Directional acoustic radiation in the strut display of male sage grouse Centrocercus urophasianus

Studies of acoustic mate-attraction displays have concentrated on the role of spectral and temporal characteristics in signal evolution. Despite the likely importance of signal directionality (Bradbury and Vehrencamp, 1998; Witkin, 1977), the fitness consequences of directional patterns of acoustic display have been largely overlooked because these patterns have been considered too difficult to map in the field. In this paper, we present measures of the acoustic radiation (beam) patterns of the male sage grouse strut display. The spectral and temporal aspects of this display have been extensively examined (Gibson, 1996; Gibson and Bradbury, 1985; Gibson et al., 1991; Hartzler, 1972; Spurrier et al., 1994; Wiley, 1973a; Young, 1994). However, despite anecdotal observations of strong directionality (Wiley, 1973a; J. W. Bradbury and M. S. Dantzker, personal observation), these directional patterns have not been quantified and their consequences have not been explored.

The strut display of the male sage grouse is highly stereotyped and elaborate (Hjorth, 1970; Honess and Allred, 1942; Wiley, 1973a; Young et al., 1994). Prior to display, a male sage grouse inflates a region of his esophagus that sits in a muscular bag at the base of his neck. The strut display is a series of three heaves of this esophageal expansion through outreached wings held stiffly at either side (Fig. 1). Sage grouse keep their beak closed for most of the display; sound production coincides instead with tandem anterior expansions of the esophageal air sac through ventral apteria (featherless regions) on the neck (Hjorth, 1970; Young et al., 1994). These dark, inflated skin patches bulge repeatedly through a dense region of white feathers to create a dramatic and unusual acoustic (Fig. 2) and visual (Fig. 1) display. While many other groups of vertebrates use external air sacs in sound production and radiation, this explosive use of dual anterior air sacs is unique to the Centrocercus grouse strut display. This mate-attraction display has evolved under the intense sexual selection of a lek mating system (Höglund and Alatalo, 1995). However, male sage grouse do not appear to direct their display efforts towards females by facing them during display (Simon, 1940; Wiley, 1973a). In this study, we examine the directionality of the acoustic signal produced by this unusual display mechanism of sage grouse.

Directional patterns of acoustic radiation, beam patterns, have usually been measured under laboratory conditions where the animal’s position and orientation are held constant and repeated measurements are made on induced or coaxed phonation. Similar methods can be used in the field when the animals of interest remain stationary during display and will tolerate movements around them (Gerhardt, 1998). Sage grouse afford neither such opportunity as they are rarely available for captive study and wild birds travel rapidly around their territories between consecutive displays. We developed a novel method for the measurement and calculation of these beam patterns which allows for the use of unrestrained and actively mobile wild individuals while correcting for heterogeneity in the acoustic environment.

Here, we present evidence that acoustic radiation patterns of the sage grouse strut display are unlike any previously measured in a vertebrate. Where vertebrate acoustic signals have been found to be directional, they are most intense anteriorly and are bilaterally symmetrical (see Discussion). Our results show that sage grouse beam patterns are variably asymmetric about the animal’s anterior–posterior axis. The beam pattern of one of the call elements, the ‘whistle’, has a deep minimum (null) directly in front of the displaying bird and is instead most intense to the sides and behind.

Sage grouse Centrocercus urophasianus (Bonaparte) travel across the mostly flat display arena (lek) on foot; therefore, we focused our efforts on characterizing acoustic directionality in the horizontal plane. The sound field surrounding an acoustic source, in this case a strutting sage grouse, is considered to be directional when the amplitudes of the induced pressure waves vary as a function of angle. This angular variation is described by a normalized function called the beam pattern (Kinsler et al., 1982, p.175). Here, we describe a method for measuring beam patterns of unrestrained animals in complicated acoustic environments.

We recorded male sage grouse using an eight-microphone array that we deployed around their display territories (described in greater detail below). The essential problem we faced in determining the beam pattern from these recordings was to separate the superimposed effects of source directionality, which were inherent to the calling bird, and directionality introduced by the environment. In our case, this environmental effect arose primarily from interfering acoustic reflections from the lek surface. To discriminate between the directional effects of the source and those that were environmentally induced, we measured the directionality of the signal received from a known, calibrated source positioned inside our array. This calibration procedure allowed us to identify and quantify the effects of the environment across the range of frequencies used by the sage grouse. We could then calculate beam patterns generated only by the directional properties of the source. The mathematical basis underlying our calibration procedure, as well as the concept of beam pattern relevant to this study, are developed in some detail below.

