Analytical Modelling of Natural Frequency of Tympanic Membrane as the Low-Frequency Limit of Hearing

  • D. John JabarajEmail author
Original Article


We examine the low-frequency limit of hearing of the mammalian ear through the analytical modelling of the natural frequency of the tympanic membrane. The resulting equation of the natural frequency of the modelled tympanic membrane is numerically verified against previous theoretical studies, and is statistically validated against the experimental data on the low-frequency limit of hearing. By utilizing the Wilcoxon signed-rank test; W-values of 29 (p value = 0.25014) and 23 (p value = 0.11642) are respectively obtained for the 0.2% and 0.3% prestrain (at 5% significance level for sample size of 13). We fail to reject the null hypothesis as the W-values are within the critical values of the test statistics, and therefore conclude that the tympanic membrane acts as a low-frequency limiter of acoustic stimulus. Based on our study, we can predict the low-frequency limit of hearing in mammals (e.g., for the whale as 3.6 Hz and for the zebra as 44.0 Hz).


Tympanic membrane Low-frequency limit of hearing Mammalian ear Analytical modelling 



This work was conducted in UniKL-MSI. D. John Jabaraj appreciates Pearl VKJ and Amber QPJ for support during this research and for introducing joy in everything. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. D. John Jabaraj declares that he has no conflict of interest. This article does not contain any studies with human or animal subjects performed by the author.


No benefits in any form have been or will be received from a commercial party related directly or indirectly to the subject of this manuscript.


