Two-Dimensional Motion of a Viscoelastic Membrane in an Incompressible Fluid: Applications to the Cochlear Mechanics

Abstract

In this work we present the two-dimensional motion of a viscoelastic membrane immersed in incompressible inviscid and viscous fluids. The motion of the fluid is modelled by two-dimensional Navier-Stokes equations, and a constitutive equation is considered for the membrane which captures along with the fluid equations the essential features of the vibrations of the fluid. By using the Fourier transform, the linearized equations are reduced to a functional equation for membrane displacement which is solved analytically and the inverse transform is evaluated asymptotically. Results obtained show some of the basic known characteristics of cochlear mechanics.

Appendix

A continuous linear functional ${{\varvec{f}}}\in \mathscr {D}'$ on the space $\mathscr {D}$ of $C^{\infty }$ test functions having compact support is called a distribution, or generalized function. The image of the test function $\phi$ under ${{\varvec{f}}}$ is denoted by $\langle {{\varvec{f}}},\phi \rangle$. Every locally integrable function f there corresponds to a distribution ${{\varvec{f}}}$ defined by

$$ \langle {{\varvec{f}}},\phi \rangle =\int _{-\infty }^{\infty }f(x)\phi (x)dx\,, \quad for \quad \phi (x)\in \mathscr {D}\,. $$

The distribution ${{\varvec{f}}}$ is said to be generated by the function f. For any distribution ${{\varvec{f}}}$, the functional ${{\varvec{f}}}':\phi \longmapsto -\langle {{\varvec{f}}},\phi '\rangle$ is a distribution. The derivative of a distribution ${{\varvec{f}}}$ is also a distribution ${{\varvec{f}}}'$ defined by $\langle {{\varvec{f}}}',\phi \rangle =-\langle {{\varvec{f}}},\phi '\rangle$ for all $\phi \in \mathscr {D}$. A $C^{\infty }$ function $\phi (x)$ is of class $\mathscr {S}(\mathbf {R})$ if

$$ x^{n}\phi ^{r}(x)\rightarrow 0 \quad \text {as} \quad x\rightarrow \pm \infty \quad \text {for}\,\,\text {all} \quad n,r\ge 0\,. $$

$\mathscr {S}$ is called the Schwartz class and its elements are called test functions too. A distribution of slow growth (tempered distribution) is a continuous linear functional on the space $\mathscr {S}$. The space of all distributions of slow growth is denoted by $\mathscr {S}'$. $\mathscr {D}$ is a subspace of $\mathscr {S}$, but the distributions of slow growth comprise a proper subspace of $\mathscr {D}$, $\mathscr {D}\subset \mathscr {S}\subset \mathscr {S}'\subset \mathscr {D}'$. If f(x) satisfies the condition $\lim \limits _{|x|\rightarrow \infty }|x|^{-N}f(x)=0$ for some integer N, then f(x) is said to be of slow growth. If f is locally integrable and of slow growth, then f defines a tempered distribution given by

$$ \langle {{\varvec{f}}},\phi \rangle =\int _{-\infty }^{\infty }f(x)\phi (x)dx\,, \quad for \quad \phi (x)\in \mathscr {S}\,. $$

Distributions that can be generated through the above expression are called regular distributions. Distributions that are not regular are called singular distributions. If ${{\varvec{f}}}$ is a tempered distribution, then the functional ${\hat{{{{\varvec{f}}}}}}:\phi \longmapsto \langle {{\varvec{f}}},\hat{\phi }\rangle$ is a tempered distribution.

Definition 1

(Generalized Fourier Transform) If ${{\varvec{f}}}$ is a tempered distribution, its Fourier transform is the tempered distribution ${\hat{{{\varvec{f}}}}}$ defined by $\langle {\hat{{{\varvec{f}}}}},\phi \rangle =\langle {{\varvec{f}}},\hat{\phi }\rangle$ for all $\phi \in \mathscr {S}$.

The inverse Fourier transform of a tempered distribution is also a tempered distribution defined by $\langle \mathscr {F}^{-1}\left\{ {{\varvec{f}}}\right\} ,\phi \rangle =\langle {{\varvec{f}}},\mathscr {F}^{-1}\left\{ \phi \right\} \rangle$ for all $\phi \in \mathscr {S}$. We now calculate the generalized Fourier transforms of some functions.

Example 1

Consider the Dirac delta function $\delta (x)$ (unsuitably called fuction). To calculate $\mathscr {F}\left\{ \delta ^{(k)}(x-a)\right\}$ we proceed as follows.

$$\begin{aligned} \left\langle \mathscr {F}\left\{ \delta ^{(k)}(x-a)\right\} ,\phi (\xi )\right\rangle= & {} \left\langle \delta ^{(k)}(x-a),\mathscr {F}\left\{ \phi \right\} \right\rangle =(-1)^{k}\left\langle \delta (x-a),\left( \mathscr {F}\left\{ \phi \right\} \right) ^{(k)}\right\rangle \nonumber \\= & {} (-1)^{k}\left\langle \delta (x-a),\mathscr {F}\left\{ (-2\pi i\xi )^{k}\phi (\xi )\right\} \right\rangle \nonumber \\= & {} (-1)^{k}\int _{-\infty }^{\infty }(-2\pi i\xi )^{k}\phi (\xi )e^{-2\pi ia\xi }d\xi \nonumber \\= & {} \left\langle (2\pi i\xi )^{k}e^{-2\pi ia\xi },\phi (\xi )\right\rangle \nonumber \end{aligned}$$

Hence $\mathscr {F}\left\{ \delta ^{(k)}(x-a)\right\} =(2\pi i\xi )^{k}e^{-2\pi ia\xi }$.

