Extreme call amplitude from near-field acoustic wave coupling in the stridulating water insect Micronecta scholtzi (Micronectinae)

1. Introduction

Micronectinae are an extremely widespread and populous subfamily of Corixidae of the sub-order Hydrocorisae, yet despite their near worldwide distribution studies on the family are sparse. From the remaining two clades, Corixinae and Diaprepocorinae, the Corixidae have extensive morphological and physiological studies, while investigations into Micronectinae are frustrated by their small size (1.5–6.0 mm) [1] and Diaprepocorinae consist of a single rare genus which occurs only in Australia and New Zealand. Recently, Micronecta scholtzi, a common aquatic heteroptera, was identified as having an extremely loud mating call in relation to its body size, reaching a peak of 99.2 dB sound pressure level (SPL) (ref 20 μPa, air equivalent level) at a distance of 1 m [2]. The average body length of M. scholtzi is not more than 2.5 mm in length; therefore, the emission of such an intense signal seems to contradict the expected correlation between body size and call amplitude [3].

In common with Corixinae, the mechanism of sound production is believed to be stridulation, although the site of the plectrum and file (pars stridens) appears significantly different. In Corixidae, striated areas and the plectrum are located on the fore femora and the head capsule, respectively [4], while in Micronecta batilla [5], the pars stridens appears to be located on the posterior abdominal segment. A striated area is observed in M. gracilis, M. australiensis and M. batille [6] on the right side of the sixth tergite and in M. scholtzi on the right paramere of the genital appendage [7]; however, no direct observation of stridulation has been achieved. The pars stridens in M. scholtzi is particularly small at only 50 μm in length, compared with 100 μm in M. batilla. Males of all species produce a distinct song which is species specific [8]. The song is produced only by the males and takes the form of a pulse train with two or three echemes, which is obligatory for copulation to be successful [9]. Males appear to be able to synchronize their calls, generating a chorus [8].

Production of high-amplitude mating calls may be facilitated by exploiting structural resonance for sound radiation, such as the wing in Gryllus bimaculatus [10]. In corixids, the resonant structure is provided by the air cushion that is maintained by the diving insect as an air reserve, evidenced by the strong correlation between the calling frequency of the animal and the air volume in the reserve [11]. Air cushions in submerged corixids are distributed over the ventral surface, maintained by hydrophobic hair structures, and on the dorsal surface of the abdomen and between the abdomen and the elytrum [12]. These air cushions make ideal resonators, producing a pure tone at the Minnaert frequency which scales inversely with the square root of the bubble volume [13]. The recorded calls of Corixa dentipes, C. punctata and M. scholtzi are all found to be pure tones [2,11], and in the case of the corixids were found to not only decrease in frequency with body size but also increase over the length of time submerged as the air reserves are depleted [11].

Corixa generate underwater sound by means of volume pulsations of this air cushion, with the mating call of one animal also stimulating the resonant behaviour of the air cushion in nearby animals [11]. Unfortunately, the pars stridens in the same animal is located on the fore femora, where it would also be enclosed by the ventral air cushion, presenting a significant problem for sound transmission due to the large impedance mismatch between the air and water. Neither is the sound power from strigilation in air likely to be strong enough to excite significant volume pulsation in the air cushion. Micronecta scholtzi's plectrum and pars stridens, being located below the fifth tergite of the abdomen, are more likely to be submerged in water, allowing not only the excitation of the bubble resonance by the more powerful free field in water but also, due to their extreme proximity to the bubble, to excite pulsation through the far higher pressures of the hydrodynamic near field. The transfer of energy from striated area to bubble in the formulation would be extremely efficient, allowing the multipole, highly directive sound field produced by a plectrum and comb to drive the far more efficient, near-ideal monopole source formed by the air cushion. Here, we present analytical models of the pressure gain from the presence of a bubble in the near field of monopole and dipole sources and compare the results with finite-element analysis simulations of M. scholtzi's body and air cushions to demonstrate that significant pressure gains can be achieved. It is proposed that the small size of M. scholtzi, reducing the separation between plectrum and pars stridens and bubble, is responsible for the high call volume and that taking into account the bubble gain the relationship between call volume and sound pressure level falls within the expected range of other stridulating insects.

