The Human Ear
2.2 MECHANICAL PROPERTIES OF THE CENTRAL PARTITION
2.2.1 Basilar Membrane Travelling Wave
The pressure fields observed at any segment of the basilar membrane consist not only of contributions due to motion of the stapes and, as shown here, to motion of the round window but, very importantly, to contributions due to the motion of all other parts of the basilar membrane as well. Here, it is proposed that the upper and lower galleries may each be modelled as identical transmission lines coupled along their entire length by the central partition, which acts as a mechanical shunt between the galleries (Bies, 2000).
Introducing the acoustic pressure, p, volume velocity, v (particle velocity multiplied by the gallery cross-sectional area), defined in Equations (2.1) and (2.2) and the acoustical inductance , LA , per unit length of either gallery, the equation of motion of the fluid in either gallery takes the following form:
Mv
Mz ' 2CA Mp
Mt (2.7)
M2p Mz2 ' 1
c2 M2p
Mt2 (2.8)
c2 ' 1
2CALA (2.9)
LA ' ρ
Sg (2.10)
u ' Mξ
Mt (2.11)
ξ ' ξ0ejωt (2.12)
& ju
CMω ' FS%FB (2.13)
The acoustical inductance is an inertial term and is defined below in Equation (2.10).
Noting that motion of the central partition, which causes a loss of volume of one gallery, causes an equal gain in volume in the other gallery, the following equation of conservation of fluid mass may be written for either gallery:
where CA is the acoustic compliance per unit length of the central partition and is defined below in Equation (2.16).
Equations (2.6) and (2.7) are the well-known transmission line equations due to Heaviside (Nahin, 1990), which may be combined to obtain the well-known wave equation.
The phase speed, c, of the travelling wave takes the form:
The acoustical inductance per unit length, LA , is given by:
where ρ is the fluid density and Sg is the gallery cross-sectional area.
The acoustical compliance, CA , per unit length of the central partition is readily obtained by rewriting Equation (2.3). It will be useful for this purpose to introduce the velocity, u, of a segment of the basilar membrane, defined as follows:
Sinusoidal time dependence of amplitude, ξ0 , will also be assumed. Thus:
Introducing the mechanical compliance, CM , Equation (2.3) may be rewritten in the following form:
The mechanical compliance, CM , is defined as follows:
CM ' (κ & mω2 % jKω)&1 (2.14)
CA ' w2∆z CM (2.15)
CA ' w2∆z
(κ&mω2%jCω) (2.16)
c ' Sg(κ&mω2%jCω)
2ρw2∆z (2.17)
ωN ' κ/m (2.18)
X ' ω/ωN (2.19)
ζ ' C
2mωN ' C
2 κm (2.20a,b)
The acoustical compliance per unit length, CA , is obtained by multiplying the mechanical compliance by the square of the area of the segment upon which the total force acts and dividing by the length of the segment in the direction of the gallery centre line. The expression for the acoustical compliance per unit length is related to the mechanical compliance as follows:
Substitution of Equation (2.14) in Equation (2.15) gives the acoustical compliance as follows:
Substitution of Equations (2.10) and (2.16) into Equation (2.9) gives the following equation for the phase speed, c, of the travelling wave on the basilar membrane:
To continue the discussion it will be advantageous to rewrite Equation (2.17) in terms of the following dimensionless variables, which characterise a mechanical oscillator. The un-damped resonance frequency or characteristic frequency of a mechanical oscillator, ωN , is related to the oscillator variables, stiffness, κ, and mass, m, as follows (Tse et al., 1979):
The frequency ratio, X, defined as the stimulus frequency, ω, divided by the characteristic frequency, ωN , will prove useful and here is the frequency of maximum response at a particular location along the basilar membrane. That is:
The critical damping ratio, ζ, defined as follows, will play a very important role in the following discussion (see Section 10.2.1, Equation (10.12)):
m ' αρSg∆z (2.21)
c ' αSgωN
w 2 1&X2%j2ζX (2.22)
c ' αSgωN
w 2 (2.23)
c ' αSgωN
w 2 jX (2.24)
c ' αSgωN
w 2 ζ(1%j) (2.25)
It will be convenient to describe the mass, m, of an oscillator as proportional to the mass of fluid in either gallery in the region of an excited segment. The proportionality constant, α, is expected to be of the order of one.
Substitution of Equations (2.18) to (2.21) in Equation (2.17) gives the following equation for the speed of sound, which will provide a convenient basis for understanding the properties of the travelling wave on the basilar membrane.
At locations on the basal side of a place of maximum response along the cochlear partition, where frequencies higher than the stimulus frequency are sensed, the partition will be driven below its frequency of maximum response. In this region the partition will be stiffness-controlled and wave propagation will take place. In this case, X < 1 and Equation (2.22) takes the following approximate form, which is real, confirming that wave propagation takes place.
At distances on the apical side of a place of maximum response, the partition will be driven above the corresponding frequency of maximum response, the shunt impedance of the basilar membrane will be mass controlled and wave propagation in this region is not possible. In this case X >> 1 and Equation (2.22) takes the following imaginary form, confirming that no real wave propagates. Any motion will be small and finally negligible, as it will be controlled by fluid inertia.
In the region of the cochlear partition where its stiffness and mass, including the fluid, are in maximum response with the stimulus frequency, the motion will be large, and only controlled by the system damping. In this case X = 1 and Equation (2.22) takes the following complex form.
As shown by Equation (2.25), at a place of maximum response on the basilar membrane, the mechanical impedance becomes complex, having real and imaginary
cg ' ωN dX
dk (2.26)
k ' w 2
αSg X(1&X2%j2ζX)&1/2 (2.27)
cg ' αSgωN 2w
(1&X2%j2ζX)3/2(1&jζX)
(1%ζ2X2) (2.28)
parts, which are equal. In this case, the upper and lower galleries are shorted together.
At the place of maximum response at low sound pressure levels when the damping ratio, ζ, is small, the basilar membrane wave travels very slowly.
Acoustic energy accumulates at the place of maximum response and is rapidly dissipated doing work transforming the acoustic stimulus into neural impulses for transmission to the brain. At the same time, the wave is rapidly attenuated and conditions for wave travel cease, so that the wave travels no further, as first observed by von Békésy (1960). The model is illustrated in Figure 2.2, where motion is shown as abruptly stopping at about the centre of the central partition.