Transduction of mechanical energy into sound energy in the cicada Cyclochila australasiae

Cicadas produce sound by rapid buckling of a pair of domed tymbals situated on the sides of the first abdominal segment (Pringle, 1954). Each tymbal is a highly specialised structure bearing an oval posterior sclerotised tymbal plate, anterior to which runs a row of narrow vertical ribs of sclerotised cuticle (Fig. 1). These sclerites are separated by and surrounded by strips of the elastic protein resilin (Weis-Fogh, 1960; Scott, 1970; Young and Bennet-Clark, 1995).

During sound production, the posterior tymbal plate is pulled inwards by a large fast muscle (Pringle, 1954; Simmons and Young, 1978). This tymbal muscle acts as the source of energy in cicada sound production. Each muscle contraction appears to provide a train of store-then-release cycles of energy to the sound-radiating system via the elastic strain of the tymbal followed by the release as each tymbal rib buckles. The tymbal muscle has no muscular antagonist, so re-extension of the muscle is brought about by the elastic strain energy in the buckled tymbal.

Initially, the inward movement is opposed by the convex tymbal ribs. As the tymbal plate is pulled in further, the tymbal rib adjacent to it buckles suddenly, becoming concave. This rapid buckling movement and the resonant vibration of the tymbal plate produce a pulse of sound. Further contraction of the tymbal muscles causes the more anterior ribs to buckle; as each subsequent rib buckles, a pulse of sound is produced (Simmons and Young, 1978; Young and Bennet-Clark, 1995; Bennet-Clark, 1997). In the Australian cicada Cyclochila australasiae, the tymbal plate plus ribs, in their elastic surrounds, act respectively as the mass and compliant elements of a mechanical resonator (Bennet-Clark, 1997).

The vibration frequencies of this resonant system in C. australasiae are close to the dominant frequency of the insect’s song, so the tymbal acts as major determinant of the song frequency. The tymbal also acts as a frequency multiplier that converts the 117 Hz contraction frequency of each of the paired tymbal muscles into the 4.3 kHz frequency of the insect’s song (Bennet-Clark, 1997; also see Michelsen, 1983, for a general discussion of the role of frequency multipliers in sound production.)

In many cicadas, the transduction of sound from mechanical energy into acoustic energy takes place in distinct stages. During the first stage, the pulses of sound produced by the tymbals cause high-pressure acoustic vibrations within the abdominal air sac. The abdominal air sac and the large thin eardrums of C. australasiae form, respectively, the compliant and inertial elements of a Helmholtz resonator tuned to the song frequency (Young, 1990; Bennet-Clark and Young, 1992). This second stage in the transduction chain maintains the purity of the song and assists in producing a smooth song pulse envelope. Because the eardrums are far larger than the tymbals, this second stage also acts as an acoustic impedance converter between the tymbals and the surrounding medium (Bennet-Clark and Young, 1992; Bennet-Clark, 1995).

In C. australasiae, the tymbal has four ribs (Fig. 1). As each rib buckles, it converts a comparatively slow muscle contraction into a brief sound pulse. Each of these sound pulses has maximum amplitude in the first cycle and thereafter decays exponentially (Bennet-Clark, 1997). This suggests that the tymbal acts as an energy storage/release mechanism which provides an impulse that starts the sympathetic vibration of an abdominal Helmholtz resonator (Young and Bennet-Clark, 1995).

The action of the tymbal muscle on the tymbal can be modelled either by pulling on its apodeme or by pushing on its insertion on the tymbal plate (Simmons and Young, 1978; Bennet-Clark, 1997). Previous studies have been concerned with the nature of the sound produced as the tymbal buckled and have been essentially qualitative. However, as the transduction process from muscle power to acoustic power in this cicada occurs in a comparatively small number of stages, it is feasible to examine the energetics of transduction of mechanical power to sound power. An earlier attempt to do this with the mole cricket Gryllotalpa vineae (Bennet-Clark, 1970) suffered from uncertainty about the available muscle power, but nonetheless suggested that the efficiency of transduction was remarkably high.

The insect used here is particularly suitable for energetic studies of this type. It is large and robust, and the sound is produced as a long series of similar discrete pulses, each of which is produced by a single muscle contraction, in contrast with the songs of many other singing insects such as crickets (Popov et al., 1974) or cicadas (e.g. Fonseca, 1991) in which far greater inter-and intra-pulse variability occur. Also, many elements in the sound-producing chain of Cyclochila australasiae have now been studied (Bennet-Clark, 1997; Bennet-Clark and Young, 1992; Josephson and Young, 1981; Young, 1990; Young and Bennet-Clark, 1995).

The present work examines the energetics of various stages in the sound-production chain of the cicada C. australasiae: the tymbal muscle, the tymbal buckling process and the sound power that is produced.

Male Cyclochila australasiae Donovan were caught at night in parkland in Melbourne, Australia, as they emerged from the last larval instar. Thereafter, they were kept in fine mesh bags on a tree outside the Zoology Department of Melbourne University or on acacia shrubs in the laboratory. In these regimens, they survived for over 2 weeks. Insects were used for experiments between 4 days and 2 weeks after eclosion; only those that produced loud protest song when handled were used.

