Approximate LG Solution for a Nonuniform Cochlea

3.2 Review of Established Solution Techniques

3.2.3 Approximate LG Solution for a Nonuniform Cochlea

dissipative systems [112]. Our approach will be (1) to nd an expression for the energy ow as a function of amplitudeA, (2) to nd an expression for the group velocity as a function of k, and (3) to combine these expressions to determineAas a function ofk. We shall develop the condition for the undamped case, which is quite simple; the validity of the result in the damped case has been shown by de Boer and Viergever [22]. An alternate approach, which appeals to Hamilton's principle and Lagrangian mechanics, was used by Steele and Taber [112], based on the general treatment by Whitham [126]. We now proceed with the simpler rst-principles energy approach.

The energy of the wave consists of three components: the kinetic energy of the membrane massKm, the potential energy of the membrane stinessVm, and the kinetic energy of the uidKf. The time-averaged kinetic energy of the membrane mass per unit area is

K_{m = 12}

Z 2

0 1

2Mv_2yd_{= 14}M!²A²: (3:21) The time-averaged potential energy of the membrane stiness per unit area is

Vm= 12

Z 2

0 1

2S²d _{= 14}SA²: (3:22)

To computeKf, we need the velocity potential in terms of the amplitude factor A. Com- bining Equations 3.7 and 3.19 yields

= i!A

ksinh(kh) cosh(ky)expi(!t kx): The time-averaged kinetic energy of the uid per unit area is

Kf = ^Z₀^h 1 2

Z 2

0 1

2(2)(v_2x+v_2y)ddy (3.23)

= !²A²

2ktanh(kh) (3.24)

= 12!²A²Q(k); (3.25)

where the 2 term accounts for the uid in both chambers, and the important function

Q(k) is dened as

Q(k) = 1 ktanh(kh): The energy balance then assumes the form

Vm =K_f +Km;

which is identical to the dispersion relation when Equations 3.21, 3.22, and 3.25 are substi- tuted [22]. Finally, the total energy density E is given by

E = Vm+K_f +Km (3.26)

= 2Vm (3.27)

= 12SA²: (3.28)

We now have an expression for the energy density E as a function of membrane dis- placement amplitude A. The second step in the derivation is to nd an expression for the group velocityU as a function ofk. The lossless dispersion relation can be written in terms of the function Q(k):

Q(k) = 1

ktanh(kh) = S M!² 2!² : Dierentiating with respect tok yields

@Q@k ⁼ @Q

@!@!

@k ^(3.29)

= U @Q@! ; ^(3.30)

which leads to

U = @Q

@k !³ S

!

: (3:31)

Energy ows at the group velocity [126]. For a constant rate of energy ow, we must have

EU = const: (3:32)

A(k) = C @Q

@k

= iCktanhkh

ptanhkh+khsech²kh;

where C is a constant of dimension (length)². This important equation is called the transport equation, since it relates to the transport of energy.

Substituting this result into Equation 3.20 yields the full equation for the membrane displacement:

(x;t) = iCktanhkh

ptanhkh+khsech²kh^expi(!t ^Z₀^xk(u)du); (3:33) where k is the local root of the dispersion relation. Combining this result with Equation 3.7 yields the expression for the velocity potential:

(x;y;t) = C!cosh(ky)

cosh(kh)^ptanhkh+khsech²kh^expi(!t ^Z₀^xk(u)du); (3:34) We can derive similar expressions for the membrane velocity vy(x;t), and uid pressure p(x;y;t), using the dening relations 3.1 and 3.6.

Rhode's data are expressed in the form of a ratio of basilar-membrane displacement to malleus displacement, which we assume is proportional to stapes displacement [91]. We must now compute the stapes displacement.

Recall that the horizontal uid velocity

v

x(x;y) at any point (x;y) in the uid is

v

x(x;y) = @

@x:

The horizontal uid displacement

d

x(x;y) is the time integral of the horizontal uid velocity.

For a sinusoidal disturbance with angular frequency!, we have

d

x(x;y) = i

!@

@x: ⁽³:35)

Dierentiating Equation 3.34 with respect to x yields

@@x ^{= (} ik+O[k⁰(x)])(x;y): (3:36) As a rst approximation, we shall ignore the terms involvingk⁰(x). Combining Equations 3.36 and 3.35 yields

d

x(x;y) = k

!⁽x;y): (3:37)

Following Steele and Taber [112], the stapes displacement dst is the value of the hori- zontal uid displacement at x= 0, averaged over the height of the ducth:

dst = 1h

Z h

0

d

x(0;y)dy (3.38)

= k0

!h

Z h

0 (0;y)dy;

wherek0 is the value of the wavenumber kevaluated atx= 0. Substituting Equation 3.34 and performing the integration yields

dst= Ctanh(k0h)exp(i!t)

h^qtanh(k0h) +k0hsech²(k0h): (3:39) Combining Equations 3.33 and 3.39 yields the ratioDof membrane to stapes displace- ment:

D(x;!) =

dst =ikh^tanh(kh) tanh(k0h)

stanh(k0h) +k0hsech²(k0h)

tanh(kh) +khsech²(kh) exp i^Z₀^xk(u)du: (3:40) Note that the above expression is only a rst approximation, since the terms involvingk⁰(x) indst have been neglected.

The general LG solution for the velocity potential given in Equation 3.34 degenerates to the following simple form at y=h under the long-wave approximation:

(x;t) = constk ¹⁼² expi(!t ^Z₀^xk(u)du); (3:41) where kis given by the simple long-wave expression of Equation 3.17. This form was rst applied to cochlear mechanics problems by Zweig, Lipes, and Pierce [134]. The validity

k1²dk dx¹:

Zweig and colleages also showed that it is possible to evaluate the integral in closed form, under the scaling assumption:

S(x) = S0exp( 2x=d); (x) = 0exp( x=d); M(x) = const:

Alternate derivations of the long-wave LG result are given by Lighthill [60, p. 189{90] and by Viergever [121, p. 103{106].

In the short-wave region, the general LG solution for the velocity potential given in Equation 3.34 degenerates to the following simple form aty =h:

(x;t) = constexpi(!t ^Z₀^xk(u)du);

where k is given by the simple short-wave expression of Equation 3.18. This form was rst applied to cochlear mechanics problems by Siebert [103]. Under the same scaling assumptions used by Zweig, Lipes and Pierce, for the long-wave case, we can evaluate the short-wave integral in closed form.

A Mathematica implementation of the two-dimensional LG algorithm is given in Ap- pendix A.

Comparison of LG and Finite-Dierence Results

Steele and Taber compared their LG results to the nite-dierence results of Neely for a number of dierent frequencies, using identical parameters for the two models [112].

Their results have been recomputed, and are presented here for comparison with the mode- coupling LG solution presented in section 3.2. The physical parameters used for their comparison were

S(x) = 1:010⁷ e ^x=d g s ² mm ²; (x) = 2 g s ¹ mm ²;

M = 1:510 ³ g mm ²; d = 5 mm;

h = 1 mm; L = 35 mm;

= 1:010 ³ g mm ³:

Note that neither Neely nor Steele and Taber regarded these parameters as physically realistic. In particular, the membrane mass M is unrealistically large.

The amplitude and phase of the basilar-membrane displacement ratios are shown in Figure 3.5 for the two methods. Clearly, the LG solution captures the general behavior of a gentle increase in amplitude toward a peak, followed by a sharp cut-o. However, the quantitative agreement is poor for the lowest frequencies, and in the cut-o region, the amplitude of the LG solution decreases much too fast, and the phase behavior is incorrect.

Both of these problems are addressed in Section 3.3.