Effective-Spring Model of Tympanic Response in Archosaurs

Whereas for smaller animals the eardrums are well-characterized as excitable membranes or drums, some animals such as several archosaurs feature, as a first approximation, a rather stiff elastic shell supported by an elastic ring. Mathematically, the theory of plates and shells is applicable but its governing equations overly complicate the modeling. Here the notion of tympanic structure is introduced as a generalization of “ordinary” tympanic membranes so as to account for sound perception as it occurs in archosaurs, such as birds and crocodilians. A mathematical model for the tympanic structure in many archosaurs called two-spring model implements this notion. The model is exactly soluble and solutions are presented in closed form and as a series expansion. Special emphasis is put onto offering an easy-to-apply model for describing experiments and performing numerical studies. The analytic treatment is supplemented by a discussion of the applicability of the two-spring model in auditory research. An elasticity-theoretic perspective of the two-spring model is given in the Appendix.


Introduction 1.Internally Coupled ears (ICE)
More than half of the land-living vertebrates possess an air-filled cavity connecting left and right eardrums.That is, they possess internally coupled ears, for short ICE [1]. Figure 1 shows the evolution of hearing in vertebrates using ICE [2] [3] [4].In the simplest models [5] [6] of this widespread mechanism the two eardrums are an elastic structure interfacing the external acoustic environment Figure 1.Left: Evolution of vertebrate hearing systems and animals using ICE.Courtesy of Anupam P. Vedurmudi for the full graphic, adapted from van Hemmen et al. [1].Right: The elastic shell of a crocodilian (youngster's) eardrum.The black bar sets a length scale of 1 cm.The diameter and height of the shell structure are 8 mm and 2 mm, respectively.The extracolumellar lever is located in the upper right quadrant of the picture.Its geometry resembles a spherical sector.Picture courtesy of Bruce A. Young.
with the interaural cavity, functioning as an acoustic wave guide [7] [8].The interaural cavity ensures that the membranes' vibrations are not independent but mutually coupled.For neuronally less developed animals, the coupling provides an easy-to-realize mechanism to generate a more precise directional hearing than without ICE; for early attempts at an explanation, see Autrum [9] [10].
Although the mechanism of internally coupled ears seems in principio universal for numerous animals, the biological realization differs from species to species and animal to animal [1].Aquatic frogs such as Xenopus [11] employ plate-like eardrums bounding an air-filled cavity whereas several birds, in particular chicken, use a flexible membrane in a cavity filled with, surprisingly, not air [3].Lizards [5] [6] [12] use flexible membranes connected by a large cylindrical cavity whereas crickets employ, possibly to comply with their exoskeletal anatomy, a complicated tracheal system featuring two membranes connected through a complicated cavity system.The latter is partitioned itself into two symmetric halves by another elastic structure called septum [13].
An increase in the area covered by the eardrums leads to a lowering of the fundamental frequency in a "clamped membrane model" because the first eigenvalue of the Laplacian (with Dirichlet boundary conditions) scales as the inverse of the square of the area covered by the eardrums.Thus a small eardrum area seems favorable to these animals.Anatomically, these requirements are realized in some archosaurs through a self-supporting spherical shell supported by an elastic ring.In reptiles and birds, an elastic ring occurs far more often [14] in the middle ear than one might naively expect.
The construction is shown schematically in Figure 2. The spherical shell itself may vibrate due to bending and is shown in vitro in Figure 1.Its net displacement in vertical direction leads to an induced vibration of the ring supporting the spherical shell elastically.This means that the notion of a tympanic membrane naturally breaks down because the shell is far less flexible than the elastically DOI: 10.4236/ojbiphy.2019.91003tion of the ring leads, once the shell is not a hemisphere, to a an excess torque applied to the boundary of the shell and thus to shell bending, x s , and has been incorporated through a coupling of the x r -oscillator to the x s -oscillator through a massless spring of spring constant k 0 .
supporting ring so that the latter is responsible most for the fundamental frequency of the "eardrum".It needs to be constantly borne in mind that the amplitudes are in the nm range whereas the ring has a thickness of mm so that the difference is at least 5 orders of magnitude.Physical auditory research [13] currently lacks both adequate jargon and formalism to capture these phenomena and indicate the direction of useful modeling.This is the gap we aspire to fill with the present treatment.