For most experiments, insects were prepared by removing the legs and wings, and then waxing the body to a 6 mm diameter support rod by the pro-and mesonotum. In addition, for force and distance measurements, the body was made stiffer by waxing the first abdominal tergite to the metanotum and the second abdominal sternite to the opercula on the thoracic metasternum using a 5 mm length of femoral cuticle.

Singing was induced by brain stimulation via a pair of 0.1 mm diameter stainless-steel insect pins inserted into the front of the head 2 mm either side of the mid-line and 45 ° above the horizontal plane. Sound production was then induced by short trains of 1 ms duration stimuli at 50 Hz and 2–5 V amplitude. Insects were mounted head up and, to stretch the abdomen and open the opercula to simulate the position found in singing insects, a 20 g weight was suspended on a 50 mm length of wire waxed to the posterior end of the abdomen.

For force measurements on the tymbals, insects were killed by placing them in a freezer at −15 °C for 30 min and then thawing them immediately prior to use. This procedure effectively detached the tymbal muscle from its apodeme and sternal origin and also made it easy to dissect out the tymbal muscles for weighing.

The area of the tymbal muscle insertion on its apodeme (see Fig. 3C) was measured from camera lucida drawings using a Zeiss MOP digital measuring table. Muscle fibre lengths were measured directly from the intact insect using Mitutoyo digital callipers.

Tymbal muscles were weighed to the nearest 0.1 mg after dissection from the previously frozen insect. Weighings were completed within 5 min of the start of the dissection.

The trajectory of the tymbal muscle fibres was measured from photographs of the abdomen, taken from behind after removal of the posterior end or from the mid-line after splitting the insect’s body along the sagittal plane. The angle of the tymbal muscle apodeme was measured from dissections of dried specimens.

Because the available commercial force and distance transducers were unsuitable, special transducers were built, tested and calibrated.

Measurements of the force–distance relationships of the tymbal were made using a stiff transducer with a resonant frequency of 1 kHz (Fig. 2). The springs were 20 mm lengths of 12.5 mm wide by 0.15 mm thick stainless-steel shim glued with Sylastic silicone rubber to either side of a 12 mm high U-shaped steel support. Two Showa F8 foil strain gauges were glued with cyanoacrylate adhesive to the central parts of each of the two outer faces of the steel shims (Fig. 2). These four strain gauges were connected as a Wheatstone bridge.

The stiff transducer was used to apply force to the tymbal via a probe rod glued into one end of the aluminium tube that connected the springs on either side of the gauge. A hook for attachment of calibration weights was attached to the other end of this tube. As the effective mass of the tymbal is less than 1 mg (Bennet-Clark, 1997), the loading of the 0.5 g mass of the force transducer by the tymbal was negligible. The compliance of this transducer was 49 μm N⁻¹, which was approximately one-twentieth of that of the tymbals. Force could be resolved to 0.01 N at 200 Hz with linearity and cross-talk from side loads of better than 2 %.

The distance moved by the probe rod of the force transducer was measured using a UGN 3503 linear Hall effect sensor placed 0.5 mm away from a permanent magnet with a flat face 2.5 mm high and 4.5 mm wide glued to the tube that held the probe rod (Fig. 2). The distance could be measured to ±5 μm over a range of 1 mm. According to the makers’ specification, the bandwidth of the distance sensor was from direct current to over 10 kHz.

A more compliant force transducer with a resonant frequency of 100 Hz was constructed for measuring the force and distance relationships of the tymbal muscle. Its compliance, 990 μm N⁻¹, was similar to that of a typical cicada tymbal (see Fig. 8). Its sensitivity, its force and distance resolution and its linearity were similar to those of the stiffer transducer. This compliant transducer was used as an elastic load into which the tymbal muscle contracted; this type of auxotonic loading has been used to measure the force–distance relationships of locust flight muscle (Neville, 1965; Neville and Weis-Fogh, 1963).

Both the force and distance transducers, as well as the preparation, were mounted on short 6 mm diameter brass or light alloy rods clamped in specially machined holders bolted to the Prior micromanipulators. All micromanipulators were mounted as close as possible to a 12 mm thick machined-steel baseplate using magnetic stands. Tests in which a force of 1 N was applied across the apparatus showed that the movement between the end supports was less than 10 μm.

Stepwise force calibrations using weights and stepwise distance calibrations were carried out during every experiment, and also provided a test of the linearity of the response of the apparatus.

One force transducer was connected to a MacLab bridge amplifier with a bandwidth of 2 kHz.

Most force and distance data were recorded on separate channels of an Analog Digital Instruments MacLab 4 12-bit data-acquisition system using Chart 3.5 software at up to 1000 samples s⁻¹ with the channels sampled alternately, not simultaneously. Some force–distance measurements were also made using Scope 3.5 software at sampling rates of 4 or 10 kilosamples s⁻¹.

The software allowed baselines and scales to be defined. Using the x–y display, force versus distance work loops were calculated directly. The work done or released was then obtained as the area under different regions of the work loops.

The insect preparation was mounted on one micromanipulator with its long axis parallel to the baseplate. The stiff force transducer (Fig. 2) was mounted on another micromanipulator with its probe rod also parallel to the baseplate. Using one protractor placed on the rod on which the insect was mounted and another on the baseplate, the position of the insect could be adjusted so that the angle of push on the tymbal plate could be varied from 110 ° to a practical maximum of 170 ° above the mid-ventral line of the insect and from 90 to 120 ° relative to the anterior-to-posterior long axis position (0 ° by 0 ° was taken as mid-ventral and anterior: see Figs 3A,B, 11). The precision of these settings was less than ±3 °.