We constructed acoustic radiation maps from recordings of three males displaying at a lek (lek 8) in Long Valley, Mono County, California, USA (37°40′N, 118°50′W) from 4 to 21 April 1997. This spanned the main period of display activity in this year. Lek 8, like most of the leks in this region, is located on a mostly flat alkali meadow surrounded by rolling hills of sagebrush. The vegetation on the meadow (Carex sp. and saltgrass) was sparse and low (as seen in Fig. 1).

We deployed recording arrays (Fig. 3) at two focal sites chosen to overlap 1–2 male display territories where female visits were common. In both these areas, we arranged the circular eight-point arrays using a Sokkia measuring system (DIO343) to measure angle and a tape measure for distance. We positioned the microphones at each point on this array at approximately female head height (approximately 20 cm) so that their diaphragms faced the center of the array. On one side of the array, 20 m from the center, we positioned a hide that served as an observation and recording station (see Appendix A for details of the acoustic recording system). We stretched alkali-resistant microphone cables from the microphones to the hide. Sage grouse avoid exposed cables so we buried the cables in shallow (4 cm) ditches. This array design minimized disturbance to the birds’ movements and activities.

Using the Sokkia measuring system, we also assessed the relief of the encircled area (Fig. 3). This was accomplished by marking the distance from a fixed plane above the lek, determined by the height of the leveled survey scope, to the ground below. We measured this distance at 1 m intervals along the eight radial paths from the array center to each microphone location. From these data, we constructed topographic maps using MATLAB’s ‘contour’ function, with the heights converted so that they were relative to the array’s lowest point.

We made synchronous video recordings for all focal studies using a Canon Hi-8mm ES2000 camera positioned in the hide approximately 0.75 m from the ground on a tripod. We synchronized the audio and video recordings by video-taping the running tape counter on the face of the audio recorder for a few seconds at the beginning of each recording session. Each day, we started recording as soon as there was sufficient light and continued until the males ceased their activity on the lek. We designated two sites within each array as measurement positions: the center of the array and a position off center where the birds were seen calling. We measured transmission loss along each radial propagation path from these points as described in Appendices A and B. To minimize disturbance to the birds, we made these measurements in the birds’ absence. Preliminary study showed that propagation properties are stable through the night and the display period (05:00–07:30 h), but change rapidly later in the morning (see Results and Fig. 4). These measurements were therefore made between 23:00 h and 03:30 h when the propagation effects are highly concurrent with those of the display period but the birds are absent.

We digitally transferred all recordings to a Power Macintosh for analysis using an eight-track synchronous acquisition system (Digidesign ProTools v.4.1 with 888 I/O coupled to a TASCAM IF-88AE digital audio-interface unit). Using this software, we identified short sections of audio for detailed analysis, then created synchronized monaural files (22.05 kHz, wave format) for each track, which we transferred to MATLAB (MathWorks, 1998) for subsequent analyses.

Our method for the mapping of acoustic radiation patterns required that vocalizations be produced near the designated measurement sites. We used acoustic localization to identify vocalizations produced in very close proximity to these calibrated sites (⩽1.5 m). Before localization, we preprocessed all signals with a highpass filter (second-order Butterworth with 200 Hz cut-off; MathWorks, 1998).

We employed a localization algorithm based on a least-squares best fit (Spiesberger and Fristrup, 1990) to equations that relate the sources’ positions to the known locations of the microphones and the measured time delays in signal arrival between the microphones (Watkins and Schevill, 1971, 1972). We modified these equations in a number of ways. First, neglecting the small height variations of the lek surface, we reduced these equations to a two-dimensional case. Second, we employed the MATLAB function ‘backslash’ for the least-squares calculation of the over-determined set of linear equations (MathWorks, 1998). Third, the time delay estimates were taken from the peak location of the time-domain cross-correlation functions generated with the MATLAB function ‘XCORR’ (MathWorks, 1998; Shure and McClellan, 1997). Fourth, and most significantly, we calculated the final location estimate as the average location of eight semi-independent calculations made using each of the eight microphones as the designated reference. The algorithm requires the designation of a reference microphone, and previous users have designated a single receiver as this reference (Freitag and Tyack, 1993; Spiesberger and Fristrup, 1990; Watkins and Schevill, 1971, 1972). Our multi-reference algorithm increases the number of time delay estimates used in the calculation by a factor of four. In addition, only calls that showed a high correspondence between all eight location estimates were used in these analyses. We found that imperfections in the data (e.g. overlapping vocalizations, low signal/noise ratio) led to disparate estimates of location. This additional step allowed us to identify and purge these vocalizations from the data set.