  1. 1.
    Aernouts, J., J. A. Soons, and J. J. Dirckx. Quantification of tympanic membrane elasticity parameters from in situ point indentation measurements: validation and preliminary study. Hear. Res. 263(1–2):177–182, 2010.CrossRefPubMedGoogle Scholar
  2. 2.
    Ahn, T., M. Baek, and D. Lee. Experimental measurement of tympanic membrane response for finite element model validation of a human middle ear. SpringerPlus 2:527, 2013.CrossRefPubMedPubMedCentralGoogle Scholar
  3. 3.
    Altiok, T., and B. Melamed. Simulation Modeling and Analysis with ARENA. New York: Elsevier Academic Press, 2007.Google Scholar
  4. 4.
    Appler, J. M., and L. V. Goodrich. Connecting the ear to the brain: molecular mechanisms of auditory circuit assembly. Progr. Neurobiol. 93(4):488–508, 2011.CrossRefGoogle Scholar
  5. 5.
    Ayela, C., and L. Nicu. Micromachined piezoelectric membranes with high nominal quality factors in newtonian liquid media: a Lamb’s model validation at the microscale. Sens. Actuators B 123(2):860–868, 2007.CrossRefGoogle Scholar
  6. 6.
    Baboolal, A. J., W. B. Green, and K. CIarke. The effects of tympanic membrane intubation on middle ear resonant frequency in children. J. Speech Lang. Pathol. Audiol. 22(1):10–14, 1998.Google Scholar
  7. 7.
    Barklow, W. E. Low-frequency sounds and amphibious communication in Hippopotamus amphibious. J. Acoust. Soc. Am. 115:2555, 2004.CrossRefGoogle Scholar
  8. 8.
    Beards, C. F. Structural Vibration: Analysis and Damping (1st ed.). Oxford: Butterworth-Heinemann, 1996.Google Scholar
  9. 9.
    Bell, A. A resonance approach to cochlear mechanics. PLoS ONE 7(11):e47918, 2012.CrossRefPubMedPubMedCentralGoogle Scholar
  10. 10.
    Blevins, R. D. Formulas for Natural Frequency and Mode Shape. New York: Van Nostrand Reinhold, p. 492, 1979.Google Scholar
  11. 11.
    Caminos, L., J. Garcia-Manrique, A. Lima-Rodriguez, and A. Gonzalez-Herrera. Analysis of the mechanical properties of the human tympanic membrane and its influence on the dynamic behaviour of the human hearing system. Appl. Bionics Biomech.. Article ID 1736957, 2018.Google Scholar
  12. 12.
    Carson, II, J. S. Model verification and validation. Proc. Winter Simul. Conf. 52–58:2002, 2002.Google Scholar
  13. 13.
    Castagno, L. A., and L. Lavinksy. Tympanic membrane healing in myringotomies performed with argon laser or microknife: an experimental study in rats. Rev. Bras. Otorrinolaringol. 72(6):794–799, 2006.CrossRefGoogle Scholar
  14. 14.
    Chasin, M. The etiology of the REUG: did we get it completely right? Hear. J. 58(12):22–24, 2005.CrossRefGoogle Scholar
  15. 15.
    Cheng, T., C. Dai, and R. Z. Gan. Viscoelastic properties of human tympanic membrane. Ann. Biomed. Eng. 35(2):305–314, 2007.CrossRefPubMedGoogle Scholar
  16. 16.
    Chole, R. A., and K. Kodama. Comparative histology of the tympanic membrane and its relationship to cholesteatoma. Ann Otol. Rhinol. Laryngol. 98(10):761–766, 1989.CrossRefPubMedGoogle Scholar
  17. 17.
    Chou, C. F., J. F. Yu, and C. K. Chen. The natural vibration characteristics of human ossicles. Chang. Gung Med. J. 34(2):160–165, 2011.PubMedGoogle Scholar
  18. 18.
    O’Connor, K. N., H. Cai, and S. Puria. The effects of varying tympanic-membrane material properties on human middle-ear sound transmission in a three-dimensional finite-element model. J. Acoust. Soc. Am. 142:2836, 2017.CrossRefPubMedPubMedCentralGoogle Scholar
  19. 19.
    Dagg, A. I. Giraffe: Biology, Behaviour and Conservation (1st ed.). Cambridge: Cambridge University Press, p. 84, 2014.CrossRefGoogle Scholar
  20. 20.