2. Theory

The acoustic output of a source can be significantly increased by the presence of nearby scatterers, particularly by resonant cavities. A submerged gas bubble can be thought of as a simple oscillatory system, with the spring represented by the compressed air within the bubble and the mass being that of the water displaced by the movement of the bubble's boundaries. For a simple volume pulsation, the natural resonance frequency of a submerged gas bubble is given by Houghton [14] as follows:

where P_W is the hydrostatic water pressure, ρ is the water density, r₀ is the radius of the bubble and γ is the ratio of specific heats in the gas. The second term takes account of the surface tension σ, while the third includes the effect of the dynamic viscosity of the liquid η. The first term gives the approximate Minnaert resonant frequency, which can be used to relate the velocity potentials from the source (Φ_s) and bubble (Φ_b) at the bubble wall (here denoted by ‘b’) (figure 1) [15]: where ω₀ is the Minnaert resonance frequency of the bubble, while δ is a damping parameter which encompasses radiative, thermal and viscous losses and ı is the imaginary unit. The Minnaert frequency has little dependence on bubble shape, so air bubbles of similar volume will have the same angular frequency of resonance with small errors [16] and is assumed, in this formulation, to produce sound as a near-ideal monopole source. This model was developed to predict behaviour in the far field of a sound source and assumes there will be no dipole motion of the bubble in the absence of a restoring force; however, in the near field of a source the relative velocity of the bubble and hydrodynamic flow around the source provides the necessary restoring force for dipole motion. In the near field, the sound field scattered from a bubble will be dominated by the monopole motion of the bubble, but a significant underestimation of the total sound field will be made if the dipole motion is not also considered. The pressure gains are calculated at an arbitrary listening point ‘P’ as the ratio of the pressure from the bubble acting as a monopole and the incident pressure from the source; the full derivation of the equations for pressure gain of a bubble in the near field of a monopole or dipole source is given in appendix A. For monopole gain on a logarithmic scale, The pressure gain A is given in terms of the bubble radius r_b, distance d between the source and bubble, the angular velocity of the source ω and the Minnaert resonance of the bubble ω₀. Thermal and viscous losses are incorporated in the term δ. Similarly, for a dipole source the increase in sound pressure as a result of the submerged gas bubble in the near field would be where in addition to the dependence on separation between the source and bubble d, bubble radius r_b and angular velocity ω there is a dependence on the angle between the source and bubble θ_b and the angle between the source and listening point P (θ₁). The distance to the listening point r also appears in this equation although if kr≫1, i.e. the listening point is in the far field, that dependence disappears. Dipole gains are significantly higher than monopole gains in the near field, being inversely proportional to the square of the distance between the source and bubble. The presence of the bubble in the near field essentially couples the low transmission efficiency of a multipole source to the high transmission efficiency of a near-ideal monopole. The equations above consider only the monopole action of the source and so underestimate the pressure gain. Calculation of the dipole motion of the bubble in the near field was provided by Ffowcs Williams [17]; however, the solution is complex and difficult to apply to more general cases than ideal sources, making finite-element analysis a more attractive strategy for determining the complete pressure gain from the bubble as a scatterer. A comparison of the pressure gain estimations from this simple monopole model and finite-element analysis is shown in figure 2.