Tympanic Structures
A tympanic structure is defined to be the (finite) set of elastic elements that 1) interface the interaural cavity of the animal under consideration with its external acoustic environment, 2) by virtue of mutual elastic coupling of the constituents function as a single elastic structure, and 3) respond in this form at least locally to auditory signals.We call a given tympanic structure N-constituent tympanic structure, if the set of elastic elements in the sense of the previous definition has cardinality N.An elastic element of a tympanic structure is also called a constituent.

Heuristic Derivation of the Two-Spring Model
The two-spring model is based on two simplifying assumptions.First, we are only interested in the impact that the vibration of a tympanic structure has on the volume in the interaural cavity.Vibrations give rise to a local mass density change in the cavity and thus trigger the formation of a pressure wave traveling between the, usually two, tympanic structures [1] [5] [6].We refer to Howe [8] DOI: 10.4236/ojbiphy.2019.91003D. T. Heider, J. L. van Hemmen [7] for an introductory treatment of sound generated by vibrating elastic structures, to Timoshenko [15] [16] for a detailed treatment of elasticity, and to van Hemmen and Leibold [17] for elasticity theory as applied to biological membranes.
Let ( ) , , q q q be coordinates such that 3 0 q = and ( ) , q q ∈ Γ define the equilibrium position of an elastic element where 2 Γ ⊂  is bounded and suitably regular domain.(For two eardrums there is, say, another element at 3 0 q L = > .)Furthermore, let u be the displacement, i.e., a small ( ) Area u Γ  perturbation from the former equilibrium.Solving elasticity equations for two-dimensional elastic structures is equivalent to the inversion of operators where O typically is the Laplacian ∆ , its square 2 ∆ , the bi-harmonic operator 2 bi ∆ or linear combinations thereof [17].Let us replace the displacement of the constituents of a tympanic structure by its average This approximation is called piston approximation [5] [12].Assuming physical behavior of the full displacement u depending on the temporal variable t as well as the spatial coordinates 1 2 , q q , we can apply the mean value theorem of integra- tion.It ensures the existence of a ( ) , , u t q q x t ′ ′ = .
We now perform a partial Taylor series expansion in the spatial variables ( ) ( ) ( ) where ( ) , , R t q q is Lagrange's remainder, i i i j j j i R t q q q q s u t s q sq The remainder is absolutely bounded from above by firstly applying Cauchy's inequality to the Euclidean inner product.We note that , diam sup , , : , , , q q q q q q q q q q q q ′ and that the function inside the integral is smaller than or equal to its maximum modulus.The remainder is now bounded by In particular, the piston approximation is dynamically accurate, if Our first assumption is that the above inequality is fulfilled.
Next we define x  and 2 0 Since O 0 − ≥ , i.e., O is positive-definite for all practical purposes, we obtain a square of an effective average frequency and an average displacement x  that DOI: 10.4236/ojbiphy.2019.91003may be different from x.By inspection of a formal eigenfunction expansion of u, it is seen that if dominantly one eigenmode is excited in the spatial vibration pattern.The dominant biological mode is the fundamental mode, which is most often the excited vibrational mode of elastic structures relevant to auditory processes.The second assumption is that x x ≈  can be utilized as x x =  .That is, only one mode of the elastic structure is excited dominantly.
Effectively, the partial differential equation that would be needed for a full description of one constituent of the tympanic structure can be written in the form of a harmonic oscillator equation, To derive the two-spring model we will use the following argument: Since the spherical shell is supported elastically by a ring underneath, we only need to store local vibrations of the shell surface in s x .
The displacement of the shell as a whole along the symmetry axis of the ring is equal to the displacement r x of the ring.
We now need to specify the driving forces First, we assume that an external pressure signal p hits the spherical shell homogeneously.Put simple, p transfers a uniform momentum to the shell's surface and thus makes the shell move up and down without deformation of its equilibrium shape.By definition of , s r x x , this corresponds to an external force ap- plied to r x because s x shall only store local shell vibrations but no global piston modes.Since the ring itself is elastic, it will vibrate and thus lead to a small bending of the shell [15].
For the elastic interaction between the two constituent elastic structures, we use the Hooke approximation.We regard the constituents as being coupled by a massless, elastic spring with spring constant where 0 k is treated as a fitting parameter.A rough estimate concerning the ratio 0 r k k can be found in the Appendix.The other parameters , , , , ,

