Acoustics of Sound Production and of Hearing in the Bladder Cicada Cystosoma Saundersii (Westwood)

The Australian bladder cicada Cystosoma saundersii (Westwood) is a remarkable insect in that the male produces a calling song which consists of a train of brief tone bursts of approximately 800 Hz sound repeated about 40 times per second.

This insect and its associated acoustical and behavioural questions have only recently been studied (Young, 1972; Young & Hill, 1977). The most striking feature of the male is its grossly enlarged, hollow, abdomen (Fig. 1), which enhances radiation of sound (Young, 1972), and also affects hearing in the male, rendering the auditory system non-directional to the song frequency (Young & Hill, 1977). In the female of C. saundersii the abdomen is not enlarged, but it contains an air sac associated with the auditory tympana, and the hearing of the female is directional at the frequency of the male song (Young & Hill, 1977).

Organs specialized for sound production are limited to the male in C. saundersii. Each sound pulse in the calling song is generated by the buckling of two tymbals, which consist of flexible cuticular membranes braced by a series of seven stiffer, parallel ribs. Each tymbal is approximately triangular in shape, about 20 mm² in area, and is convex outwards in the resting state (Young, 1972). The mechanics of tymbal buckling and its neuromuscular control are complex and are the subject of a paper in preparation by P. J. Simmons and D. Young.

Briefly, during production of the calling song the tymbal is held in outward tension by the tensor muscle, contraction of the large tymbal muscle overcomes this tension and the tymbal buckles inward (see also Young, 1972), generating a pulse of sound. As about six of the ribs are bent as the tymbal buckles, a series of shocks associated with sequential bending of the ribs may correspond with the 800 Hz carrier frequency of the song produced.

The enlarged abdomen of the male appears to be essential for effective sound radiation. During production of the calling song, the male abdomen is extended, possibly as the result of an increase in static internal air pressure. The singing behaviour of C. saundersii during pair formation is the subject of a paper in preparation by K. G. Hill and C. E. Hill.

The sound pressure of the emitted signal falls if the abdomen is removed (Young, 1972). The size of the male abdomen severely limits the insect’s flight speed and agility. However, concentration of sound energy into a narrow frequency band and efficient radiation into the air medium are essential characteristics of the sound producing organs of terrestrial insects if communication signals are to be propagated over appreciable distances (Bennet-Clark, 1971 ; Michelsen & Nocke, 1974). In the acoustical analysis which follows, we show that the male abdomen in C. saundersii acts as a resonator to amplify the song and stabilize its frequency, and optimizes radiation efficiency of the signal. Although C. saundersii may represent an extreme case acoustically, the principles discussed may have general applicability in insects.

As with most quantitative acoustic discussions, the most convenient methods of analysis are those based on analogues derived from the more familiar field of electric circuit theory. The bases of these analogies are discussed in standard texts (Morse, 1948; Olson, 1957; Skudrzyk, 1968) which should be consulted for details. For the present we simply state some of the concepts involved in our later discussions.

Acoustics is concerned with fluctuations in atmospheric pressure and with the oscillatory flow of volumes of air through pipes and apertures, sometimes impeded by flexible membranes. Electric circuits, on the other hand, are concerned with currents and voltages and with the way in which current flow is impeded by inductances, capacitances and resistances. The most fruitful analogy for our present purpose identifies acoustic pressure with electrical potential and acoustic volume flow with electrical current. In such a system of analogies, mass-like acoustic impedances are identified with inductances, spring-like acoustic compliances with capacitances, and viscous or other similar resistances with electrical resistances. We shall discuss these analogies in greater detail as we proceed but point out that, since we are concerned with oscillatory quantities, it is customary to use angular frequencies ω = 2πv rather than the common frequencies v expressed in hertz, and that the oscillatory quantities are represented through a factor exp(jωt) where j = √ – 1. Details of this approach are given in the references listed above or can be found in standard texts on electric circuits.

The abdomen in the male of C. saundersii is segmented (Fig. 1), and segments 3-7 may be considered as stiff rings joined together by lighter, flexible membrane (schematically illustrated in Fig. 2). The internal air cavity (an enlarged tracheal air sac) communicates with the exterior through a narrow, closable spiracle situated on the metathorax (Young & Hill, 1977). At ordinary sound frequencies, therefore, the abdomen may be considered sealed. During calling song production, the abdomen is extended so that the intersegmental membranes are in tension. The two tymbals in the first abdominal segment form part of the wall of the abdominal cavity (Young & Hill, 1977).

The acoustic circuit has three elements. The tymbals, being relatively stiff and driven by powerful muscles, act together always to displace a certain volume of air, almost irrespective of the acoustic pressure acting on them. They are thus represented by a high-impedance (constant-current) generator G as shown in Fig. 3.

