Chamber music: an unusual Helmholtz resonator for song amplification in a Neotropical bush-cricket (Orthoptera, Tettigoniidae)

Acoustic communication plays a vital role in the life of many animals as a means to attract and locate potential mating partners or deter rivals (e.g. Drosopoulos and Claridge, 2006; Hauser, 2000; Hedwig, 2014). Loud signals are often positively selected for because high amplitude sounds can reach more potential mating partners and contain information about the sender's fitness (e.g. Marten et al., 1977; Prestwich, 1994). For small vertebrates and insects, the production of high amplitude communication signals is often a morphological challenge owing to the power and size of their sound production mechanisms (Bennet-Clark, 1998; Prestwich, 1994). Many animals therefore produce sounds by coupling the initial sound-producing structures to mechanical resonators that increase the amplitude of the generated sound at and around their resonant frequencies (Fletcher, 2007). This also serves to increase the sound radiating area, which increases impedance matching between the structure and the surrounding medium (Bennet-Clark, 2001). Common examples of these kinds of primary resonators are the avian syrinx (Fletcher and Tarnopolsky, 1999) or the cicada tymbal (Bennet-Clark, 1999). In addition to primary resonators, some animals have developed morphological or behavioural adaptations that act as secondary resonators, further amplifying acoustical vibrations and facilitating sound radiation. The Bornean tree-hole frog Metaphrynella sundana, for example, adapts the frequency of his calls to the resonance of the tree trunk cavities it sings from, thereby increasing call amplitude (Lardner and bin Lakim, 2002), whereas the bladder cicada Cystosoma saundersii uses an internal air sac as a secondary resonator (Bennet-Clark and Young, 1994) and mole crickets (Neoscapteriscus borellii, formerly Scapteriscus acletus) sing from horn-shaped burrows to amplify their songs (Bennet-Clark, 1987).

In the Ensifera (crickets, bush-crickets and allies), it is generally the male that produces acoustic mating signals by stridulating with their forewings, whereas the (mostly) silent female receives the song and performs phonotactic movements towards the sender (e.g. Kalmring et al., 2003; Robinson and Hall, 2002). In bush-crickets, stridulation involves a serrated vein (the file) on the underside of the left wing which is swept over a hardened part of the anal edge (the plectrum) of the right wing, creating a series of oscillatory impacts that are sustained and amplified by specialised resonating wing areas (the primary resonators), such as the mirror (Montealegre-Z, 2009; Montealegre-Z and Postles, 2010; Morris, 1999; Robinson and Hall, 2002). Whereas crickets and humpbacked crickets use coupled radiators in symmetric wings for sound radiation (Bennet-Clark, 2003; Chivers et al., 2016; Masaki et al., 1987; Montealegre-Z et al., 2011), male bush-crickets exhibit conspicuous wing asymmetry, with the sound radiators of the file-bearing wing being thicker and acoustically more damped than the plectrum-bearing wing (Chivers et al., 2014; Montealegre-Z and Mason, 2005; Sarria et al., 2016). This morphological and functional asymmetry results in the loss of an acoustic radiator, and different bush-cricket species have evolved various forms to compensate for this situation (Bennet-Clark, 1998), e.g. wing inflations (Heller et al., 2010), thorax depression (Braun et al., 2009; Morris et al., 2016) and pronotal covers (Morris and Mason, 1995).

In most species of tettigonids, the pronotum is flat and either leaves the stridulum on the forewings free or only partly covers them, so the produced sound signals are radiated unhindered or slightly baffled [although see Rentz (1988) for notable exceptions in the Arytropteridini of the Tettigoniinae]. In contrast to this, males of the Neotropical bush-cricket Acanthacara acuta (Tettigoniidae, Conocephalinae, Cestrophorini) possess an unusually inflated pronotum that stretches posteriorly to cover the entirety of the short, brachypterous wings and the first abdominal segment (Fig. 1). The extent of this pronotal inflation is such that it covers the dorsal half of the thorax laterally and either encloses the wings completely or only leaves a small opening at the posterior part, depending on posture. This inflation thus resembles a cave or chamber and is therefore called the pronotal chamber (PC) from here on.