Two-Spring Model
The deliberations of the previous paragraph are summarized by the governing We assume . This means that the coupling through the massless spring does not dominate the spring constants corresponding to eigenmodes of the individual constituents.
Since the tympanic structure is supposed to be at rest in the absence of an incident acoustic pressure signal on the shell, the initial conditions at 0 t = , i.e., when a signal is about to hit the shell, are fully homogeneous: Let us denote the surface of the shell by ( ) . An incident pressure wave is described as ( ) , p t x where x are the coordinates in the surrounding space.It becomes a vector quantity when we multiply it by the direction of wave propagation, viz., the normal to the wave front.Then the scalar net force exerted on the shell is given by In practice, a time harmonic signal ( ) ( ) proves to give decent results for an incident low-frequency plane-wave-like signal with uniform scalar amplitude 0 p due to a sourced at a position in the acoustic far field.0 Σ is the area of the equilibrium shape of the shell constituent, ω denotes the frequency carried by the acoustic signal, and φ symbolizes where to place an additive phase to account for directional information stored in the signal.

Solution of the Two-Spring Model
We solve the two-spring model by reduction to a first-order system.
We have employed the following definitions of squares of reduced frequencies: The system is linear so that we can invoke the variation-of-constants or Duhamel formula [18].We denote by [ ] kl A the ( ) , k l -entry of a matrix A .Conveniently, the initial conditions to be satisfied by , r s x x are homogeneous so that the vari- ation of constants method yields

Solution to the Eigenvalue Problem
The matrix exponential requires the solution of the eigenvalue problem for the matrix M and, in particular, knowledge of the zeros of the characteristic po- lynomial ( ) ( ) More precisely, we need the roots of the following quartic equation, The above equation can be analytically solved for ω by combining the me- thods of Cardano and Ferrari [19] [20] for the solution of a generic cubic (Cardano) and quartic (Ferrari) polynomial equations by algebraic completeness of  .By a somewhat lengthy but straightforward calculations we can specify the four, possibly multiple, roots k ( ) ( ) ( ) ( ) ) ( ) .

Damping
It is interesting to note that damping enters directly through the average of the individual damping coefficients r γ , s γ as specified in Eq- uation (11) and Equation (12).Under physical assumptions regarding the measurement parameters, , , , , , and the fit parameter 0 k , we may assume that all four roots (18), ( 19), (20), and ( 21) are pairwise distinct.By a small variation of the parameters within the range of measurement uncertainty, this assumption can be fulfilled trivially.We denote by  the resulting set of the four solutions ω to the eigenvalue problem of the matrix M .

Calculation of the Matrix Exponentials
To obtain s x and r x from ( 15) and ( 16), we need to calculate the matrix ex- ponential.The calculation is performed with the aid of Mathematica.
We find ( ) ( ) ( ) where the sum runs over the four solutions .
Assuming regularity of the input force F and thus of r a and by the assumption of distinctness of the k ω ∈  , the sum and the integral in the solution formulas ( 26) and ( 27) have been interchanged.Using the computational methods summarized above, we find the convolution kernels G ω and H ω to be given by ( ) Needless to say that although the model is simple and captures the physics of a combined movement of the spherical shell membrane due to bending and the elastic deformation of the supporting ring, the solution is somewhat cumbersome.

