Impact of the ear canal motion on the impedance boundary conditions in models of the occlusion effect

The occlusion effect (OE) denotes the increased low-frequency perception of bone-conducted (BC) sounds, resulting from the occlusion or blockage of the ear canal (EC) with devices such as earplugs or hearing aids.¹ This phenomenon manifests in the amplification of physiological noise and alterations of one's own voice perception. The OE contributes to the discomfort experienced by workers who wear earplugs,² and it can diminish the overall acceptance of hearing aids among their users.³ 

The OE is objectively measurable as the sound pressure level difference in the EC due to the occlusion. The OE exhibits a low-pass characteristic^4–6 and mainly appears at frequencies below 1 kHz.⁷ In contrast to air-conducted sound, which enters the EC at its entrance, the OE is a result of the structural transmission of BC sound. Hence, the sound pressure in the EC is in the open and occluded case induced by the moving EC boundaries. Influencing factors on this phenomenon are as follows:

• the frequency-dependent distribution of the EC wall vibrations in magnitude and phase,^8,9 which varies between the elastic soft tissue/cartilage part and bony part of the EC;⁵ 

• the different areas and orientations of the EC wall, EC entrance, and tympanic membrane (TM)^10,11 in relation to the direction of the motion;

• the source of the BC stimulation, e.g., BC transducers,^5,12–14 one's own voice,^15,16 or mastication;¹⁶ 

• the variation of the vibrating area of the wall with the occluding device's insertion;^3,5

• the mechanical load of the occluding device to the vibration of the EC wall;^3,15,17

• the acoustical load to the vibration of the EC wall due to the occlusion;¹⁸ and

• the boundary and loading conditions within numerical simulations, e.g., at the temporal bone and soft tissue of an outer ear model⁸ or at the base of a truncated head model.¹⁹ 

Researchers have employed various modeling approaches to simulate the OE, ranging from one-dimensional circuit models (e.g., Refs. 5, 6, 8, 15, 20, and 21) to more complex finite element (FE) models (e.g., Refs. 8, 13, 18, 19, and 22). It is a common approach within OE models to represent the sound radiation at the open EC entrance by an acoustic impedance.^{5,6,8,13,15,18,20,22} Similarly, also the occluding device can be considered as an impedance at the EC entrance, which differs from the open-ear case. In several OE models, such an impedance is incorporated explicitly in at least one of the investigated conditions.^{5,6,8,13,15,18,20} At the TM, the impedance represents the load by the membrane itself, the middle ear (ME) cavity, the ME ossicles, and inner ear (see, e.g., Shaw and Stinson²³ or Hudde and Engel^24–26) Using such an impedance allows bypassing the necessity of modeling the ME structures in detail, therefore, it is the most common approach for OE models.^{5,6,8,13,15,18–22}

Conceptually, the impedance approach assumes a piston-like motion at the entrance and TM.²⁷ Such a simplification is justified because the frequency range of interest extends up to a maximum of 2 kHz, hence, the wavelength in air is significantly larger than the radial dimensions of the EC and TM [e.g., with a speed of sound in air of 343 m/s, the wavelength at 2 kHz exceeds a typical TM diameter of about 9 mm (Ref. 28) by 19 times]. Furthermore, experimental observations suggest a “simple vibration pattern”²⁹ of the TM below 2 kHz. This allows treating its motion by means of a volume velocity instead of considering the spatially distributed point velocity.^29,30 The corresponding sound pressure level close to the TM varies spatially by about 1 dB within the considered frequency range.³¹ The TM acoustic impedance can then be defined as the average sound pressure on the TM divided by the volume velocity of the TM during air-conducted stimulation.²⁸ It is worth noting that for BC stimulation, the TM may also play an active role by radiating sound originating from the ME into the EC. Experiments suggest that this reverse excitation does not significantly contribute to the EC sound pressure except at the EC resonance at approximately 3 kHz.⁷ 