Because male A. acuta are small (∼2 cm) but produce very loud calling songs [85 dB SPL re. 20 µPa at 10 cm (Morris and Mason, 1995)] and females do not exhibit an inflated pronotum (Gurney, 1972), it seems very likely that the PC is connected to sound production and serves to amplify and/or direct the song to increase male courtship success. Indeed, Morris and Mason (1995) investigated the role of the PC and the possibility that it could function as a (secondary) Helmholtz resonator, thereby greatly amplifying and filtering the acoustic output. Although Morris and Mason's research showed that the PC of A. acuta does amplify and sharpen the produced songs when compared with songs that were produced by males with the chamber removed, they only provided estimates for morphological properties of the chamber and therefore only concluded that its ‘function as a Helmholtz resonator is plausible’ (Morris and Mason, 1995).

In a Helmholtz resonator, a volume of gas (usually air) in a confined space with a small opening (e.g. a neck or a hole) is set into motion by a driving force whereby the gas is either compressed or rarefied. This creates either a high or low pressure field, expelling or pulling in air through the opening, respectively. Owing to the momentum of the flowing gas, the pressure field does not equalise to the surrounding pressure but reverses, creating a low pressure field where gas is pushed out and a high pressure field where it is pulled into the confined space. Once set in motion, this process repeats in an oscillatory fashion, like a mass on a spring, where the frequency of oscillation – the resonance of the system – is dependent on the volume and density of the gas and the size and height of the opening, and largely independent of the structural material (Bennet-Clark, 1999; Fletcher and Rossing, 1998). Well known examples of Helmholtz resonant systems are the air cavities in musical instruments such as the guitar and violin, and when blowing air across the top of an empty bottle to create low frequency tones (von Helmholtz, 1863).

If the PC of A. acuta is indeed a Helmholtz resonator for the calling song, it should show resonance at the song's carrier frequency (CF) when stimulated non-invasively with broadband sound. This resonance frequency should match with theoretical Helmholtz resonance calculations using the chamber's geometrical dimensions, and should be independent of the chamber material. Furthermore, removing the PC from the insect should destroy the Helmholtz resonator and change the chamber's resonance, revealing the material's vibratory properties.

Using scanning laser Doppler vibrometry (LDV), microcomputed tomography (µ-CT), 3D-printed models of the insects and finite element modelling (FEM), this research is designed to show that the pronotal chamber of A. acuta not only resonates at the male's song frequencies but also acts as a Helmholtz resonator owing to its morphological dimensions and structure.

Five Acanthacara acuta Scudder 1869 males were collected in Ecuador (Santa Lucía Cloud Forest Reserve, Barrio La Delicia, Nanegal, Pichincha, Ecuador, 10–20 September 2014) and transported to Lincoln, UK, where they were kept in climate-controlled chambers (16°C, 70% relative humidity, 12 h:12 h day:night cycle) until experiments were performed. Animals were fed ad libitum with apple, baby corn and fish flakes (Sinclair Animal & Household Care Ltd, Gainsborough, UK). One pinned specimen (provided by Glenn Morris, University of Toronto, Canada) was collected in Bellavista, Ecuador in 1988. All experiments were carried out on these six specimens.

Of the five specimens collected, songs of four freely behaving males were recorded in the field (18°C) using a Tascam DR-05 Version 2 Dictaphone Linear PCM Portable Recorder (TEAC Corporation, Tokyo, Japan) and saved at a 96 kHz sampling rate.

