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Re: AUDITORY Digest - 27 Dec 2011 to 28 Dec 2011 (#2011-304)
Dear list -
The note from Prof. Frosch reminds me that I am looking for a source for the
film that Bekesy made in 1955 or 1956 of his large hanging-pendulum model.
He did one with uncoupled oscillators driven by the beam from which they
were hung, and a second with coupled oscillators (weighted strings linking
adjacent pendulums). This was driven in one case at the "base" and in
another by the "fluid", the beam. The moving image was most compelling.
When asked why he built the model instead of calculating the expected
behavior of the system, Bekesy replied, "Because I can not do the
mathematics." (!)
If anyone knows of the film, whether it still exists and whether copies have
been made I would be most appreciative.
Doug Creelman, Psychology, University of Toronto
C. Douglas Creelman 416-690-9407 (phone & fax)
9 Fernwood Park Ave. 416-708-9407 (cell)
Toronto, ON Canada creelman@xxxxxxxxxxxxxxxxx
M4E 3E8
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From: AUDITORY - Research in Auditory Perception
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Sent: December 29, 2011 12:04 AM
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Subject: AUDITORY Digest - 27 Dec 2011 to 28 Dec 2011 (#2011-304)
There is 1 message totalling 167 lines in this issue.
Topics of the day:
1. Apparent travelling waves in cochlear models.
----------------------------------------------------------------------
Date: Wed, 28 Dec 2011 15:02:26 +0000
From: "reinifrosch@xxxxxxxxxx" <reinifrosch@xxxxxxxxxx>
Subject: Apparent travelling waves in cochlear models.
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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@xxxxxxxxxx .
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</head><body><div class=3D'bwmail'><P><FONT size=3D2>Dear colleagues,</FONT=
></P>
<P><FONT size=3D2>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."</FONT></P>
<P><FONT size=3D2>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></P>
<P><FONT size=3D2>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;</FONT></P>
<P><FONT size=3D2>at t =3D 0.45 sec, that maximum has moved to x=
=3D 15 cm and has shrunk slightly [y =3D +1.70 mm];</FONT></P>
<P><FONT size=3D2>at t =3D 0.50 sec, the maximum has moved to x =3D 19=
.5 cm [y =3D +1.61 mm]. </FONT></P>
<P><FONT size=3D2>The propagation velocity of that maximum is seen to be ab=
out dx/dt =3D 85 cm / sec.</FONT></P>
<P><FONT size=3D2>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). </FONT></P>
<P><FONT size=3D2>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=
></P>
<P><FONT size=3D2>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];</FONT></P>
<P><FONT size=3D2>at t =3D 300.5 sec, that maximum has moved to x =3D 24.85=
cm and has grown [y =3D + 10 cm];</FONT></P>
<P><FONT size=3D2>at t =3D 300.6 sec, the maximum has moved to x =3D 24.95 =
cm and has shrunk again [x =3D +9 cm].</FONT></P>
<P><FONT size=3D2>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.</FONT></P>
<P><FONT size=3D2>Reinhart Frosch,<BR>CH-5200 Brugg.<BR>reinifrosch@bluewin=
.ch . </FONT></P></div></body></html>
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End of AUDITORY Digest - 27 Dec 2011 to 28 Dec 2011 (#2011-304)
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