The topology of the analogue circuit, which is shown in Fig. 3, is determined by considerations of pressure transmission and volume flow through the various acoustic elements. If the impedances of the two tympana, represented by Z_T, are large enougn that the acoustic volume flow through them can be neglected, and if the membrane compliance C_M is greater than that of the enclosed air C_B, then the circuit approximates a parallel resonant system fed from a constant-current generator G. The radiated sound power is represented by the power dissipated in the radiation resistance R_A, which is a maximum at the resonance frequency.

If the membrane joining the abdominal segments were perfectly elastic (R_M = o) so that damping was entirely by radiation, then numerical values inserted in (11) would give Q_B ≃ 200. It is likely, however, that R_M ⪢ R_A so that the internal friction of the membrane constitutes the major damping mechanism, and indeed most biological compliances have Q < 10.

Independent evidence is available on this point from the work of Young (1972), who observed a decrease of 8-10 dB in sound pressure level in the distress call when the abdominal sac was removed, and from Young & Hill (1977) who measured a 24 dB decrease in auditory sensitivity at 800 Hz under similar conditions. Interpretation is not simple because the bare tymbals and tympana are exposed on both surfaces and so act as dipoles. More direct is the observation of Young and Hill that the sound pressure at 800 Hz inside the male abdomen is about 11 dB higher than in tl free field. Again, as we see later, interpretation is not simple but the implication of all these results is that Q_B is between about 3 and 10. In a more normal situation oscillograms of the free song (Fig. 5 a ofYoung 1972) show that each tone burst has an exponential decay with a characteristic decay time for sound pressure of about 8 ms. This implies Q_B ≃ 18, which may be an overestimate since there may be some energy input from the tymbals. It is probably not too far from correct to suggest Q_B ≃ 10, giving a 20 dB gain in radiated power and implying R_M ≃ 3 c.g.s. acoustic ohms.

For ease of reference and comparison, values of analogous impedances for both male and female are collected in Table 1.

The two auditory tympana in the male form part of the wall of the abdominal cavity as shown schematically in Fig. 4. Each tympanum is about 4 mm² in area and has two large masses of amorphous material attached to it. A cuticular bar connects the tympanum to the receptors located laterally in the auditory capsule (Young and Hill, 1977). When the animal is singing, tension in the tympanum is removed by the detensor tympani muscle (Young, 1975).

The stiffness of the tympanum is provided by the elasticity of its membrane, together with some contribution from the amorphous material. It is difficult to decide unambiguously on the mode of motion of the tympanum until more is known about the elastic properties of the amorphous loading material. The most likely vibration is one in which the whole tympanum moves with most of its stiffness being provided by the tympanic membrane rather than by the amorphous load. By comparison with the female tympanum, which we discuss later, it is likely then that C_T ≃ 1 × 10⁻⁷ which implies a resonance frequency near 500 Hz, or well below the song frequency. If the Q for the tympanum resonance has a rather low value, say 3, because of damping in the load, then R_T≃ 1000 c.g.s. units. From these values, which are not critical in our later analysis, it is clear that the impedance of the tympana at the song frequency is at least a factor of 100 greater than that of the abdomen, thus justifying the assumption made in the previous section.

Consideration of the paths of action of the acoustic pressures and the relationships between the acoustic flows now leads to the analogue circuit shown in Fig. 5. Note that there are some changes from the circuit in Fig. 3 because one of the excitations, p₃, is now applied as a pressure acting on the outside of the abdomen rather than as a volume flow to its interior.

Before presenting the results of a calculation on this circuit it is helpful to analyse it qualitatively. From our discussion above and the numerical values summarized in Table 1 it is clear that the impedance of the branches AB and CD ( > 1000 acoustic Q) is greater than that of EF and GH ( ∼ 30 acoustic Q) so that to a first approximation we can consider just the low-impedance circuit EFGH. From the viewpoint of the pressure generator p₃ this is a series circuit resonant at the song frequency and the resonance leads to an enhanced pressure αQ_Bp₃ across C_B, where a = C_M/(C_M +C_B) lies between and 1. This is the enhanced pressure inside the abdomen measured by Young & Hill (1977). The phase of the abdominal pressure is, however, in quadrature with the external pressurep₃ and hence, since the phase shifts ωδ_i are small, essentially in quadrature with p₁ and p₂ as well.

This analysis thus accounts for the non-directionality of hearing sensitivity in the male and provides an estimate that seems not too unrealistic for the absolute value of the deflection of the tympanum. To complete the analysis involves straightforward calculation of the currents in the branches AB and CD of Fig. 5 and their integration to give tympanum displacement as a function of frequency. The results of such a calculation, using the values given in Table 1, are shown in Fig. 6. For easy comparison with published experimental results (e.g. Young & Hill, 1977) the results are given as a threshold sensitivity curve assuming that a tympanum displacement of 10 Å is required at the auditory threshold. The curve can be simply shifted up or down for different assumed thresholds or inverted to give tympanum deflexion for a given sound pressure level.

