The physics of the ear Comps II Presentation Jed Whittaker December 5, 2006

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

Ear anatomy Outer Ear (Resonator) http://www.nidcd.nih.gov/StaticReso urces/health/hearing/images/normal_ear.asp

Ear anatomy Middle Ear (Impedance Matching) Outer Ear http://www.nidcd.nih.gov/StaticReso (Resonator) urces/health/hearing/images/normal_ ear.asp

Ear anatomy Middle Ear (Impedance Matching) Inner Ear (Fourier Analyzer) Outer Ear http://www.nidcd.nih.gov/StaticReso (Resonator) urces/health/hearing/images/normal_ ear.asp

Ear anatomy substructures Semicircular Canals Ossicles (Balance) Cochlea Ear Drum

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

The outer ear The human ear is most responsive at about 3,000 Hz Most speech occurs at about 3,000 Hz

Partial y closed pipe resonator model speed of frequency sound mode

Outer ear resonator The length of the human auditory canal is This gives a fundamental mode of

Outer ear Q Outer ear is a low-Q cavity; other frequencies pass through also W. J. Mul in, W. J. George, J. P. Mestre, and S. L. Vel eman, F undam entals of sound with applicationsto speech and hear ing (Al yn and Bacon, Boston, 2003)

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

The middle ear There is an impedance mismatch between the outer and inner ears Air Fluid Without the middle ear there would be large attenuation at the air-fluid boundary

Transmission and reflection Reflection Transmission Coefficient Coefficient D. T. Blackstock, F undam entals of P hysical A coustics (Wiley, New York, 2000)

Power transmission Doing some math gives the power transmission coefficient Plugging in numbers gives the attenuation W. J. Mul in, W. J. George, J. P. Mestre, and S. L. Vel eman, F undam entals of sound with applicationsto speech and hear ing (Al yn and Bacon, Boston, 2003)

Ossicles as levers Ligaments act Levers increase as fulcrums force by 30% To oval window http://hyperphysics.phy-astr.gsu.edu/hbase/sound/imgsou/oss3.gif

Stapes footprint Oval window Ear drum http://www.ssc.education.ed.ac.uk/courses/pictures/hearing8.gif

Impedance match Since Sound intensity the sound intensity increases 625 times, or 28 dB

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

The inner ear The cochlea transduces sound into electrochemical signals To brain Sound The cochlea is a Fourier analyzer that separates frequency information for the brain

Cochlear schematic W. Yost, F undam entals of H earing (Academic Press, San Diego, 2000)

Waves on the basilar membrane Wave amplitude h(x,t ) 0 Basilar h(x,t ) 1 membrane W. R. Zemiln, S peech and H earing S cience (Al yn and Bacon, Boston, 1998)

Cochlear cross-section Basilar membrane motion transduces sound signal Hair cel shearing releases neurotransmitters W. R. Zemiln, S peech and H earing S cience (Al yn and Bacon, Boston, 1998)

Hair cell shearing Tectoral membraneHair cel s Basilar membrane Sheared hairs W. R. Zemiln, S peech and H earing S cience (Al yn and Bacon, Boston, 1998)

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

Helmholtz’s resonance theory (1857) One hair cell = one resonant frequency The sum signal of the Hairs vibrating hairs reproduces the sound Fourier analysis http://www.vimm.it/cochlea/cochleapages/overview/helmholtz/helm.htm H. von Helmholtz, On the sensations of tone as a physiological basis for the theory of m usic, (translated by A. J. El is) (Longmans, Green, and Co., London, 1895)

Problems with resonance theory Needed 6,000+ hair cells to explain human hearing Only ~3,000 hair cells were found Later revisions used coupled membrane fibers as resonators Cochlear anatomy was not well known E. G. Wever, T heory of H earing (Wiley, New York, 1949)

Competing theories Telephone theory required each hair cell to reproduce all frequencies Standing wave theory had hair cells detecting patterns on the basilar membrane Traveling wave theory described hair cells as detecting the amplitude of a wave traveling along the basilar membrane E. G. Wever, T heory of H earing (Wiley, New York, 1949)

von Bèkèsy’s physical model The model mimicked each of the theories Needle G. von Bèkèsy, E xperim ents in H earing (McGraw-Hil , New York, 1960)

von Bèkèsy’s observations Saw traveling waves in mammalian cochleae Position of peak depends on frequency Spatial frequency separation G. von Bèkèsy, E xperim ents in H earing (McGraw-Hil , New York, 1960)

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

Traveling wave description relative pressure relative current; induced by pressure gradients

Pressure-current relation Changing the relative current in time changes the relative pressure in space fluid viscosity fluid density scala height