Our system was well suited for acoustic localization using this algorithm. The method assumes that the speed of sound is known, that there is little wind and that receiver positions are accurately known (Spiesberger and Fristrup, 1990). Sound speed was calculated from air temperature measurements. Concurrent wind measurement allowed us to exclude data recorded during periods of wind in excess of 3 km h⁻¹. Receiver positions were definitely fixed and precisely known. Localization accuracy is high when the sound source is located near the microphone array (McGregor et al., 1997) and is best when the animal is completely encircled by receivers (Watkins and Schevill, 1971, 1972). Both were true of our experimental design. Also, the number of receivers, eight, far outnumbered the minimum (four) necessary to estimate the location unambiguously in two dimensions using this algorithm. In addition, the abrupt frequency and amplitude modulations of the sage grouse vocalization (Fig. 2) yielded sharp unambiguous peaks in the time-domain cross-correlation functions.

For each vocalization, the bird’s location was combined with the orientation of the male relative to the receiver array to determine each receivers’ value of θ for that call. The orientation could not be determined acoustically and was therefore estimated from the synchronous video recordings.

Band-averaged beam patterns, b_band(θ), were then calculated for those calls that were generated at the measurement locations. The propagation delays, identified as part of the localization algorithm, were removed from the time series data in order to synchronize the signals. We then Fourier-transformed the data using a 128-point fast Fourier transform (with a zero-overlap Hanning window) and calculated the power spectral density, |p_r(f,r,θ)|². This, along with the measures of transmission loss, TL(f,r,θ), calculated as in Appendix B, were then substituted into equations 7 and 8 to yield the band-averaged beam pattern over a 5.8 ms time slice. The transmission loss corrections were truncated at the measurement-system noise floor, corresponding to TL=−45 dB.

The directivity index (DI) measures the degree to which a radiation pattern is concentrated. This index compares the intensity of the maximum lobe of a beam pattern with the intensity of a uniform source radiating the same total power output. In this way, the DI incorporates aspects of the radiation field off the maximum lobe whereas beam width measures do not. Derivations of DI are available elsewhere for three-dimensional radiation patterns (e.g. Kinsler et al., 1982). In Appendix C, we derive an analogous measure for radiation patterns resolved only in a single plane.

Transmission loss fluctuated with time of day but was stable from 22:30 to 07:30 h (Fig. 4). Therefore, late-night measurements made within a bird’s territory while the animal was away were highly concurrent with propagation during the period of display. Below 1.2 kHz, these transmission loss measurements were extremely consistent, differing in value by a mean of 1.1±0.5 dB (N=5). The higher frequencies, 1.2–5.0 kHz, show more variation, 3.7±1.8 dB (N=5), across the time window of calibration and bird display. Because of this variation in the transmission loss curves over time, we estimate our error of measurement for beam pattern mapping to be ±2 dB below 1200 Hz and ±4 dB above 1200 Hz.

The transmission loss curves for sound propagation over the different paths of the lek (Fig. 5) show many similarities but also important differences. Measurements made from the center of each array are easiest to compare since the path lengths are equal (Fig. 5C,D). In both arrays, transmission loss curves for each path are highly concordant below 500 Hz. At these low frequencies, sound propagation was reliable across paths and less acoustic pressure was lost in transmission than in free-field spherical spreading. The low-frequency components of the sage grouse display (coos and pop 2) have most of their energy in this well-propagated frequency range (Fig. 2). Therefore, these components of the strut display must propagate reliably across fairly long distances.

In contrast, the higher-frequency whistle spans a less reliable frequency band from 0.6 to 3.2 kHz (Fig. 2). Sound in this range propagated far less effectively than free-field spherical spreading. Also in this frequency range we find a deep propagation minimum (notch) in the transmission loss curves of all paths. The exact position of this notch is variable between paths but is always between 1 and 2 kHz. Even outside the path-specific propagation notch, there are 10–15 dB differences in transmission loss between the paths. Long-distance communication in this frequency range is very unreliable, suggesting that the whistle component of the call might be an exclusively close-range component of sage grouse communication. Shorter paths show shallower notches and a concomitant decrease in the frequency-dependence of propagation (Fig. 5D,E).

The major features of our transmission loss curves (Figs 4, 5) are consistent with the measured curves and models of previous acoustic studies of sound propagation close to the ground (Embleton, 1996; Embleton et al., 1976, 1983). These major features include (1) low levels of loss for the low frequencies, (2) a stop-band notch in the intermediate frequencies due to interference between the direct and reflected paths and (3) lower levels of loss again for the higher frequencies where the reflected paths are less coherent. These features are expected for propagation near most surfaces.

