, Botai Seiri Bijuaru Mappu 4 (Igaku Shoin, Toko, ) p. 37 [in Japanese]. 3) M. Fatemi, L. E. Wold, A. Aliod, and J. F. Greenleaf, IEEE Trans. Med. Imaging, (). 4) M. Fatemi and J. F. Greenleaf, Proc. Natl. Acad. Sci. U.S.A. 9, 3 (999). 5) K. Nightingale, M. S. Soo, R. Nightingale, and G. Trahe, Ultrasound Med. Biol., 7 (). ) G. E. Trahe, M. L. Palmeri, R. C. Bentle, and K. R. Nightingale, Ultrasound Med. Biol. 3, 3 (4). 7) B. J. Fahe, K. R. Nightingale, R. C. Nelson, M. L. Palmeri, and G. E. Trahe, Ultrasound Med. Biol. 3, 5 (5). ) Japan Societ of Ultrasonics in Medicine, Cho-onpa Igaku, 4 (94) [in Japanese]. 9) J. Bercoff, M. Tanter, and M. Fink, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 5, 39 (4). ) K. Masuda, R. Nakamoto, N. Watarai, R. Koda, Y. Taguchi, T. Kouka, Y. Miamoto, T. Kakimoto, S. Enosawa, and T. Chiba, Jpn. J. Appl. Phs. 5, 7HF (). ) H. Hasegawa, M. Takahashi, Y. Nishio, and H. Kanai, Jpn. J. Appl. Phs. 45, 47 (). ) Y. Odagiri, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phs. 47, 493 (). 3) Z. Wu, K. Hot, D. J. Rubens, and K. J. Parker, J. Acoust. Soc. Am., 535 (). 4) G. R. Torr, Am. J. Phs. 5, 4 (94). 5) A. J. Livett, F. W. Emer, and S. Leeman, J. Sound Vib. 7, (9). ) S. Catheline, J.-L. Gennisson, G. Delon, M. Fink, R. Sinkus, S. Abouelkaram, and J. Culioli, J. Acoust. Soc. Am., 3734 (4). 7) Y. Yamakoshi, J. Sato, and T. Sato, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 45 (99). ) H. Kanai, M. Sato, Y. Koiwa, and N. Chubachi, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 79 (99). 9) H. Kanai, H. Hasegawa, N. Chubachi, Y. Koiwa, and M. Tanaka, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 44, 75 (997). ) H. Kanai, K. Sugimura, Y. Koiwa, and Y. Tsukahara, Electron. Lett. 35, 949 (999). ) K. Ikeshita, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phs. 5, 7GF4 (). ) K. Kitamura, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phs. 5, 7GF (). 3) Y. Honjo, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phs. 5, 7GF (). 4) H. Shida, H. Hasegawa, and H. Kanai, Jpn. J. Appl. Phs. 5, 7GF5 (). 5) S. Catheline, F. Wu, and M. A. Fink, J. Acoust. Soc. Am. 5, 94 (999). ) E. R. Fitgerald, E. Ackerman, and J. W. Fitgerald, J. Acoust. Soc. Am. 9, (957). 7) H. L. Oestreicher, J. Acoust. Soc. Am. 3, 77 (95). 4 The Japan Societ of Applied Phsics