Numerical Results
For the numerical simulation we use a time scale with units  For the same choice of parameters, we have plotted the ratio in the bottom-left plot of Figure 3 and in the bottom-right plot of Figure 3.By definition, both ratios are smaller than unity.This is indicated by the orange lines in the two bottom plots.The ratios .Yet, and most of the time, the bending of the spherical shell is small compared to the vertical displacement of the ring.
, rounded mathematically to one counting digit.This agrees with the reasoning of the introduction that the elastic ring is the preferred constituent of the tympanic structure of multitudinous archosaurs to generate the fundamental frequency that is processed further by ICE.As for the extracolumella that is wired to the cochlea, the relative smallness of the bending of the shell compared to the elastic stretching of the ring implies that the its deflection from equilibrium, s r x x + , can be approximated as r x for sufficiently small 0 k compared to r k and s k .

Physical Discussion
The quantity of keen interest to model builders in auditory research is the velocity of the tympanic membrane, or, more generally, of the tympanic structure [13].
While for geckos and frogs the interaural cavity can be and has been approximated by a cylinderical acoustic waveguide of equivalent volume [3] [4] [6], the archosaur counterpart has the topology of a, sometimes even higher-genus, torus [21].In view of ongoing analytic approaches to the study of hearing in "icy" animals, the geometric complexity as well as the number of equations required for mathematical description both necessitate approximations for efficient solutions.The piston approximation in conjunction with approximate modeling of the entire interaural cavity as effectively one-dimensional wave guide form the base of the preferential toolkit.
For comparison with biological data, quantities of the structure denote the (total) speed of the left and right tympanic membrane.In the new case of a tympanic structure with more than one constituent, a similar measure can be obtained by considering the total displacement of the tympanic structure, denoted by L U and R U .In the case of the present two-spring model, we iden- tify for the total speed at the left and right tympanic structure.The substitutions , , ILD 20log .
The primary objective of this work is to provide auditory research with a practically applicable model for the tympanic structure of numerous archosaurs.
We discuss a simpler treatment of the two-spring model in Section 4.1.

Iterative Solution of the Two-Spring Model
Since the constituents of the tympanic structure-see ( 11) and ( 12)-are damped, an iterative solution is to be favored instead of the lengthy closed form solution derived in the previous paragraph.By iterative solution, we mean a decoupling of the equations of motion for the full tympanic structure into the individual equations of motion for its constituents using the assumption Because of damping, the impact of higher-order coupled contributions to the displacement of either constituent from the other constituent will decrease in time.We refer to Hassani [18] for a mathematical introduction to operators as used in physics.For the sake of notational brevity, we introduce the following constant-coefficient differential operators They are defined on the space ( ) . Moreover, we require the driving force F to be smooth.Both operators are accompanied by homogeneous initial conditions.We have . That is, the operators defined in (34) and (35) commute for high enough regular functions.
The governing equations of the two-spring model read in operator notation Using one of Schwarz' representation theorems, we can represent the inverse operator by a convolution-type integral [18].Let ( ) . If we are able to find such s x and r x , the existence and uniqueness theorems from the theory of ordinary differential equations guarantee that we have also found the only solution to the inhomogeneous initial value problem.Using the inverse operators (37) and (38), we recast (36) into the equivalent system of integral equations The Banach fixed-point theorem entails a construction to obtain the fixed-point by iteration.In order to satisfy the model's initial conditions, we choose We introduce the notation for subdivisions ( ) ( ) with the properties declared above The indices have been inserted in the definition of the A operators to make the nesting of integrals explicit.They indicate the integration variables in the nested integrals.
For 0 t > , we find the following series representation for the components of ( ) The above expression is not a Neumann series because of the appearance of the nested integrals.It cannot be summed up to a closed-form solution.Recalling the fixed-point argument from above, we see that the maximum norm of the series converges.The regularity properties of , s r x x are the result of smoothness of the input F and the fact that the operators (37) and (38) have smooth integration kernels and thus do not decrease the regularity of their arguments.Equations (41) and (42) can be applied in practice by truncating the series.In the next paragraph, we compare truncations of the series solutions (41) and (42) with the exact solutions of the two-spring model for a given parameter set.

Discussion of the Applicability in Auditory Research
We state the first two iterations.In the notation of the previous paragraph, this corresponds to ( ) ( ) , From the phenomenological point of view, the "strength" of coupling can be determined from the ratios .By mathematical induction it follows The series solution gradually builds up the coupling between the two constituents of the tympanic structure.