Yet, predicting the OE and assessing the effectiveness of measures to reduce it, such as vents,³² open-fitting hearing aids,³³ or sound-absorbing earplugs,^34,35 remains challenging.^8,19 Comparisons between experimental data and simulations^18,19,22 reveal an overestimation of the OE, particularly at frequencies below 250 Hz. Interestingly, experimental results also exhibit a higher variance of the OE and stronger excitation dependence in this frequency range.^12,16 Using a FE model of a human head, Xu et al.¹⁹ highlighted that the radiation of the BC-stimulated head vibrations into the open EC could account for some of these observations. However, existing discrepancies between experimental and simulation data remain unexplained.

This study investigates whether considering a more comprehensive view on the three-dimensional structural motion of the EC could, at least partially, elucidate the discrepancies at low frequencies between the calculated and measured OEs. Below 300 Hz, the skull exhibits a rigid body motion,³⁶ which causes the part of the EC surrounded by bone to follow. Additionally, the wavelength in the surrounding soft tissue and cartilage is larger or similar to the dimensions of the EC. For instance, when employing the material parameters referred to as “reference” in Xu et al.,¹⁹ the wavelength of shear waves in soft tissue at 100 Hz exceeds the EC dimensions by a factor of 3. In the case of cartilage, the wavelength surpasses the EC dimensions even by a factor of 20–40 at this frequency. Consequently, EC segments partially exhibit a rigid body motion with a displacement which is characterized by being in phase, of similar magnitude, and spatially in the same direction along all points on the EC wall within a segment. This phenomenon will be referred to as nondeforming motions of parts of the EC throughout this paper. Particularly, such a nondeforming motion of the EC parts at the EC entrance and TM causes the entrance plane and TM to change their positions in space, respectively, as a consequence of their connections with the surrounding structures. The motions of these boundaries complement the vibration of the EC wall, which has been widely accepted to primarily generate the sound pressure in the EC. The two cases of considering the motions of the EC entrance and TM due to nondeforming motion of adjacent EC parts and solely accounting for the vibration of the EC wall are illustrated in Fig. 1.

To the best of our knowledge, the effect of accounting for the motions of the EC entrance and TM due to nondeforming motion of the adjacent structures on the sound pressure in the open and occluded ear has not been quantified, yet. The objective of this paper is to study the impact of this phenomenon on the simulation of the OE, specifically focusing on effects related to the acoustic impedances at the EC entrance and TM. Influencing factors and underlying mechanisms are discussed based on circuit calculations through examples using an EC motion extracted from a FE model of a human head.

Two examples highlight the validity of Eq. (1). For an acoustically rigid boundary (⁠ $Z → ∞$⁠), v_a = v_s, which means it moves with the adjacent solid. Neglecting v_s would misleadingly cause v_a to be zero in this case, implying that the acoustic boundary is fixed in position regardless of the surrounding structure's motion. If the structure is fixed (⁠ $v s = 0$⁠), Eq. (1) leads to the impedance definition $Z = p / v a$⁠, which is commonly used within OE models.

As depicted in Fig. 1, the two boundaries often assigned with acoustic impedances for the simulation of the OE are the EC entrance and the TM (e.g., in Refs. 5, 6, 8, 13, 15, 18, and 20–22) The question is how to implement the impedance boundary condition from Eq. (1) into the simulation models to account for a nondeforming motion of the surrounding structures. Because it is a general phenomenon of structure-fluid interaction, it may affect both numerical approaches, e.g., using the FE method, and simplified approaches, such as equivalent circuit calculations. We begin the investigation by employing the circuit model depicted in Fig. 2 to provide a “visual image of the system.”⁴⁰ It gives an overview of the physical mechanisms involved and allows separation of the various contributions to the TM sound pressure. Subsequently, an application of Eq. (1) to FE models is proposed.