    Dawood, M. R. Frequency dependence hearing loss evaluation in perforated tympanic membrane. Int. Arch. Otorhinolaryngol. 21(4):336–342, 2017.CrossRefPubMedPubMedCentralGoogle Scholar
  21. 21.
    Decraemer, W. F., and W. R. J. Funnell. Anatomical and Mechanical Properties of the Tympanic Membrane Chronic Otitis Media. Pathogenesis-Oriented Therapeutic Management. Amsterdam: Kugler Publications, 2008.Google Scholar
  22. 22.
    Dempster, J. H., and K. Mackenzie. The resonance frequency of the external auditory canal in children. Ear Hear 11(4):296–298, 1990.CrossRefPubMedGoogle Scholar
  23. 23.
    Deserno, M. Chapter 2: Membrane Elasticity and Mediated Interactions in Continuum Theory: A Differential Geometric Approach (Biomembrane Frontiers: Nanostructures, Models, and the Design of Life [Handbook of Modern Biophysics]). London: Springer, 2009.Google Scholar
  24. 24.
    Fay, J. P., S. Puria, and S. R. Steele. The discordant eardrum. Proc. Natl. Acad. Sci. 103(52):19743–19748, 2006.CrossRefPubMedGoogle Scholar
  25. 25.
    Fleischer, G. Tympanic Membrane and Tympanic Plate. In: Evolutionary Principles of the Mammalian Middle Ear. Advances in Anatomy, Embryology and Cell Biology (ergebnisse der anatomie und entwicklungsgeschichte/revues d’anatomie et de morphologie expérimentale), Vol. 55/5. Berlin: Springer, 1978.Google Scholar
  26. 26.
    Funnell, W. R. J. On the undamped natural frequencies and mode shapes of a finite-element model of the cat eardrum. J. Acoust. Soc. Am. 73:1657–1661, 1983.CrossRefPubMedGoogle Scholar
  27. 27.
    Gan, R. Z., B. P. Reeves, and X. Wang. Modeling of sound transmission from ear canal to cochlea. Ann. Biomed. Eng. 35(12):2180–2195, 2007.CrossRefPubMedGoogle Scholar
  28. 28.
    Gefen, A., D. Elad, and R. J. Shiner. Analysis of stress distribution in the alveolar septa of normal and simulated emphysematic lungs. J. Biomech. 32(9):891–897, 1999.CrossRefPubMedGoogle Scholar
  29. 29.
    Goldsmith, T. H. Optimization, constraint, and history in the evolution of eyes. Q. Rev. Biol. 65:281–322, 1990.CrossRefPubMedGoogle Scholar
  30. 30.
    Gonzalez-Herrera, A., and E. S. Olson. A study of sound transmission in an abstract middle ear using physical and finite element models. J. Acoust. Soc. Am. 138(5):2972–2985, 2015.CrossRefPubMedPubMedCentralGoogle Scholar
  31. 31.
    Heffner, H. E. Hearing in large and small dogs: absolute thresholds and size of the tympanic membrane. Behav. Neurosci. 97(2):310–318, 1983.CrossRefGoogle Scholar
  32. 32.
    Heffner, R. S., and H. E. Heffner. Hearing range of the domestic cat. Hear. Res. 19(1):85–88, 1985.CrossRefPubMedGoogle Scholar
  33. 33.
    Heffner, R. S., and H. E. Heffner. Hearing in domestic pigs (Sus scrofa) and goats (Capra hircus). Hear. Res. 48(3):231–240, 1990.CrossRefPubMedGoogle Scholar
  34. 34.
    Heffner, H. E., and R. S. Heffner. Hearing ranges of laboratory animals. J. Am. Assoc. Lab. Anim. Sci. 46(1):20–22, 2007.PubMedGoogle Scholar
  35. 35.
    Heffner, H., and B. Masterton. Hearing in Glires: domestic rabbit, cotton rat, feral house mouse, and kangaroo rat. J. Acoust. Soc. Am. 68(6):1584–1599, 1980.CrossRefGoogle Scholar
  36. 36.
    Herbst, K., and C. Humphrey. Prevalence of hearing impairment in the elderly living at home. J. R. Coll. General Pract. 31:155–160, 1981.Google Scholar
  37. 37.
    Herman, I. P. Physics of the Human Body. Berlin: Springer, 2008.Google Scholar
  38. 38.
    Jabaraj, D. J., and M. S. Jaafar. Vibration analysis of circular membrane model of the alveolar wall in examining ultrasound-induced lung haemorrhage. J. Med. Ultrasound 21(2):81–91, 2013.CrossRefGoogle Scholar
