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Derived from the respiratory tracheae, bush-crickets' acoustic tracheae (or ear canals) are hollow tubes evolved to transmit sounds from the external environment to the interior ear. Due to the location of the ears in the forelegs, the acoustic trachea serves as a structural element that can withstand large stresses during locomotion. In this study, we report a new Atomic Force Microscopy Force Spectroscopy (AFM-FS) approach to quantify the mechanics of taenidia in the bush-cricket Mecopoda elongata. Mechanical properties were examined over the longitudinal axis of hydrated taenidia, by indenting single fibres using precision hyperbolic tips. Analysis of the force-displacement (F-d) extension curves at low strains using the Hertzian contact model showed an Elastic modulus distribution between 13.9 MPa to 26.5 GPa, with a mean of 5.2 ± 7 GPa and median 1.03 GPa. Although chitin is the primary component of stiffness, variation of elasticity in the nanoscale suggests that resilin significantly affects the mechanical properties of single taenidia fibres (38% of total data). For indentations up to 400 nm, an intricate chitin-resilin response was observed, suggesting structural optimization between compliance and rigidity. Finite-element analysis on composite materials demonstrated that the Elastic modulus is sensitive to the percentage of resilin and chitin content, their location and structural configuration. Based on our results, we propose that the distinct moduli of taenidia fibres indicate sophisticated evolution with elasticity playing a key role in optimization. STATEMENT OF SIGNIFICANCE: In crickets and bush-crickets, the foreleg tracheae have evolved into acoustic canals, which transport sound to the ears located on the tibia of each leg. Tracheae are held open by spiral cuticular micro-fibres called taenidia, which are the primary elements of mechanical reinforcement. We developed an AFM-based method to indent individual taenidia at the nanometre level, to quantify local mechanical properties of the interior acoustic canal of the bush-cricket Mecopoda elongata, a model species in hearing research. Taenidia fibres were immobilized on a hard substrate and the indenter directly approached the epicuticle surface. This is the first characterization of the nano-structure of unfixed tracheal taenidia, and should pave the way for further in vivo mechanical investigations of auditory structures.

Bush-crickets (or katydids) have sophisticated and ultrasonic ears located in the tibia of their forelegs, with a working mechanism analogous to the mammalian auditory system. Their inner-ears are endowed with an easily accessible hearing organ, thecrista acustica(CA), possessing a spatial organisation that allows for different frequencies to be processed at specific graded locations within the structure. Similar to the basilar membrane in the mammalian ear, the CA contains mechanosensory receptors which are activated through the frequency dependent displacement of the CA. While this tonotopical arrangement is generally attributed to the gradual stiffness and mass changes along the hearing organ, the mechanisms behind it have not been analysed in detail. In this study, we take a numerical approach to investigate this mechanism in theCopiphora gorgonensisear. In addition, we propose and test the effect of the different vibration transmission mechanisms on the displacement of the CA. The investigation was carried out by conducting finite-element analysis on a three-dimensional, idealised geometry of theC. gorgonensisinner-ear, which was based on precise measurements. The numerical results suggested that(i)even the mildest assumptions about stiffness and mass gradients allow for tonotopy to emerge, and(ii)the loading area and location for the transmission of the acoustic vibrations play a major role in the formation of tonotopy.

Early predator detection is a key component of the predator-prey arms race, and has driven the evolution of multiple animal hearing systems. Katydids (Insecta) have sophisticated ears, each consisting of paired tympana on each foreleg that receive sound directly externally, and internally via a narrowing ear canal through the acoustic spiracle. These ears are pressure-time difference receivers capable of sensitive and accurate directional hearing across a wide frequency range, despite the small size of katydids. Many katydid species have cuticular pinnae which form cavities around the outer tympanal surfaces, but their function is unknown. We investigated pinnal function in the katydid Copiphora gorgonensis by combining experimental biophysics and numerical modelling using 3D ear geometries. Results show that the pinnae in C. gorgonensis do not assist in directional hearing for specific call frequencies, but instead pinnae act as ultrasound detector devices. Pinnae induced large sound pressure gains that enhanced sound detection at high ultrasonic frequencies (> 60 kHz), matching the echolocation range of co-occurring insectivorous bats. Comparing pinnal mechanics of sympatric katydid species supports these findings, and suggests that pinnae evolved in katydids primarily for enhanced predator detection. Audiograms (both behavioural and neural) and tympanal cavity resonances obtained from living specimens corroborate our findings.

