# Contribution of Even/Odd Sound Wave Modes in Human Cochlear Model on Excitation of Traveling Waves and Determination of Cochlear Input Impedance

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Compressible Perilymph–Filled Cochlea Model

#### 2.1. Modeling

_{0}= 1.5 MPa s/m

^{3}) to an air–filled middle ear cavity (Z

_{0}= 440 Pa s/m

^{3}), where Z

_{0}is the characteristic acoustic impedance of the medium. The SV, ST, and helicotrema were filled with compressible perilymph with a viscosity of 0.7027 mPa s, a velocity of 1520 m/s, and a mass density of 994.6 kg/m

^{3}. A Young’s modulus of 1 MPa, Poisson’s ratio of 0.49, and a mass density of 1200 kg/m

^{3}were used for the BM [16]. Other structural parameters are given in the caption of Figure 1.

**u**stands for the fluid velocity,

**I**is the identity tensor, p is the fluid pressure, ρ is the fluid density, and μ is the viscosity.

#### 2.2. Frequency Domain Analysis

#### 2.3. Time Domain Analysis

- The SPL of the ST became zero all the time at the base of the cochlea (at 0 mm position) due to the free–end reflection condition of the RW.
- As shown by the SPLs of the SV and ST, a small vibration can be seen in each graph at the position from 0 mm to 12 mm.
- The SPLs of the SV and ST, corresponding to the graphs of the P
_{SV}and P_{ST}, completely overlapped from 12 mm to 35 mm. - The displacement of the BM became the maximum at around 12 mm, at the position where the tonotopy predicts that humans hear 5000 Hz sounds.

_{AV}= (P

_{SV}+ P

_{ST})/2, and the pressure difference, P

_{Diff_SV}= P

_{SV}– P

_{AV}and P

_{Diff_ST}= P

_{ST}– P

_{AV}. The results are shown in the lower graphs in Figure 4. Interestingly, the graphs (P

_{Diff_SV}and P

_{Diff_ST}) show perfect symmetric against the zero axis and P

_{AV}draws a smoothly changing waveform. In other words, this means that the sound wave traveling in the cochlea is composed of an even symmetric sound wave (even mode) and an odd symmetric sound wave (odd mode).

## 3. Even/Odd Mode Analysis

_{g}, and the inner impedance of the source, Z

_{g}, were connected in series to design an excitation circuit, and the input impedance of the cochlea, Z

_{c}, wa expressed by a parallel connection of an even mode impedance, Z

_{even}, and an odd mode impedance, Z

_{odd}. Since the even and odd modes behave independently in the SV and ST, the coupling of both modes was ignored. Then, Z

_{even}and Z

_{odd}were written as (see Appendix A):

^{2}= −1), Z

_{0}is the acoustic characteristic impedance of the perilymph, ρ is the mass density, κ is the bulk modulus, f is the sound wave frequency, β is the phase constant, and L

_{c}is the cochlear length, respectively.

_{c}was expressed by the parallel connection of the Z

_{even}and Z

_{odd}as follows:

_{0}was calculated as:

_{c}) to the Z

_{even}and Z

_{odd}were estimated by:

_{even}and Z

_{odd}were given by:

_{even}and PD

_{odd}, and the total power dissipation by the cochlea, PD

_{c}, were expressed as:

_{g}= 1.0 V (corresponding to a 1 Pa sound wave excitation), Z

_{0}= 1.5 MPa s/m

^{3}, ρ = 994.6 kg/m

^{3}, κ = 2.30e9 Pa, f = 5000 Hz, and L

_{c}= 35 mm were used.

_{even}and Z

_{odd}, are shown in Figure 9a,b. Blue solid curves and red dashed curves denote the real part and imaginary part of the impedance, respectively. As shown by Z

_{even}, the real part of Z

_{even}became zero for all frequencies because the even mode sound wave was perfectly reflected at the apex and went back to the voltage source again without power dissipation. On the other hand, at lower frequencies, below 10,600 Hz, the imaginary part of the Z

_{even}became negative, which meant that the cochlea became capacitive. Contrarily, at higher frequencies beyond 10,600 Hz, it became positive, and the cochlea became inductive. This result tells us an important fact in that the Z

_{even}reached zero at around 10,600 Hz. In this case, since Z

_{even}was connected to Z

_{odd}in parallel, as shown in Figure 8b, the input impedance of the cochlea, Z

_{c}, also reached zero, and sound stimuli could not enter the cochlea. As a result, hearing loss will occur at this frequency. In fact, the displacement of the BM shown in Figure 2f was also confined around there. Furthermore, note that hearing deterioration is confirmed at the measured equal–loudness contours around 10,000 Hz [22]. We need to compare both results carefully because the equal loudness contours include psychological effects on the test subjects. However, our simulation result implies that the hearing deterioration at this frequency might occur due to the structure of the cochlea.

_{even}and PD

_{odd}, are presented in Figure 9c,d. The graph of the PD

_{even}indicates that only the reactive power was stored in the cochlea and there was no power dissipation at any frequency. In contrast, the graph of the PD

_{odd}shows that the cochlea dissipated active power to displace the BM and generate the traveling waves. However, the power dissipation was reduced around 10,600 Hz, because the even mode impedance, Z

_{even}= 0, affected the odd mode properties.