Numerical Results
In order to assess the accuracy of the iterative solution developed in this section, In each of Figure 4 and Figure 5, the left column depicts results for the real parts ( ) The right columns of Figure 4 and Figure 5 show the results for In the top row of Figure 4, is shown as solid blue curves, while , represented by the solid orange curves.The same color code has been employed in the top row of Figure 5 with ( ) x and ( ) x used in place of ( )      relative deviation compared to the amplitudes is below 1% for both real and imaginary part for both the ring and the shell displacement.This is lower than the relative deviation compared to the amplitude of the oscillation of the real and imaginary part of r x and s x found for the first iterates from the bottom row of Figure 4.In this case, we find the ratio in question being under 6% for

Summary
For the first time in auditory research, we have articulated a physically implementable definition of tympanic structures that generalizes the notion of a tympanic membrane.In the terminology introduced in the main body of the present article, a tympanic membrane is seen to be a special case of the wider notion of "tympanic structure".The case of archosaurs as "icy" animals exemplified the necessity to define a more general notion in order to model the tympanic responses in these animals in a physically meaningful way.Favoring applicability of the ICE model in auditory research over mathematical complexity, the two-spring model has been derived from the piston approximation applied to the elastic constituents of the 2-constituent tympanic structure as observed in numerous archosaurs.Two damped harmonic oscillators, interacting via a Hooke-elastic coupling, were found to account for the combined motion of the tympanic structure.That is, the displacement of the spherical shell membrane of the animals without local bending and bending of the shell.
By numerical simulations with parameters chosen from the expected parameter range, the biologically motivated hypothesis that the overall displacement of the extracolumella is mainly due to uniform, piston-like displacement of the shell as a whole without large contributions due to local bending, could be supported.Only a maximum of 20% of the extracolumella displacement's amplitude was found to be on the part of shell bending in the two-spring model with the chosen simulation parameters.
Because of the length of the analytic solution formulas, a simpler iterative treatment has been developed.Treating the two-spring model as a set of two damped, suitably weakly coupled harmonic oscillators, a Banach fixed-point iteration scheme has been derived.Simulations of the first two non-trivial iterates for the model solutions were performed using the same parameters as in the exact solution of the two-spring model.For sufficiently weak coupling of the two constituents of the tympanic structure, the second iterates were found to be sufficiently accurate and efficient-to-calculate at the same time so as to be favored over the exact solution, if the computational power available is limited.The strength of the coupling is stored in the parameter , which requires specification through up-to-now unavailable experimental data.

Appendix
The two-spring model is an effective model, a strong simplification of both biological and mathematical reality.In order to ensure the overall applicability of the model as well as the readability for a less mathematically inclined audience, a more complete, yet mathematically more advanced model, has been postponed to this appendix.Here spheroidal wave functions will appear at the end.Unfortunately, the state of the art in the numerics of spheroidal wave functions obstructs a detailed numerical evaluation while further analytic investigations are ruled out by the non-existence of closed form representations of spheroidal wave functions.

Geometry of the configuration
As explained in the main text, the two constituents of the tympanic structure ∈ −  of the shell.The extracolumella effectively prevents vibrations on the shell surface itself.Yet, in the present model the shell is allowed to move up and down, the entire shell moving uniformly as a piston.
For the ensuing treatment, the extracolumella is assumed to be only connected to the shell but not to the underneath ring.The ring itself has radius 0 R and height H, extending in negative z-direction.We denote by 0 0 3 , s r Ω Ω ⊂  the equilibrium surfaces of shell and ring, For the elasto-dynamic analysis, it is useful to account for the small but finite thickness of the elastic constituents as well.Let us denote the corresponding compacta by  λ µ .Since, even in the absence of a physical cause for deformations, the shell plus extracolumella has the shape of a hemisphere, it is treated as self-supporting: If stimulated by an pressure signal incident from the z-direction, the shell undergoes elastic deformations only along the z-direction.