Note that Eq. (2) accounts for the vibration of the EC wall in general, which could include nondeforming motion of the parts. Yet, a nondeforming motion of an EC segment does not contribute to $q w , s$⁠. For such a motion, $v s$ remains constant across all positions on the wall within that segment. Consequently, the contributions of opposing boundaries within this segment to the overall volume velocity nullify each other as a result of the differing signs of $v s ( x ) d S ( x )$⁠. One should be aware that describing the EC wall vibration as source using the integral quantity $q w , s$ is only valid as long as the wavelength in air is much larger than the dimensions of the radiating wall area. This is the case, at least, in the considered frequency range below 1 kHz.

Equation (1) already gives the boundary condition for acoustic impedances within FE models, and a more detailed formulation is, e.g., given in Ref. 39. However, implementing Eq. (1) raises the question on how to obtain the structural velocity v_s.

Obtaining the corresponding structural velocity $v e , s$ at the EC entrance is feasible following the same approach by averaging via the mutual edge of the entrance plane with the surrounding structures. For instance, with a rigid impedance representing the occluding device, the resulting acoustic boundary condition in Eq. (1) locally accommodates the piston-like motion of the device with its surroundings. Thus, nonrigid devices and open-ear radiation just result in different impedance terms p / Z in Eq. (1), but the method for determining $v e , s$ remains unchanged. However, it is essential to acknowledge that Eq. (1) solely accounts for sources inside the EC. To also encompass the BC-excited radiation from the head into the EC via the EC entrance,¹⁹ an additional term would need to be incorporated.

We now demonstrate the effect of considering the EC wall vibration and the motions of the EC entrance and TM on the simulation of the OE using the circuit model in Fig. 2. The problem, however, is that each specific contribution of the boundaries involved depend on the type of excitation. As no general data on the spatially distributed motion of the EC for BC stimulation and the corresponding volume velocities are available yet, an example is taken as a case study.

The motions were extracted from the FE model of a human head presented by Xu et al.¹⁹ The excitation was a BC stimulation of 1 N at the ipsilateral mastoid similar to a BC transducer.^5,12,13 The head base was fixed, and the tissue material properties denoted as “reference” in Ref. 19 were applied. Despite the adjustments at the EC entrance and TM described in  Appendixes A and  B, which serve the purpose of ensuring that the circuit calculations are representative in highlighting the mechanisms involved, the FE model was kept as documented in detail by Xu et al.¹⁹ The volume velocities $q w , s , q e , s$⁠, and $q tm , s$⁠, were extracted from the FE simulations following Eqs. (2) and (6) as reported in  Appendix A. Subsequently, the volume velocity transfer functions were calculated based on the circuit in Fig. 2. Using the extracted volume velocities as input then allowed calculation of the terms $T w q w , s , T e q e , s$⁠, and $T tm q tm , s$ from Eq. (7) as well as the resulting OE according to Eqs. (8) and (9).

To obtain the volume velocity transfer functions, the sound propagation within the EC needs to be considered. Therefore, the EC acoustics were modeled using a simplified circular cross section, cylindrical geometry with an EC length, l_ec, of 27.7 mm and radius, r_ec, to 3.85 mm. These dimensions lead to the same length and volume of the cylinder as those for the EC in the FE model.¹⁹ The cylindrical geometry neglects the variability of the radius along the EC axis. The comparison between the circuit- and FE-calculated sound pressure in  Appendix B shows that the circuit calculations can be considered to appropriately represent the mechanisms related to the motions of the EC entrance plane and TM.

At the TM, the impedance, Z_tm, was taken from Shaw and Stinson.²³ It was implemented according to the values given in Ref. 20. Z_tm exhibits a resonance at about 900 Hz, and below it acts as a compliance. Two occlusion conditions were considered at the EC entrance. For the first condition, a perfect occlusion was modeled by setting $Z e occl$ to infinity. The second occluding condition represents a vented earplug with a hole of 0.9 mm radius and 21 mm length. This condition was already used for the circuit calculations by Carillo et al.⁶ for comparison with experimental data by Hansen¹⁵ (here called “vented” condition is termed “partially occluded” in Ref. 6). The air in the vent, the air in the EC, and the TM compliance form a Helmholtz resonator with a resonance frequency of about 400 Hz.⁶ For the corresponding calculations, the same volume velocities $q w , s occl , q e , s occl$⁠, and $q tm , s occl$⁠, as employed for the perfectly occluded case were used.