After the force and distance transducers had been zeroed, the insect was brought towards the probe rod of the force transducer until contact was detected as a positive force transducer reading. The vertical and horizontal positions of the insect were adjusted so that the tip of the probe rod entered the apodeme pit on the tymbal plate (Figs 1, 3C). Force and distance recordings were then made as the probe rod was moved forwards and backwards rapidly over a distance of 0.5–0.8 mm; with practice, the complete in–out movement was achieved in 0.3–0.4 s. Force versus distance loops were then plotted using the MacChart or MacScope software; these loops were highly consistent (see Fig. 6).

At the end of each series of measurements, the angle of the probe rod was changed by 10 ° relative to the coordinates of the insect body, and another series of measurements was made. In most cases, 10 measurements were made at 200 samples s⁻¹ followed by a further 10 at 1 kilosample s⁻¹. Some measurements were also made using MacScope to provide greater temporal resolution.

Tymbal muscle preparations were used in situ and as far as possible in their normal orientation. Using a live animal prepared for brain stimulation, a stainless-steel wire stirrup was attached with cyanoacrylate glue to the outside of the tymbal plate, with the centre of the stirrup set to pull along the line of the tymbal apodeme. After the glue had set, a ring of cuticle was cut away round the stirrup, detaching the apodeme and stirrup from the tymbal. The stirrup was then connected to a force transducer via a 20 mm long loop of 0.3 mm diameter stainless-steel wire.

Force and distance measurements were made using the more compliant force transducer. Muscle contraction was elicited by brain stimulation (as described above): typically, a brief stimulus resulted in a train of between 5 and 20 muscle contractions.

The temperature of the preparation was measured using a 0.2 mm diameter thermocouple placed inside the abdominal air sac of the cicada. The thermocouple was connected to a Bailey BAT 12 thermometer. The insect’s internal body temperature was raised from the ambient temperature of 24–25 °C to a maximum of 39 °C using a 60 W bench lamp. Repeat experiments at 28 °C before and after heating to 39 °C produced closely similar force and distance recordings.

Sound pressure levels were measured using a Bruel and Kjaer 2230 sound level meter and Bruel and Kjaer 4155 microphone. The sound level meter was set to measure the impulse peak maximum of the sound (which, for a continuous pure-tone signal, is 3 dB higher than the root mean square value that is normally quoted as the sound pressure level).

Checks on the validity of the readings obtained were carried out as follows. The 1 kHz waveform produced by a Bruel and Kjaer type 4230 calibrator, giving 94 dB root mean square sound pressure, was recorded via the alternating current output socket of the sound level meter using MacScope. The peak voltage of this recording was then compared with the peak voltage of recordings of the insect sounds produced by brain stimulation. This procedure gave values that were within 0.2 dB of those shown by the sound level meter.

The microphone was positioned 100±1 mm from the ventral surface of the insect at the opening of the tympanal opercula. The insect was rotated in a series of 45 ° steps, first around the transverse plane, then in planes 45 ° anterior and 45 ° posterior to the transverse plane, and finally readings were taken 100 mm straight in front of and behind the insect on the long axis. After a set of readings around the insect had been completed, a further set of readings was taken with the microphone in the starting position to check that the insect was still producing the same sound level.

The preparation was placed above an 85 mm thick sheet of Sonex anechoic foam. Further sheets of foam were inserted between the preparation and the support stands, and around the sides and over the top of the preparation. There was no evidence of echoes in our recordings of song made in these conditions.

Calculations of the ratio of peak to mean power in the song were made from field recordings of singing cicadas made in 1988 by D. Young, using a Nagra IVS tape recorder and Sennheiser MKH816 microphone. Portions of song containing two or complete three sound pulses were recorded onto MacScope at 100 kilosamples s⁻¹. Pulse period was measured from the MacScope recordings and used to obtain pulse frequency. Pulse duration was taken as the time from the start of the pulse to its decay to 10 dB below the peak level (Fig. 4A). The mean sound intensity of the pulses was calculated from the same recordings, following the procedure described below and illustrated in Fig. 4B. (This method of calculating mean sound intensity is similar to that used in Bennet-Clark, 1970.) (1) The sound level meter reading (in dB at 100 mm range) gave the amplitude of the loudest cycle in the song pulse as a sound pressure (as given by the peak impulse reading). (2) A digitised oscillogram of two pulses of song was made (see Fig. 4A). (3) The data from stage 2 were squared (see Fig. 4B). (4) The mean of the data set obtained in stage 3 was calculated (this gives a relative measure of the mean sound power in the pulse). (5) The ratio of the mean value of the sound power to the square of the amplitude of the largest cycle in the pulse was calculated (the amplitude of the largest cycle is obtained from stage 3). Stages 2–5 were all calculated by the MacScope software. (6) The fractional peak power to mean power ratio was converted into a ratio in dB (a power ratio of 10:1=10 dB). (7) The mean sound intensity, integrated thoughout the pulse (in dB) was obtained by subtraction of the values obtained in stages 1 and 6. 90 dB is a sound intensity of 1 mW m⁻² equivalent, in a plane wave, to a sound pressure of 0.63 N m⁻². (8) The mean sound intensity at 100 mm was converted to the range to a 90 dB isobar using equation 1. (9) The mean sound power in the song (in mW) is given by the area of the 90 dB isobar (in m²).