  39. 39.
    Jackson, L. L., R. S. Heffner, and H. E. Heffner. Free-field audiogram of the Japanese macaque (Macaca fuscata). J. Acoust. Soc. Am. 106(5):3017–3023, 1999.CrossRefPubMedGoogle Scholar
  40. 40.
    Johnson, F. R., R. M. McMinn, and G. N. Atfield. Ultrastructural and biochemical observations on the tympanic membrane. J. Anat. 103(Pt 2):297–310, 1968.PubMedPubMedCentralGoogle Scholar
  41. 41.
    King, R. D., and C. D. Tunitsa. The landscape of assumptions. Proc. Spring Simul. Multiconf. 81–88:2008, 2008.Google Scholar
  42. 42.
    Lim, D. J. Tympanic membrane. Electron microscopic observation. Part I: pars tensa. Acta Otolaryngol. 66:181–198, 1968.CrossRefPubMedGoogle Scholar
  43. 43.
    Lim, D. J. Structure and function of the tympanic membrane: a review. Acta. Otorhinolaryngol. Belg. 49(2):101–115, 1995.PubMedGoogle Scholar
  44. 44.
    Liu, J., K. S. Agrawal, H. M. Ladak, and W. Wan. Fiber arrangement in the rat tympanic membrane. Anat. Rec. 299(11):1531–1539, 2016.CrossRefGoogle Scholar
  45. 45.
    Manley, G. A. Nature (London) 230:506–509, 1971.CrossRefGoogle Scholar
  46. 46.
    Manley, G. A. Evolution (Lawrence, Kans.) 26:608–621, 1972.CrossRefGoogle Scholar
  47. 47.
    Masterton, R. B., H. E. Heffner, and R. Ravizza. Evolution of human hearing. J. Acoust. Soc. Am. 45:966–985, 1969.CrossRefPubMedGoogle Scholar
  48. 48.
    Nummela, S. Scaling of the mammalian middle ear. Hear. Res. 85(1–2):18–30, 1995.CrossRefPubMedGoogle Scholar
  49. 49.
    Overstreet, E. H., and M. A. Ruggero. Development of wide-band middle ear transmission in the Mongolian gerbil. J. Acoust. Soc. Am. 111:261–270, 2002.CrossRefPubMedPubMedCentralGoogle Scholar
  50. 50.
    Païdoussis, M. P. Fluid-Structure Interactions: Slender Structures and Axial Flow. London: Elsevier Science, 1998.Google Scholar
  51. 51.
    Pickles, J. O. Introduction to the Physiology of Hearing. Cambridge: Academic Press, 2008.Google Scholar
  52. 52.
    Pierson, L. L., K. J. Gerhardt, G. P. Rodriguez, and R. B. Yanke. Relationship between outer ear resonance and permanent noise-induced hearing loss. Am. J. Otolaryngol. 15(1):37–40, 1994.CrossRefPubMedGoogle Scholar
  53. 53.
    Pritchett, K. R., and B. F. Corning. Biology and medicine of rats. In: Laboratory Animal Medicine and Management, edited by J. D. Reuter, and M. A. Suckow. New York: Elsevier, 2004.Google Scholar
  54. 54.
    Ravicz, M. E. Gerbil middle-ear sound transmission from 100 Hz to 60 kHz. J. Acoust. Soc. Am. 124:363, 2008.CrossRefPubMedPubMedCentralGoogle Scholar
  55. 55.
    Rosowski, J. J. The effects of external- and middle-ear filtering on auditory threshold and noise-induced hearing loss. J. Acoust. Soc. Am. 90:124–135, 1991.CrossRefPubMedGoogle Scholar
  56. 56.
    Rosowski, J. J., J. T. Cheng, and M. E. Ravicz. Computerassisted time-averaged holograms of the motion of the surface of the mammalian tympanic membrane with sound stimuli of 0.4–25 kHz. Hear. Res. 253:1–2, 2009.CrossRefGoogle Scholar
  57. 57.
    Ruggero, M. A., and A. N. Temchin. The roles of the external, middle, and inner ears in determining the bandwidth of hearing. Proc. Natl. Acad. Sci. USA 99(20):13206–13210, 2002.CrossRefPubMedGoogle Scholar
  58. 58.
    Shabana, A. A. Theory of Vibration: An Introduction (Mechanical Engineering Series Vol 1) (2nd ed.). New York: Springer, 1995.Google Scholar
  59. 59.
    Shahnaz, N., and L. Polka. Standard and multifrequency tympanometry in normal and otosclerotic ears. Ear. Hear. 18(4):326–341, 1997.CrossRefPubMedGoogle Scholar
  60. 60.