There is, however, much heterogeneity between the transmission loss curves, even those with identical path lengths (Fig. 5C,D). Differences between these curves reveal propagation effects that arose from differences in topography (Fig. 5A,B) and soil composition. No extant acoustic model sufficiently explains the heterogeneity in the propagation curves that we have presented here. However, using our method, we were still able to reconstruct beam patterns for calls produced at sites where we had made empirical measures of the transmission loss. All these sites and their associated transmission loss curves are shown in Fig. 5.

We identified 39 vocalizations produced within 1.5 m of our calibration sites: 30 vocalizations from male A, seven from male B and two from male C. Although this sample is skewed towards a single individual, the range of directionality exhibited by male A subsumes the range of values exhibited by the other two males. Therefore, we consider each vocalization to be independent for statistical analysis. Where appropriate, we also present directionality measures for each individual. In our subsequent studies, we will use an omnidirectional speaker that allows us to calibrate more sites within each array and thereby achieve sample sizes sufficient for comparisons between males.

We found that the acoustic component of the sage grouse strut display is directional. This directionality showed high between-note variability in all measured vocalizations. However, we found that the coos and pops showed high within-note stability in their beam patterns. These notes, therefore, were best examined by calculating time-averaged beam patterns, b_band,note(θ). In contrast, the beam patterns of the frequency-modulated whistles varied significantly in time and consequently were examined as a series of discrete time slices, b_band(θ)?????

The coo notes had directional beam patterns (Table 1) which were consistent between the two coos of a single call but variable between vocalizations (Fig. 6). The beam patterns of the coos tended to be loudest to the front and quietest at the sides, with a variable amount of energy radiated behind the displaying bird. The orientations of the major lobes of the coo beam patterns were variable with respect to the anterior–posterior axis of the birds. In many vocalizations, the maximum lobe was rotated by up to 45 ° to either side of the bird, and in some examples the posterior lobe was actually slightly louder than the anterior lobe. Fig. 7 illustrates these trends as histograms of the maximum and minimum points on each beam pattern, θ_max and θ_min. These maxima and minima had significantly non-random angular distributions (Table 2). The two pops showed markedly different beam patterns from the coos and from each other. Pop 1 was strongly directional (Table 1) but the angular distribution of the beam pattern maxima and minima was not distinguishable from random (Rayleigh test: combined data set, N=39, P>0.1). Pop 2 was much less directional (Table 1). The beam pattern range for many vocalizations fell below the noise floor of our technique, 4 dB in this frequency band. However, when we restricted our examination of the directionality to those pop 2 notes with a beam pattern range greater than the potential measurement error, we found that, when the beam pattern was significantly directional, this pop tended to be louder in front of the bird and quieter behind (Table 2).

During the duration of the whistle, both the dominant frequency and the air-sac geometry change extensively. This variation should be accompanied by fluctuations in the directionality of acoustic radiation. Such variability was seen in all whistles (Fig. 8). Despite this variability, all whistles showed a deep anterior null in the beam pattern. This null was not present for the entire whistle but was always present for at least part of it. In contrast to the beam patterns of the coo and pop notes described above, the modal positions of the whistles’ beam pattern minima, mode[θ_min], were typically in front of the strutting bird (Fig. 9; Table 2). In addition, the modal maxima, mode[θ_max], of the whistles were never in front of the bird but were oriented to either side or behind. Thus, the angular distribution of the whistles’ modal maxima and minima are the inverse of the coos’ distributions. On average, the whistle was highly directional and extremely variable (Table 3). At its most directional time slice, the average whistle was 22.9±3.4 dB less intense in front of the calling bird than it was at the sides (Table 3).

The results described above show that sage grouse strut displays have strikingly directional patterns of acoustic radiation. Here, we compare these directional patterns with the patterns observed in other vertebrates and show that the sage grouse radiates sound in a fashion that is fundamentally different from that for any previously measured vertebrate (see Gerhardt, 1998, and below for review). To measure these patterns and map them accurately, we had to develop a new method for the measurement of acoustic directionality in the field. We start our discussion by explaining the major novelties and advantages of our method that allowed us to make these discoveries.

There are two features of our method for measuring acoustic emission patterns that are advantageous relative to previous methods used in both laboratory and field studies: (1) synchronous recordings from multiple angles around the focal individual, (2) explicit measurement of environmental effects on propagation. Here, we examine the specific advantages associated with each of these features.

