ACOUSTICS OF A SMALL AUSTRALIAN BURROWING CRICKET : THE CONTROL OF LOW-FREQUENCY PURE-TONE SONGS
1
Department of Zoology, University of Western Australia, Nedlands, WA 6907,
Australia
2
Department of Zoology, University of Oxford, South Parks Road, Oxford OX1
3PS, UK
3
Department of Electronic Materials, Engineering Research School of
Physical Sciences and Engineering, Australian National University, Canberra,
ACT 0200, Australia
*
Author for correspondence (e-mail:
henry.bennet-clark{at}zoo.ox.ac.uk
)
Accepted June 4, 2001
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Summary |
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The song has a mean centre frequency of 3.2 kHz and is made up of variable-length trills of pulses of mean duration 15.8 ms. Many song pulses had smooth envelopes and their frequency did not vary by more than ±40 Hz from the centre frequency, with a relative bandwidth Q-3dB of over 50. Other pulses showed considerable amplitude and frequency modulation within the pulse.
When driven by external sound, burrows resonated at a mean frequency of 3.5 kHz with a mean quality factor Q of 7.4. Natural-size model burrows resonated at similar frequencies with similar Q values. One cricket, which had previously called from its own burrow at 2.95 kHz, sang at 3.27 kHz from a burrow that resonated at the same frequency.
Life-size model burrows driven by external sound resonated at similar frequencies to the actual burrows; models three times life size resonated at one-third of this frequency. In all models, the sound pressure was more-or-less constant throughout the top chamber but fell rapidly in the neck of the burrow; the phase of the sound was effectively constant in the top chamber and neck and fell through approximately 180° in passing from the neck into the lower chamber. A numerical model of the sound flow from region to region gave essentially similar results.
A resonant electrical model fed from a high-impedance source with discrete tone bursts at different frequencies showed similar amplitude and frequency modulation to the various types of song pulses that were observed. It is suggested that the high purity of the songs results from close entrainment of the sound-producing mechanism of the insect's wings to the sharply resonant burrow.
Key words: acoustics, cricket, song, tuned burrow, frequency, song purity, Rufocephalus sp., communication, sound
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Introduction |
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Insects have exploited three strategies to achieve effective sound
transmission. First, concentrating the sound power within a narrow frequency
band improves the effective signal-to-noise ratio and is achieved by the use
of resonant sound-producing structures. The second requirement is for
efficient coupling or impedance matching between the sound-producing
structures and the surrounding air. Finally, related to both sound power and
increasing the insect's effective size, insects have exploited external
structures that are larger than themselves (Forrest,
1982; Bennet-Clark,
1998
).
The forewings of male crickets are specialised for sound production: a
plectrum on the posterior edge of one wing engages with the teeth of a file on
the underside of the opposite wing. Crickets sing by closing the forewings and
thereby creating a series of catch-and-release actions between the plectrum
and file, which in turn distorts specialised veins of the wings (the harp),
exciting their resonant vibration (Nocke,
1971; for reviews, see
Bennet-Clark, 1989
;
Bennet-Clark, 1999b
). Each
wing-closure causes a series of excitations of the harp resonators, which
merge into a sustained coherent vibration that is radiated as a sound
pulse.
Because the resonant frequencies of the harp regions of the two wings are
similar and sharply tuned, cricket songs are musical and most songs are
confined to a narrow frequency band corresponding closely to the resonant
frequency of the harps (Nocke,
1971). However, this rather
simplistically suggests that crickets produce pure-tone calls, although this
is seldom observed: in many crickets, the frequency of the signal decreases
within each pulse (Leroy,
1966
; Simmons and Ritchie,
1996
); the effective resonant
frequency appears to decrease throughout the closing stroke (Bennet-Clark,
1999b
). In this paper, we
examine the extent to which a pure-tone carrier frequency can be maintained by
a calling insect.
Efficient impedance matching between a vibrating structure and the
surrounding fluid medium depends on the effective size of the sound-producing
structure. For example, the radius of a monopole source that radiates sound
from only one side should exceed one-sixth of the carrier frequency
wavelength; if the source is a doublet, radiating sound from both sides, the
effective radius should exceed one-quarter wavelength (Olson,
1957; Fletcher,
1992
). Not surprisingly, where
there has been strong selection on low-frequency pure-tone signals, insects
have evolved second-stage transduction devices that effectively increase the
size of the sound source.
Many insects that produce pure-tone sounds increase the power of the signal
by coupling the sound-producing organ to a resonator (Forrest,
1982). These resonators can
form part of the sound-producing structures themselves, but the insect may
also use external structures to enhance the signal further. Such devices may
include leaves as baffles (Forrest,
1982
) or the specially shaped
burrows of mole crickets that are tuned to the resonant frequency of the
sound-producing wings (Bennet-Clark,
1970
; Bennet-Clark,
1987
; Daws et al.,
1996
). In cicadas, the primary
resonators are small paired tymbals on the side of the abdomen which, in many
cicadas, drive a Helmholtz resonator consisting of the abdominal air sac and
ear drums (Young, 1990
;
Bennet-Clark and Young, 1992
).
A feature of both the mole cricket and the cicada systems is that the second
stage of impedance matching increases the effective size of the sound source
to close to the optimum and, since it is tuned to the carrier frequency of the
song, tonal purity is maintained.