The total surface area of the sound field of the three-dimensional map that was built up was then used to calculate the peak sound power output of the insects. This was converted to an estimate of the mean power output using the peak power to mean power ratio obtained for field records of calling song (see Table 2).

Although our sound level measurements were recorded to the nearest 0.1 dB or ±2.4 %, the realisable precision of these measurements is unlikely to be better than ±0.5 dB or ±12 % because of inaccuracies in other parts of the measuring chain.

Sound pressure levels are quoted throughout this study in decibels (dB) relative to the accepted threshold: 0 dB= 20 ×10⁻⁶ N m⁻² (or relative to 20 μPa). This sound pressure level is equivalent, in the plane wave conditions that prevail here, to a sound intensity (power per unit area) of 10⁻¹² W m⁻².

Relative power is measured on a logarithmic scale in decibels. 10 dB is a 10-fold power ratio (=10¹). A 1 dB ratio=10^0.1 or a power ratio of 1.26. Sound power is proportional to the square of sound pressure level. Therefore, a 10 dB power (or sound intensity) ratio is equivalent to a sound pressure ratio of √10 or 3.16. Whether the power ratio is multiplicative or divisive is indicated by the sign: −3dB or ‘3 dB below’ indicates a power ratio of 10^−0.3 (1:0.5) or half power. For a fuller discussion of these relationships, see Olson (1957) or Fletcher (1992).

The anatomy of the tymbal of C. australasiae has been described previously in some detail (Young and Bennet-Clark, 1995; Bennet-Clark, 1997) (Fig. 1). The tymbal muscle inserts on the apodeme, which is connected to the tymbal plate via a short flexible strap-like length of the apodeme (Fig. 3C). The dorsal region on the tymbal plate from which the tymbal apodeme invaginates forms a discrete pit approximately 0.15 mm in diameter. This apodeme pit acts as the focus for the force produced by the tymbal muscle and also forms a convenient site for external application of force via a probe rod (Fig. 1).

The tymbal muscle consists of a bundle of long fibres which extend from the sternal origin to their insertion on the tymbal apodeme. At the ventral origin on the chitinous V of the first abdominal sternite (Fig. 3B,C), the cross section of the muscle is approximately oblong, but the cross section becomes approximately circular at the tymbal apodeme (Fig. 3C). The muscle has the following measurements (mean ± S.D.): muscle mass 87.1±14.3 mg (N=10); muscle fibre length 7.59±0.37 mm (N=8); and apodeme insertion area 13.0±1.77 mm² (N=7).

Viewed from normal to the sagittal plane (Fig. 3A), the fibres form a truncated triangle which is broader at the ventral origin of the muscle on the chitinous V. The most anterior fibres run upwards and backwards at 110 ° to the animal’s mid-ventral line, and the most posterior run upwards and forwards at 60–65 ° to the mid-ventral line, with the centre of the muscle block running at approximately 85 °. Viewed from behind (Fig. 3B), the fibres of the tymbal muscles run from close to the midline at their ventral origins to their dorso-lateral insertions on the tymbal apodemes at angles between approximately 152 ° and 165 ° to the mid-ventral line, with the centre of the muscle block running at approximately 160 ° to the mid-ventral line.

Viewed from the posterior, the dorsal regions of the tymbal plates lie at an angle of approximately 20 ° either side of the sagittal plane. The strap-like distal part of the apodeme of the tymbal muscle, which is aligned with the central axis of the tymbal muscle, meets the tymbal plate at approximately 155 ° to the sagittal plane and thus at approximately 45 ° to the dorso-ventral axis of the tymbal plate (Fig. 3D). Viewed from above, the tymbal plates lie at angles of approximately 45 ° either side of the insect’s long axis, and the strap-like regions of the apodemes run at 95 ° to the long axis, meeting the tymbal plates at approximately 50 ° to the horizontal axis of the plate (Fig. 3E). Taking the angles of 45 ° to the dorso-ventral axis and 50 ° to the horizontal axis of the tymbal plate, we calculate, by simple trigonometry, that the apodeme runs at an angle of approximately 36 ° to the plane of the tymbal plate.

The force required to push the tymbal plate inwards rises steadily until, suddenly, the tymbal ribs buckle inwards, releasing energy. The force–distance relationships of the inward movement were affected both by the position of the probe rod on the tymbal plate relative to the apodeme pit and by the angle or direction of the push of the probe rod.

Because the tymbal muscle pulls in a linear manner via its apodeme on the apodeme pit on the tymbal plate (Fig. 1) along the main trajectory of the muscle (Fig. 3), we tested the effect of changing both the position on the tymbal plate and the angle at the apodeme pit at which force was applied by the probe rod. The effect of moving the probe rod vertically on the tymbal plate from 0.4 mm dorsal to the pit (the dorsal edge of the tymbal plate) via the apodeme pit to 2 mm ventral to the apodeme pit is shown in Fig. 5A for a push approximately normal to the surface of the tymbal plate, at 120 ° behind the insect’s anterior in the horizontal plane and 140 ° above the mid-ventral line in the sagittal plane. Both the force required and the distance through which the probe rod had to be moved to bring about tymbal buckling were maximal when the tip of the probe rod was positioned in the apodeme pit.