Previous studies have made detailed beam pattern maps using two microphones by repeatedly repositioning a single receiver at various angles while another remains in a fixed position to correct for fluctuations in overall amplitude. This asynchronous, two-microphone method allows for fine spatial resolution in the mapping of acoustic fields and has been used to reconstruct some very complicated radiation patterns (Au et al., 1995; Bennet-Clark, 1987; Forrest, 1991; Hartley and Suthers, 1987, 1989, 1990; Hunter et al., 1986; Larsen and Dabelsteen, 1990). The only limit to the spatial resolution is the number of measurement sites chosen by the researcher. However, this method can only be used when the focal animal’s position is fixed by behavior or design and the same vocalization can be coaxed or induced for repeated measures. In addition, using asynchronous measures to reconstruct radiation patterns assumes that beam patterns remain constant between vocalizations of the same type. Our synchronous recording technique allowed us to test this assumption in sage grouse. Our results show that the beam patterns for each call note vary between vocalizations. Therefore, this assumption may be inappropriate for some organisms, and synchronous recording is therefore preferred (Gerhardt, 1998). The trade-off associated with synchronous recording is that the spatial resolution of radiation maps is limited by the number of receivers deployed. This number has practical limitations, especially in the field. However, our eight-microphone design was sufficient to describe the major features of the radiation patterns produced by sage grouse.

Laboratory study of acoustic emission fields has one principal advantage over our method: a controlled environment. Anechoic recording chambers obviate detailed environmental calibration since the measures of the acoustic field are direct measures of the emission field. For many organisms, laboratory study is impossible or impractical and measures must be made in the field. Field study also allows for the measurement of acoustic directionality in the context of natural display behavior. Field studies must account for the complexities of propagation in natural environments. Previous field studies of acoustic directionality have dealt with the environment by either assuming or demonstrating (a) that sound propagation is not frequency-dependent over the measurement distance and (b) that environmental effects are homogeneous across the measurement paths. These simplifications allow the treatment of field measurements as if they were conducted in an anechoic chamber and thus require no explicit corrections for propagation effects. These simplifications may be appropriate (a) when the measurement propagation paths are very short, (b) when there are no nearby surface boundaries, and (c) when all paths are free from heterogeneous features. To make measurements on unrestrained and mobile sage grouse in the wild, our method used longer measurement distances over a heterogeneous surface boundary. Therefore, neither simplification was appropriate. Instead, we measured the precise transmission loss curve for each measurement path and explicitly corrected for propagation heterogeneity in frequency and space. In this way, our method makes acoustic-directionality research possible in a wider variety of organisms and environments.

Our measurements showed that sage grouse strut displays have markedly directional acoustic beam patterns. Although the directionality of these patterns varied both within a vocalization and between calls, clear patterns emerged. First, the coos were always intense in front of the animal and sometimes intense behind as well. The coos were least intense to either side. This maximal lobe rotated up to 45 ° to either side between vocalizations. The whistles showed a distinctly different beam pattern. While variable across the length of the note, the principal beam pattern that dominated the radiation of the whistle was laterally bilobate with a strong anterior null in the beam pattern. Pop 1 was strongly directional but highly variable in its orientation, whereas pop 2 had a much lower magnitude of directionality but was fairly stable in its directional pattern.

The acoustic radiation patterns of a number of other vertebrate vocalizations have been measured and they conform to a general theme. Directional vertebrate vocalizations are bilaterally symmetrical with their maximal radiation point anterior to their site of acoustic radiation (mouth, nares or vocal sac). This is consistent with models of single-site acoustic radiation. The degree to which these fields are directional varies from a narrow beam to nearly or completely omnidirectional. The most directional vocalizations, with approximately 20–36 dB in beam pattern range, are the echolocation pulses of bats and of toothed whales (Au et al., 1995; Fuzessery et al., 1992; Hartley and Suthers, 1987, 1989, 1990; Pilleri et al., 1976; Purves and Pilleri, 1983; Schnitzler and Henson, 1980; Wotton et al., 1997). Few of the above studies resolved these patterns beyond the forward-facing 180 °, so these range values are estimates and it would be inappropriate to estimate DIs. Intermediate directionality, with beam pattern ranges of 6–15 dB and DI between 2.0 and 3.5 depending on frequency, has been found in human speech and blackbird vocalizations (Flanagan, 1972; Larsen and Dabelsteen, 1990). Anuran calls are at the low end; the most directional of the measured frog calls has a beam pattern which ranges 6 dB from front to back (DI≈1.5) (Gerhardt, 1975). Most frogs vocalizations are much less directional and some have been found to exhibit a nearly omnidirectional pattern, DI≈0 (Gerhardt, 1975; Passmore, 1981; Prestwich et al., 1989).