Example 2

Let ${{\varvec{f}}}$ be the distribution generated by the function $f(x)=\ln |x|$. Note that $\lim \limits _{\varepsilon \rightarrow 0}\frac{\left( 1-|x|^{-\varepsilon }\right) }{\varepsilon }=\ln |x|$. Then

$$\begin{aligned} \left\langle \mathscr {F}\left\{ \ln \left| x\right| \right\} ,\phi (\xi )\right\rangle= & {} \int _{-\infty }^{\infty }\ln \left| x\right| \mathscr {F}\left\{ \phi (\xi )\right\} dx=\int _{-\infty }^{\infty }\ln \left| x\right| \left\{ \int _{-\infty }^{\infty }\phi (\xi )e^{-2\pi ix\xi }d\xi \right\} dx \\= & {} \int _{-\infty }^{\infty }\phi (\xi )\left\{ \int _{-\infty }^{\infty }\lim _{\varepsilon \rightarrow 0}\frac{\left( 1-\left| x\right| ^{-\varepsilon }\right) }{\varepsilon }e^{-2\pi ix\xi }dx\right\} d\xi \\= & {} \int _{-\infty }^{\infty }\phi (\xi )\lim _{\varepsilon \rightarrow 0}\left\{ \frac{1}{\varepsilon }\left[ \int _{-\infty }^{\infty }e^{-2\pi ix\xi }dx-\int _{-\infty }^{\infty }\left| x\right| ^{-\varepsilon }e^{-2\pi ix\xi }dx\right] \right\} d\xi \\= & {} \int _{-\infty }^{\infty }\phi (\xi )\lim _{\varepsilon \rightarrow 0}\left\{ \frac{1}{\varepsilon }\left[ \delta (\xi )-\varGamma (1-\varepsilon )\sin (\frac{\varepsilon \pi }{2})2^{\varepsilon }\pi ^{(-1+\varepsilon )}\left| \xi \right| ^{(-1+\varepsilon )}\right] \right\} d\xi \,, \\= & {} \left\langle -\frac{1}{2}\left| \xi \right| ^{-1}-\left( \gamma +\ln 2\pi \right) \delta (\xi ),\phi (\xi )\right\rangle \,, \end{aligned}$$

by considering that $\varGamma (1-\varepsilon )\sin (\frac{\varepsilon \pi }{2})2^{\varepsilon }\pi ^{(-1+\varepsilon )}=\frac{\varepsilon }{2}+\left( \frac{\gamma }{2}+\frac{\ln 2\pi }{2}\right) \varepsilon ^{2}+O(\varepsilon ^{3})$, $\lim \limits _{\varepsilon \rightarrow 0}\varepsilon \left| \xi \right| ^{\varepsilon -1}=2\delta (\xi )$, and $\lim \limits _{\varepsilon \rightarrow 0}\left\{ \left| \xi \right| ^{(-1+\varepsilon )}-2\varepsilon ^{-1}\delta (\xi )\right\} =\left| \xi \right| ^{-1}$.

Example 3

We consider the unit step function defined by

$$ U(\xi ) =\frac{1+ \text{ sgn }(\xi )}{2}= \left\{ \begin{array}{cl} 1 &{} \quad \text{ si } \quad \xi \ge 0 \\ 0 &{} \quad \text{ si } \quad \xi < 0 \end{array}\right. \,. $$

We have

$$\begin{aligned} \left\langle \mathscr {F}^{-1}\left\{ U(\xi )\right\} , \phi (x)\right\rangle= & {} \left\langle U(\xi ),\mathscr {F}^{-1}\left\{ \phi (x)\right\} \right\rangle =\int _{-\infty }^{\infty }U(\xi )\left\{ \int _{-\infty }^{\infty }\phi (x)e^{2\pi ix\xi }dx\right\} d\xi \\= & {} \int _{-\infty }^{\infty }\phi (x)\left\{ \int _{-\infty }^{\infty }U(\xi )e^{2\pi ix\xi }d\xi \right\} dx \\= & {} \int _{-\infty }^{\infty }\phi (x)\left\{ \frac{1}{2}\int _{-\infty }^{\infty }\left( 1+\text{ sgn }(\xi )\right) e^{2\pi ix\xi }d\xi \right\} dx \\= & {} \int _{-\infty }^{\infty }\phi (x)\frac{1}{2}\left( \delta (x)+\frac{i}{\pi x}\right) dx=\left\langle \frac{1}{2}\left( \delta (x)+\frac{i}{\pi x}\right) ,\phi (x)\right\rangle \,. \end{aligned}$$

Hence, $\mathscr {F}^{-1}\left\{ U(\xi )\right\} =\frac{1}{2}\left( \delta (x)+\frac{i}{\pi x}\right)$.