The work required to bring about inward buckling of the tymbal, assuming that the tymbal obeys Hooke’s law (see Figs 6, 7), is given by 0.5 (force at buckling × distance at buckling). Fig. 5A shows that the work required was maximal when the tymbal plate was pushed inwards at the apodeme pit, but that only one-tenth as much work was required when the tymbal plate was pushed inwards 2 mm ventral to the apodeme pit.

The effect of altering the angle of push on the apodeme pit is shown in Fig. 5B. It was difficult to make measurements at angles of push of over 170 ° to the mid-ventral line or at angles of push anterior to the insect’s transverse plane because the probe rod tended to slip out of the apodeme pit. At angles of push of more than approximately 130 ° behind the insect’s anterior, the probe rod tended to be obstructed by the cuticle of the tymbal frame (see Fig. 3C). It also proved difficult to obtain consistent results, probably because of the difficulty of positioning the tip of the probe rod in the same place on the tymbal plate after changing the angle of the probe rod. However, the same general trend was observed with 10 tymbals: the force required to cause buckling and the distance through which the probe rod had to be moved before buckling occurred both approximately doubled as the angle of push was increased from 110 ° to 170 ° above the mid-ventral line. This is shown for one tymbal in Fig. 5B. The effect of altering the direction of push relative to the insect’s anterior–posterior axis was less marked (results not shown).

The force required to move the tymbal plate inwards increased steadily as the tymbal plate was pushed, until the tymbal ribs buckled (Figs 6, 7), when the force fell rapidly to less than half its peak value. Further inward movement was accompanied by a further increase in the force. As the probe rod was withdrawn, the force decreased more or less steadily, increased slightly as the tymbal ribs clicked outwards, then decreased again until the probe rod was removed from the tymbal plate. For any one preparation, the work (force–distance) loops that were obtained were highly repeatable (Fig. 6).

In summary, the work loops contain the following stages after contact between the probe rod and the apodeme pit: these are numbered on Fig. 6. (1) Energy is stored elastically during the initial phase of the inward movement; during this phase, the force rises approximately linearly. (2) The force decreases in a stepwise fashion as successive tymbal ribs buckle inwards. (3) After all the ribs have buckled, further inward movement of the tymbal plate causes further elastic distortion of the tymbal. (4) The initial part of the outward movement of the tymbal shows an elastic, but non-linear, release of the remaining stored energy. (5) During the latter part of the outward movement, the force increases as the tymbal ribs buckle outwards. (6) Finally, the force decreases to zero as the probe rod is removed from the tymbal plate.

The course of the storage and release of energy is shown in detail in Fig. 7 for one tymbal which was pushed at 150 ° above the mid-ventral line and at 110 ° to the anterior. The time course of the changes in force together with the distance moved inwards then outwards is shown in Fig. 7A, and the force–distance work loop is shown in Fig. 7B. The initial phase of energy storage for this particular tymbal was almost linear. Slightly concave and convex force–distance curves were recorded from other tymbals. Peak forces before tymbal buckling of between 0.28 and 0.55 N were recorded from 11 tymbals (Table 1).

This initial phase of movement of the tymbal plate was essentially elastic. Experiments in which the tymbal was pushed inwards and released over distances that allowed the tymbal plate to return outwards before buckling had occurred showed a force versus distance pattern that closely mirrored that during the inward phase (Fig. 8). This suggests that, during the first part of the inward movement that precedes rib buckling, the elastic surround (Fig. 1) acts as a simple spring controlling the movement of the tymbal plate.

Previous work has shown that tymbal buckling occurs in a series of steps as successive ribs buckle inwards (Simmons and Young, 1978; Young and Bennet-Clark, 1995; Bennet-Clark, 1997). Typically, the inward buckling of the first two or three ribs occurred over a time span of 1–3 ms, each of which produced a pulse of sound.

Buckling of the tymbal ribs was always accompanied by a decrease in force and thus a release of energy by the elastic regions of the tymbal. In most cases, it was difficult to measure the precise contribution of the buckling of each individual rib to the course of the release of energy, but Fig. 7B shows a work loop in which the stages of buckling can be distinguished. In this example, the force fell to approximately two-thirds of the maximum value as rib 1 buckled and then to approximately half the maximum as rib 2 buckled.

The force–distance relationships of the outward recovery movement after buckling tended to be markedly non-linear (Figs 6, 7). In many cases, the force remained almost constant for most of the movement but showed a small increase, near the end of the movement, as the tymbal ribs buckled outwards.

The areas of the force–distance loops give the energy stored and dissipated in different stages of the inward and outward movement. These areas and the associated storage and release of energy are shown for a typical force–distance loop in Fig. 7C. Initially, the force rises until the first tymbal rib becomes unstable and buckles inwards suddenly, with an accompanying rapid decrease in force and release of stored energy, followed in sequence by buckling of the other tymbal ribs. Of the total 69 μJ energy stored during the initial inward movement, approximately two-thirds (46 μJ) was dissipated as the tymbal buckled inwards, leaving the remaining one-third (23 μJ) available to re-extend the tymbal muscle and restore the tymbal to its resting position with the ribs buckled outwards.