From Fig. 6 it is clear that the model reproduces the general features of the behaviour measured by Young & Hill (1977). The auditory system is most sensitive near the song frequency of 800 Hz, where it shows no distinction between ipsilateral and contralateral stimulation and is thus completely insensitive to sound direction. At frequencies near 500 Hz the system shows appreciable directional discrimination. This frequency is not determined by the tympanum resonance but rather by a series resonance in the circuit branch EF, while the extent of the discrimination is influenced by the membrane resistance R_M. The small directionality noted by Young & Hill (1977) at other frequencies away from 800 Hz may possibly be due to a diffraction effect making p₂ not quite equal to p₁

In the female of the species, as we have already pointed out, the abdominal cavity is smaller than in the male, only about 0·2 cm³, and the abdominal walls are relatively massive (Young & Hill, 1977). This implies a very large value for the inductance L_B in Fig. 5 so that the pressure p₃ is effectively insulated from action on the system. The analogous circuit thus reduces to the simpler network shown in Fig. 7. The capacitance C_B representing the cavity volume is, from (1), approximately

As we shall see presently, the bare tympanum is probably resonant near the song frequency which implies C_T ≃ 7 × 10⁻⁷. A loaded Q value near 5 is probably about right for the membrane, although this could be less because of the ridging and the layer of amorphous material noted by Young & Hill (1977). This implies R_T ≃ 50 though it might be 100 or a little more. Here, as before, c.g.s. units are implied. (Later we shall see that we need to assume R_T ≃ 150 in order to achieve sufficient directional discrimination. This is within the uncertainty of our estimates.)

The factor B in (19) affects the general shape of the response curve and has a maximum near the frequency for which K = 0. This cannot be brought to coincidence with the song frequency except by assuming that the tympanum is much heavier than seems possible. In fact this peak frequency seems likely to lie above 2000 Hz for the dimensional values discussed above.

Evolutionary processes have presumably optimized the parameters of the acoustic system for the female cicada. Our calculations, shown in Fig. 8, display ipsilateral and contralateral behaviour for one such set of parameters consistent with our limited knowledge of the system, the values being those given in Table 1. As before, the results have been plotted as threshold sound pressure level for a deflexion of 10A at the tympanum.

From Fig. 8 it is clear that our model is incomplete. There is, certainly, a discrimination of about 10 dB between ipsilateral and contralateral sound at the song frequency and this discrimination, which has a cardioid pattern, reduces at higher or lower frequencies. The ipsilateral response, too, is peaked near the song frequency, but this sensitivity is very much less prominent than the neurophysiologically measured peak.

Several mechanisms might be proposed to account for this discrepancy. The simplest is to suggest that the lever system connecting the tympanum to the auditory capsule is mechanically resonant at a frequency near 800 Hz. Alternatively it is possible that the auditory capsule behaves like a mass load coupled in such a way as to reduce its high-frequency response, while the spiracle vents may constitute a resistive path to the abdominal cavity of such a magnitude as to reduce the low-frequency response. Some auxiliary mechanism such as this seems necessary since it is not possible, by reasonable variation of the available parameters, to reproduce accurately both the frequency response and directional discrimination of the system.

We have examined an acoustic model for sound production and for hearing in C. saundersii and shown, in general terms, that it is capable of giving a semi-quantitative account of the measured behaviour. Agreement with experiment is not good in some features. These defects in the treatment can be attributed to uncertainties in some of the physical quantities involved, to oversimplifications in the mechanical models for sub-systems like the tympana, or to effects arising from the neglect of the physical dimensions of some of the elements compared with the wavelength of sound. It also seems probable that some secondary resonant system is interposed between the tympana and the neural transducers in the female cicada, and possibly also in the male, or that the transducers themselves exhibit a response that is sharply peaked near the song frequency of 800 Hz.

Despite these discrepancies, the general agreement between theory and experiment is sufficiently good that the treatment may be accepted as broadly correct. This then allows us to appreciate the acoustical functions of the various anatomical features that have been described, and to see how the magnitudes of the acoustical impedances with which they are associated affect the performance of the whole system.

C. saundersii is an insect unusually well suited for the sort of analysis present above, for its song is low enough in frequency that simple lumped-parameter electrical analogues can be used to elucidate its acoustic behaviour. For many insects with songs of higher frequency such simplifications will no longer be possible, but clues to the function of various system elements can perhaps be found by analogy with the present study.

This project was supported, in part, by the Australian Research Grants Committee. We are grateful also to Suszanne Thwaites for computing assistance.