The remaining male was recorded in the lab (temperature 23–26°C) with a 1/8″ Brüel & Kjær Type 4138 condenser microphone, connected to a Brüel & Kjær 2633 preamplifier (Brüel & Kjær, Nærum, Denmark). Signals were recorded via the LDV internal data acquisition board (National Instruments PCI-4451; Austin, TX, USA) and stored as text files (96 kHz sampling rate) for later analysis with MATLAB (R2014a, MathWorks, Inc., Natick, USA). Spectral analysis was performed on recorded song pulses using power spectral density estimates (discrete Fourier transform, nfft=1024 points, Δf=94 Hz) over each pulse. Averages for the CF were calculated for each individual and the resulting means were pooled to gain an overall mean and s.d. (see Table S1).

To obtain the resonance properties of the wings and the PC, we used micro-scanning laser Doppler vibrometry [PSV-F-500 laser Doppler vibrometer, Polytec, Waldbronn, Germany; see Montealegre-Z et al. (2011) for a detailed description] to measure the vibration velocities and displacements of these structures in response to stimulation with broadband periodic chirps. In short, animals were positioned and fixed with metal clamps on a small, custom-built holder with the thorax bent forward so that the entire chamber entrance is exposed, as when singing naturally. The entire dorsal and lateral surfaces of the PC were scanned while moving the laser beam around the specimen and keeping the speaker in a fixed position dorsal to the specimen. Broadband audio chirps (2–100 kHz) were generated by the PSV-500 acquisition board and transmitted to an ultrasound loudspeaker (Avisoft Vifa, Avisoft, Glienicke, Germany) positioned 30 cm away from the animal via an amplifier (A-400, Pioneer, Kawasaki, Japan). The speaker output was flattened to broadcast all frequencies at equal amplitudes (±3 dB SPL) at the wings as measured by a calibrated reference microphone (see above). Vibrations, reference stimulus and internally calculated transfer functions of the measured vibration relative to the measured sound pressure were recorded at a 256 kHz sampling rate [Δf=15.6 Hz; see Montealegre-Z et al. (2011) for details], and analysed and stored using PSV 9.0 (Polytec, Waldbronn, Germany). Experiments were carried out on four of the collected animals and one preserved specimen at temperatures between 23 and 26°C.

After these recordings, the chamber was removed using micro-dissection tools, exposing the wings fully. Wings were elevated and fixed in position with a drop of wax at the base of the axillary sclerites and scanned using LDV as described above (see also Chivers et al., 2017; Montealegre-Z et al., 2011; Sarria et al., 2016).

To investigate the vibrational properties of the PC itself, two pronota dissected from two living specimens were scanned using LDV. Removed pronota were fixed at the anterior end on a small pin using bees wax. Results should provide vibrational characteristics of the open cuticular structure, unimpeded by thorax and abdomen. If results from the intact PC are to some degree due to Helmholtz resonance, the removed chamber vibrations should show marked differences. Also, because Helmholtz resonance is not influenced by material properties (here, the hard cuticle of the pronotum and the softer cuticle of the thorax and abdomen), it should be possible to restore the original vibrational patterns by artificially sealing the ventral side (or groove) of the removed PC, recreating the original dimensions of the resonator. In order to test this, the ventral side of the PC was covered with Blu Tack (Bostik Ltd, Leicester, UK) to approximately achieve a similar internal volume and opening as in the natural condition. This sealed PC was then rescanned using LDV.

In addition, one 1:1 scale model of A. acuta was 3D printed and subjected to LDV recordings to investigate the resonances of the PC when made of artificial materials (see below). The model was fixed to a holder and scanned using LDV and acoustic stimulation as described above.

X-ray µ-CT of one adult A. acuta male was performed using a SkyScan 1172 µ-CT scanner (Bruker Corporation, Billerica, MA, USA) with a resolution of 10 µm (50 kV source voltage, 200 µA source current, 420 ms exposure and 0.6-deg rotation steps). Before the µ-CT scan, the animal was asphyxiated with ethyl ethanoate (ethyl acetate; Sigma-Aldrich Company Ltd, Dorset, UK), placed upright in a small Eppendorf tube and positioned in a custom-built holder in the CT scanner. µ-CT projection images were reconstructed to produce stacks of orthogonal slices with NRecon (v. 1.6.9.18, Bruker Corporation, Billerica, MA, USA), stacks were processed with CTAn (v. 1.15.4, Bruker Corporation) and virtual 3D models were built from the resulting images using Avizo (v. 8.0, FEI, Hillsboro, OR, USA). Models were 3D printed on a Form 1+ (Formlabs, Inc., MA, USA) with a layer thickness of 25 µm, using white resin (White Resin Version 2, Formlabs, Inc.).