_{c}, is expressed by Equation (6), and the frequency dependence of Z

_{c}is summarized in Figure 9e. As pointed out, the zero impedance of the even mode Z

_{even}= 0 at 10,600 Hz affected the whole hearing performance of the cochlea; namely, the input impedance of the cochlea also became zero, and a significant impedance mismatch between the source and the cochlea occurred. As a result, although the cochlea as excited by the 1 Pa (v

_{g}= 1.0 V) sound source, the actual pressure applied to the cochlea became smaller. The red point shows an applied pressure at 5000 Hz, calculated by the FEM–based structural simulation of the even and odd mode sound waves in Figure 4. Although there are some errors between the results obtained by the structural analysis and the equivalent circuit analysis, we consider that they are within an acceptable range.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**a**) Ladder network of a transmission line, composed of series inductors L and shunt capacitors C. (

**b**) Z

_{l}–terminated transmission line. (

**c**) Open–terminated transmission line.

_{in}of the Z

_{l}–terminated transmission line with length d, shown in Figure A1b, was written as [14]:

_{in}could be derived as [14]:

_{even}, could be determined as follows by analogy from the input impedance of the open–terminated transmission line in Equation (A22).

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**Figure 1.**A straight–shaped cochlear model. The SV, ST and helicotrema are filled with a compressible perilymph. The fan–shaped BM is embedded in the SM. L

_{c}= 35 mm, W

_{c}= 1.2 mm, H

_{bv}= H

_{bt}= 1.235 mm, H

_{av}= H

_{at}= 1.245 mm, W

_{bm}= 100 µm, H

_{bm}= 30 µm, W

_{am}= 500 µm, and H

_{am}= 10 µm [17,18,19,20].

**Figure 2.**(

**a**–

**e**) Frequency domain analyses of the sound pressure level (SPL) of the SV (blue) and ST (green), and the displacement of the BM (red) when the OW plane is excited by a 1 Pa sinusoidal sound wave. The horizontal axis shows the internal position of the cochlea. (

**f**) Frequency characteristic of the maximum displacement of the BM, extracted from the maximum displacements of the BM such as those in (

**a**–

**e**).

**Figure 3.**Time domain analysis of the sound pressure level (SPL) of the SV and ST and the displacement of the BM (D–BM) when the OW is excited by a 1 Pa, 5000 Hz pure tone. (

**a**) t = 0.03 ms, (

**b**) t = 0.06 ms, (

**c**) t = 0.17 ms, and (

**d**) t = 0.46 ms.

**Figure 4.**The upper graphs show the sound pressure levels (SPLs) in the SV (P

_{SV}in green) and ST (P

_{ST}in blue) and the displacement of the BM (BM in red) when a 1 Pa, 5000 Hz pure tone is applied to the OW. The lower graphs present the SPLs of the even symmetric sound mode (P

_{AV}in black) in the SV and ST, and odd symmetric sound mode in the SV (P

_{Diff_SV}in green) and ST (P

_{Diff_ST}in blue). The horizontal axis shows the internal position of the cochlea, and all graphs reach the steady state (t = 38.94 ms).

**Figure 5.**A coupler composed of two transmission lines. (

**a**) Geometry and Port 1 excitation. (

**b**) Port 1 and Port 2 excitation with the in–phase signals (even mode). (

**c**) Port 1 and Port 2 excitation with the anti–phase signals (odd mode).

**Figure 6.**A cochlea–based symmetric model for even and odd mode analyses. Two input planes “input 1” and “input 2” are excited by in–phase (even mode) and anti–phase (odd mode) sound waves. The helicotrema is removed from the original model. Other structural parameters are completely the same as those of the cochlea model in Figure 1.

**Figure 7.**Even and odd mode simulation results for the cochlea–based symmetric model presented in Figure 6. The horizontal axis shows the internal position of the cochlea. (

**a**) Odd mode and (

**b**) even mode.

**Figure 8.**An equivalent circuit of the cochlea. (

**a**) Contributions of the even and odd sound wave modes to the cochlear acoustics. (

**b**) an equivalent circuit of the cochlea designed based on transmission line theory.

**Figure 9.**Equivalent circuit simulation results when a 1 Pa input (v

_{g}= 1.0 V) is applied to the voltage source in Figure 8b. (

**a**) Input impedance of the even mode. (

**b**) Input impedance of the odd mode. (

**c**) Power dissipation of the even mode. (

**d**) Power dissipation of the even mode. (

**e**) Input impedance of the cochlea. (

**f**) The actual pressure applied to the cochlea under the impedance mismatch between the source and cochlea. The red point shows an applied pressure at 5000 Hz, which is calculated by the FEM–based structural simulation of the even and odd mode sound waves in Figure 4.

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**MDPI and ACS Style**

Hong, W.; Horii, Y.
Contribution of Even/Odd Sound Wave Modes in Human Cochlear Model on Excitation of Traveling Waves and Determination of Cochlear Input Impedance. *Acoustics* **2022**, *4*, 168-182.
https://doi.org/10.3390/acoustics4010011

**AMA Style**

Hong W, Horii Y.
Contribution of Even/Odd Sound Wave Modes in Human Cochlear Model on Excitation of Traveling Waves and Determination of Cochlear Input Impedance. *Acoustics*. 2022; 4(1):168-182.
https://doi.org/10.3390/acoustics4010011

**Chicago/Turabian Style**

Hong, Wenjia, and Yasushi Horii.
2022. "Contribution of Even/Odd Sound Wave Modes in Human Cochlear Model on Excitation of Traveling Waves and Determination of Cochlear Input Impedance" *Acoustics* 4, no. 1: 168-182.
https://doi.org/10.3390/acoustics4010011