This paper describes a grylline system analogous to that of the mole cricket (Gryllotalpidae). The remarkably low frequency of the song, with its almost pure tone, is produced by a very small cricket, thus challenging the notion that small insects can produce only high-frequency calls. This example of an insect that uses external means to enhance its calling performance represents, to the best of our knowledge, the first description of a member of the Gryllidae using a complex tuned burrow for this purpose. We test the acoustics of such a system by natural experiment and also by developing both model burrows and model crickets. Finally, we draw comparisons between the subject of this paper, an undescribed species of Rufocephalus, and the acoustics of mole crickets.
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Materials and methods |
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Male insects were collected from the Coalseam Nature Reserve on the Irwin River north of Mingenew in Western Australia on three occasions in October and November 1999. Insects were located by their song during the night and dug up immediately after song recordings had been made. The insects made their burrows in red alluvial sandstone clays where the top 20-30 mm hardens to a crust-like cap. Some burrows were photographed while the insect was calling, enabling a measurement of the exposed length of their antennae. Insects were brought back live to Perth in separate specimen jars and were subsequently preserved in 70% ethanol. Body dimensions of preserved insects were measured to the nearest 0.1 mm using a graticule eyepiece in an Olympus binocular microscope.
Measurements of song parameters
After the burrow opening had been located, recordings were made from a
distance of approximately 150 mm. The microphone was a Tandy tie-clip
microphone (catalogue no. 33-1052) fastened to the end of a 400 mm long 7 mm
diameter plastic pipe; this was calibrated against a Bruel and Kjaer
microphone (type 4165) and Bruel and Kjaer sound level meter (type 2209); its
response was flat ±2 dB between 1 and 20 kHz. Recordings were made at
12 dB below peak level into one channel of a Sony TCD-D8 digital audio tape
recorder, with a specified and measured frequency response that was flat
±1 dB from 20 Hz to 22 kHz. Recordings were voice-cued with the date
and serial number of the insect. Between 30 and 60s of recording were made for
each insect.
Sound level measurements were made in the field using a Bruel and Kjaer (type 2231) sound level meter with integrating module (type BZ7100) and microphone (type 4155). Both mean sound pressure and peak sound pressure levels were measured with a distance of 200 mm between the mouth of the burrow and the face of the microphone.
Tape recordings were analysed either using Signal (Engineering Design, Belmont, MA, USA) data-acquisition and analysis software or using the sound-acquisition input of an Apple Macintosh Powerbook 3400C and Canary 1.2.1 sound-analysis software sampling at 44.1 kilosamples s-1. Pulse duration was measured directly from oscillograms; the dominant frequency was measured by Fast Fourier Transform (FFT), and cycle-by-cycle frequency was measured by zerocrossing analysis.
Most FFT analyses using the Canary software were made using a Hamming window and a 2048-point analysis window giving a filter bandwidth of 87 Hz. Most recordings were analysed without filtering, but in some cases where there was intrusive wind noise, frequencies below 500 Hz and above 12 kHz were removed before signal analysis with the `filter out' tool in the Canary software; this filters by 30 dB in 50 Hz, regardless of the frequency. The harmonic content of different parts of song pulses was observed by similar filtering-out of the frequency band from 1 to 4.5 kHz and comparing the filtered pulse with the unfiltered one. The frequency spectra of short sections of sound pulses were analysed by copying part of the pulse and pasting it into the middle of a 0.5 s duration blank recording. To improve the resolution of the FFT analysis, single song pulses were analysed by copying them into a 0.5 s duration blank recording and making a 16 384-point analysis, giving a filter bandwidth of 10.9 Hz.
Resolution of the cycle-by-cycle or instantaneous frequency within the pulses was performed using ZeroCrossing ZC version 3 software written by K. N. Prestwich, which measures the time of zero crossing of the wave form to a precision of approximately 0.5 µs by exploiting the interpolation between sample points provided by the Canary software. Tests with 3 kHz sine-wave signals showed that the frequency of individual cycles was measured to less than ±5 Hz. Further tests showed that the method can give anomalous results with signals with any direct current component or where the signal-to-noise ratio is poor: the analyses reported here were made after using the software to filter out all signal components below 500 Hz and normally only with signals for which the signal-to-noise ratio exceeded 40 dB.
Burrows
Casts of burrows were made in the field by pouring dental plaster made up
to a fluid consistency through the surface hole of the burrow. A 15 mm high
ring of plastic water pipe was used to confine the plaster to the region of
the mouth of the burrow. The casts were dug up the next day, and adherent soil
was washed away.
Most dimensions of the washed plaster casts were measured to the nearest 0.1 mm using Mitutoyo digital callipers. Actual burrows in the soil were fragile, so the dimensions of the surface holes and other regions were measured optically in the laboratory or from photographs taken in the field with a reference scale. Further measurements were made directly from casts of burrows, which were dug up and then sectioned in the laboratory.
The porosity of different regions of the burrows was assessed by painting the inside of the burrow with cellulose paint and subsequent washing to remove soil particles. Surface roughness was measured from sections of the burrow cut after embedding the parts of the burrow in epoxy resin.
For acoustic tests, blocks of approximately the top 50-100 mm of soil, each containing a burrow, were dug up and transported to the laboratory. Acoustic testing was carried out after placing the block of soil containing the burrow on a 10 mm thick layer of cotton wool on top of a 100 mm thick layer of acoustic foam.