After creating models of the animal, its wings and PC, the surface geometry of the latter was saved in STL format and post-processed in 3ds Max 2015 (Autodesk, Inc., San Rafael, CA, USA) to ensure fidelity of the geometry. The model was then used to carry out the FEM to resolve the PC eigenfrequencies and frequency responses using Comsol (v. 5.2a, Comsol, Inc., Burlington, VT, USA).

Here, the chamber was modelled as a simple shell structure coupled to an acoustic pressure air domain with the following material properties: Young's modulus E=8 GPa; Poisson's ratio ν=0.33; cuticle density ρ=1200 kg m⁻³ (values taken from Klocke and Schmitz, 2011; Malkin et al., 2014; Vincent and Wegst, 2004). Values for Young's modulus for insect cuticle in the literature range from several 100 MPa to ca. 50 GPa, depending on the species, the structure investigated and method used (Klocke and Schmitz, 2011; Smith et al., 2000; Vincent and Wegst, 2004). Because no data for the pronotum of any bush-cricket or comparable insect could be found, the value of 8 GPa was chosen as reported for tibia cuticle in Vincent and Wegst (2004).

Frequency responses of the chamber to sound were modelled by placing a dipole point sound source (with a fixed 1e–6 N force vector pointing dorsally, resulting in an SPL of ca. 98 dB at the source) into the middle of the chamber model, approximately where the wing resonators would be localised in the animal. Using parametric sweeps, the response of the chamber to sine wave pure tones at a range of frequencies (2–40 kHz at 50-Hz intervals) was analysed.

The finished model had 1894 meshed faces with an average element quality of 0.95±0.05 and an average face size of 0.040±0.009 mm². An increase in the number of meshed elements for the model did not significantly alter the computational results (<0.1% for 5218 faces); we therefore assumed that element quality and quantity was sufficient to produce stable solutions.

Spectral analysis of the male calling song (Fig. 2E; Fig. S1) shows that most of the song's energy is contained within a relatively narrow frequency band between ca. 8 and 11 kHz (pooled mean CF=9.2±0.2 kHz, n=5; see also Table S1). Recordings under lab conditions exhibit an additional low-amplitude frequency peak at 26–29 kHz that is not present in field recordings. As similar ultrasonic frequency components were reported by Morris and Mason (1995), we assume that the higher frequencies are present in all songs but attenuate quickly under field conditions.

LDV of the intact pronotal chamber of two living A. acuta males revealed two main resonance peaks at frequencies around 5 and 11 kHz (4.7 and 6 kHz, 10.4 and 11.6 kHz for Male1 and Male2, respectively; see Fig. 2A,B,E and Fig. S2), with vibration amplitudes of the latter being 11–22% lower than the former. Vibratory responses decline sharply after ca. 13 kHz and no additional responses can be observed in the remainder of the investigated frequency range (2–100 kHz; only frequencies up to 40 kHz are shown here). LDV of three preserved specimens present a similar picture, each showing a resonance peak between 10 and 12 kHz (Fig. S2). In addition, the preserved specimens show one or more lower-frequency peaks, although the number and location of these are variable.

Removal of the PC of one male (Male1) exposed the animal's forewings. As is the case in most Tettigoniidae (Chivers et al., 2014; Morris, 1999; Richards and Davies, 1977), the left forewing lies on top of the right one and the wings are morphologically asymmetrical, with the main difference being the enlarged stridulatory file (on vein CuPb) on the left wing, which is hardly developed [or ‘vestigial’ (Morris et al., 2016)] on the right (Fig. 2C,D).