The radiation impedance, $Z e open$⁠, at the EC entrance for the open ear was obtained from the FE model by an accompanying simulation with a piston-like excitation at the entrance plane toward the exterior of the head. For this simulation, the “perfectly matched layer,” which allows a free field radiation without reflection, was used as described in Ref. 19, and the head surfaces were modeled as acoustically rigid. $Z e open$ is inertia dominated within the inspected frequency range.

In Fig. 3, the OE calculations according to Sec. II D for an EC motion, which accounts for the vibration of the EC wall and the motions of the EC entrance and TM, and when only considering the EC wall contribution are shown for a perfectly occluded (left) and vented EC (right). To illustrate the influence on the sound pressure at the TM itself, the level differences between the calculations of the sound pressure, including both mechanisms and exclusively accounting for the wall vibration, are displayed in Fig. 4. These were computed from the ratio $q tm / ( T w q w , s )$ for the open (left), perfectly occluded (center), and vented (right) EC entrance conditions. In Fig. 5, the relative contributions $T w q w , s$ (orange), $T e q e , s$ (green), and $T tm q tm , s$ (purple) are shown normalized to their resulting sum, q_tm [see Eq. (7)]. These curves represent the contributions of the motion associated with the wall, the EC entrance, and TM to the sound pressure at the TM.

The narrowband data extracted from the FE simulations exhibit strong resonances, especially at the lowest frequencies below 300 Hz (see Fig. 6 in  Appendix A). To better illustrate the general trends, the results were calculated from third octave band averages of the overall volume velocities, $q tm open$ and $q tm occl$⁠, in Fig. 3 [the numerator and denominator in Eq. (8)] and the contributors $T w q w , s , T e q e , s$⁠, and $T tm q tm , s$⁠, as well as their sum, q_tm, in Figs. 4 and 5. The center frequencies were chosen according to the IEC 61260-1:2014 standard.⁴⁶ 

The results for the OE only based on the vibration of the EC wall depicted in Fig. 3 (orange lines) were calculated according to Eq. (9). The curves resemble the results in Ref. 6 (Fig. 5) with the small distinction that, here, the difference between the volume velocities $q w , s open$ and $q w , s occl$⁠, which was below 2 dB throughout the whole frequency range of interest, is accounted for. The related mechanisms are briefly summarized in the following. This will help to understand the additional effects when the contributions of the EC entrance and TM are accounted for in Sec. III B.

For the perfectly occluded EC entrance (left), the OE exhibits the well-known second order low-pass filter characteristic.⁶ It can be explained by the variation of the volume velocity transfer from the wall toward the TM, which determines the OE according to Eq. (9) for the case of $q w , s open ≈ q w , s occl$ given here. In the open case, the wall's volume velocity is mainly transferred toward the EC entrance as a result of the low impedance of the inertia of the air in the upstream section and the EC entrance. With increasing frequency, this transfer is reduced because of the increase in the inertial impedance and the decreasing impedance of the TM and downstream air compliance. In the occluded case, the transfer of the volume velocity toward the TM is constant with frequency caused by the pressure chamber effect of the compliant EC volume, resulting in the low-pass characteristic of the OE.⁶ 

The vent at the EC entrance causes a reduction of the OE compared to the perfect occlusion because it allows for a volume velocity transfer from the EC wall through the vent instead of toward the TM below the Helmholtz resonance (cf. right panel in Fig. 3).⁶ The OE is constant with frequency within this frequency range. Above the resonance, the air in the vent occludes the EC entrance, hence, the OE for vented EC entrance and the case of perfect occlusion follow a similar trend.