Tests with model burrows
Natural-size models of the burrow cavity were carved from paraffin wax to
dimensions taken from representative plaster casts of the insects' burrows;
any rough carved marks were smoothed with a warm soldering iron. The models
were carved so that the dimensions of the surface hole, upper and lower
chambers and neck (see Fig. 1)
were within ±0.2 mm of the dimensions of the plaster casts of burrows,
but the models were made with a longer neck at the surface hole and a long
exit tunnel, giving a total length of 150 mm and without the natural and
occasional deformations that occurred in the casts of the burrows. The wax
models were cast in the centre of a 30 mm square block of plaster, so the wall
thickness of the cast exceeded 10 mm. After the cast had set, it was dried and
heated to melt out the wax. Holes 1.3 mm in diameter were drilled through the
walls of the cast into the model cavity to allow the insertion of probe
microphones; when not in use, these holes were plugged with wooden pegs. The
sharp lip of the surface hole of the real burrow was simulated by grinding
away any excess plaster left after the casting process.
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A model cricket body was made from a cylinder (10 mm long and 3 mm in diameter) of modelling clay mounted on a straight length of 0.2 mm wire, which was inserted through the bottom end of the plaster model. Using a scale attached to the wire, this was moved to measured positions along the length of the burrow.
A series of model burrows approximately three times natural size was made from 30 mm internal diameter by 45 mm internal length polyethylene film canisters, joined by lengths of 14.5 mm internal diameter polystyrene tubing. A variety of surface holes was provided by washers in which the hole had been chamfered to give a sharp edge. Holes 1.5 mm in diameter were drilled along the length of the model for the insertion of probe microphones. A 3 mm wide slot cut along the length of the top chamber of the model allowed the sound source to be moved along the chamber, where it was sealed in place with modelling clay. The modular construction allowed the dimensions of the models to be altered. Termination in the models was provided by a 300 mm length of 14 mm internal diameter plastic tubing filled with cotton wool. Variable porosity was provided by rows of 1.5 mm diameter holes in the walls of the model which could be sealed with modelling clay.
The relative amplitudes and phases of the inputs and responses of the models were measured directly from an oscilloscope. Phase was measured in degrees, and phase lead relative to that at the datum was, for convenience, given a positive value.
An electrical model of the burrow was made from a parallel LCR (inductor,
capacitor and resistor: Fig.
12C, inset) resonant circuit in which a 20 mH inductor was tuned
by a 0.15 µF capacitor to 2.9 kHz and adjusted with a 20 k shunt
resistor to a quality factor Q of 10. This was driven via a
22 k
resistor, which provided a source of similar impedance to that of
the resonant circuit at its resonant frequency, from a Tektronix FG 501A
function generator gated to produce tone bursts by pulses from a Digitimer
DS9A stimulator, and measurements were made of the response across the
inductor. The frequency within the tone bursts was set to the nearest 10 Hz
using a Testlab M2365 multimeter.
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Sound sources
The wings of crickets are held above the body, so there is sound leakage
from the two sides around the edges of the wings. An appropriate model is a
doublet source consisting of a moving diaphragm open on both sides so that the
antiphase sound radiated from one side interacts with that from the other
side. A high-quality Sony 15 mm diameter earphone was removed from its housing
and glued to a 2 mm square support rod parallel to the plane of its diaphragm.
The back of the diaphragm radiates sound through small ports in the earphone
surround anti-phase with that radiated from the front of the diaphragm. To
balance the output from the two sides of the diaphragm, a 12 mm diameter disc
of adhesive tape was attached to occlude the central part of the front face of
the earphone. The following tests confirmed that the source acted as a
suitable dipole or doublet source: first, the phase of the sound output at the
front of the source differed by 180° from that at the back; second,
measurements made by moving a probe microphone in a straight line normal to
the plane of the diaphragm, 1 mm away from the edge of the earphone, showed a
nearly symmetrical V-shaped distribution of sound pressure on either side of
the diaphragm with a null at the plane of the diaphragm and a 180° phase
shift either side of the null.
Burrows and models were also driven externally by an unmodified Sony earphone mounted on a 2 mm square support rod at a range of 50 mm from the surface hole with the centre of the diaphragm on the axis of the burrow.
Both sound sources were driven directly by a Hewlett Packard 8111A pulse and function generator gated to produce discrete tone bursts of complete sinusoidal cycles by pulses generated by a Neurolog pulse and delay-width generator. Both sources were effectively aperiodic when driven by tone bursts between 0.8 and 5 kHz; in other words, there were no significant resonances in the frequency band studied here and, when the source was driven by a tone burst of 10 cycles, it produced 10 cycles of sound.
Probe microphones
Measurements of the acoustics of burrows, casts and models were made using
a pair of probe microphones similar to those described by Young and
Bennet-Clark (Young and Bennet-Clark,
1995) made from Realistic
Electret tie-pin microphones (Tandy catalogue 33-1052) but with 18 mm long
probe tubes of 1.25 mm external diameter and 0.8 mm internal diameter. The
microphones were used in conjunction with a specially constructed two-channel
preamplifier and, overall, the responses were amplitude-matched to less than
±0.5 dB from 200 Hz to 6 kHz with a time difference of less than
±5 µs equivalent to ±5° at 3 kHz.
