A Scalar Acoustic Equation for Gases, Liquids, and Solids, Including Viscoelastic Media

Eugen Mamontov^1*, Viktor Berbyuk²

¹Utilization Group, Department of Research and Development, Foundation Chalmers Industrial Technology, Gothenburg, Sweden.
²Division of Dynamics, Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden.

DOI: 10.4236/jamp.2014.210109                

Abstract

The work deals with a mathematical model for real-time acoustic monitoring of material parameters of media in multi-state viscoelastic engineering systems continuously operating in irregular external environments (e.g., wind turbines in cold climate areas, aircrafts, etc.). This monitoring is a high-reliability time-critical task. The work consistently derives a scalar wave PDE of the Stokes type for the non-equilibrium part (NEP) of the average normal stress in a medium. The explicit expression for the NEP of the corresponding pressure and the solution-adequateness condition are also obtained. The derived Stokes-type wave equation includes the stress relaxation time and is applicable to gases, liquids, and solids.

Keywords

Acoustic Monitoring, Gas, Liquid, or Solid, Acoustic Equation, Visoelastic Media, Stress Relaxation Time, Average Normal Stress, the Stokes-Type Wave Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Rose, J.L. (1999) Ultrasonic Waves in Solid Media. Cambridge University Press, Cambridge.
[2]	Datta, S.K. and Shah, A.H. (2009) Elastic Waves in Composite Media and Structures: With Applications to Ultrasonic Nondestructive Evaluation. CRC Press, Boca Raton.
[3]	Grad, H. (1958) Principles of the Kinetic theory of Gases. In: Flügge, S. Ed., Handbuch der Physik, Band XII, Springer-Verlag, Berlin, 205-294.
[4]	Balescu, R. (1997) Statistical Dynamics: Matter out of Equilibrium. Imperial College Press, London.
[5]	Pollard, H.F. (1977) Sound Waves in Solids. Pion, London.
[6]	Landau, L.D. and Lifshitz, E.M. (1986) Theory of Elasticity. Pergamon Press, Oxford.
[7]	Stokes, G.G. (1845) On the Theories of the Internal Friction of Fluids in Motion and of the Equilibrium and Motion of Elastic Solids. Trans. Cambridge Philos. Soc., 8, 287-319.
[8]	Merkel, H. (2006) Apparatus and a Method for Determining the Spatial Distribution of Physical Parameters in an Object. US Patent Application 11/375,133, 20.
[9]	Landau, L.D. and Lifshitz, E.M. (1987) Fluid Mechanics. Pergamon Press, Oxford.
[10]	Koshlyakov, N.S., Smirnov, M.M. and Gliner, E.B. (1964) Differential Equations of Mathematical Physics. North-Holland Publishing, Amsterdam.
[11]	Longman, I.M. (1980) Wave Propagation in a Viscoelastic Solid. Journal of Computational Physics, 37, 171-182. http://dx.doi.org/10.1016/0021-9991(80)90019-4
[12]	Ricker, N.H. (1977) Transient Waves in Visco-Elastic Media. Elsevier, Amsterdam.
[13]	Sedov, L.I. (1971) A Course in Continuum Mechanics. Vol. 1, Wolters-Noordhoff, Groningen.
[14]	Trigg, G.L. (1991) Encyclopedia of Applied Physics, Vol. 1, VCB, New York.
[15]	Dukhin, A., et al. (2014) Volume Viscosity, Wikipedia. Wikimedia Foundation, Inc. http://en.wikipedia.org/wiki
[16]	Goldstein, H. (1980) Classical Mechanics. Addison-Wesley, Reading.