Navier-Cauchy equation
The continuum momentum balance equations simplify to linear partial differential equations if only small elastic deformations are to be modeled.For the shell and the ring, they read quite generally where , s r ρ ρ denote the constant mass per volume density of shell and ring material, respectively, s f and r f stand for an external force density applied to the shell and ring, and r σ and r σ denote the (symmetric) stress tensors.
The Navier-Cauchy equations [23] are obtained as a consequence of the assumption of a linear stress-strain relation.The linearized strain tensors σ and s σ are themselves symmetric.They can only depend linearly on the respective strain tensor and the trace of the respective strain tensors.A textbook calculation [23] demonstrates that the Equation (51) and Equation ( 52) reduce to the elasto-dynamic equations, also known as the Navier-Cauchy equations, as the force-per-surface density which is responsible for driving the system.
The ICE model has proven quite successful in auditory research [1] [12].Due to the superposition principle, the investigation is confined to a pure tone, ( ) ( ) with a fixed positive stimulus angular frequency 0 ω > .
The ring only interacts with the shell but receives no external stimulation, i.e., r = f 0 .The ring-shell interaction via boundary conditions will be discussed lat- er on.
Our focus is on the quasi-stationary state.That is, the tympanic structure is assessed at a time where tympana's undulations generated previously have decayed as a result of damping.Mathematically, this amounts to working in frequency-domain space where linearity ensures that the tympana undulate with the same angular frequency ω as carried by the stimulus but in general with a phase shift for stimulus directions ≠ 0. The self-supporting assumption reduces the components of the displacement vector field s u from 3 to only 1, namely, to the component aligned in parallel to the external stimulus: Upon insertion of the vector operators in cylindrical coordinates [22], (53) and (54) simplify, , . and extracolumella is driven as one point mass by the incident pressure signal needs to be considered in the above equations.

Force-collector model
Let us suppose that the shell is placed on an incompressible iron floor instead of an elastic ring.Since the shell is self-supporting and the pressure uniform, only forces in z-direction need to be considered.By Newton's third law, the iron floor enacts a excess normal force on the annular interface region of floor and shell as a reaction to the total of excess pressure driving the membrane.The resulting reaction force equates to the reaction pressure int p times the area of the interface int A .Since the shell obviously does not oscillate up and down through the "iron" force, the reaction force needs to compensate the total force generated by the incident pressure p on the area of the shell plus extracolumella as seen from a point on the z-axis, . We note The global force balance according to Newton's third law (actio = -reactio) translates into ( ) The difference between the iron floor and the elastic ring is that the latter experiences the interface pressure as an additional pressure.We observe that (59) needs no correction, if the piston mode is subtracted from ( ) s z u since ( ) s z u only quantifies dislocations generated by the compressional coupling between shell and ring.
One technical issue needs to be fixed.In the shell model (55) the pressure difference along the shell interface enters the force-per-volume balance rather than only an incident pressure gradient.The shell collects the pressure difference, the physical motivation for the nomenclature "force-collector model".It needs to be shown that the interface region Since we are only interested in displacement averaged over the shell's thickness s d , the variables r and z are no longer independent but subject to the constraint ( ) µ on the one hand and the Young's modulus E and the Pois- son number ν on the other hand becomes ill-defined.Furthermore, the Pois- son number is typically between 0 and 1/2.Biological tissue does not increase its volume in such a way that it gets thicker under application of an external pulling force.Rather, its volume stays constant or increases while the shell becomes thinner [23].
In view of the above issues, a different method to localize the shell on In the new coordinates ( ) , , r z φ ′ , the physical hemisphere defined through .In view of (67), oblate spheroidal coordinates are useful.For their definition, we refer to Spencer and Moon [22].We need to solve the equations   The function p depends on η and q in a well-behaved way.The dependence

Figure 2 .
Figure 2. Left: Schematic physical decomposition of the vibration of the tympanic structure due to an incident pressure signal.Right: Conceptual representation of the two-spring model.The pressure signal causes a uniform displacement of the spherical shell or, equivalently, an elastic deformation x r of the supporting ring underneath.The vibra- from experimental measurements.The above choice of source terms defines the two-spring model.Its underlying physical picture is summarized for the reader's convenience in Figure2, on the right.
diagonalization of M and defined such that