LDV recordings of the forewings also demonstrate a clear vibratory asymmetry between left and right wings, as the right wing exhibits strongest vibratory responses at 11.5 kHz, whereas the left wing is highly damped at this frequency, resulting in only minimal vibrations on this side (Fig. 2C, Movie 1). In contrast, at the slightly lower frequency of 10.1 kHz, the resonance properties are reversed so that the left wing is active, whereas the right is damped (Fig. 2D). In both the left and right wings, the area of vibrations is a field delineated by veins CuPb anally, CuPa costally and CuPaα laterally, and bisected by vein CuPaβ, resulting in a heart-shaped vibratory pattern [venation based on Béthoux and Nel (2001) and Chivers et al. (2017)]. As is the case in the PC, the left wing also shows a lower resonance peak at 6.2 kHz (Fig. 2E). The results of the LDV experiments of both PC and wings therefore show matched resonances of these two systems that coincide with the song's CF of 9.2 kHz.

To test the hypothesis of the PC functioning as a Helmholtz resonator, the pronotum of one specimen (Male3, see Fig. S2C) was removed from the animal and scanned both on its own and with its ventral side covered with Blu Tack to mimic the presence of the abdomen closing the chamber and forming a confined space. Fig. 3 shows the vibrational responses of the dorsal part of the PC intact, removed from the animal and covered ventrally. The intact PC exhibits, like mentioned above, pronounced resonances around 3 kHz and 10 kHz (Fig. 3, black line) that are greatly diminished when the pronotum is removed from the animal (Fig. 3, red line). Vibrational responses for the removed chamber show various lower-amplitude frequency peaks from ca. 3 to 11 kHz and a broader resonance from ca. 17 to 20 kHz that are all absent in the responses from the intact chamber. When the ventral side of the PC is subsequently covered with Blu Tack to recreate the internal space previously formed by the chamber and the thorax and abdomen, the original vibratory pattern is recovered, albeit with lower amplitudes (Fig. 3, blue line). This suggests that the recorded vibrational responses from the PC of intact animals are mostly due to the chamber's internal space rather than a result of the chamber's inherent material properties alone.

To further investigate the above hypothesis, a 1:1 scale model of A. acuta was printed in 3D and examined with LDV as stated before for real specimens. Recorded vibrations were not as consistent over the whole area of the PC as in animal specimens but LDV results identified an area – laterally at the posterior end, similar to the one shown in Fig. 2A – that exhibited a pronounced, sharp resonance at 11.7 kHz, closely resembling results obtained from living animals (Figs 2,3) and a lower-amplitude broader frequency response at ca. 25 kHz [Fig. 4; low-frequency peaks at 3 kHz were subsequently identified as external noise through control measurements (Fig. S3)].

These results provide further evidence that the resonances recorded from the PC in living animals is due to Helmholtz resonance, as the same results can be obtained with a model of the resonator that does not exhibit any of the original's material properties.

A virtual 3D model of one A. acuta male was produced using data obtained from µ-CT (Fig. 5A, Movie 2). From this model, the volume of free space within the PC that is not occupied by the abdomen, thorax or wings was measured to be 64.8 mm³, with the opening of the chamber having an area of approximately 5.9 mm² (Fig. 5B) with an equivalent radius of 1.4 mm.

Here, c is speed of sound in air (343 m s⁻¹), A is cross-sectional area of the opening with radius r, l is the length of the neck and V denotes the volume of the resonator (Fletcher and Rossing, 1998; Rossing and Fletcher, 2012).

Using values derived from the µ-CT model in the latter equation, the resonance frequency for the PC was calculated as f_h=10.8 kHz. This value fits both with the experimental LDV results of the freely vibrating PC in response to broadband sound stimuli (10.4 and 11.6 kHz, respectively) and with the calling song CF (9.2 kHz). It also corresponds very well to resonance values obtained from experiments with isolated but ventrally closed and artificial chambers (Figs 4,5). Therefore, these theoretical results further support the hypothesis of the PC functioning as a Helmholtz generator.