Terminology
The sharpness of tuning of resonances is quoted as the quality factor (or
Q), given by /loge(decrement) of the free decay of an
oscillation. The Q value is also the ratio of the resonance frequency
of a system to the width of the response curve at 50% amplitude or at 3 dB
below the peak response (Morse,
1948
).
The relative bandwidth of frequency spectra is quoted as Q-3dB, given by the frequency at which amplitude is maximal divided by the bandwidth at 3 dB below the maximum.
Two separate terms are used here because they concern different aspects of
the sound signals we describe. The contexts in which the use of different
measurements of Q are appropriate are discussed in Bennet-Clark
(Bennet-Clark, 1999a).
The frequency at which the acoustic response to a sinusoidal drive is maximal is termed the resonant frequency, Fo. The frequency of maximum power in a frequency versus relative power spectrum is termed the best or centre frequency, Fc.
In power spectra, the relative power is given in decibels (dB) relative to the peak power: 10 dB is a 10-fold change in power; 20 dB is a 10-fold change in amplitude.
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Results |
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Insects sang from within their burrows, which were often situated on flat regions of bare or sparsely covered hardened red clay. The burrow is complex and has an extremely consistent structure (Fig. 1). The surface hole opens into an ovoid flask-like top chamber connected via a narrower neck to a roughly ovoid lower chamber. An exit tunnel extends downwards from the lower chamber often to 200-300 mm below the soil surface. The sharp-edged surface hole expands into the top chamber at a depth of between 1 and 2 mm; the diameter of the surface hole is only slightly larger than the width of the insect's head (Table 1, Table 2). The dimensions of the top chamber and neck of the burrow were less variable than those of the lower chamber (Table 2). The interior of the top chamber, neck and top half of the lower chamber were smoothed or plastered to a surface roughness of less than 0.2 mm, compared with a roughness of more than 0.5 mm in the lower regions of the burrow. The smooth regions appeared to have been coated with some secretion, such as saliva, which consolidated and sealed the surface (Fig. 1); cellulose paint applied to these regions did not penetrate deeper than 0.2 mm but penetrated to over 0.5 mm in the rougher unsmoothed lower regions of the burrow. In some burrows, the lower parts of the lower chamber appeared to link up with larger cavities or chambers made by termites.
It was not possible to see where the insects were situated while singing, but we are able to estimate the singing position indirectly: while singing, the tips of the insect's antennae were visible extending from the surface hole. The length extending outside the burrow, measured from photographs, was 4.1±1.3 mm (mean ± S.D., N=5). The mean length of the antennae is 18.9 mm (Table 1) so, using the dimensions of the surface hole and top chamber, we estimate that the anterior of the insect's head was probably between 13 and 14 mm below the soil surface and between 4 and 5 mm above the burrow neck, with the posterior half of the body extending into the neck and the wings extended across the lower end of the top chamber (Fig. 1).
Songs
Insects sang in three periods, first for approximately 1 h starting
approximately 30 min after sunset, second for approximately 2 h starting just
before midnight and third for approximately 1 h starting 1 h before dawn.
The songs of more than 40 insects were recorded. Overall, the songs had a remarkable bell-like purity. The song is made up of a long-duration series of trills with a variable number of pulses and variable intervals between the trills, ranging from trills made up of from two to over 50 pulses, with intervals lasting from 0.4 s to over 1 s (Fig. 2). Within each trill, the first two or three pulses were approximately 2 dB quieter than the later pulses (Fig. 2), which usually did not vary in amplitude by more than 1-2 dB. Different insects produced songs with different pulse duration and pulse period, but each individual tended to produce trills of similar pulses. Mean values of certain song parameters are shown in Table 3.
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The songs of many insects showed smooth pulse envelopes
(Fig. 3A) and more-or-less
constant pulse durations. In such songs, the level of the second and third
harmonics was typically at least 30 dB below that of the best frequency,
Fc (Fig.
3B, Fig. 4C). Fc showed some inter-individual variation
(Table 3), but tended to be
constant to less than ±40 Hz in successive pulses of the song of an
individual (Fig. 3C). The song
pulses of most crickets show considerable frequency modulation (Leroy,
1966; Simmons and Ritchie,
1996
; Bennet-Clark,
1999b
) but, in many of the
songs analysed here, particularly where the pulse envelope was smooth (e.g.
Fig. 4A), the frequency varied
by as little as ±40 Hz for the major part of the pulse
(Fig. 4B). Frequency spectra of
this type of song showed that the song power was confined to a narrow
frequency band: the example shown in Fig.
4C,D has a Q-3dB of 55. Q-3dB values of over 30
were measured from the song pulses of 10 other insects. In many pulses, the
cycle-by-cycle frequency varied rapidly at the start and at the end of the
pulse. Similar phenomena are described below for the results of tests using an
electrical model.
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The decay of amplitude at the end of the pulse in many cases appeared to be nearly linear (Fig. 2D, Fig. 5A pulses 1, 2 and 3), but in others the decay appeared to be exponential (e.g. Fig. 3A, Fig. 6B). In certain cases, the frequency during this exponential decay differed markedly from the frequency in the adjacent parts of the pulse (Fig. 6C). The mean Q for the decay of the pulses from 10 insects was 6.91±2.33 (mean ± S.D.; maximum 11.0, minimum 4.1), which is similar to the measured Q for the burrows (see below and Fig. 8) and suggests that these exponential decay components are due to the resonance of the burrow. During the exponential decay of the pulse, the amplitude of the second and third harmonics in the sound fell by 10-15 dB; this is consistent with the behaviour of a freely decaying resonant system.
