Formulation of the Passive Two-Dimensional Problem

Membrane Fluid Fluid

Hard Wall

Helico-trema

(a)

(b) Hard Wall

Fluid Membrane

x=0

x=L y=0

y=h y

x

Stapes Round Window

Figure 3.1 The physical two-dimensional model of the cochlea. (a) The model showing both chambers. Since the uid is incompressible, uid move- ment in the two chambers is complementary. (b) An equivalent model with only one chamber. Since waves normally cut o before reaching the apex, the helicotrema is usually ignored in the single-chamber model.

We begin with a description of the two-dimensional problem. Numerical solutions are described as a standard of comparison for the analytical methods. We then review the simple LG solution in detail, to provide necessary background for understanding the present work.

Finally, we develop the higher-order formula for stapes displacement and the mode-coupling LG solution, and compare the results to the numerical simulations.

resting pressure have opposite signs for corresponding points in the two chambers. Because the solution is symmetrical in the two chambers, we may consider only one chamber, as shown in Figure 3.1(b); however, we must account for the missing uid mass. Since waves normally cut o before reaching the apex, the helicotrema is usually ignored in the single- chamber model. The length dimension of the model runs from x = 0 to x = L, and the height dimension runs fromy= 0 toy=h, as shown.

3.1.1 Hydrodynamics

The development of the hydrodynamics given in this section follows Lyon and Mead [67].

In general, the uid velocity vector

v

at any point (x;y) will havex and y components

v

x

and

v

y, respectively. It is convenient to dene a velocity potential, such that

v

x = @

@x ^and

v

y= @

@y^{; (3}:1)

or,

v

= ^r:

For an incompressible uid, there is no net ow into or out of any small region, so

r

v

= @

v

x

@x ⁺@

v

y

@y ^{= 0 or r2}= @²

@x^{2 +}@²

@y^{2 = 0}: (3:2) Thus, the velocity potentialobeys Laplace's equation.

The hard-wall boundary conditions at the right and bottom sides of the model imply that there is no uid ow in a direction normal to the boundary. The boundary conditions are thus

@@x ^{= 0 at} x=L;

and @

@y ^{= 0 at} y= 0: (3:3)

At x = 0, the motion of the uid is determined by the motion of the stapes, so the

boundary condition is

@@x ⁼f(t) at x= 0: (3:4)

By considering a small element of uid and the forces acting on it, we can show that the pressurepin the incompressible uid is related to the velocity of the uid

v

by the relations

@p@x ⁼@

v

x

@t ^and @p

@y ⁼@

v

y

@t ; ⁽³:5)

where is the density of the uid. Substituting Equation 3.1 into Equation 3.5, we get the relationship between the pressure and the velocity potential at any point in the uid:

p=@@t ; ⁽³:6)

wherep now represents the deviation from the pressure at rest.

3.1.2 Basilar-Membrane Boundary Condition

To complete the description of the problem, we must specify the boundary condition corre- sponding to the basilar membrane. The displacement of the membrane in the positivey direction is related to the vertical uid velocity aty =h:

@@t ⁼

v

y = @

@y: ⁽³:7)

Application of Newton's second law to an element of the membrane leads to the basilar- membrane boundary condition [67]:

2@@t ⁼S(x)+(x)@

@t ⁺M(x)@²

@t^{2 at} y=h; (3:8)

whereS(x),(x), andM(x) are the stiness, damping, and mass of the membrane, respec- tively, all of which may vary as a function of position along the membrane. The stiness term S(x)has its form because the membrane acts like sti uncoupled beams running across the width of the membrane, as described in Chapter 2; hence, in the two-dimensional model, the beams exert a restoring force that is only proportional to their displacement [120]. The factor of 2 on the left side of the equation accounts for the complementary motion of the

r 2

=0

@@y ⁼⁰

@@x ⁼⁰

@@x ^=f(t)

@t @y @y@t @y@t

x

L 0

0 h

Figure 3.2 The mathematical two-dimensional model of passive cochlear mechanics. The contribution of the active outer hair cells is not included.

uid mass on the other side of the partition.

Some authors include a \tension" term, corresponding to longitudinal coupling between the beamlike laments of the basilar membrane. However, any signicant tension term destroys the high-frequency cut-o observed in real cochleas [67, 3, 60], and therefore most authors neglect it.

Nearly all authors include membrane mass [137, 2, 78, 120, 112, 59, 22]; Lighthill argues that membrane mass is necessary to account for the existence of a critical-layer- absorption phenomenon, which is suggested by the sharp high-frequency cut-o observed in real cochleas. However, Lyon and Mead argue that the membrane mass can be neglected if the wave energy is dissipated before the point of resonance [67]. In the present work, we shall include the membrane mass as a free parameter.

Finally, there is the question of the active behavior of the outer hair cells, which aects the partition boundary condition. A treatment of the active case is deferred until Chapter 5.

Dierentiating both sides of Equation 3.8 with respect tot, and eliminatingvia Equa- tion 3.7, yields the basilar-membrane boundary condition:

2@²

@t^{2 =}S(x)@

@y ⁺(x) @²

@y@t ⁺M(x) @³

@y@t^{2 at} y=h: (3:9) Figure 3.2 summarizes the two-dimensional boundary-value problem, corresponding to the passive cochlear mechanics. It may be surprising that wave behavior is expected in this problem, since the uid is incompressible; the wave behavior is made possible by the coupled movement of the uid and membrane, rather than by compression of the uid itself.

Waves are, of course, expected in hyperbolic systems, but Whitham points out that elliptic systems, such as those governed by Laplace's equation, are also capable of propagating waves because of the interesting eects of the boundary conditions [126, p. 432].