The 3D mesh created above (Fig. 5B) served as basis for two separate, simplified linear elastic and isotropic shell models, one ventrally open (Fig. 5C) and one closed (Fig. S4). These were used in FEM to simulate the vibratory responses of both open and sealed PCs in response to sound (Fig. 6).

Simulated frequency responses of the open chamber show high vibrations at 3.55 kHz, reduced vibrations at 6.15 and 10.55 kHz, and a series of smaller responses upwards from ∼16 kHz (Fig. 6, red line; Fig. 5C for vibration pattern at 10.55 kHz). This behaviour not only follows the general response patterns seen for the removed chamber in Fig. 3 (red line), but also shares similarities with measurements taken from intact PCs, in which lower-frequency resonances are followed by a resonance close to song frequency (Fig. S2), being followed by either minor further responses or none. Therefore, this result illustrates a good match between the experimental measurements and the FEM simulation.

In contrast, simulated results of the closed chamber show no vibrations in the lower frequency ranges but three clear resonance peaks with decreasing amplitudes at 13.3, 22.9 and 31 kHz (Fig. 6, blue line). The observed loss of the lower-frequency resonances and the abolishment of the spurious higher resonances suggest that the closed chamber approaches the function of a Helmholtz resonator with multiple, non-harmonic resonances. The simulated first frequency peak here is 13.3 kHz, a higher frequency than the song frequency and resonances of intact chambers (9.2 and 10–12 kHz, respectively). This might be a result of the chamber opening being too big or of mechanical constraints in the model increasing overall stiffness of the system, such as the rigid seal used to close the chamber dorsally. These pronotal frequency responses as a result of FEM again match the experimentally determined resonances (Fig. 2A,B) and, although to a lesser degree, the mathematically calculated resonance and the general behaviour of a neckless Helmholtz resonator with similar dimensions.

Our results show that the production of loud calling songs of A. acuta relies on the use of not only two, but three mechanical oscillators – the wings, the PC (possessing structural resonance), and the Helmholtz resonator formed by the chamber and the animal's body – that are set into motion and driven at or close to resonance by the animal's stridulatory movements.

The first oscillator is activated during stridulation: the movement of the scraper across the teeth on the file-bearing wing creates soundwaves through a frequency multiplication process (Bennet-Clark, 1999) at frequencies between 8 and 11 kHz (mean CF=9.22±0.22 kHz; Fig. 2). The sound is amplified by the wings through activation of vibratory modes close to its CF (the wing's resonance frequency; here, 10.1 and 11.5 kHz for the left and right wing, respectively; Fig. 2). It is rather unusual for tettigonids to exhibit two main active resonators in their wings because the highly asymmetric damping of the file-bearing left wing usually renders this wing ineffective for the radiation of sound (Montealegre-Z, 2012; Montealegre-Z and Mason, 2005; Montealegre-Z and Postles, 2010). However, Sarria et al. (2016) described the resonances of another South-American bush-cricket, Copiphora vigorosa, that shows a similar shift in the resonators between right and left wings (with the resonance of the file-bearing wing being at a higher frequency) and reduced amplitudes (by ∼50–75%) in the left wing (Sarria et al., 2016). In both cases, the observed wing asymmetries are not as pronounced as in other tettigonids, leading to – in comparison – less damped left wings. This tendency to wing symmetry seems to be a common feature of Tettigoniidae species with wings encapsulated by the pronotum (Morris et al., 1975). The lower resonance of the file-bearing left wing in A. acuta could be explained by the added mass from the inflated file vein, whereas the decreased amplitude is most likely due to increased damping of the vibratory field.