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In many recordings, one or two brief quiet sound pulses occurred during the interval between the normal song pulses Fig. 7A,B). The interval between these pulses was similar to the duration of the normal song pulses. The Fc of these pulses was similar to that of the normal pulses but, because the pulses were brief (Fig. 7B), the bandwidth of the frequency versus power spectra was far broader that those of the normal pulses (cf. Fig. 4C and Fig. 7C). These quiet pulses are probably caused during the wing opening movement; typically one or two are seen before the first pulse in a trill and another after the last pulse.
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Acoustics of burrows and the effect of the burrow on the insect's
song
Five burrows were used: of these, three included the whole of the lower
chamber and part of the exit tunnel and the other two were broken off
approximately halfway down the lower chamber. The pressure and phase of the
sound were measured 5 mm inside the top chamber (see
Fig. 1). All burrows showed
resonant properties: the mean Fo was 3.48±0.18 kHz
(mean ± S.D., N=5; range 3.22-3.67 kHz) and the mean
Q was 7.44±2.56 (mean ± S.D., N=5; range
4.0-9.7). The resonant response of one complete burrow (8/23Nov99) to a
discrete 11-cycle tone burst at 3.27 kHz is shown in
Fig. 8. The sound pressure
within the burrow builds up and decays exponentially; the Q of the
decay of the response was 5.7. After the initial build-up of the response, the
phase of the internal sound lags by approximately 90° that of the driving
sound measured outside the burrow. Because the burrows were fragile, more
elaborate tests were not carried out.
Two insects for which both song recordings and complete burrows were obtained showed a close correspondence between the song Fc and burrow Fo: for the first, the song Fc was 3.43 kHz and the burrow Fo was 3.47 kHz; for the second, the song Fc was 3.15 kHz and the burrow Fo was 3.22 kHz.
Another cricket which had previously been recorded in its own burrow was dug up (destroying its burrow) and transferred to burrow 8/23Nov99, which was placed in a large bucket of earth. This insect subsequently sang on three successive evenings following its re-location. In both burrows, the insect was able to produce song pulses with smooth envelopes (Fig. 9A). In its own burrow, the Fc of the song was 2.95 kHz, but in burrow 8/23Nov99 the Fc of its song was close to the measured Fo of the burrow, 3.27 kHz (Fig. 9B,C), and the lack of overlap in the cycle-by-cycle frequency of the song pulses in the two burrows suggests that the burrow is a major determinant of the song frequency. Unfortunately, because of the lateness of the season and the extreme fragility of the dry soil surrounding the burrows, we were unable to extend these important experiments.
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Tests with model burrows
In total, five natural-size plaster model burrows were tested. All showed
resonant properties with Fo between 3.35 and 3.8 kHz and
Q values between 5.8 and 9.5, which are similar to the values
measured for the natural burrows. The sound pressure and sound phase for one
model, relative to that of the external sound drive, are shown in
Fig. 10A. The sound pressure
rose approximately threefold and remained approximately constant within the
top chamber, then fell within the length of the neck to a value similar to
that outside the model in the lower chamber. The phase fell outside the
surface hole as the hole was approached then fell to -90° within the top
chamber; there was a further rapid phase lag of approximately 135° in the
length of the neck followed by a further slower phase lag in the lower chamber
and exit tunnel. From these measurements, it appears that the top chamber acts
as a unit and that rapid phase and amplitude changes occur at both ends. The
sound pressure and phase changes that occurred outside the model close to the
surface hole are probably due to near-field effects that are likely to occur
in the vicinity of a cavity of this type.
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The effect of moving a model cricket along the length of the model burrow is shown in Fig. 11. When the model was moved from the top chamber into the neck, the resonant frequency fell and then rose again as the model was moved into the lower chamber. The insect appears to sing with the tip of its abdomen within the neck of the burrow (see above). This experiment shows that the resonant frequency of the model burrow is sensitive to the effective diameter of the neck and that the insect may be able to tune the burrow by ±0.1 kHz by moving 2.5 mm up or down the burrow.
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Further tests used models approximately three times natural size made from film canisters, which were driven internally by a doublet source. The model shown in Fig. 10B had an Fo of 1.06 kHz and a Q of 9.9. The sound pressure and phase of the sound are shown in Fig. 10B, with the `surface hole' both open and closed. With the hole open, the sound pressure and phase are approximately constant throughout the major part of the `top chamber', but the sound pressure drops in the length of the neck. Within the `lower chamber', the sound pressure rises but the phase falls through approximately 180°. When plotted against the sound wavelength, the behaviour within this model closely parallels that within the natural-size plaster model shown in Fig. 10A; with the plaster model, driven externally, there is a 90° phase lag between the outside and immediate inside of the model. With the `surface hole' closed, the sound pressure within all regions of the film canister model when driven at 1.06 kHz was greatly reduced; the phase, in this configuration, changed through 180° either side of the plane of the doublet source, but there was a similar phase lag of approximately 180° between the neck and the end of the `lower chamber'. With the surface hole open or closed, the phase of the sound was inverted if the polarity of the doublet source was reversed by rotating it through 180° in the plane of the diaphragm.