The OE results in Fig. 3 (blue lines) represent an illustrative example considering the wall vibration and the motions of the EC entrance plane and TM. In contrast to the prevalent assumption in OE simulations, which predominantly considers wall vibration alone (orange lines), the OE is reduced in the 100, 160, and 200 Hz third octave bands while the curves are similar at other frequencies. This OE reduction can, in principle, be attributed to changes in the open or occluded ear sound pressure. Therefore, Fig. 4 shows the effect of the contributions of the EC entrance and TM to the TM sound pressure itself, examined for the scenarios with open EC (left), perfectly occluded EC (center), and vented EC (right). In contrast to sound pressure calculated solely from the vibration of the EC wall, a reduction is observed in the lowest frequency range. However, this reduction is more pronounced in the occluded conditions compared to the open ear, which results in the reduction of the OE displayed in Fig. 3.

Figure 5 helps to identify the underlying mechanisms which explain the difference of the effect between the open and occluding conditions on the TM sound pressure and shows the relative contributions of the three distinct volume velocities associated with the wall (depicted in orange), the EC entrance (green), and TM (purple) to the resulting sound pressure at the TM. In the case of the open ear, the wall's contribution predominantly influences the TM pressure above 250 Hz (as shown in the left plot of Fig. 5) as it closely approaches 0 dB. In the 125, 160, and 200 Hz third octave bands, the wall's contribution exhibits a magnitude slightly greater than 0 dB, which indicates a partial cancellation by the TM's contribution at 125 Hz and by the entrance's contribution at 160 and 200 Hz. Interestingly, the entrance's and TM's contributions cancel each other at 100 Hz, resulting in the wall's contribution to be dominant. Consequently, the difference in Fig. 4 (left panel) is relatively small except between 125 and 200 Hz, where accounting for motion of EC entrance and TM reduces the TM sound pressure by approximately 3 dB.

For the perfectly occluded and vented conditions, the wall's contribution predominates only for frequencies higher than approximately 250 Hz (right and center plots in Fig. 5). Below 250 Hz, the entrance's and the wall's contributions are similar in magnitude. What is noteworthy here is that both contributions exhibit a magnitude greater than 0 dB, signifying their partial cancellation in this frequency range. Conceptually, this can be visualized as the rigid EC entrance plane moving together with the attached wall but with different surface orientations. This type of motion leads to an overall reduction of the volume velocity when these contributions are summed [see also explanations on Eq. (2)]. Consequentially, the occluded TM pressure is reduced. This phenomenon is very similar for the two cases of perfect occlusion and a vented EC entrance considered here. Accordingly, when only accounting for the wall's contribution for the OE (orange line in Fig. 3), this effect, stemming from the reduction of the wall's contribution resulting from the motion of the entrance plane, is omitted. This omission leads to an overestimation of the predicted occluded sound pressure.

Notably, in all three cases, the contribution linked to the motion of the TM with the surrounding bone remains small, except at the lowest considered frequencies below 125 Hz (purple lines in Fig. 5). This phenomenon can be attributed to the different materials surrounding the EC. At the lowest frequencies inspected, the EC exhibits a rigid body motion as a whole. However, already above approximately 125 Hz, the EC section surrounded by soft tissue and cartilage exhibits a higher amplitude of the motion than the bony part attached to the TM. Consequently, the resulting volume velocity associated with the TM remains significantly smaller in magnitude when compared to the volume velocities linked to the wall and EC entrance plane. The only exception is the 125 Hz band with open ear, where the volume velocity associated with the motion of the TM partially cancels the effect of the wall vibration on the TM pressure.