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3. Material and methods

Micronecta schlotzi used in this work were provided by Dr Jérôme Sueur of Muséum national d'Histoire naturelle, Département Systématique et Evolution, paris, France, and were collected from Paris and Morsang-sur-Orge. The sample M. scholtzi were preserved in 70% ethanol, and were unsexed at the time of scanning. x-Ray microscale computed tomography (μCT) scans were performed using a Bruker Skyscan 1172. Each insect was prepared by removing it from the ethanol and drying for a period of 8 h, then mounting on a block of dental wax with ventral side up and encasing the insect in a plastic straw. The scans shown here were conducted using a voltage of 50 kV on the X-ray source, with no filter applied to obtain the maximum contrast between the often uniform X-ray attenuation coefficients of the insect's soft body. Images were generated with 2664 × 4000 pixels at a resolution of 1.36 μm per pixel, with four frame averages taken at each 0.3° increment around one hemisphere of the insect's body. Three-dimensional volumetric reconstruction was performed using Bruker's CTvol software, and the image resized and noise reduced using the same suite's CTAn program. Six M. scholtzi were scanned in this way with the best image of a representative male being used to generate a three-dimensional model of the insect's body for finite-element analysis simulations.

The acoustic scattering problem was simulated in COMSOL Multiphysics v. 5.3.1 using a three-dimensional pressure acoustics model of a 326 μm radius air bubble contained within a 20 mm radius spherical water domain, the boundaries of which are modelled as a perfectly matched layer mimicking an infinite water domain. The minimum mesh element size was constrained to 9 μm due to the size of the source and fine detail on the model M. scholtzi body, while the source frequency was swept in 200 Hz steps from 1 to 20 kHz with a minimum of 543 mesh elements per wavelength at high frequencies. The plectrum and pars stridens are located on the last segment of the abdomen, which is split, and the exact nature of the vibrations and resulting sound source are unknown. Here, we assume the majority of the bodily vibrations are concentrated on the last segment, and simulate the source as the sound field on a spherical surface of 50 μm radius which encompasses the site of strigilation, treating the boundary as an acoustic hologram which is continuous with the water domain to omit the consideration of backscatter. The insect's body itself was modelled as an acoustically rigid boundary (Neumann boundary condition). The locations of the bubbles were taken as being on the ventral side of the insect's body, omitting the site of stridulation and the head capsule, and on the dorsal side underneath the elytrum in line with previous estimates of bubble locations [11]. The bubble dimensions were chosen to obtain a volume that would produce a Minnaert resonance at 10 kHz to match the frequency of the call of M. scholtzi.

To simulate the sound field around M. scholtzi's body, a model of the insect in a normal swimming posture was required, which was considered in the simplified case here to be a model of the abdomen, thorax and head capsule omitting the legs. Here, the coronal cross-section images generated by the μCT scans were used, as generating a full body model directly from the floating point mesh of the insect proved too computationally expensive. Slices of M. scholtzi were taken every 20 μm along the insect length and traced in a series of 67 work planes in Autodesk Inventor, generating a wireframe model of the body. The slices were joined via a series of 20-μm-thick lofts, and the resulting solid body imported into COMSOL Multiphysics (figure 3). Two air domains were considered, one on the ventral surface and one on the dorsal surface. The ventral bubble was considered bounded only by the ventral surface of the insect's body, while the dorsal bubble was treated as being nearly totally enclosed by the insect's abdomen and elytrum save for a 100 μm wide belt around the minor axis of the bubble. The attachment to the insect's body would be expected to change the resonance frequencies due to the unbalanced hydrodynamic field around the bubbles, resulting in a more damped resonance frequency and allowing a coarser frequency sweep and reduced computation times. The sound source in the free field was treated as a simple piston with a prescribed acceleration of 0.1 m s⁻², which, in a free field, generated 7.35 dB SPL (ref 1 μPa) at the water domain boundaries 30 mm from the centre of the bubble along the axis of maximum amplitude. The source strength was chosen arbitrarily as we were looking for the relative gain with the presence of the bubble in the near field, which is expected to be linear with amplitude. Hydrostatic water pressure was taken as 100 kPa, equivalent to 1 m depth.