In field crickets (Gryllus bimaculatus), where wings are more symmetric in general morphology than in tettigonids, an asymmetry in the vibration patterns of the wings can be observed as well, with lower resonances seen in the left wing than in the right (Montealegre-Z et al., 2011). However, this difference in resonance between the wings disappears when the animal is actively stridulating, with both wings converging on the lower resonance of the left wing (Montealegre-Z et al., 2011). Because the basic process of stridulation is mechanically very similar in crickets and bush-crickets, it is conceivable that the wings of A. acuta will likewise converge on one resonant frequency during song production.

The second resonator in the song production system is the PC itself. Using LDV recordings on isolated PCs and FEM, we have experimentally and theoretically shown that the PC exhibits a wide range of lower-frequency resonances, including one in the frequency range of the song. It is this vibratory mode that, during stridulation, will be activated to further increase the amplification of the soundwaves produced by the wings.

Furthermore, we have shown, through comparisons of the intact chamber's vibratory responses with those of chambers removed from the animal's body (Figs 3,4), vibratory responses of artificial models (Fig. 4), theoretical calculations and FEM (Fig. 6), that the resonance pattern of the PC changes dramatically from its intact to its removed form. When intact, it resonates as a whole at frequencies nearly identical to the resonant frequencies of the animal's left wing (10.4 kHz and 10.1 kHz for PC and left wing, respectively; Fig. 3, Fig. S2), which is close to the song's CF and within its bandwidth. Examination of a 3D printed artificial A. acuta model using LDV produced similar results (Fig. 4), whereas removal of the chamber from the animal resulted in the breakdown of observed resonances and emergence of frequency responses as described above. Artificially closing the PC to mimic the state of an intact chamber was used to recreate the original frequency responses of the system. These results suggest that the overall resonant behaviour of the chamber is strongly dependent on the cavity that is formed by the chamber and the animal's thorax and abdomen (Fig. 5B), and that this behaviour is, to a certain extent, independent of the chamber's material properties, which suggests a Helmholtz resonator as the basis. The theoretical calculations using the inner geometry of the chamber further document that the observed behaviour can be explained with Helmholtz resonance, and FEM substantiated this by showing that the closed model does indeed produce frequency responses like an harmonic oscillator with a fundamental resonance only slightly higher than the song frequency. These higher resonances, normally not seen in classical Helmholtz resonators, could be a result of the FEM approach, where the full numerical solution produces additional eigenfrequencies that might not be stable under real conditions (Kobayashi et al., 2009). Therefore, we propose that the third resonator in this system is the PC functioning as a Helmholtz resonator, adding more amplification and bandpass filtering to the produced song (the latter being demonstrated conclusively by Morris and Mason, 1995).

Having more than one resonating system for sound production is highly advantageous for small insects wanting to produce loud sounds for mate attraction because the secondary resonator does not only amplify the sound further but it also provides improved impedance matching to the surrounding air owing to its increased size, and therefore better sound radiation (Bennet-Clark, 1999, 2001; Fletcher, 1992). A secondary resonator can also act as a bandpass filter, restricting the frequency range of the sound coming from the first resonator. Although there are multiple examples of animals using primary and secondary resonators in sound production (see Introduction), the use of a tertiary resonator in animal acoustic communication has not been reported thus far. Likewise, the effect such a resonator – here in the form of the chamber resonating at the song frequency owing to its own resonant properties – has on the overall amplitude and frequency component of the resulting song is largely unknown. However, for musical instruments such as violins, it is well known that the effects of different cavity and wood resonances can be combined and coupled to produce, if not necessarily louder, richer tones (Benade, 1990; Hutchins, 1990).

To conclude, using modern imaging techniques, LDV and FEM, we show that the PC is indeed acting as a Helmholtz resonator at the frequencies of the animals' calling song. In addition, we also show that the vibratory pattern of the PC alone exhibits resonances in the frequency range of the calling song, which allows the animal to utilise not two but three coupled resonators for sound production, amplification and radiation.

The authors thank the Editor and two anonymous reviewers for their valuable comments that greatly improved the quality and clarity of the paper.