Incompressibility Assuming the fluid is incompressible means that a change in relative current gives a change in basilar membrane displacement h(x,t) basilar membrane displacement

The wave equation Stiffness relates pressure and displacement Plug this in to get the wave equation

Stiffness The stiffness of the basilar membrane is described by Plug this in to get G. Ehret, J . A coust. S oc. A m . 64, 1723 (1978) T. Duke and F. Jülicher, P hys. R ev. L ett. 90, 158101 (2003)

Solution This equation separates into separation constant We can go no further analytically Usually the equation is solved numerically

Numerical y generated plots T. Duke and F. Jülicher, P hys. R ev. L ett. 90, 158101 (2003)

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

Recent cochlear model Life size Si fabrication Mass producible R. D. White and K. Grosh, P N A S 102, 1296 (2005)

Data R. D. White and K. Grosh, P N A S 102, 1296 (2005)

Outline I. Ear anatomyII. Outer ear resonatorIII. Middle ear impedance matchingIV. Inner ear a. Anatomyb. Theories of cochlear functionc. Traveling wave theory VII. Current work a. Physical modelb. Mathematical model

Recent math model Why is the cochlea spiraled? Previously though to have no function Data correlate number of turns to low frequency sensitivity in various mammals C. West, J . A coust. S oc. A m . 77, 1091 (1985) D. Manoussaki, E. K. Dimitriadis, and R. Chadwick, P hys. R ev. L ett. 96, 088701 (2006)

Rayleigh’s whispering gal ery Sound Sound ray confined to Source boundary L. Rayleigh, T he T heory of S ound, vol. II (MacMil an and Co., New York, 1895)

Spiral Boundary b a > b a http://biomed.brown.edu/Courses/BI108/2006-108websites/group10cochlearimplant/images/greenwood_function.png

Low frequency amplification Sound concentrated at boundary Low frequencies are transduced at the apex D. Manoussaki, E. K. Dimitriadis, and R. Chadwick, P hys. R ev. L ett. 96, 088701 (2006)

Math model Solving the traveling wave equation through a spiral cochlea shows that the sound concentrates on the outer wall of the cochlea D. Manoussaki, E. K. Dimitriadis, and R. Chadwick, P hys. R ev. L ett. 96, 088701 (2006)

Conclusion The ear uses resonance, impedance matching, and Fourier analysis Traveling wave theory most accurately describes cochlear dynamics Theories of cochlear function are still advancing with physical and mathematical models

Appendix A Transmission

Transmission and reflection Reflection Transmission Coefficient Coefficient D. T. Blackstock, F undam entals of P hysical A coustics (Wiley, New York, 2000)

Transmission and reflection Use and to get

Power Sound intensity is the time averaged integral of pressure times velocity The power transmission coefficient is then

Power transmission coefficient Since (using the definition of T) the power transmission coefficient is

Impedance values

Power transmission Doing some math gives the power transmission coefficient Plugging in numbers gives the attenuation W. J. Mul in, W. J. George, J. P. Mestre, and S. L. Vel eman, F undam entals of sound with applicationsto speech and hear ing (Al yn and Bacon, Boston, 2003)

Appendix B Spiral cochlea

Cochlear equations Start with an irrotational fluid of velocity velocity potential and whose mass has no source or sink

Navier-Stokes Use a linearized momentum equation derived from the Navier-Stokes equation for an incompressible fluid with viscosity μ and negligible body forces (like gravity)

Navier-Stokes Spatially integrate and use the mass conservation relation to get

Continuity Change in the velocity potential gives a change in the membrane displacement The equation of motion for the membrane is stiffness mass damping

Appendix C Traveling wave derivation

Traveling wave description relative pressure relative current

Pressure-current relation Changing the relative current in time changes the relative pressure in space fluid viscosity fluid density scala height

Incompressibility Assuming the fluid is incompressible means that a change in relative current gives a change in basilar membrane displacement h(x,t) basilar membrane displacement

Time derivative Take the time derivative Plug in the pressure current relation

The wave equation Stiffness relates pressure and displacement Plug this in to get the wave equation

Stiffness The stiffness of the basilar membrane is described by Plug this in to get G. Ehret, J . A coust. S oc. A m . 64, 1723 (1978) T. Duke and F. Jülicher, P hys. R ev. L ett. 90, 158101 (2003)

Solution This equation separates into separation constant We can go no further analytically Usually the equation is solved numerically

Numerical y generated plots Instantaneous position h(x,t) Wave amplitude T. Duke and F. Jülicher, P hys. R ev. L ett. 90, 158101 (2003)