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4. Results

The volume and location of the air cushions for the COMSOL models were taken from the μCT scans of M. scholtzi (figure 4). The bubble volumes were treated as ellipsoids intersected by the insect body, with the depth and wetting area on the insect body being determined by the resonance frequency. Both the volume underneath the elytrum and that on the ventral surface of the insect resonated at 10 kHz with a wetting area that comprised the entire abdomen in the case of the dorsal bubble and a wetting area that spanned from the head capsule to the lower abdomen in the case of the ventral bubble (figure 5). Bubble depth in the ventral bubble was 180 μm, while the dorsal bubble depth was 150 μm. For a simple dipole source placed at the posterior side of the abdomen at the approximate location of the pars stridens, pressure gains of +42.27 dB (in comparison with 7.35 dB free field) could be observed at resonance for the ventral bubble only and +39.41 dB for the dorsal bubble only, the change probably reflecting the increased distance from the simulated stridulation site to the ventral bubble (figure 6). A compiled list of the scenarios tested and the resulting pressure gains from the presence of the bubble are given in table 1. When both bubbles are present, however, their oscillations become linked by the fluid domain between them and two resonance peaks occur: the first as the bubbles oscillate in phase and the second at higher frequency as the bubbles oscillate out of phase giving a dipole sound field (figures 7 and 8). It is equally possible to have the two air domains linked around the neck underneath the wing hinges where the system will act as a single bubble; however, the depth of the air cushion on the ventral surface then decreases significantly. This remains the most probable scenario given the absence of a secondary peak in recordings of M. scholtzi's mating call [2].

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In comparison with the source placed inside the air cushion, a resonant peak at 10 kHz was also observed; however, the resulting field was still considerably attenuated by this process, reaching a peak pressure gain of −19.76 dB in the case of a monopole source and −10.51 dB in the case of a dipole source. Additional simulations were performed with the M. scholtzi body model using a sawtooth waveform stimulus from the source to approximate the clicks of stridulation. The fundamental frequency was set to 5 kHz, 10 kHz and 15 kHz for three total simulations in the time domain, which is a considerably larger range than might be expected from temperature variations in plectrum stroke speed. Resonant peaks remained constant at 10 kHz from the bubble resonance with pressure gains from the 5 kHz and 15 kHz sawtooth waves calculated as 75% and 22%, respectively, of the matched frequency pressure gain at 10 kHz.

5. Discussion

Previous research on the mating call of aquatic Heteroptera has suggested that the striated area lies within the air cushion, which then acts as a sound box amplifying the signal; however, even when stimulated with a signal matched to the natural resonance frequency of the air cushion, the signal is considerably attenuated rather than amplified. In the bubble as an oscillatory system, the mass is provided by the water surrounding the bubble, which the relatively small amplitude of pressure variations in air is unable to significantly affect. In the near field of any acoustic source, the acoustic impedance is primarily reactive—the force exerted on the medium accelerates rather than compresses it. Energy is not transmitted but merely transferred back and forth in the flow around the source. Perhaps counterintuitively for an acoustic source in the near field, we can treat the fluid motion as incompressible, since when the source is small in comparison with the wavelength the transmitted energy is negligible in comparison with the energy of the fluid flow. The bubble placed within this reactive field can then be driven by these much higher hydrodynamic pressures, using the static field to drive a near-ideal monopole source. The system might be described as an acoustic analogy of evanescent near-field coupling in electromagnetic waves with the maximum energy transfer occurring when the source frequency matches the bubble resonance [18]. The apparent gains in sound power and sound pressure are the result of the bubble's increased efficiency at generating an acoustic wave in comparison with the source, hence the far greater gains from dipole or higher order sources than those observed with monopole sources. In essence, a poor acoustic radiator which is relatively easy to drive is coupled with a very efficient radiator.

The high call volume of M. scholtzi song is potentially explained by the inverse relationship between the separation between the source and bubble and the sound power gain. The insect's small size and necessary proximity between the striated area and bubble resonator, simulated here as a separation of 120 μm, offers a high coupling efficiency. Previous work on M. scholtzi [2] has speculated that the volume of the call was an example of runaway selection in the absence of suitable predators; however, the model presented here allows high call volume from closer proximity of the striated area and air cushion. The call volume is possibly a by-product of pressures towards a smaller body size, or the location of hydrophobic hairs on the insect, rather than sexual selection for higher call amplitude.