The Fo and Q of the film canister model were affected by the diameter of the surface hole, varying from an Fo of 1.35 kHz and Q of 7.2 with an 18 mm diameter surface hole through an Fo of 1.06 kHz and maximum Q of 9.9 with a 10 mm diameter hole to an Fo of 1.01 kHz and Q of 6 with a 6 mm diameter hole. With a hole 4 mm in diameter or smaller, the resonance weakened and became hard to measure. When the porosity of the top chamber was increased by progressively unsealing 1.5 mm diameter holes previously drilled along its length, Fo increased and Q decreased; this effect was reversed by re-sealing the holes. Increasing the porosity of the lower chamber had only a small effect on the acoustics of the top chamber.
These models show similar resonant properties to those of the burrow and also show that Fo scales inversely with burrow size. The models also show that Fo and the sharpness of tuning depend of the dimensions of the surface hole as well as on the porosity of the walls of the top chamber.
Tests with an electrical model
The electrical model, which was tuned to 2.9 kHz with a Q of 10
(Fig. 12C inset), was driven
with tone bursts at either 2.9 and 3.2 kHz
(Fig. 12A). With 2.9 kHz tone
bursts of 40 cycles, the pulse envelope of the output built up exponentially
to a sustained maximum and then decayed exponentially
(Fig. 12A): the envelope was
similar to those shown in Fig.
3 or Fig. 5 pulse
2, and the frequency measured within the pulse and during the decay remained
at that of the input ±5 Hz (Fig.
12C). When the model was driven by two tone bursts at 2.9 kHz,
separated by a gap of half a cycle duration
(Fig. 12D), the pulse envelope
of the output showed an interruption and there was a decrease in the
cycle-by-cycle frequency for several cycles after the time of interruption but
otherwise, throughout both tone bursts and the decay, the output frequency was
close to that of the input (Fig.
12F). However, following the interruption, the amplitude of the
output fell more rapidly than at the end of the second tone burst; the second
exponential build-up, following the interruption, was similar to the build-up
at the start of the first tone burst; the final decay was slower and
exponential (Fig. 12D). The
effects on the amplitude of the output depended on whether the interruption to
the input caused the second tone burst to start in-phase or anti-phase with
the existing response of the model: gaps of 0.5, 1.5, etc. cycles had the
largest effects.
When the electrical model was driven with 3.2 kHz tone bursts of 40 cycles, the amplitude of the response initially rose rapidly, then decreased (Fig. 12B), then rose slightly after 5 ms to a steady value until 12 ms, after which the amplitude decayed exponentially. The cycle-by-cycle frequency rose over the first 3 ms from approximately 3.05 kHz to over 3.2 kHz before decreasing to a steady 3.2 kHz for the remainder of the driven part of the pulse (Fig. 12C). Throughout the exponential decay at the end of the pulse, the frequency was approximately the 2.9 kHz to which the model was tuned (Fig. 12C). When driven at 2.6 kHz, the envelope showed a similar bulge at the beginning of the pulse, but the cycle-by-cycle frequency showed a mirror image of that when the frequency of the input was higher than the Fo of the model. These wave forms resemble the irregular unimodal envelopes seen in, for example, Fig. 3A, Fig. 5 pulse 10 or Fig. 6A.
The electrical model was also driven by 3.2 kHz 20-cycle tone bursts separated by brief intervals. Fig. 12E shows the input and output of the model to a pair of tone bursts separated by half a cycle of the driving frequency. The output at the start of the first tone burst resembles that to the 40-cycle tone burst (Fig. 12A) but, after the gap between the first and second tone burst, there is an abrupt decrease in amplitude and in frequency, followed by a build-up in amplitude and change in frequency similar to those at the start of the first tone burst. The decay of the output at the end of the second tone burst resembles that of the 40-cycle tone burst. The wave forms and changes in frequency seen when this resonant system is driven by a series of short, interrupted tone bursts at different frequencies resemble those of some of the song pulses shown in Fig. 5.
With lower source impedances with a 3.2 kHz input, the amplitude changes at the start and end of the output occurred more rapidly and were smaller than those shown in Fig. 12, and the cycle-by-cycle frequency reached that of the input more rapidly; in contrast, with higher source impedances, the amplitude changes occurred more slowly, were larger in amplitude and the cycle-by-cycle frequency reached that of the input less rapidly.
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Discussion |
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Crickets calling from the ground or from exposed foliage typically have
calls with their Fc between 4 and 8 kHz (Dumortier,
1963; Leroy,
1966
), while the
Fc of mole crickets is between 1.6 and 3.5 kHz
(Bennet-Clark, 1970
). So a
general feature of burrow-calling insects is that physical factors that
normally constrain song frequency to vary with body size are relaxed. The
reason for this is that the acoustic transformation that occurs in the burrow
allows small sound-producing structures to be coupled via the burrow
to the far larger area of the mouth of the burrow from which the sound is
radiated (Bennet-Clark, 1995
;
Bennet-Clark, 1998
). The
present study describes for the first time a comparable situation in
burrow-calling crickets (Gryllidae, Gryllinae).
