eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2021): 1.024
Shock wave propagation in nonlinear microstructured wool felt; pp. 361–367
PDF | doi: 10.3176/proc.2015.3S.06

Anatoli Stulov, Vladimir Erofeev

On the basis of experimental data from the piano hammers study a one-dimensional constitutive equation of wool felt material is proposed and used to study compression pulse propagation in microstructured felt. One-dimensional strain wave propagation in wool felt is considered. It is revealed that stiffness of microstructured wool felt is a nonlinear function of the felt compression, and it is strongly determined by the rate of the felt loading. This means that the speed of the compression wave that propagates in such medium depends on the form of the wave and its amplitude. It is shown that a pulse of a smooth form that has no discontinuity on its front propagates with a constant speed up to the moment when the accumulation of nonlinear effects results in the eventual continuous wave breaking. After that moment, a shock wave will be formed, and the velocity of the shock wave propagation depends on the value of its amplitude jump discontinuity across the wave front. It is shown that the front velocity of the shock wave is greater than the velocity of sound in a linear medium.



  1. Hall, D. E. Piano string excitation III: general solution for a soft narrow hammer. J. Acoust. Soc. Am., 1987, 81, 547–555.

  2. Suzuki, H. and Nakamura, I. Acoustics of pianos. Appl. Acoust., 1990, 30, 147–205.

  3. Fletcher, N. H. and Rossing, T. D. The Physics of Musical Instruments. Springer-Verlag, New York, 1991, 319–321.

  4. Hall, D. E. Piano string excitation in the case of small hammer mass. J. Acoust. Soc. Am., 1986, 79, 141–147.

  5. Askenfelt, A. and Jansson, E. V. From touch to string vibrations. III: string motion and spectra. J. Acoust. Soc. Am., 1993, 93, 2181–2196.

  6. Stulov, A. Piano hammer--string interaction. In Proceedings of ICA 2004. Kyoto, Japan, 2004, Vol. III, 2127–2130.

  7. Stulov, A. Hysteretic model of the grand piano hammer felt. J. Acoust. Soc. Am., 1995, 97, 2577–2585.

  8. Stulov, A. Dynamic behavior and mechanical features of wool felt. Acta Mech., 2004, 169, 13–21.

  9. Stulov, A. Experimental and computational studies of piano hammers. Acta Acust. united Ac., 2005, 91, 1086–1097.

10. Yanagisawa, T. and Nakamura, K. Dynamic compression characteristics of piano hammer. Transactions of Musical Acoustics Technical Group Meeting of the Acoustic Society of Japan, 1982, 1, 14–18.

11. Kartofelev, D. and Stulov, A. Propagation of deformation waves in wool felt. Acta Mech., 2014, 225, 3103–3113.

12. Kartofelev, D. and Stulov, A. Wave propagation and dispersion in microstructured wool felt. Wave Motion, 2015, 57, 23–33.

13. Whitham, G. Linear and Nonlinear Waves. Wiley-Interscience, New York, 1974.


Back to Issue