Both the ventral and dorsal air cushions have the potential to act as resonators for the mating call; however, if both are present as separate bubbles the resulting call would show two resonance peaks because they are essentially being ‘rung’ be the near-field pressure. Recordings of M. scholtzi show only a single peak, suggesting that the air cushions on the ventral and dorsal surfaces are linked. The measurements of M. scholtzi's call volume were given in previous research as reaching a peak of 99.2 dB SPL with reference to 20 μPa [2]. The figure quoted has been converted to an air equivalent level in order to make a direct comparison with the SPL at 1 m of other animals' mating calls, taking the average of the ratio of SPL (dB)/(3 × log₁₀ body length) to draw ordinary least-squares regression lines. In water, at 1 m the equivalent peak SPL of M. scholtzi would be approximately 126 dB re 1 μPa. With a potential dipole gain of between 39 and 45 dB from the near-field coupling to the bubble in order to reach the extreme amplitude recorded by M. scholtzi, a simple dipole source of 50 μm radius would have to be capable of generating an SPL of 80–85 dB (re 1 μPa) at 1 m, placing it within the regression lines for body size and peak SPL for other striating insects [2]. Unfortunately, without direct measurements of the source without the air cushion or the energy used in the call, it is not possible to say whether near-field coupling to the air cushion completely explains the large call volume.

Micronecta scholtzi, in common with other Hydrocorisae, would be expected to be a poikilothermic creature with the stroke speed of the plectrum and resultant frequency being dependent upon temperature. Maximum pressure amplitudes will naturally occur when the click frequency of the plectrum on the pars stridens is matched to the 10 kHz Minnaert resonance of the air cushion, raising the question of whether M. scholtzi uses a similar escapement mechanism to regulate stroke speed as male crickets (the ‘clockwork cricket’) [19]. Here, it is unclear that there is any selection advantage to louder calls and the resonator is still well, although not optimally, stimulated by the sawtooth waveform at 5 kHz. In addition the resonance frequency of the air cushion in Hydrocorisae changes with diving time as the oxygen is depleted, varying as much as 1 kHz between surfacings [11], again suggesting that the loud call volume is an unintended by-product of the system and not something which contains useful information to a mate.

Near-field acoustic coupling of the bubble to the source allows the transfer of energy from a high-input efficiency but low-radiative-efficiency source to a low-input efficiency and high-radiative-efficiency source. The system has potential applications in very-low-frequency underwater transducers, where energy losses from existing systems are extremely high [20,21] due to the requirement that they work around their fundamental resonance frequency in order to generate enough acoustic power. For transducers working below 500 Hz, this can make the working area necessarily large, requiring high electrical and mechanical excitation levels with consequent problems of piezoelectric efficiency (in the case of piezoelectric transducers) and mechanical fatigue. In separating the acoustic radiator (the bubble) from the source, a much smaller and more electrically efficient piezoelectric source can drive a flexural bubble through near-field coupling and maintain the desirable radiation characteristics of a surface in volume pulsation.

Data accessibility

All data created during this research are openly available from the University of Strathclyde Pure/KnowledgeBase at http://dx.doi.org/10.15129/46c7cb9b-7966-40fa-947f-b9f65757f6a5 [22].

Authors' contributions

A.R. carried out the mathematical analysis, carried out the finite-element analysis, participated in the design of the study and drafted the manuscript; D.J.W.H. helped carry out the mathematical analysis, helped the conception of the study, and helped design the study; D.M. carried out the X-ray scans; J.C.J. helped carry out the mathematical analysis, and helped draft the manuscript; J.F.C.W. conceived of the study, designed the study, coordinated the study and helped draft the manuscript. All the authors gave their final approval for publication.

Competing interests

We have no competing interests.

Funding

Research at the University of Strathclyde leading to these results received funding from the European Research Council under the European Union's Seventh Framework Programme FP/2007-2013/ERC grant agreement no. 615030.