Consideration of the impedance matching between sound sources and the
surrounding air (see, for example, Olson,
1957; Fletcher,
1992
) shows that the 3.2 mm
diameter hole from which the song of Rufocephalus is radiated is a
far smaller source than that which would provide optimal impedance matching of
the 3.2 kHz song to the air. For optimum impedance matching, the source
diameter should exceed one-third of the sound wavelength. For the
approximately 3 kHz song of Rufocephalus, the wavelength,
,
is close to 100 mm, which requires a burrow with a mouth diameter of
approximately 35 mm. However, if the insect were to sing in the open air, the
wings would act as a dipole or doublet source with a diameter of approximately
5 mm, or 0.05
, with a specific acoustic resistance approximately
2x10-4 times that of air. By singing from within the burrow
and radiating sound from the surface hole, the source becomes a monopole with
a specific acoustic resistance 2x10-2 times that of air. The
burrow thus provides a 100-fold improvement in the coupling between the insect
and the air, permitting far louder and more efficient sound production;
similar advantages have been proposed for the singing burrows of mole crickets
(Bennet-Clark, 1987
) or the
abdominal resonator of cicadas (Bennet-Clark and Young,
1992
).
In contrast to the tuned low-Q singing burrow of mole crickets
(Bennet-Clark, 1987; Daws et
al., 1996
), that of
Rufocephalus is far more sharply tuned, typically with a Q
of over 7. This is achieved by radiating a relatively small proportion of the
sound energy in the burrow via the small surface hole, in contrast to
the far larger burrows (and hence far larger resonant mass) of mole crickets,
which radiate sound from surface holes that approach the optimal size for
impedance matching with the surrounding air (Bennet-Clark,
1995
). Mole crickets are large
robust insects that weigh between 10 and 50 times as much as
Rufocephalus and are, hence, capable of transducing a commensurately
greater mechanical power into sound. Also, by virtue of their far greater
size, mole crickets are able to build large burrows with surface hole
dimensions approaching that of the sound wavelength, which would be extremely
energy-demanding for an insect as small as Rufocephalus.
A numerical model of the burrow
A theoretical model is helpful in understanding the observed tonal purity
of the insects' songs and the resonant properties of the burrow. For this
purpose, we adopt the simplest model possible and require that it have no
adjustable parameters.
The geometry of the real cricket burrow is complex, so the theoretical model is constructed using the dimensions of the real burrows but with simplified geometry, as shown in Fig. 13A (and resembling that in Fig. 10B). The form of the theory provides simple scaling. Its results are unchanged if all dimensions are altered by a factor K and the frequency is changed by a factor 1/K; this accords with the measurements for two different sizes of physical model, as reported above.
|
The theory to be formulated is essentially one-dimensional, by which it is implied that all parameters are averaged over the cross section of the appropriate part of the burrow, so that they depend only upon a single parameter, the distance from the burrow mouth. Given these constraints, the major difficulty is then the modelling of the cricket sound source. In reality, there will be a complex flow of air around the vibrating wings, and a detailed solution of this aerodynamic problem is far beyond the demands of the present study. Instead, it is assumed that the cricket behaves as a simple dipole source, generating an oscillating pressure step, and hence an acoustic volume flow U along the axis of the burrow, and presenting no obstacle to other flow. The limitations of this model will be discussed below.
The acoustic pressure waves generated by the wings of the cricket are partially reflected at each discontinuity in the burrow, so that, in addition to the initial flow U, allowance must be made for two counter-propagating acoustic flows in each burrow chamber. If the amplitude and phase of each of these flows are initially taken as independent variables, then the model contains 12 such variables, four for each chamber of the burrow. On each of the four planes A, B, C and D of Fig. 13A, however, there are equations relating these variables. At A the connection is through the impedance of the surface hole, at D through the impedance of the exit tunnel, assumed to be infinitely long, while at B and C there are continuity conditions on both acoustic pressure and acoustic flow. This makes 12 equations in all, when both amplitude and phase are taken into account, which is a sufficient number to determine all the variables. The loss of energy as sound passes through the surface hole is actually very small, and most acoustic energy is dissipated as losses to the rough walls and as propagation into the non-reflecting lower large exit tunnel.
The results of the calculations show good agreement with experiment in most respects. Fig. 13B shows the calculated variation of acoustic pressure along the burrow and the calculated variation of acoustic phase for a cricket-source position in the middle of the top chamber; these results can be compared with those shown in Fig. 10. Of more interest is the calculated acoustic output from the mouth of the burrow as a function of frequency, as shown in Fig. 13C. There is a sharp resonance at 3.25 kHz, which is near the measured song frequencies, suggesting that the cricket must adapt its song frequency to the burrow in which it is singing (or, of course, the other way round). Changing the diameter and wall thickness of the surface hole can vary the frequency of the resonance. The Q of the resonance depends upon the value assumed for the wall absorption. However, even with the rather large wall damping coefficient of 10, which has been assumed in these calculations, the resonance is still quite sharp, with a Q of approximately 7, which again agrees with the measured Q values for real burrows. A further calculation shows that the acoustic output from the burrow is not greatly affected by the length of the lower chamber CD. The calculation further shows a minimum in the output power for a lower chamber length of approximately 7 mm, but only a small increase in output for both longer and shorter chambers. This insensitivity accords with the observed variability in the length of the lower chamber (Table 2). If acoustics were the only function of the burrow, one might have expected the insect to opt for a very short lower chamber in the interests of economy, but the extended burrow may provide the insect with access to water in a very dry environment or may be used as an oviposition site for mated females.
