Apparent travelling waves in cochlear models. ("reinifrosch@xxxxxxxx" )

Subject: Apparent travelling waves in cochlear models. From: "reinifrosch@xxxxxxxx" <reinifrosch@xxxxxxxx> Date: Wed, 28 Dec 2011 15:02:26 +0000 List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY> ------=_Part_1631_29893451.1325084546413 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Dear colleagues, The first paper of the recently published proceedings volume of the 11th In= ternational Mechanics-of-Hearing Workshop (July 2011, Williamstown, MA, USA= ; "What Fire Is in Mine Ears: Progress in Auditory Biomechanics") is entitl= ed "MoH 101: Basic concepts in the mechanics of hearing". In the section "W= hat are the requirements for traveling waves in the ear?" of that first pap= er, the authors (Bergevin, Epp, and Meenderink) have written: "Note that th= e longitudinal fluid coupling makes this inherently different from a series= of uncoupled oscillators, which von B=C3=A9k=C3=A9sy demonstrated can give= rise to an apparent traveling wave." The just mentioned series of oscillators is shown in Fig. 13-10 of von B=C3= =A9k=C3=A9sy's book "Experiments in Hearing", and the apparent travelling w= ave is shown in his Fig. 13-11. The oscillators (pendula) are not really un= coupled, since they are suspended from a common rigid horizontal driving ro= d, which is set in motion by a heavy pendulum clamped to the rod. In order = to study the motion of those pendula, I assumed that the points of suspensi= on are on a straight line parallel to the x-axis, and that the length of an= y given pendulum is equal to the x-coordinate of its point of suspension. A= t time t < 0, the points of suspension are assumed to be at rest at y =3D 0= ; at time t > 0 all those points of suspension are assumed to oscillate hor= izontally so that their common y-coordinate varies as y(t) =3D a_y * sin(om= ega*t), where a_y =3D 1 mm, omega =3D 2pi * f, and f =3D 1 Hz. The damping = of the oscillations is assumed to be fairly weak, namely so weak that in th= e case of a free oscillation the amplitude of each of the pendula would tak= e as many as 32 cycles to decay to a(t) =3D a(0)/e (where e=3D2.72). If one= solves the equations for the transient response of the series of pendula d= escribed above, one indeed finds "apparent traveling waves":=20 a snapshot of the line y(x) formed by the suspended spheres at t =3D 0.40 s= ec shows a maximum of y [y =3D +1.75 mm] at x =3D 11 cm; at t =3D 0.45 sec, that maximum has moved to x =3D 15 cm and has shrunk sl= ightly [y =3D +1.70 mm]; at t =3D 0.50 sec, the maximum has moved to x =3D 19.5 cm [y =3D +1.61 mm].= =20 The propagation velocity of that maximum is seen to be about dx/dt =3D 85 c= m / sec. At time t =3D 0.8 sec [1.0 sec], a minimum of the curve y(x) formed by the = suspended spheres is found at x =3D 13 cm [x =3D 23 cm]; in comparison with= the above-mentioned maximum of y(x), that minimum has a greater absolute v= alue (y =3D-3 mm, approximately) and a lower propagation velocity (about 50= cm / sec).=20 A stationary state is reached at about t =3D 60 sec. Now the pendula at x b= elow 20 cm oscillate appproximately in phase with the driving rod; the reso= nating pendulum at x =3D 25 cm lags behind by 0.25 sec, and the pendula at = x above 30 cm lag behind by 0.5 sec. That stationary state, too, involves a= pparent travelling waves;=20 for instance, the line y(x) formed by the suspended spheres at time t =3D 3= 00.4 sec exhibits a maximum at x =3D 24.75 cm [y =3D +9 cm]; at t =3D 300.5 sec, that maximum has moved to x =3D 24.85 cm and has grown = [y =3D + 10 cm]; at t =3D 300.6 sec, the maximum has moved to x =3D 24.95 cm and has shrunk = again [x =3D +9 cm]. As stated by Bergevin, Epp, and Meenderink, those apparent travelling waves= differ inherently from the travelling waves in more realistic cochlear mod= els; those latter waves are similar to surface gravity waves on a mass-load= ed lake, e.g. on a lake covered by floating pieces of wood or ice. Reinhart Frosch, CH-5200 Brugg. reinifrosch@xxxxxxxx . ------=_Part_1631_29893451.1325084546413 Content-Type: text/html;charset="UTF-8" Content-Transfer-Encoding: quoted-printable <html><head><style type=3D'text/css'>  </style> </head><body><div class=3D'bwmail'>Dear colleagues,</FONT= > The first paper of the recently published proceedings vol= ume of the 11th International Mechanics-of-Hearing Workshop (July 2011, Wil= liamstown, MA, USA; "What Fire Is in Mine Ears: Progress in Auditory B= iomechanics") is entitled "MoH 101: Basic concepts in the mechanics of hear= ing". In the section "What are the requirements for traveling waves in the = ear?" of that first paper, the authors (Bergevin, Epp, and Meenderink) have= written: "Note that the longitudinal fluid coupling makes this inherently = different from a series of uncoupled oscillators, which von B=C3=A9k=C3=A9s= y demonstrated can give rise to an apparent traveling wave." The just mentioned series of oscillators is shown in Fig.= 13-10 of von B=C3=A9k=C3=A9sy's book "Experiments in Hearing", and the app= arent travelling wave is shown in his Fig. 13-11. The oscillators (pendula)= are not really uncoupled, since they are suspended from a common rigid&nbs= p;horizontal driving rod, which is set in motion by a heavy pendulum clampe= d to the rod. In order to study the motion of those pendula, I assumed that= the points of suspension are on a straight line parallel to the x-axis, an= d that the length of any given pendulum is equal to the x-coordinate o= f its point of suspension. At time t < 0, the points of suspension = are assumed to be at rest at y =3D 0; at time t > 0 all those points of = suspension are assumed to oscillate horizontally so that their common = y-coordinate varies as y(t) =3D a_y * sin(omega*t), where a_y =3D 1 mm= , omega =3D 2pi * f, and f =3D 1 Hz. The damping of the oscillations&n= bsp;is assumed to be fairly weak, namely so weak that in the case of a=  free oscillation the amplitude of each of the pendula would take= as many as 32 cycles to decay to a(t) =3D a(0)/e (where e=3D2.72). If= one solves the equations for the transient response of the series of = pendula described above, one indeed finds "apparent traveling waves": </FON= T> a snapshot of the line y(x) formed by the suspended spher= es at t =3D 0.40 sec shows a maximum of y [y =3D +1.75 mm] at x =3D 11= cm; at t =3D 0.45 sec, that maximum has moved to  x= =3D 15 cm and has shrunk slightly [y =3D +1.70 mm]; at t =3D 0.50 sec, the maximum has moved to x =3D 19= .5 cm [y =3D +1.61 mm]. The propagation velocity of that maximum is seen to be ab= out dx/dt =3D 85 cm / sec. At time t =3D 0.8 sec [1.0 sec], a minimum of t= he curve y(x) formed by the suspended spheres is found at x =3D 13 cm = [x =3D 23 cm]; in comparison with the above-mentioned maximum of y(x),= that minimum has a greater absolute value (y =3D-3 mm, approximately)= and a lower propagation velocity (about 50 cm / sec). A stationary state is reached at about t =3D 60 sec. Now = the pendula at x below 20 cm oscillate appproximately in phase with th= e driving rod; the resonating pendulum at x =3D 25 cm lags behind by 0.25 s= ec, and the pendula at x above 30 cm lag behind by 0.5 sec.&= nbsp;That stationary state, too, involves apparent travelling waves; </FONT= > for instance, the line y(x) formed by the suspended spher= es at time t =3D 300.4 sec exhibits a maximum at x =3D 24.75 cm [y =3D +9 c= m]; at t =3D 300.5 sec, that maximum has moved to x =3D 24.85= cm and has grown [y =3D + 10 cm]; at t =3D 300.6 sec, the maximum has moved to x =3D 24.95 = cm and has shrunk again [x =3D +9 cm]. As stated by Bergevin, Epp, and Meenderink, those apparen= t travelling waves differ inherently from the travelling waves in more real= istic cochlear models; those latter waves are similar to surface gravity wa= ves on a mass-loaded lake, e.g. on a lake covered by floating pieces of woo= d or ice. Reinhart Frosch, CH-5200 Brugg. reinifrosch@xxxxxxxx= .ch . </div></body></html> ------=_Part_1631_29893451.1325084546413--

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