F
m ω .The remaining parameters are fixed by = .The fit constant 0 k is chosen such that top row of Figure 3, one sees that the elastic ring vibrates at a larger amplitude than the amplitude of bending of the spherical shell for both the real part (left) and the imaginary part (right) of the solution for the input ( ) exp 0.8 i t in the unit system specified above.The color code employed in Figure 3 is as follows for the ring's vertical elastic displacement r x .The real part [ ] r x ℜ in the top left plot and the imaginary part [ ] r x ℑ in the top right plot are shown as solid blue curves.As for the shell's bending displacement s x , the following color code is employed in Figure 3.The real part [ ] s x ℜ in the top left plot and the imaginary part [ ] s x ℑ in the top right plot are shown is solid orange curves.Varying the parameter 0 k in the model simulation, it is found that the smaller 0 k , the smaller the amplitude of s x .

Figure 3 .
Figure 3. Top-left plot: [ ] r x ℜ (blue) and [ ] s x ℜ (orange) in units of proved useful[3] [4].Such quantities are inspired by the definition of the sound pressure level in acoustics.Via no-penetration boundary conditions eardrums, these quantities compare the acoustic pressure at the left to the one at the right ear.In Equation (32), L u  and R u  tympanic structures.By the assumptions regarding 0 k , both ratios Open Journal of Biophysics D. T. Heider, J. L. van Hemmen are smaller than unity.A suitable smallness parameter is defined through we simulate the Equation (43) and Equation (44), respectively (45) and (46), for both the parameter choice and the input model used for the simulation of the full two-spring model presented the previous section.The numerical results for (43) and (44) are shown in Figure 4.The axis units therein have been chosen in accordance with Figure 3 and the unit system introduced at the beginning of the previous section.The numerical results for (45) and (46) are shown in Figure 5.
of the two figures are devoted to quantifying the absolute deviations from the full solution obtained by means of the simulation presented in the previous section.In the two bottom plots of Figure4, the differences shown as the solid blue and solid orange curve on the right.blue and solid orange curve in the left plot and the differences shown as the solid blue and solid orange curve in the right plot of the bottom row in Figure5.At a first sight, the top rows of Figure4and Figure5look like their "exact" pendants in the top row of Figure5.We see, however, that the deviations of the moduli of the real and imaginary part of the first iterates, i.e., (43) and (44), from the moduli of the real and imaginary part of the exact solution of the two-spring model do not equilibrate fast into the quasi-stationary state behavior; cf. the bottom row of Figure4.In contrast, the bottom row of Figure5shows that, for the specified parameter values, the moduli of the real and imaginary part of the exact solution of the two-spring model and the absolute values (45) and (46) of the real and imaginary part of the second iterates match well, as compared with the amplitude of the exact solution, after equilibration into the quasi-stationary state.For .4236/ojbiphy.2019.91003

Figure 4 .
Figure 4. Top-left: Approximate solution − .Bottom-left: Comparison of the solu- tion for the full two-spring model with the approximate solutions: bottom right: Analogous plots as in the left column with the corresponding imaginary parts used in place of the real parts.

Figure 5 .
Figure 5. Top-left: Approximate solution (orange) in units of Bottom-left:   Comparison of the solution for the full two-spring model with the approximate solutions: t ∈ in units of 1 r ω − .Top-and bottom-right: Analogous plots as in the left row with the cor- responding imaginary parts used in place of the real parts.DOI: 10.4236/ojbiphy.2019.9100335 Open Journal of Biophysics D. T. Heider, J. L. van Hemmen

in multitudinous archosaurs are ( 1 )
an elastic sectorial spherical shell and (2) an elastic circular symmetric ring.In spherical coordinates, the shell has radius 0 R , .The breaking of the full rotational symmetry of the tympanic structure results from the presence of the extracolumella; cf. Figure 1.It is a bony structure placed in the sector [ ) ( ]