Calculations suggest that the optimal singing position of the cricket would be close to one end or the other of the top chamber, which agrees well with our observations on where the insect sings (see Fig. 1); the variation in output is quite large. However, the model source adopted here was that of a very small dipole oscillating with a rather larger amplitude, whereas the source provided by the cricket's wings may be better approximated by a much larger source oscillating with small amplitude. It is not simple to introduce this refinement into the numerical model, but an argument on general physical grounds suggests that an increase in source size will move the optimum singing position progressively towards the middle of the chamber, where the acoustic flow is smallest.
Song frequency and song purity
As has been indicated, the songs of Rufocephalus are, for their
size, unusually low in frequency, while the loudness
(Table 3) compares,
size-for-size, favourably with that of the far larger G. campestris,
which is approximately 95 dB at the same distance (from data in Bennet-Clark,
1970). In many examples, the
songs were also unusually constant in frequency, both within and between
pulses, with Q-3dB of greater than 30 and with harmonics
at levels of less than 3% of the amplitude at Fo. In
Rufocephalus, the Q of the burrow is greater than that
reported for mole crickets (Bennet-Clark,
1987
). Because of this high
Q, the burrow will be most easily driven at its resonant frequency.
And, as a consequence, the high Q and the close coupling between the
insect and the burrow will tend to entrain the frequency of catch-and-release
of the plectrum-and-file mechanism to that of the burrow; the system acts as a
resonator with its frequency set by the burrow. Such a system appears to
maintain the song frequency and coherence, as can be seen in
Fig. 3 and
Fig. 4. Indeed, if the coupling
between the insect and the burrow is tight and the difference between the
natural frequency of the insect's wings and the Fo of its
burrow is small, there may be no alternative. This appears to have been the
case with the burrow exchange experiment shown in
Fig. 9.
However, several of the songs we measured contained pulses with gaps, and showed both frequency and amplitude modulation. From the measurements made with the electrical model shown in Fig. 12, pulses of this type may result from a frequency mismatch between the insect and its burrow that causes an erratic cycle of catch-and-release of the plectrum against the file. The results with the electrical model (Fig. 12) suggest that the file-and-plectrum mechanism may act in two different ways during the production of the pulse.
First, the cycle of catch-and-release of the file teeth may become entrained to the Fo of the burrow. In this case, the pulse envelope will be comparatively smooth and the cycle-by-cycle frequency will be fairly constant. The sound pulses shown in Fig. 4A,B are examples of this phenomenon and are associated with the very high Q-3dB for this type of song pulse. However, the initial process of entrainment at the start of the pulse may be accompanied by rapid amplitude modulation of the waveform of the pulse. This amplitude modulation may rise rapidly and then fall (see Fig. 3, Fig. 4A, Fig. 12A) and be accompanied by frequency modulation. This is shown in the first 2 ms of the pulse in Fig. 4 or the first 4 ms of that in Fig. 12A. What is not clear is whether the insect's wings are being entrained to the Fo of the burrow or the burrow is being driven by the cycle-by-cycle frequency determined by the file-and-plectrum mechanism of the wings. The latter would explain the sudden jump in frequency that occurs at the end of pulses of the type shown in Fig. 6, where the amplitude decays exponentially at a nearly constant but different frequency from that in the preceding part of the pulse and with a lower harmonic content.
The second alternative is that the file-and-plectrum mechanism does not become entrained or does not remain entrained to the Fo of the burrow. This was modelled in Fig. 12B, which shows that even a brief irregularity in the cycle-by-cycle input to a resonant load can result in rapid amplitude and frequency modulation of the output, particularly when the frequency of the input differs from the Fo of the load. An initial lack of entrainment would explain the brevity of the first sub-pulse of pulse 9 in Fig. 5, which occurs at a different cycle-by-cycle frequency from that of pulse 2 from the same insect, with a slow amplitude modulation and comparative constancy of frequency. Subsequent loss or failure of entrainment may then result in an interruption to the input from the insect's wings to the burrow. In this case, if entrainment fails, causing irregular movement of the insect's wings, the frequency generated by the catch-and-release of the file-and-plectrum mechanism of the wings may show considerable variation during the wing-closing movement. Our preliminary studies here, and those with the songs of other species of cricket, show a clear relationship between rapid changes in amplitude or frequency and the relative amplitude of harmonics of the Fc of the sound (H. C. Bennet-Clark and W. J. Bailey, in preparation).
The comparatively sharply tuned burrow appears to be an important
determinant of the song frequency (Fig.
9) and it appears that even very brief sounds, such as the `ticks'
produced during the wing-opening movements, are capable of exciting the
resonance (Fig. 7).
Transduction from the small wings of the insect to the surrounding air
via two resonant stages offers advantages in impedance matching and
maintenance of song purity that have been considered elsewhere (e.g.
Bennet-Clark, 1995). The
resonant singing burrow of mole crickets (Bennet-Clark,
1987
; Daws et al.,
1996
) offers similar
advantages but, as described above, its acoustics differs.
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Acknowledgments |
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References |
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