Dear colleagues,
The first paper of the recently published proceedings volume of the 11th International Mechanics-of-Hearing Workshop (July 2011, Williamstown, MA, USA; "What Fire Is in Mine Ears: Progress in Auditory Biomechanics") is entitled "MoH 101: Basic concepts in the mechanics of hearing". 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 BÃkÃsy demonstrated can give rise to an apparent traveling wave."
The just mentioned series of oscillators is shown in Fig. 13-10 of von BÃkÃsy's book "Experiments in Hearing", and the apparent travelling wave is shown in his Fig. 13-11. The oscillators (pendula) are not really uncoupled, since they are suspended from a common rigid horizontal driving rod, 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 suspension are on a straight line parallel to the x-axis, and that the length of any given pendulum is equal to the x-coordinate of its point of suspension. At time t < 0, the points of suspension are assumed to be at rest at y = 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) = a_y * sin(omega*t), where a_y = 1 mm, omega = 2pi * f, and f = 1 Hz. The damping of the oscillations 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) = a(0)/e (where e=2.72). If one solves the equations for the transient response of the series of pendula described above, one indeed finds "apparent traveling waves":
a snapshot of the line y(x) formed by the suspended spheres at t = 0.40 sec shows a maximum of y [y = +1.75 mm] at x = 11 cm;
at t = 0.45 sec, that maximum has moved to x = 15 cm and has shrunk slightly [y = +1.70 mm];
at t = 0.50 sec, the maximum has moved to x = 19.5 cm [y = +1.61 mm].
The propagation velocity of that maximum is seen to be about dx/dt = 85 cm / sec.
At time t = 0.8 sec [1.0 sec], a minimum of the curve y(x) formed by the suspended spheres is found at x = 13 cm [x = 23 cm]; in comparison with the above-mentioned maximum of y(x), that minimum has a greater absolute value (y =-3 mm, approximately) and a lower propagation velocity (about 50 cm / sec).
A stationary state is reached at about t = 60 sec. Now the pendula at x below 20 cm oscillate appproximately in phase with the driving rod; the resonating pendulum at x = 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 apparent travelling waves;
for instance, the line y(x) formed by the suspended spheres at time t = 300.4 sec exhibits a maximum at x = 24.75 cm [y = +9 cm];
at t = 300.5 sec, that maximum has moved to x = 24.85 cm and has grown [y = + 10 cm];
at t = 300.6 sec, the maximum has moved to x = 24.95 cm and has shrunk again [x = +9 cm].
As stated by Bergevin, Epp, and Meenderink, those apparent travelling waves differ inherently from the travelling waves in more realistic cochlear models; those latter waves are similar to surface gravity waves on a mass-loaded lake, e.g. on a lake covered by floating pieces of wood or ice.
Reinhart Frosch,
CH-5200 Brugg.
reinifrosch@xxxxxxxxxx .