)
Cl M denotes the topological closure operation on the set M in brackets.Letting 0 s d ≥ tend to zero from above, we reproduce 0 s Ω and 0 r Ω as the maximal set contained in all the s d s Ω 's and s d rΩ 's, respectively.The interface between the two constituents is given the intersection the problem Each of the two constituents of the tympanic structure is an elastic material that satisfies approximately material isotropy and homogeneity assumptions.That is, shell and ring have respective Lamé constants ( ) denotes double contraction in engineers' notation.The material homogeneity and isotropy assumptions reduce the number of components from a priori4  3 81 = to 2 for both of the rank 4 tensors ( ) r Λ and ( ) s Λ .The stress tensors r

.
Lamé constants to other elastic parameters Let λ and µ indicate Lamé's constants.The literature denotes µ occa- sionally by G and calls G the shear modulus.The expression( )defines the Poisson number ν and is smaller than 0.5.The Young's modulus E satisfies .Heider, J. L. van HemmenModel specificationThe shell's force-density drive is assumed to take place only in the z-direction due to the self-supporting property of the shell.The prototypical external stimulus used in auditory research is a pressure signal emanating from a point far away from the shell surface so that a plane is the more appropriate shape for the wave front than a shell.Since the propagation speed of sound times a few milliseconds still exceeds the geometric dimensions of the tympanic structure, arrival time differences on the shell surfaces are negligible.Thus the net force-per-volume density on the shell is spatially uniform and purely time-dependent, and only has non-trivial z-component: the acoustic pressure signal is established by considering the z-component of force-per-volume density to be the pressure difference along the thickness of the shell: Indeed, integrating over the homogeneous membrane's thickness s d , we reproduce the pressure difference ( )p t

..
As mentioned before, the shell rests on top of the ring.If piston-like motion is excited, the entire ring vibrates as if it were a point mass under the influence of Newton's law in vertical direction.Neglecting back-couplings of the ring's displacement to spatially dependent vibrations on the shell, the ring's displacement is aligned in parallel to s The configuration hints at using cylindrical coordinates for the spatial arguments of the functions( ) r ∆ denote the Laplace-Beltrami operators in cylindrical coordinates on the three-dimensional equilibria volumes of shell and ring.That is, on ∇ ⋅ − ∇ × ∇ × can be utilized to convert (54) DOI: 10.4236/ojbiphy.2019.91003 oscillations of the sphere are prohibited by the self-supporting assumption.The interface region s d Γ between shell and floor is annular and has the area

Γ
is subject to the collected pressure difference.Namely, the force-collector model demonstrates that an additional force-per-volume density acts on the ring at the boundary namely extend in radial direction by s d .Let us replace the fields ( )

(
clamping due to the presence of the extraco- lumella is unproblematic and only settles is a cue to try transforming the differential operator of the spatial variables in (55) into spherical coordinates.The problem, however, .The relation between the shell's Lamé con- stants s λ and s 0z ≥ is what we look for.One possibility starts with redefining z z′ ≡ ⋅ℵ where 0 ℵ > is a suitable re-scaling factor involving the Lamé con- stants.It is then chosen so that the problematic partial differential operator with spatial variables in (55) equals the Laplacian in the new coordinates.
the deliberations outlined above, (55) reads in ( ) thickness of the membrane.This is needed because the Helmholtz operator cannot be restricted to the 0 η η = iso-surface in oblate spheroidal coordinates so that a solvable eigenvalue problem is maintained.This is in contrast to spherical coordinates where we can restrict them to the r R = iso-surface.The coordinate transform we aim at is established as cosh sin r a of frequency ω .Qua- si-stationarity permits the assumption that the shell oscillates with the same frequency.The resulting Helmholtz equation( ) ( ) separation parameters.They will be fixed during the discussion to follow.The thickness ( s d ≈ ) of the elastic shell in η-direction enters by noting that be a non-zero constant in the interval.This corresponds to requiring that the whole shell moves along its thickness in η-direction.Atπ 2 θ = , the η-average is consistent with the radial average performed to obtain Equation (64) and Equation (63).Physically, the average under consideration formalizes the requirement that only constant elastic deformations with respect to η occur.The equation (70) then gives rise to an equation for the separation parameter p, can be solved for p and the correct sign in front of the square root of the discriminant is fixed by the requirement 0 p > .The result for p is then ( ) -determined functions O, P & A. Along the lines indicated elsewhere ([22], p. 36), the following equations result,