Resonators in Insect Sound Production: How Insects Produce Loud Pure-Tone Songs
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The Journal of Experimental Biology 202, 3347–3357 (1999) 3347 Printed in Great Britain © The Company of Biologists Limited 1999 JEB2136 RESONATORS IN INSECT SOUND PRODUCTION: HOW INSECTS PRODUCE LOUD PURE-TONE SONGS H. C. BENNET-CLARK* Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK *e-mail: [email protected] Accepted 15 June; published on WWW 16 November 1999 Summary In a resonant vibration, two reactive elements, such as a produced by the excitation of a simple resonator, the song mass and a spring, interact: the resonant frequency frequency may not be constant, suggesting that other depends on the magnitude of these two elements. The build- factors, such as the mechanism of excitation, or variation up and decay of the vibration depend on the way the of the effective mass or elasticity of the system during resonator is driven and on the damping in the system. sound production, may be additional determinants of the The evidence for the existence of resonators in insect song frequency. sound production is assessed. The mechanics of different Loud, and hence efficient, transduction of the energy of types of sound-producing system found in insects is a mechanical resonator into sound may involve a second described. Mechanical frequency-multiplier mechanisms, stage of transduction which, by damping the resonator, which convert the relatively slow contraction of muscles to may compromise tonal purity. Some insect singers resolve the higher frequency of the sound, are commonly used to this problem by tuning both stages of transduction to the convert the comparatively slow muscle contraction rate to same frequency, thereby maintaining tonal purity. the higher frequency of the sound. The phasing and rate of mechanical excitation may also affect the frequency and duration of the sound that is produced. Key words: insect, sound production, bioacoustics, resonator, cicada, Although in many insects the song may appear to be cricket. Introduction The songs of insects may be shrill, raucous or musical, brief power, as occurs in a trumpet mouthpiece or a clock or sustained but, whether loud or quiet, they tend to convey escapement. information within a narrow band of sound frequencies. The Simple resonators rely on the interplay between two reactive narrow frequency band implies that the sound-producing elements. Consider the simple case of a mass swinging on a system is in some way tuned. In this review, I shall address the spring (Fig. 1): at the extreme of the swing of the mass, it has mechanisms by which sharply tuned sounds are produced by zero velocity and thus zero kinetic energy, but the strain energy insects and explore the acoustical and biomechanical in the spring is maximal; as the mass swings through the centre mechanisms by which insects produce narrow-band sound of its arc, its velocity is maximal so its kinetic energy is signals. I shall also examine the transduction chains involved maximal, but the stored or strain energy in the straight spring in the conversion of muscle power to sound power. is zero; the mass then decelerates to zero velocity as its kinetic energy is absorbed by the spring; the mass then re-accelerates in the opposite direction and so on. What is a resonator? The frequency of the vibration or resonant frequency, Fo, Resonators occur in many everyday forms such as whistles determined by the mass and the compliance of the spring, is and bells, pipes and drums, the pendulum of a clock, the given by: balance spring-and-wheel or the quartz crystal of a watch. All these resonant systems are employed for similar reasons: they 1 1 1 s Fo = = , (1) vibrate at or near a single frequency, whether that is a musical 2π Ί m × c 2π Ί m note or is used as a basic timing mechanism; their vibration can be excited by an impulse, whether by a puff of air, a where m is the mass, c is the compliance and s is the stiffness mechanical impact or an electrical pulse; and their vibration (the compliance of a spring is the reciprocal of its stiffness). can be sustained by continued excitation, by supplying A similar process of transfer of energy between two appropriately phased acoustic, electrical or mechanical components occurs in the simple electrical series resonant 3348 H. C. BENNET-CLARK A A practical resonator, vibrating freely, will come to rest Mass stops, because the energy of vibration is lost by friction, as heat, or Energy is stored then reverses is otherwise dissipated, for instance as sound. Ways in which in spring direction this energy loss might occur are shown in the model systems of Fig. 2. If very little of the energy of vibration of the system is lost, the vibration will, by Newton’s first law, persist; this will occur because the resistive damping is small. Stored