To our knowledge, there are only two previously reported minor exceptions to the vertebrate theme that vocalizations are loudest in front and quieter behind. One species of toad, Bufo americanus, radiates sound in two equally intense lobes 180 ° apart, one in front and one behind, separated by a notch of approximately 3–4 dB (Gerhardt, 1975). The fairly shallow lateral null in this pattern is an interesting deviation from the standard vertebrate pattern and remains unexplained. Also, the call of the African frog Cacosternum boettgeri was reported to be 1–1.5 dB louder directly posterior (Passmore, 1981). This effect was extremely small, within the range of likely measurement error for working in the field. In addition, since this study did not account for propagation effects, it is possible that the skew was environmentally induced.

Portions of the sage grouse whistle show the greatest directionality of any strictly communicative vertebrate signal so far measured. Only vocalizations used for echolocation are known to be more directional. The sage grouse coos and pop 1 show the level of directionality commonly found in the communication signals of larger vertebrates, while pop 2 is more similar to the omnidirectional sounds of frogs. We expected that lower-frequency sounds, those with large wavelength-to-body size ratios, would be less directional than higher-frequency sounds (Bradbury and Vehrencamp, 1998; Prestwich, 1994). This held generally true in sage grouse since the whistle contains energy principally in wavelengths shorter than those of the less-directional coo and pop notes. However, despite having a similar spectral make-up, pop 2 is markedly less directional than the two coos. Also, the lower frequencies of the whistle, 500–1400 Hz, often exhibited the deepest anterior nulls and the strongest directionality. Therefore, we propose that the rapid movement of the esophageal air sac or associated structures is responsible for much of this between-note variation in magnitude of directionality.

While the magnitude of the directionality is not completely unprecedented, the shape of the beam patterns of the sage grouse strut is unlike any previously seen in vertebrates. The two most surprising results are the deep anterior null in the radiation pattern of the whistle and the bilateral asymmetry and variable direction of the peak in the radiation pattern of the coos and pop 1. Previous attempts to understand the patterns generated by vertebrate vocalization have used simple acoustic models of single-source radiation to arrive at a realistic estimate of radiation mechanisms. A model of the vocal apparatus as a vibrating piston suspended in a baffle has adequately explained the basic patterns observed in birds, bats, cetaceans and humans (Flanagan, 1972; Hartley and Suthers, 1989, 1990; Hunter et al., 1986; Larsen and Dabelsteen, 1990; Mogensen and Møhl, 1979; Pilleri, 1990; Stevens, 1997; Strother and Mogus, 1970). Alternative models of single-site radiation have come closer to explaining the smaller side lobes in the acoustic field of bats (Hartley and Suthers, 1989, 1990). One dual-site radiation model argued that emission through a paired structure could lead to an increase in directionality for bats that call through their noses (Möhres, 1967; Sokolov and Makarov, 1971).

The beam patterns seen in the sage grouse do not fit those predicted by any of these models. The shape of the beam patterns for the coos and pops superficially resemble some of these patterns. However, since all the potential radiation sites (mouth and air sacs) face forwards for the duration of the vocalization, none of these models can explain the variability in the location of the peak or the intensity of the rearward-facing lobe. Furthermore, none of the models can explain the anterior null and lateral projection of the whistle component. Because of this lack of congruence with existing models, we suggest that the sound emission mechanism of the sage grouse is unlike that of any other vertebrate so far examined.

Even directionality as low as 3–6 dB is likely to be behaviorally relevant because, given spherical spreading, this translates to a 40–100 % greater distance of propagation in the direction of the maximum lobe before the signal is lost. Directionality is a defining component of an acoustic signal’s active space, although it is often overlooked (e.g. Brenowitz, 1982). When a sender knows (or can estimate) the location of its intended receiver, the signaler can reduce the cost of eavesdropping by predators and competitors by turning so that the peak of its beam pattern corresponds to the position of the receiver. The signaler can then produce just enough acoustic power to signal effectively to that individual (Klump and Shalter, 1984; Larsen and Dabelsteen, 1990). In addition, when competitors have overlapping signals, the sender can combine display orientation with a more powerful vocalization to decrease acoustic interference from its competitors and maximize the probability of response by the receiver. If the receiver can perceive the vocalization’s directionality, then appropriate posture might also help to specify the intended recipient of the signal (McGregor and Krebs, 1984; Richards, 1981). In these ways, directional patterns of acoustic radiation can strongly influence the choice of display posture, allowing for more effective or efficient communication to receivers of known location. When the intended receiver’s position is unknown, however, directional display can be a liability necessitating continual modification of display posture to maximize the probability of reception (Forrest, 1991). In situations like this, an omnidirectional vocalization may be favored. The directionality of vertebrate vocalizations is likely to be tuned to the social and environmental context of the display, and display posture is likely to reflect the degree and nature of that directionality.