Values for the force–distance relationships of 11 tymbals are given in Table 1. The force–distance relationships and the work loops of the tymbal shown in Fig. 7 are taken from a tymbal with properties close to the mean values reported in Table 1. Note that the measurements of distance assumed that the body of the cicada did not move when force was applied to the tymbal plate. In reality, it is likely that a small, but hard-to-measure, part of the inward movements that we recorded occurred as a result of distortion of the cicada abdomen. Consequently, the distances and energy values reported in Table 1 are likely to be slight overestimates.

The experiments reported below were carried out at the end of the season when fewer animals were available. The force, distance and work output of the tymbal muscle were measured to determine whether these variables were compatible with the mechanical properties of the tymbal. Preliminary results are presented here, although incomplete, because of their relevance to our other findings.

The activity of the tymbal muscles was recorded in response to contractions elicited by brain stimulation. In all cases, the muscle contraction rate was lower than the rate of 117 Hz for each muscle that occurs in calling song, but in three preparations we recorded rates of 75–80 Hz at 28 °C. Force and distance were recorded simultaneously: Fig. 9 shows an example of the force–length curves that were obtained with the muscle driving the compliant force transducer. The mean value of the peak active force recorded from seven preparations was 0.31±0.04 N (mean ± S.D.) and the mean change in length or strain of the work loop was 295±41 μm. This force is comparable with the value of 0.295 N obtained by Josephson and Young (1981) using C. australasiae (using an area of 13 mm² for the tymbal muscle apodeme) and the values for force per unit area cited by Josephson and Young (1981). The mean distance of shortening obtained here is 3.9 % of the muscle fibre length. The total work produced by the contraction of the muscle is given by the area of the approximately triangular region below the ends of the length–distance plot (Fig. 9). The mean work was 47.0±11.2 μJ (N=7) (mean ± S.D., range 30–61.3 μJ).

Note that the measurements of distance have assumed that the body of the cicada does not move when force is produced by the tymbal muscle. However, it is likely that a small, hard-to-measure, part of the inward pull by the tymbal muscle brought about an inward distortion of the cicada abdomen.

Consequently, the distances shortened and the energy values calculated above are likely to be underestimates.

We measured the effect of pre-stressing the muscle in three preparations. Passive stresses of 0.6 N could be applied reversibly. With passive stresses between 0.05 and 0.3 N, we found that the muscles produced force–distance plots that were closely similar in shape and area; in other words, over this range of passive stresses, the muscle appeared to produce a similar active stress over a similar distance of active shortening. With passive stresses less than 0.05 N or greater than 0.4 N, the active force became smaller. It thus appears that the muscle can contract over a range of passive stresses and lengths and still produce similar amounts of work per cycle of contraction.

In one preparation, raising the internal body temperature from 27 to 39 °C caused the rate of activation and contraction of the muscle following brain stimulation to increase from 73 to 97 Hz. Extrapolating from these data, the contraction rate of 117 Hz observed during singing would require a muscle temperature of approximately 42 °C. This temperature is comparable to the temperatures of 41–45 °C recorded for the tymbal muscles of the cicada Okanagana vanduzeei during singing (Josephson and Young, 1985).

The work areas we obtained from the tymbal muscles are broadly compatible with the work required for buckling of the tymbal (c.f. Fig. 9 and Table 1). Taking a mean muscle contraction rate of 117 Hz during singing and a mean muscle mass of 87 mg, we can calculate the specific muscle power that is required to buckle the tymbal. Fig. 10 shows how the force required for tymbal buckling and the inward strain of the tymbal equate with the mass-specific muscle power of the tymbal muscle. From Fig. 10, it appears that the tymbal muscle must produce between 75 and 125 W kg⁻¹ to account for the observed performance.

Using recordings of the calling song made in the field by D. Young, the structure of the songs of seven C. australasiae was measured and analysed. Variables describing the temporal structure of the song were calculated from oscillograms (see Fig. 4A for terminology) and are given in Table 2. The ratio of peak power to mean power in the song waveform was calculated according to stages 2–6 of the procedure laid out in Materials and methods and illustrated in Fig. 4B.

Sound fields were measured around three insects in which sound production was elicited by brain stimulation. The sound was loudest mid-ventrally and quietest along the body axis in the horizontal plane either directly anterior or directly posterior to the insect but, overall, the sound radiation pattern only showed a difference of 3 dB between the loudest and quietest directions.

These measurements were converted to give the effective size of the 90 dB sound pressure isobar as if the insect were producing normal calling song. The values for the peak impulse maximum sound pressure that had been made at 100 mm range were converted first by subtraction of 9.2 dB to give the mean sound intensity at that range. This value then was used to calculate the range to 90 dB sound pressure isobars (Fig. 11).

For the example shown in Fig. 11, the 90 dB sound isobar (equivalent to an intensity of 1 mW m⁻²) is approximately equivalent to an ellipsoid of radii 0.50 m ×0.50 m ×0.55 m. The surface area of this ellipsoid is 3.45 m². Thus, the mean sound power output of this particular insect was 3.45 mW; estimates from two other insects were 3.15 mW and 7.0 mW.