Significance The katydid tympanal ears have outer, middle, and inner ear components analogous to mammalian ears. Unlike mammals, each ear has two tympana exposed to sound both externally and internally, with a delayed internal version arriving via the gas-filled ear canal (EC). The two combined inputs in each ear play a significant role in directional hearing. Here, we demonstrate that the major factor causing the internal delay is the EC geometry. The EC bifurcates asymmetrically, producing two additional internal paths that impose different sound velocities for each tympanum. Therefore, various versions of the same signal reach the ears at various times, increasing the chance to pinpoint the sound source. Findings could inspire algorithms for accurate acoustic triangulation in detection sensors., Located in the forelegs, katydid ears are unique among arthropods in having outer, middle, and inner components, analogous to the mammalian ear. Unlike mammals, sound is received externally via two tympanic membranes in each ear and internally via a narrow ear canal (EC) derived from the respiratory tracheal system. Inside the EC, sound travels slower than in free air, causing temporal and pressure differences between external and internal inputs. The delay was suspected to arise as a consequence of the narrowing EC geometry. If true, a reduction in sound velocity should persist independently of the gas composition in the EC (e.g., air, CO2). Integrating laser Doppler vibrometry, microcomputed tomography, and numerical analysis on precise three-dimensional geometries of each experimental animal EC, we demonstrate that the narrowing radius of the EC is the main factor reducing sound velocity. Both experimental and numerical data also show that sound velocity is reduced further when excess CO2 fills the EC. Likewise, the EC bifurcates at the tympanal level (one branch for each tympanic membrane), creating two additional narrow internal sound paths and imposing different sound velocities for each tympanic membrane. Therefore, external and internal inputs total to four sound paths for each ear (only one for the human ear). Research paths and implication of findings in avian directional hearing are discussed.

Bush crickets have tympanal ears located in the forelegs. Their ears are elaborate, as they have outer-, middle-, and inner-ear components. The outer ear comprises an air-filled tube derived from the respiratory trachea, the acoustic trachea (AT), which transfers sound from the mesothoracic acoustic spiracle to the internal side of the ear drums in the legs. A key feature of the AT is its capacity to reduce the velocity of sound propagation and alter the acoustic driving forces of the tympanum (the ear drum), producing differences in sound pressure and time between the left and right sides, therefore aiding the directional hearing of the animal. It has been demonstrated experimentally that the tracheal sound transmission generates a gain of ∼15 dB and a propagation velocity of 255 ms−1, an approximately 25% reduction from free-field propagation. However, the mechanism responsible for this change in sound pressure level and velocity remains elusive. In this study, we investigate the mechanical processes behind the sound pressure gain in the AT by numerically modeling the tracheal acoustic behavior using the finite-element method and real three-dimensional geometries of the tracheae of the bush cricket Copiphora gorgonensis. Taking into account the thermoviscous acoustic-shell interaction on the propagation of sound, we analyze the effects of the horn-shaped domain, material properties of the tracheal wall, and the thermal processes on the change in sound pressure level in the AT. Through the numerical results obtained, it is discerned that the tracheal geometry is the main factor contributing to the observed pressure gain.

An interpolation operator is proposed using the cubic grid solution of order O(h4), h is the mesh size, of the Dirichlet problem for Laplace’s equation in a rectangular paralellepiped. It is proved that when the boundary functions on the faces of the rectangular parallelepiped are from the Holder classes C4,λ, λ ∈ (0, 1), and their second and fourth derivatives obey compatibility conditions implied by Laplace’s equation on the edges, the solution obtained by the constructed operator also has fourth-order accuracy with respect to mesh size.

Katydids (bush-crickets) are endowed with tympanal ears located in their forelegs' tibiae. The tympana are backed by an air-filled tube, the acoustic trachea, which transfers the sound stimulus from a spiracular opening on the thorax to the internal side of the tympanic membranes (TM). In katydids the sound stimulus reaches both the external and internal side of the membranes, and the tympanal vibrations are then transferred to the hearing organ crista acustica (CA) that contains the fluid-immersed mechanoreceptors. Hence the tympana are principally involved in transmitting and converting airborne sound into fluid vibrations that stimulate the auditory sensilla. Consequently, what is the transmission power to the CA? Are the TM tuned to a certain frequency? To investigate this, the surface normal acoustic impedance of the TM is calculated using finite-element analysis in the katydid Copiphora gorgonensis. From this, the reflectance and transmittance are obtained at the TM. Based on the impedance results obtained from the pressure recordings at TM and the velocity field calculations in the AT, in the frequency range 5-40 kHz, it is concluded that the tympana have considerably higher transmission around 23 kHz, corresponding to the dominant frequency of the male pure-tone calling song in this species.

The hexagonal grid version of the block-grid method, which is a difference-analytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are from the Holder classes $C^{4,\lambda}$ , $0

The fourth order matching operator on the hexagonal grid is constructed. Its application to the interpolation problem of the numerical solution obtained by hexagonal grid approximation of Laplace’s equation on a rectangular domain is investigated. Furthermore, the constructed matching operator is applied to justify a hexagonal version of the combined Block-Grid method for the Dirichlet problem with corner singularity. Numerical examples are illustrated to support the analysis made. MSC:35A35, 35A40, 35C15, 65N06, 65N15, 65N22, 65N99.