Sage grouse males do not face females head on. Early observers of the mating display of the male sage grouse noticed that, although the display was dramatic, the efforts seemed to be unrelated to the locations of potential female mates. Simon (1940, p. 470) wrote that the males, ‘seemed disinterested in the hens…[and] in strutting paid very little attention to [them]’. Indeed, while males rotate to different directions for successive calls, they rarely face a female head on (J. W. Bradbury and M. S. Dantzker, personal observation). Females turn with the male to help maintain an oblique angle between them (Wiley, 1973a). It is this indirect approach that fooled Simon (1940) into his incorrect inference that the sage cock was not in ‘actual pursuit’ of hens. We now know that lekking sage grouse are under intense sexual selection driven by female choice (Gibson and Bradbury, 1985; Hartzler, 1972; Hjorth, 1970; Lumsden, 1968; Patterson, 1952; Scott, 1942; Wiley, 1973b). The results of this study suggest that, despite maintaining an oblique orientation, male sage grouse might still reach females with the loudest portion of his acoustic signal. Beyond this, however, these data do not reveal the adaptive significance of the unusual radiation patterns we have described. While it is possible that these beam patterns evolved under direct selection, it is also possible that the patterns are an epiphenomenon of selection for loud or impulsive sounds. Alternatively, the patterns of acoustic radiation might have evolved to improve the efficacy of an already lateralized display that evolved first for other reasons.

Investigations of female sage grouse mate-choice have identified a number of spectral and temporal aspects of the strut display that are correlated with mating success (Gibson, 1996; Gibson and Bradbury, 1985; Gibson et al., 1991). While amplitude is likely to be an important determinant of mate attraction (Forrest and Raspet, 1994), these studies could not examine signal intensity since field measurements were not repeatable (J. W. Bradbury, personal observation). The results of the present study suggest that the unusual and variable beam patterns of the sage grouse display combined with the leks’ heterogeneous transmission loss blocked these previous efforts from accurately measuring signal amplitude. These same factors, directionality and heterogeneous transmission loss, also stymie efforts to measure the amplitude of other species’ acoustic displays (Gerhardt, 1998). Techniques such as the one we have described here facilitate the study of acoustic amplitude in the field and thereby allow more thorough investigation into the importance of amplitude and directionality in signal evolution.

The acoustic system, described below, had its own characteristic acoustic properties. These properties had to be identified and quantified before the system could be used to measure the transmission properties of the lek and the radiation pattern of the sage grouse. This Appendix outlines how this was accomplished. In what follows, we will assume that all time variables have been Fourier-transformed so that we are dealing with the steady-state response of the system.

The acoustic system consisted of source and receiver components. The sound source was a Kudelski powered speaker driven by a Stanford Research DS340 signal generator. A single receiver channel consisted of an Audiotechnica MB1000L microphone coupled to a custom-built amplifier through an impedance-matching transformer. The output from the amplifier was recorded on a TASCAM DA88 digital recorder, which provides 96 dB of dynamic range from 10 Hz to 22.5 kHz.

Acoustic transmissions close to the ground are characterized by frequency-dependent structure in the transmission loss caused by interference between airborne and ground-reflected arrivals. Absorption and refraction due to sound speed profiles in the air are not expected to be important over the short ranges and low frequencies of interest here (Piercy and Embelton, 1977). The ground-induced interference structure depends sensitively on the composition of the ground (Embleton et al., 1983) and ground relief. As the ground relief varied considerably along the transmission paths studied, the transmission loss along each path needed to be measured.

The subject of acoustic transmissions over ground is a subject area in its own right, and a full discussion of the phenomenon lies beyond the scope of the present paper. The interested reader is referred to the work of Embleton et al. (1976, 1983) and recent reviews by Embleton (1996) and Forrest (1994).

The actual measurement methodology for a given stretch of the lek was as follows. The measurement system was composed of the instruments described in Appendix A with the exception of the Stanford Research signal generator, which was replaced by a recording of the signal generator output. The Kudelski speaker was placed within the microphone array with its center 25 cm above the ground using a leveled stand and driven with a sinusoidal chirp linearly modulated with a triangle-wave from 200 Hz to 5 kHz. A complete chirp took 10 s to complete, providing sufficient time at any given frequency to allow for all multi-path arrivals. For each measurement path, we recorded four complete chirps of both the source excitation voltage and the microphone response voltage. At each acoustic measurement site, we repeated this measurement procedure eight times, rotating the speaker each time so that the speaker faced directly towards each microphone in the recording array. Like most loudspeakers, the Kudelski speaker is directional; however, the rotation of the speaker allowed us to calibrate our system with an effectively omni-directional source.

