# Mathematical modeling of the effects of attenuation and system electronic coupling on the determination of speed of sound in ultrasonic interferometry

## Abstract

Ultrasonic interferometry is an indispensable tool in molecular chemistry and imaging, inclusive of liquid state studies where the standard theory is used to determine many physico-chemical parameters, such as the isentropic compressibility and adiabatic bulk modulus. The first principle analysis conducted here augments the standard model with potentially significant consequences in the interpretation of these parameters and the output spectrum. The effect of attenuation of a wave on the observed separation between peaks in acoustic interferometry is a focus of the investigation. Important aspects of the theory of Hubbard and others were collated to derive two mathematical models that were used to fit experimental spectra. The first model does not assume fictitious quantities found in Hubbard’s theory and fits the experimental data well. The second model includes the effects of the electronics of the measuring system and is in excellent agreement with the experimental data. Theoretical and numerical analyses were performed to validate the two models. Numerically, the attenuation of a wave is shown to cause the peaks to deviate either positively or negatively from the otherwise ideal half-wavelength of \(\lambda /2\) and exact equations governing such deviations are derived that could have significant implications in theory and applications.

## Keywords

Wave motion analysis Ultrasonic speed Ultrasonic interferometry Attenuation of sound## Mathematics Subject Classification

76D33## Notes

### Acknowledgements

We are grateful to our colleague, Dr. Thorsten Heidelberg for his expertise in German language and also for the following Grants that have funded our research: UMRG RG293-14AFR and PPP PV006/2012A.