    Sheskin, D. J. Handbook of Parametric and Nonparametric Statistical Procedures (2nd ed.). Boca Raton: Taylor & Francis-Mathematics, 2000.Google Scholar
  61. 61.
    Sivakumar, V., J. A. P. Bodhika, R. Jayatillaka, C. Pathiraja, S. K. Pathiratne, S. R. B. Dissanayake, S. Wijeyamohan, and C. Santiapillai. Decibel level of firecrackers and its possible impact on the hearing of marauding elephants in Sri Lanka. Int. J. Sci. Environ. Technol. 2(4):593–600, 2013.Google Scholar
  62. 62.
    Skrodzka, S., and J. Modławska. Modal analysis of the human tympanic membrane of middle ear using the finite-element method. Arch. Acoust. 31(4 (supplement)):23–28, 2006.Google Scholar
  63. 63.
    Smith, S. W. The Scientist & Engineer’s Guide to Digital Signal Processing (1st ed.). San Diego: California Technical Publishing, 1997.Google Scholar
  64. 64.
    Stenfeldt, K., C. Johansson, and S. Hellström. The collagen structure of the tympanic membrane: Collagen types I, II, and III in the healthy tympanic membrane, during healing of a perforation, and during infection. Arch. Otolaryngol. Head Neck Surg. 132(3):293–298, 2006.CrossRefPubMedGoogle Scholar
  65. 65.
    Test, T., A. Canfi, A. Eyal, I. Shoam-Vardi, and E. K. Sheiner. The influence of hearing impairment on sleep quality among workers exposed to harmful noise. Sleep 34(1):25–30, 2011.CrossRefPubMedPubMedCentralGoogle Scholar
  66. 66.
    von Békésy, G. Experiments in Hearing. New York: McGraw-Hill, 1960.Google Scholar
  67. 67.
    von Muggenthaler, E. Infrasonic and low-frequency vocalizations from Siberian and Bengal tigers. J. Acoust. Soc. Am. 108:2541, 2000.CrossRefGoogle Scholar
  68. 68.
    von Muggenthaler, E. Giraffe Helmholtz resonance. Proc. Mtgs. Acoust. 19:010012, 2013.CrossRefGoogle Scholar
  69. 69.
    von Unge, M., J. A. N. Buytaert, and J. J. J. Dirckx. Anatomical Boundary of the Tympanic Membrane Pars Flaccida of the Meriones unguiculatus (Mongolian Gerbil). Anat. Rec. 294:987–995, 2011.CrossRefGoogle Scholar
  70. 70.
    Webster, W. B., and R. R. Fay. The Evolutionary Biology of Hearing, Science. New York: Springer, p. 619, 2012.Google Scholar
  71. 71.
    Willott, J. F., and C. Thomas. The Auditory Psychobiology of the Mouse. Springfield: Charles C Thomas Pub Ltd, 1983.Google Scholar
  72. 72.
    Wu, C., Y. Chen, M. S. H. Al-Furjan, J. Ni, and X. Yang. Free vibration model and theoretical solution of the tympanic membrane. Comput. Assist. Surg. 21(sup1):61–68, 2016.CrossRefGoogle Scholar
  73. 73.
    Wysocki, J. Topographical anatomy and morphometry of the temporal bone of the macaque. Folia Morphol. 68(1):13–22, 2008.Google Scholar
  74. 74.
    Yao, W., Y. Chen, J. Ma, C. Gan, and D. Wang. Numerical simulation on the dynamic behavior of the basilar membrane in the spiral cochlea. Biomed. Res. 27(3):977–984, 2016.Google Scholar
  75. 75.
    Yeowart, N. S., M. E. Bryan, and W. Tempest. The monaural MAP threshold of hearing at frequencies from 1.5 to 100c/s. J. Sound Vib. 6:335–342, 1967.CrossRefGoogle Scholar
  76. 76.
    Zwislocki, J. J. The role of external and middle ear in sound transmission. In: The Nervous System, edited by D. B. Tower, and E. L. Eagles. New York: Raven, 1975, pp. 45–55.Google Scholar
  77. 77.
    Zwislocki, J. J. Auditory Sound Transmission—An Autobiographical Perspective. Mahwah, NJ: Erlbaum, 2002.CrossRefGoogle Scholar

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© Biomedical Engineering Society 2019

Authors and Affiliations

  1. 1.Universiti Kuala Lumpur Malaysian Spanish InstituteKulimMalaysia

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