The most straightforward way to determine the ratio of voltages that appears in equation B2 would be to compute the ratio of the Fourier-transformed time series. However, in practice, it was found that this methodology was too sensitive to contamination from sources of ambient noise, which included wind and biological sources. Therefore, we employed a noise-reduction algorithm to improve the ratio estimates. The algorithm consisted of the following. For each transmission path, 30 s or three complete chirps of source excitation and microphone response voltages were digitally transferred from the tape to a computer. The digital data streams were pre-processed by cross-correlating the source and microphone voltages to determine the time delay in the transit of the airborne signal between the source and microphone. This time delay was then backed out of the microphone recording to synchronize the source and microphone time series so that the dominant frequency, f_j, within a given segment of source and microphone data was the same. The voltage ratio at f_j was then determined by taking the ratio of the spectral levels at f_j while ignoring all other spectral components. Thus, the ratio estimates were formed from data segments corresponding to the largest available signal-to-noise ratios.

For simple beam patterns, this approximation should not overestimate the value of DI. The degree to which the calculation approximates the true value of DI is dependent on where the eight samples of the beam pattern are taken. To illustrate this, we used this equation to calculate the DI of the dipole radiation pattern, H²(θ)=cos²θ, 10 000 times choosing the position of the first of eight equally spaced samples randomly from 0⩽θ<π/4. This yielded a range of values (median 2.84 dB; range 2.32 dB<DI<3.01 dB). No values exceed the value calculated analytically above. We therefore suggest that our approximation of the directivity index is a conservative measure of the directionality of the true beam pattern.

 
• b(θ)

  the beam pattern, the decibel equivalent of the directional factor H (dB)

 
• b_band(θ)

  band-averaged beam pattern (dB)

 
• b_band,note(θ)

  time- and band-averaged beam pattern (dB)

 
• c

  speed of sound (m s⁻¹)

 
• D

  directivity, a measure of the directionality of the sound field (dimensionless)

 
• DI

  directivity index, the decibel equivalent of D (dB)

 
• f

  frequency of vibration (Hz)

 
• H(θ)

  the directional factor, a normalized function (dimensionless)

 
• H_band²(θ)

  frequency-band average of the squared

 
• directional factor

  (dimensionless)

 
• H_band,note²(θ)

  time- and frequency-band average of the squared directional factor (dimensionless)

 
• H₀

  point where H(θ)=1

 
• mode[θ_max]

  beam pattern modal maximum (degrees) mode[θ_min] beam pattern modal minimum (degrees) n number of time slices

 
• p_r

  spectral density of the acoustic pressure measured at the receivers (Pa H^−1/2)

 
• p_s

  spectral density of the acoustic pressure generated by a source (Pa H^−1/2)

 
• p_s,axis

  spectral density of the acoustic pressure on the acoustic axis (Pa H^−1/2)

 
• p_u

  spectral density of the acoustic field generated by a uniform source (Pa H^−1/2)

 
• r

  distance from source (m)

 
• TL

  transmission loss (dB)

 
• V_g

  signal generator voltage (V)

 
• V_r

  output voltage from the microphone amplifiers (V)

 
• z

  vertical distance (m)

 
• α_s

  source frequency response, a frequency-dependent constant (Pa V⁻¹)

 
• α_r

  receiver response, a frequency-independent constant (Pa V⁻¹)

 
• П

  acoustic power (W)

 
• П_band

  frequency-band average of the acoustic power (W)

 
• П_u

  acoustic power generated by a uniform source (W)

 
• θ

  horizontal angle relative to the front of the source (degrees; radians in Appendix C)

 
• θ_max

  value of θ at which b(θ) is a maximum (degrees)

 
• θ_min

  value of θ at which b(θ) is a minimum (degrees)

 
• ρ_o

  density (kg m⁻³)

This work was supported by the National Science Foundation through a Graduate Research Fellowship (M.S.D.), a Doctoral Dissertation Research Grant IBN-9701201 (M.S.D. and J.W.B.) and grant IBN-9406217 (J.W.B.). We received additional field study support from a Mildred E. Mathias Graduate Student Research Grant through the University of California’s Natural Reserve System (M.S.D.). Additional support was provided by NIH training grant T32GM-07240 (M.S.D.) and the US Office of Naval Research (G.B.D.). We thank the Sierra Nevada Aquatic Research Laboratory and Dan Dawson for providing field facilities and the Bureau of Land Management (Bishop, CA, USA) and Terry Russi for access to the field site. We thank Dahlia Chazan for her excellent assistance in the field. We thank Catherine DeRivera, Darren Irwin, Dr Jules Jaffe and two anonymous reviewers for critical comments on the manuscript and also Kathryn Cortopassi, Jami Dantzker and Alejandro Purgue for helpful discussions.