## Supplementary material

## References

- 1.B.S. Finn, ISIS
**55**(179), 7 (1964)CrossRefGoogle Scholar - 2.T.M. Aminabhavi, B. Gopalakrishna, J. Chem. Eng. Data
**40**, 856 (1995)CrossRefGoogle Scholar - 3.J.C.R. Reis, I.M.S. Lampreia, J. Mol. Liq.
**179**, 130 (2013)CrossRefGoogle Scholar - 4.C.K. Yau, F. Nabi, C.G. Jesudason, J. Mol. Liq.
**212**, 79 (2015)CrossRefGoogle Scholar - 5.A. Nabi, C.K. Yau, C.G. Jesudason, J. Mol. Liq.
**224**, 551 (2016)CrossRefGoogle Scholar - 6.K.H.A.E. Alkhaldi, A.S. Al-Jimaz, M.S. AlTuwaim, J. Chem. Thermodyn.
**103**, 249 (2016)CrossRefGoogle Scholar - 7.H.S. Ju, E.J. Gottlieb, D.R. Augenstein, G.J. Brown, B.R. Tittmann, IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**57**(7), 1612 (2010)CrossRefPubMedGoogle Scholar - 8.K.A. Wear, I.E.E.E. Trans, Ultrason. Ferroelectr. Freq. Control
**47**(1), 265 (2000)CrossRefGoogle Scholar - 9.T. Motegi, K. Mizutani, N. Wakatsuki, Jpn. J. Appl. Phys.
**52**, 07HC05 (2013)CrossRefGoogle Scholar - 10.C.M. Sehgal, Ultrasonics
**33**(2), 155 (1995)CrossRefGoogle Scholar - 11.E.P. Papadakis, in:
*Physical Acoustics: Principles and Methods*, vol 12, ed. by W.P. Mason, R.N. Thurston (Academic Press, New York, NY, USA, 1976), chap. 5, pp. 277–374Google Scholar - 12.E.P. Papadakis, in:
*Physical Acoustics: Ultrasonic Measurement Methods*, vol. 19, ed. by R.N. Thurston, A.D. Pierce (Academic Press, San Diego, CA, USA, 1990), chap. 2, pp. 81–106Google Scholar - 13.G.W. Pierce, Proc. Am. Acad. Arts Sci.
**60**(5), 271 (1925)CrossRefGoogle Scholar - 14.J.C. Hubbard, Phys. Rev.
**38**, 1011 (1931)CrossRefGoogle Scholar - 15.F.E. Fox, Phys. Rev.
**52**, 973 (1937)CrossRefGoogle Scholar - 16.T. Kishimoto, O. Nomoto, J. Phys. Soc. Jpn.
**9**(4), 620 (1954)CrossRefGoogle Scholar - 17.K.F. Herzfeld, T.A. Litovitz,
*Absorption and Dispersion of Ultrasonic Waves*(Academic Press, New York, NY, 1959)Google Scholar - 18.J.J. Markham, R.T. Beyer, R.B. Lindsay, Rev. Modern Phys.
**23**(4), 353 (1951)CrossRefGoogle Scholar - 19.W. Han, D.N. Sinha, K.N. Springer, D.C. Lizon, in
*Nondestructive Characterization of Materials VIII*, ed. by R.E. Green Jr. (Plenum Press, New York, NY, USA, 1998), pp. 393–399CrossRefGoogle Scholar - 20.F. Eggers, U. Kaatze, Meas. Sci. Technol.
**7**, 1 (1996)CrossRefGoogle Scholar - 21.K.W. Katahara, C.S. Rai, M.H. Manghnani, J. Balogh, J. Geophys. Res.
**86**(B12), 779 (1981)CrossRefGoogle Scholar - 22.J.C.R. Reis, Â.F.S. Santos, I.M.S. Lampreia, ChemPhysChem
**11**, 508 (2010)CrossRefPubMedGoogle Scholar - 23.M.R. Moldover, R.M. Gavioso, J.B. Mehl, L. Pitre, M. de Podesta, J.T. Zhang, Metrologia
**51**, R1 (2014)CrossRefGoogle Scholar - 24.R.M. Gavioso, D.M. Ripa, P.P.M. Steur, C. Gaiser, T. Zandt, B. Fellmuth, M. de Podesta, R. Underwood, G. Sutton, L. Pitre, F. Sparasci, L. Risegari, L. Gianfrani, A. Castrillo, G. Machin, Phil. Trans. R. Soc. A
**374**(2064), 20150046 (2016)CrossRefPubMedGoogle Scholar - 25.F. Simonetti, P. Cawley, J. Acoust. Soc. Am.
**115**(1), 157 (2004)CrossRefPubMedGoogle Scholar - 26.J.C. Hubbard, Phys. Rev.
**46**, 525 (1934)CrossRefGoogle Scholar - 27.W.G. Cady, Proc. I. R. E.
**10**, 83 (1922)CrossRefGoogle Scholar - 28.K.S.V. Dyke, Proc. I. R. E.
**16**, 724 (1928)Google Scholar - 29.J.C. Hubbard, Phys. Rev.
**41**, 523 (1932)CrossRefGoogle Scholar - 30.H.J. Pain,
*The Physics of Vibrations and Waves*, 5th edn. (John Wiley & Sons, Singapore, 2003)Google Scholar - 31.L.E. Kinsler, A.R. Frey,
*Fundamentals of Acoustics*(John Wiley & Sons, New York, NY, USA, 1959)Google Scholar - 32.H. Lamb,
*The Dynamical Theory of Sound*(Dover Publications, New York, NY, 1960)Google Scholar - 33.V.A.D. Grosso, C.W. Mader, J. Acoust. Soc. Am.
**52**(5), 1442 (1972)CrossRefGoogle Scholar - 34.T. Takagi, H. Teranishi, J. Chem. Thermodyn.
**19**, 1299 (1987)CrossRefGoogle Scholar - 35.G. Tardajos, M.D. Peña, E. Aicart, J. Chem. Thermodyn.
**18**, 683 (1986)CrossRefGoogle Scholar - 36.K.C. Karla, K.C. Singh, D.C. Spah, J. Chem. Thermodyn.
**21**, 1243 (1989)CrossRefGoogle Scholar - 37.C.E. Teeter Jr., J. Acoust. Soc. Am.
**18**(2), 488 (1946)CrossRefGoogle Scholar - 38.C. Sörensen, Ann. der Physik
**418**(2), 121 (1936)CrossRefGoogle Scholar - 39.J. Claeys, J. Errera, H. Sack, C.R. Acad, C. R. Acad. Sci.
**202**, 1493 (1936)Google Scholar - 40.W. Buß, Ann. der Physik
**33**, 143 (1938)CrossRefGoogle Scholar - 41.G.K. Hartmann, A.B. Focke, Phys. Rev.
**57**, 221 (1940)CrossRefGoogle Scholar - 42.L.N. Liebermann, J. Acoust. Soc. Am.
**20**(6), 868 (1948)CrossRefGoogle Scholar - 43.G.S. Verma, J. Chem. Phys.
**18**(10), 1352 (1950)CrossRefGoogle Scholar - 44.C.J. Moen, J. Acoust. Soc. Am.
**23**(1), 62 (1951)CrossRefGoogle Scholar - 45.M.C. Smith, R.E. Barrett, R.T. Beyer, J. Acoust. Soc. Am.
**23**(1), 71 (1951)CrossRefGoogle Scholar - 46.R. Martinez, L. Leija, A. Vera, in
*Health Care Exchange (PAHCE), 2010 Pan America*(2010), pp. 81–84Google Scholar - 47.M.J. Holmes, N.G. Parker, M.J.W. Povey, J. Phys.: Conf. Ser.
**269**, 012011 (2011)Google Scholar - 48.M.E. Baumgardt, C. R. Acad. Sci.
**204**, 416 (1937)Google Scholar - 49.E.C. Gregg Jr., Rev. Sci. Inst.
**12**, 149 (1941)CrossRefGoogle Scholar - 50.J.R. Pellam, J.K. Galt, J. Chem. Phys.
**14**(10), 608 (1946)CrossRefGoogle Scholar - 51.Y. Wada, S. Shimbo, J. Acoust. Soc. Am.
**24**(2), 199 (1952)CrossRefGoogle Scholar - 52.J.L. Hunter, W.H. Nichols, J. Haus, Phys. Lett.
**37A**(2), 127 (1971)CrossRefGoogle Scholar