Another illustration of the mechanisms is provided with the help of the circuit in Fig. 2. The three sources are connected to two branches each with one portion of the volume velocity flowing toward the TM—encompassed by the volume velocity transfer functions—and another portion directed toward the EC entrance. The high impedance at the EC entrance for the two occluding conditions causes the portion toward the EC entrance to be relatively small. This leads to $| T w occl | ≈ | T e occl | ≈ | T t m occl |$⁠, allowing the partial cancellation of the wall's contribution by the entrance's contribution below 250 Hz, because of their phase differences related to the surface orientations. In contrast, for the open-ear condition, the air in the EC is divided into an upstream and a downstream section^6,8,15,20 connected to the low radiation impedance at the EC entrance. This division results in different ratios of the volume velocity portions transmitted toward the EC entrance and TM for each of the sources. Most significantly, it leads to $| T e open | < | T w open |$⁠. Consequently, the effect of the entrance's contribution to cancel the wall's contribution is smaller for the open compared to the occluded conditions, as shown when comparing the green lines in Fig. 5.

The differences for the perfectly occluded and vented EC entrance highlight that the magnitude and filter characteristic of the OE potentially varies depending on the EC motion. Especially, the considered example indicates that a reduction of the OE toward the lowest frequencies compared to the second order low-pass filter characteristic⁶ can be related to nondeforming motion of parts of the EC (cf. Fig. 3)—which was not accounted for at the EC entrance and TM in previous studies of the topic. Indeed, comparisons between experimental data obtained from EC sound pressure measurements and OE simulations^18,19,22 reveal an overestimation of the OE, in particular, at frequencies below 250 Hz.

Reinfeldt et al.¹² found a higher variance of the OE obtained from EC sound pressure measurements for ipsilateral mastoid stimulation with a BC transducer at lowest frequencies compared to other positions on the skull and stated that this could be related to the elastic part of the EC being closer to the ipsilateral position. Saint-Gaudens et al.¹⁶ reported the same effect comparing ipsi- and contralateral stimulation, whereas the variance was comparable with mastication as BC stimulation. The OE calculation with stimulation at the mastoid reveals the influence in the same frequency range where the higher variance is observed in the measurements. It is likely that the numerous factors listed in Sec. III B result in variations of the EC motion. Accordingly, it is plausible that the mechanisms considered within this study, which are related to the EC motion, contribute to the observed variance of the open or occluded sound pressure (cf. Fig. 4) and the OE (cf. Fig. 3).

Figure 3 highlights the effect of considering the motions of the EC entrance and TM on the OE simulation. Accordingly, additional parameters $q e , s$ and $q tm , s$ [cf. Eq. (8)] could improve the prediction of the sound pressure in the EC and OE with circuit models. However, the values of $q w , s , q e , s$⁠, and $q tm , s$ depend on the EC geometry, the orientation and magnitude of the excitation, the areas associated with the volume velocity sources, and the material properties of the EC surroundings. More experimental and numerical studies are needed to investigate the EC motion in more detail in the future, e.g., in terms of the spatially distributed EC wall vibration (as already pointed out by Carillo et al.²²) This is also indicated by a remaining difference of the open-ear pressure between the FE simulation and circuit calculation given in Fig. 7 in  Appendix B. This difference can potentially be mitigated by considering the frequency-dependent nature of the velocity distribution, particularly in terms of its centroid position.²² For the circuit calculations conducted within this study, a constant centroid position with frequency was assumed.

Within numerical models, one should be aware that applying the TM impedance locally as specific acoustic impedance deviates from the original one-dimensional formulation (see, e.g., detailed discussion in Ref. 24). Also, the pressure distribution in the EC (especially close to the TM)^31,43 and the deformation of the TM is generally complex at higher frequencies than about 1–1.5 kHz.^29,30 Therefore, it is reasonable to replace the impedances at the EC entrance and the TM, e.g., considering the coupling of the EC cavity with the external air,¹⁹ or using middle and inner ear structures at the TM.⁴⁴ Although this allows extending the frequency range of the OE simulations, it increases the model complexity and computational effort. Yet, circuit and FE models have been shown to be useful to investigate the OE using acoustic impedances (see, e.g., Ref. 8 for a comparison). The proposed extension of the impedance boundary condition could help to improve such OE simulations by accounting for nondeforming motion of the EC.