energy In a heavily damped system, the vibration will decay rapidly. accelerates mass Kinetic energy of In either case, the amplitude of vibration tends to decay mass will store exponentially. energy in spring A useful dimensionless index of the damping of a resonant B system is the quality factor, Q, which can be measured in Energy stored in spring is maximal various related ways (Morse, 1948; Fletcher, 1992). Spring has zero The Q of any of the resonators described above can be Mass has stored energy zero kinetic obtained by measuring the amplitude of successive cycles of energy the free decay following a driven oscillation. Q is then given by the slope of a plot of the natural logarithm of the amplitude per cycle during the decay (known as the loge decrement): π Kinetic energy of Q =.(2) mass is maximal loge decrement Fig. 1. Diagrams of a simple resonator composed of a mass and This method of measurement of Q has been used in a biological spring. (A) The kinetics of the system showing how elastic strain context to describe the resonators on cricket wings (Nocke, energy in the spring accelerates the mass and the kinetic energy of 1971) or those involved in cicada sound production (Bennet- the mass is converted into strain energy in the spring. (B) How Clark and Young, 1992; Young and Bennet-Clark, 1995). energy is transferred from the spring to the mass and vice versa. The decaying transient that follows impulsive excitation of a resonance is found in many sound-producing systems, such as bells or the succession of pulses that make up the songs of circuit shown in Fig. 2A, in which the capacitor acts many bush crickets. However, many biological and musical compliantly, in a manner analogous to a spring, and the sounds are produced as driven resonances, in which the inductor behaves inertially, in a manner analogous to a mass. vibration is sustained by regular re-excitation at an appropriate Likewise, in an acoustic resonator, such as the Helmholtz frequency and phase. An everyday example of this is the resonator in Fig. 2B, the two reactive elements that determine maintenance of the swing of a clock pendulum using packets the resonant frequency are analogues of a compliance or spring of energy liberated at an appropriate phase by the escapement, and an inertance or mass. which is coupled to the pendulum (Gazeley, 1956). The AB C Driver a.c. source Source of vibration Capacitance Fluid-filled Spring or C (compliance) cavity elastic element Fig. 2. Diagrams comparing electrical, acoustic and mechanical resonators to L Inductance show that each acts by the interaction (mass-like) between two reactive elements: an Neck Mass analogue of mass, and an analogue of compliance or elasticity. In practical Resistive load Load where energy oscillators, the energy of oscillation is R where energy Mouth is dissipated Liquid dissipated by a viscous or resistive is dissipated (viscous damping) load, which causes the amplitude of (damping) oscillation to decay exponentially Electrical Acoustic Mechanical mass, (see Fig. 7B,C). C, capacitance; L, resonant Helmholtz spring and dashpot inductance; R, resistance. LCR circuit resonator resonator Resonators in insect sound production 3349 A frequency or the phase of a forced vibration lags that of the ≈ 0 Fo 4.85 kHz driving force by 90 °, a simple resonator may be implicated. In −3 dB practice, it is easier to measure amplitude than phase so the -5 phase information tends to be ignored. -10 The songs of many insects, such as crickets (Elliott and Koch, 1985; Koch et al., 1988), some bush crickets (Bailey, -15 1970) and certain cicadas (Bennet-Clark and Young, 1992), appear to be produced as forced vibrations of comparatively -20 F Q = o simple resonators (Table 1). For detailed treatments of − -25 Bandwidth bandwidth at 3 dB different types of resonators, their parameters and how their Relative amplitude (dB) − 4.85 at 3 dB Q = = 24 resonant frequencies scale, see, for example, Fletcher, 1992; ≈ -30 0.2 kHz 0.2 Morse, 1948; Olson, 1957. 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Frequency (kHz) Types of resonators implicated in insect sound production There have been comparatively few formal studies of the 0 B Phase angle is −90° reactive elements that determine the frequency in insect sound- ≈ at Fo 4.85 kHz producing mechanisms. Because biological structures rarely -45 have a simple geometry with discrete disposition of the mass F and compliance (or its acoustic analogues), the studies that Q = o exist have resorted to models based on known types of -90 bandwidth at ±45° 4.85 mechanical or acoustic resonator. Q = = 24 There is abundant evidence in a few cases for the existence Bandwidth 0.2 -135 at ±45° of a resonator, such as the detailed and elegant analysis of the relative to Fo vibrations of the harp area of the forewings of male Gryllus ≈ 0.2 kHz campestris by Nocke (1971).