The peak sound pressures we measured are comparable with the values reported by Young (1990) for the same species. We found peak sound pressures of 116.2, 116.8 and 118.9 dB at 100 mm range for the three animals for which we had complete recordings. These are equivalent to 110.2–112.9 dB at 200 mm range. At 200 mm range, Young reported mean values of 109.9 dB +1.8 or −2.3 dB (mean ± S.D., N=5) for the protest song and 112.9 dB +2.9 or −4.4 dB (N=8) for the calling song of C. australasiae. The equivalent mean sound powers are 3.15 mW for the protest song and 5.1 mW for the calling song, assuming the same radiation pattern as in the present study, suggesting that the sound powers we measured in response to brain stimulation may be approximately two-thirds of those obtained under natural conditions (or approximately 2 dB lower).

The energy in each song pulse is the mean sound power divided by the song pulse rate (14.7 μJ assuming a pulse rate of 234 Hz, range 13.5–30 μJ in the present study and 13.5 μJ and 21.5 μJ for protest song and calling song from Young, 1990). These values can be compared with the work released by the buckling of all tymbal ribs in each cycle of tymbal buckling (Fig. 7; Table 1) for which a mean value of 45.1 μJ was obtained, which is substantially greater than that calculated for the song pulses.

Mechanisms in insects by which energy is stored slowly and released rapidly include the generation of sound pulses in cicadas (Pringle, 1954) and moths (Blest et al., 1963) as well as the catapult mechanism of jumping fleas (Bennet-Clark and Lucey, 1967) and many other fast-acting systems (for a review, see Gronenberg, 1996).

Previous experiments have shown that the tymbals of cicadas produce discrete clicks of sound as the ribs buckle inwards (Pringle, 1954; Simmons and Young, 1978; Bennet-Clark, 1997). The present work confirms these findings and provides an energy budget for the elastic distortion of the tymbal and the release of energy that accompanies the buckling of the tymbal ribs.

Previous work on the mechanical properties of the tymbal has shown that the thick resilin pad at the dorsal end of the tymbal (see Fig. 1) is a major elastic determinant of the resonant properties of the tymbal (Bennet-Clark, 1997). The tymbal apodeme attaches close to the dorsal end of the tymbal plate. The direction of pull along the apodeme is likely to cause both an inward movement of the tymbal plate as a whole and also an inward distortion of the dorsal resilin pad. The tymbal plate can be modelled crudely as if it were a rigid lever pivoted at its dorsal end in the resilin pad; its inward movement is opposed by the stiffness of the resilin pad and also by the resistance to buckling of the tymbal ribs.

If this tymbal plate lever is pushed inwards (or pulled inwards by its muscle), the applied force will have two main effects: to distort the resilin pad and to produce a turning moment about the effective pivot. The response of such a system will be affected both by the position at which force is applied and by the angle at which the force is applied. These are modelled in Fig. 12.

Consider two cases in which force is applied to different regions of the tymbal plate: first at the apodeme pit where the force is applied close to the pivot; and, second, ventral to the apodeme pit, close to the region in which the tymbal ribs buckle (Fig. 12A). In the first case, a major effect will be distortion of the resilin pad (Fig. 12A, upper right) and comparatively little inward force will be applied to the tymbal ribs, which will only buckle inwards after the application of a large force on the tymbal plate. In the second case, there will be a more direct effect on the tymbal ribs, which will buckle inwards with a smaller force (Fig. 12A, lower right). We observed a similar relationship between the force and distance required to cause buckling of the tymbal (Fig. 5A), which suggests that the tymbal is designed to maximise the amount of energy that can be stored prior to energy release by buckling of the ribs.

Now consider the effects of altering the angle at which the force is applied to the tymbal plate. Extreme cases are when the force is applied nearly parallel to the tymbal plate and when the force is applied nearly normal to the plate. In the first case, the turning moment will be small, but the distorting force acting on the resilin pad will be large; in the second case, the turning moment will be larger, and the distorting force acting on the resilin pad will be smaller. These situations are modelled in Fig. 12B. The resultant force and distance required to cause buckling of the tymbal ribs were found to be smaller when the tymbal plate was pushed at angles close to the horizontal (or nearly normal to the plane of the tymbal plate) than when the angle of push was more nearly vertical and hence at an acute angle to the plane of the tymbal plate (Fig. 5B). Here, also, it appears that the elastic regions at the dorsal end of the tymbal are designed to be distorted by the initial action of the tymbal muscle and thus to store energy for release by the buckling of the tymbal ribs.

The tymbal ribs, however, are light-weight, thin structures (Young and Bennet-Clark, 1995; Bennet-Clark, 1997). Because buckling occurs approximately midway along their lengths (Fig. 12A), buckling requires that a far greater force be applied at the dorsal ends of the ribs than is required at the points of buckling; thus, the ribs act as a trigger mechanism that is capable of retaining and then releasing large amounts of stored energy.

The light weight and compliant nature of this trigger mechanism ensure that the resonant properties of the tymbal will be dominated by the stiffness of the elastic elements of the tymbal and by the mass of the tymbal plate (which exceeds that of the heaviest rib by a factor of five; Bennet-Clark, 1997); the lightness and compliance also ensure that the force required to reset the trigger, by the buckling of the ribs back into their convex resting position, is far smaller than the force that must be applied to the tymbal apodeme to bring about inward buckling (Figs 6, 7), so the major part of the energy that is stored in the elastic elements of the tymbal is available for transduction into sound.