Within the FE simulations and circuit calculations conducted for this study, an impedance was applied at the EC entrance plane instead of including the earplug as solid material. Thereby, the effects of occluding devices listed in Sec. I, which are not directly related to its impedance, such as the variation of the radiating area,^3,5 were excluded. Although this is a limitation of the study regarding the prediction of the occluded sound pressure in the EC, it allowed focusing on the consideration of the motion of the earplug's medial surface due to its connection with the surroundings throughout the analysis. Similarly, also the open EC entrance was only considered using a radiation impedance. However, Xu et al.¹⁹ compared OE simulations with the same FE model of a human head as employed here between a setup with a radiation impedance at the EC entrance and an infinite surrounding acoustic domain directly coupled to the EC cavity (see their Fig. 8). Additional simulations with incorporating the impedance boundary condition according to Eq. (1) confirmed their finding that the sound radiation of the head vibrations into the open EC via the EC entrance can contribute to TM sound pressure, especially at the lowest frequencies (refer to  Appendix C for more details).¹⁹ 

Accordingly, the present considerations highlight that at least three influencing factors, namely (a) vents of the occluding device or leaks at the EC entrance, (b) the radiation of BC-stimulated sound into the open EC, and (c) the nondeforming motion of parts of the EC, contribute to the deviation of the OE from a second order low-pass filter characteristic. Comparable mechanisms, which could additionally influence the TM sound pressure, are the radiation of sound by the earplug²² and BC sound transmitted through the ME into the EC.⁷ These effects associated with the EC entrance and TM were excluded here in favor of only investigating the role of the structural motion of the EC. Yet, the considered example indicates the necessity to distinguish between the various mechanisms. However, it remains challenging to predict their effect on the resulting EC sound pressure and OE in advance because these mechanisms contribute to the TM pressure differently depending on whether the EC is open or occluded (as exemplified in Fig. 5). Also, the present analysis considered the EC motion only in a specific head model for a particular BC stimulation at the mastoid. The variability of the aforementioned mechanisms in relation to the factors listed in Sec. I, such as the type of stimulation or the individual EC geometry, remains unexplored. A more comprehensive investigation into these complexities could be a valuable pursuit for future research.

This study highlighted the impact of a nondeforming motion of parts of the EC on the sound pressure at the TM and the resulting OE. This type of motion causes the EC entrance and the TM to move with their surrounding structures. As a result, these boundaries' contributions complement the vibration of the EC wall in serving as the source of the BC-stimulated sound pressure in the EC, particularly at the lowest frequencies. An impedance boundary condition was proposed, which incorporates this phenomenon by including a term to consider the structural motion of the adjacent solids at EC entrance and TM. When applied to OE models, the effect of this motion depends on two key factors: first, the volume velocity transfer from the boundaries represented by acoustic impedances to the TM, which varies based on the open or occluded condition at the EC entrance and, second, the frequency-dependent, three-dimensional motion of the EC itself. To illustrate the related mechanisms, an example motion with BC stimulation at the mastoid was extracted from a FE model of a human head. This motion was employed as input for circuit calculations, considering perfectly occluded and vented EC scenarios.

The results revealed that the motions of the EC entrance and TM caused by nondeforming adjacent EC parts—which is addressed by the proposed extension—can contribute to a reduction of the TM sound pressure at the lowest frequencies compared to when solely accounting for the vibration of the EC wall. This impact was found to be more pronounced when the EC is occluded, leading to a concurrent reduction of the OE. It is reasonable to consider that the mechanisms associated with the motions of the occluding device and TM offer a plausible explanation of the higher variance of experimental OE data for stimulation positions close to the EC. In addition, they can contribute to the deviation between OE simulations and experimental data below 250 Hz.

