Dear colleagues,
Since the List is fairly quiet at present: In March I submitted several postings on the stiffness of the basilar membrane. That quantity, which has the physical dimension of a spring constant per BM surface unit, is discussed, e.g., in Section 3.2 of the review "Mechanics of the Cochlea" by de Boer [in "The Cochlea", Dallos et al., Eds., Springer, 1996]. In his Eq. (3.2.1), defining the BM impedance, de Boer used the stiffness function S = S_0 * exp(-alfa*x), where (for homo) S_0 = 10^10 N/m^3 and alfa = 300 m-^1; x is the distance from base, measured along the BM. That function implies that the stiffness at the base (i.e., at x = 0), is greater than that at x = 32 mm by a factor of 1.5*10^4.
Today I found that the just mentioned large factor is consistent, approximately, with an idealized model in which the BM is assumed to be composed of elastic beams oriented in the y-direction, having rectangular cross sections dx*h(x), and only loosely connected to their neighbour-beams. Beam length = BM width = w(x); vertical beam diameter (in z-direction) = h(x). Stiffness S(x) = delta-p / delta-z, where delta-p = liquid-pressure difference "below" minus "above", and delta-z = effective vertical displacement of beam caused by the liquid-pressure difference delta-p. The beams are assumed to be fixed at y = -w(x)/2 and at y = +w(x)/2 so that at these y-coordinates they cannot move vertically (details of these fixations will be given below). If delta-p differs from zero, then the beams are bent so that there is, in the considered y-z-plane, an area A(x) between the beam centerline and the straight centerline observed if delta-p vanishes. The effective vertical displacement of the beam is defined to be delta-z = A(x)/w(x).
I found the following formula for the just defined BM stiffness function:
S(x) = n * Y *[h(x)]^3 / [w(x)]^4 ; (1)
in Eq. (1), Y is Young's modulus (elasticity modulus, N / m^2) of the beam material, and n is a number which depends on how the beams are fixed at their ends (i.e., at y = -w(x)/2 and at y = +w(x)/2). If they are constrained to stay horizontal at the ends, then n = 60; if their orientation at the ends is not constrained, then the mentioned quantities A(x) and delta-z are greater by a factor of six, so that n = 10. In homo, w(0)=0.1mm and w(32mm)=0.5mm. If one assumes the effective BM thickness ratio h(0)/h(32mm) = 2.9, then Eq. (1) yields the above-mentioned BM stiffness factor of 1.5*10^4. I have not found measurements of the just mentioned human BM thickness ratio, but in several mammals the corresponding ratio (base/apex) is indeed considerably greater than one, e.g. 2.7 in chinchilla and 5.5 in guinea-pig.
The human (apex/base) ratios given above [5 for w(x), (1/2.9) for h(x)] imply a factor of 5^5 * (2.9)^3 = 8*10^4 for the areas A(x). The corresponding experimental factor according to Fig. 11-73 of von Békésy's book "Experiments in Hearing" (1960) is 10^2 only. That strong disagreement has already been discussed on the List in March 2010.
Reinhart.
Reinhart Frosch,
Dr. phil. nat.,
r. PSI and ETH Zurich,
Sommerhaldenstr. 5B,
CH-5200 Brugg.
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch@xxxxxxxxxx .