To fulfil its role in sound production, the tymbal should provide two types of loading to its muscle: it should store and then dissipate the major part of the work done in each cycle of muscular contraction; it should also provide sufficient residual strain energy to re-elongate the relaxing muscle. It seems likely that the major elastic energy store is the resilin pad (Fig. 1), which is a major determinant of the resonant frequency of the tymbal (Bennet-Clark, 1997), but the highly stressed tymbal apodeme may also store a proportion of the muscle energy. The strap-like region of the apodeme is relatively short and thin so its contribution to the energetics of the tymbal cycle is probably minor.

The muscle contraction cycles described here suggest that the muscle is capable of producing an appropriate force over an appropriate distance to distort the tymbal to the stage at which it buckles inwards (c.f. Figs 7C and 9). The probable power output from the muscle (Fig. 10) is approximately half the highest values reported for the sustained power output from striated muscle (see, for example, Weis-Fogh and Alexander, 1977; Askew and Marsh, 1997), suggesting that the loading provided by the tymbal may be very different from the simple elastic load used here. In this context, it should be noted that the singing insect adjusts the position and dimensions of its abdominal resonator (Young, 1990), presumably to exploit its muscle power maximally; it is also able to alter the curvature of the tymbal, and presumably the work required to buckle it, by the activity of the tymbal tensor muscle (Pringle, 1954; Fonseca and Hennig, 1996), which will have similar effects.

Sound production in a cicada such as C. australasiae occurs as a series of links in a chain: a neural pattern initiates muscle contraction; the muscle contractions are converted into mechanical work; the mechanical work is transduced into sound.

From the measurements reported in the present study, we can derive approximate values for the energy involved in the last two links of this chain. A typical muscle contraction through 0.30 mm produces a peak force of 0.32 N and thereby produces 47 μJ of energy: these values are broadly compatible with, although somewhat lower than, the mean values required to bring about tymbal buckling (Table 2). As the tymbal buckles, it produces a train of pulses of increased air pressure within the abdominal air sac of the cicada, which excites and sustains a sympathetic resonance in the abdominal Helmholtz resonator (Bennet-Clark and Young, 1992; Young and Bennet-Clark, 1995), from which sound is radiated via the large ventral tympana (Young, 1990) (see Fig. 3B,C). Each pulse of the song is produced by a single muscle contraction and consequent inward buckling of one of the two tymbals. The mean energy in a pulse of the calling song is 21.8 μJ and the mean energy released by the buckling of all tymbal ribs is 45.1 μJ, but both these values show variations of approximately ±35 %.

Taking the mean values for the energy of tymbal buckling (47 μJ) and the energy per pulse of calling song (21.8 μJ), the efficiency of transduction from mechanical energy to sound energy appears to be approximately 46 %. Even using the maximum value for the energy that can be released by tymbal buckling (74.2 μJ) and the lowest values for the energy per sound pulse (13.5 μJ) gives a transduction efficiency of 18 %. The very high efficiency that we ascribe to the mechanical-to-sound transduction process is similar to that suggested for a similar process of transduction in the mole cricket Gryllotalpa vineae (Bennet-Clark, 1970). This efficiency is far higher than the overall efficiency of the whole animal reported for the sound production of Gryllotalpa australis and the cricket Teleogryllus commodus (Kavanagh, 1987), which was found to be 1.05 % and 0.05 % respectively. This reflects the fact that we are only examining the energetics of one or two links in the chain, rather than the energetics of the whole animal, as measured by Kavanagh (1987), which includes the energetics of its physiological support systems.

Nonetheless, such a high apparent transduction efficiency deserves further comment. Although the tymbal provides the pressure drive to the abdominal resonator (Young and Bennet-Clark, 1995), the sound is radiated through the large tympana, which are extremely thin (Young, 1990) and extend across the full width of the abdomen. As such, the tympana provide a sound source that can be modelled as a piston in an infinite baffle. The coupling of a sound source to the fluid medium into which it is radiating depends on the specific acoustic resistance of the source relative to that of the fluid medium (see, for example, Olson, 1957; Fletcher, 1992). Calculations based on the dimensions of the tympana suggest that their specific acoustic resistance is close to that of the air into which they are radiating sound (Bennet-Clark, 1995). Thus, the extreme specialisation of the abdomen in male cicadas can be seen to result in efficient mechanical to sound energy transduction and, for their size, the production of extremely loud sounds.

The experimental work reported here was largely carried out at the University of Melbourne. This work was supported in part by an Australian Research Council grant to Dr David Young. We thank David Young for his support and encouragement throughout this work, for the loan of space and equipment and for allowing us to analyse his song records. This work was made possible by an Overseas Study Visit Grant from the Royal Society to H.C.B.-C. and permission to take sabbatical leave from Oxford University and St Catherine’s College, Oxford; this generosity is gratefully acknowledged. Special thanks are due once more to the Department of Zoology, University of Melbourne, and to Professor M. B. Renfree for hospitality throughout H.C.B-C’s stay in Australia, without which this work could not have been undertaken. We also thank two anonymous referees for constructive comments on an early version of this paper.