The considered EC motion highlighted the importance of discerning between multiple mechanisms that contribute to the sound pressure at the TM in open or occluded scenarios. Yet, a more detailed knowledge on the EC motion and its influencing factors is needed to determine the relative contributions of the various effects and their degree of variability before, e.g., including the contributions by the different surfaces within improved circuit simulations of the OE.

See the supplementary material for a Python script (Python Software Foundation)⁴⁵ with the circuit calculations, and the corresponding data extracted from the FE simulations.

The authors would like to thank Steffen Marburg for pointing to the literature to base the considerations on. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 352015383—SFB 1330 A4 and supported through a Mitacs Globalink Research Award (Ref. No. IT34758).

All authors have no conflicts to disclose.

The data that support the findings of this study are available within the article and its supplementary material.

Open and perfectly occluded conditions were applied at the EC entrance in the FE model (at the position termed “E1” in Ref. 19) using the same impedances as were used for the circuit calculations. Likewise, the TM impedance after Shaw and Stinson^20,23 was used at the TM. The circuit impedances (in Pas/m³) were multiplied with the area of the corresponding boundary surfaces to apply them locally as specific acoustic impedances (in Pas/m). To account for the structural motion at the boundaries represented by acoustic impedances, the underlying equations in the software COMSOL Multiphysics® version 6.1 (COMSOL AB, Stockholm, Sweden), which was used for the FE simulations, were manually modified as described in Sec. II B 2. Given these simulation parameters, the volume velocities $q w , s open$ and $q w , s occl$ were extracted according to Eq. (2), and $q e , s open , q e , s occl , q tm , s open$⁠, and $q tm , s occl$ were obtained according to Eq. (6). The volume velocities are depicted in Fig. 6. The inspected frequency range included the third octave bands from 100 Hz to 1 kHz. The frequency resolution was set to 24 frequencies per octave.

To ensure that the circuit calculations are representative in highlighting the mechanisms involved, the calculations were verified by simulating the TM pressure for the open-ear condition with radiation impedance and perfectly occluded condition with infinite impedance at the EC entrance. The comparison is displayed in Fig. 7.

The sound pressure level difference between the FE simulation and circuit calculation with perfectly occluded EC is generally negligible within the examined frequency range. The level difference for the open EC is relatively small except in the 200 Hz third octave band, where the circuit-calculated sound pressure is about 6 dB higher than that in the FE simulation. This difference can be potentially attributed to the assumption of a constant length for the EC segments (8 and 19.7 mm) across all frequencies. This is a simplification of the potentially frequency-dependent effect where the air in the EC is split up into an inertia-dominated part toward the EC entrance and a compliant section toward the TM.⁶ However, it is crucial to note that this simplification does not compromise the fundamental findings regarding the impact of the motions of the EC entrance plane and TM on TM sound pressure. This is best illustrated by the negligible difference visible between circuit calculation and FE simulation for the perfectly occluded EC in Fig. 7, whereas the effects of the different contributors to the TM sound pressure are most pronounced in this case (see Fig. 4, center plot). Therefore, the circuit calculations can be considered to appropriately represent the relevant mechanisms involved.

In addition to the radiation impedance at the EC entrance, a FE simulation with a perfectly matched layer around the head was conducted in the same way as the corresponding setup by Xu et al.,¹⁹ but including the impedance boundary condition at the TM according to Sec. II B 2. Compared to the radiation impedance at the EC entrance, this simulation condition also accounts for the exterior radiation of the BC-stimulated head vibrations into the EC. The difference between the resulting TM pressure simulations is shown in Fig. 8. Accordingly, the main effect of the sound propagation from the exterior into the EC is an increase in the TM sound pressure below approximately 300 Hz. This effect further reduces the resulting OE compared to Fig. 3. Note that this contribution to the TM sound pressure in the open ear is not represented within the circuit model used for the present study.