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A numerical study of the role of the tragus in the big brown bat

Rolf Mu ¨ller

The Maersk Institute, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

?Received 30 April 2004; revised 15 September 2004; accepted 19 September 2004?

A comprehensive characterization of the spatial sensitivity of an outer ear from a big brown bat

?Eptesicus fuscus? has been obtained using numerical methods and visualization techniques. Pinna

shape information was acquired through x-ray microtomography. It was used to set up a

finite-element model of diffraction from which directivities were predicted by virtue of forward

wave-field projections based on a Kirchhoff integral formulation. Digital shape manipulation was

used to study the role of the tragus in detailed numerical experiments. The relative position between

tragus and pinna aperture was found to control the strength of an extensive asymmetric sidelobe

which points in a frequency-dependent direction.An upright tragus position resulted in the strongest

sidelobe sensitivity. Using a bootstrap validation paradigm, the results were found to be robust

against small perturbations of the finite-element mesh boundaries. Furthermore, it was established

that a major aspect of the tragus effect ?position dependence? can be studied in a simple shape

model, an obliquely truncated horn augmented by a flap representing the tragus. In the simulated

wave field around the outer-ear structure, strong correlates of the tragus rotation were identified,

which provide a direct link to the underlying physical mechanism. © 2004 Acoustical Society of

America. ?DOI: 10.1121/1.1815133?

PACS numbers: 43.80.Ka, 43.64.Ha ?JMS?

Pages: 3701–3712

I. INTRODUCTION

Bats have evolved the common anatomical layout of a

mammal’s outer ear to yield prominent structural features

which presumably serve special functions in biosonar sens-

ing. One such feature is the tragus, a frontal prominence,

which is highly conspicuous in many bat species. Previous

work ?Lawrence and Simmons, 1982; Wotton et al., 1995;

Wotton and Jenison, 1997? has characterized the sensitivity

of the outer ear of the big brown bat ?Eptesicus fuscus? as a

function of direction ?directivity? by presenting data on im-

pulse responses or transfer functions. Such results have typi-

cally been obtained over a limited angular range and with

limited angular resolution. Directivity gain as a function of

both of the two angles ?azimuth and elevation? which char-

acterize a direction in three-dimensional space has been

shown for two frequencies only ?Wotton and Jenison, 1997?.

Similar limitations apply to work performed on other bat

species ?Grinnell and Schnitzler, 1977; Jen and Chen, 1988;

Coles et al., 1989; Obrist et al., 1993; Fuzessery, 1996; Fir-

zlaff and Schuller, 2003?. Despite limitations in the scope of

the data obtained so far, the research on the big brown bat

has revealed interesting functional properties relevant to

source localization in elevation. In particular, it has demon-

strated the importance of the tragus for elevation discrimina-

tion by comparing the performance of animals with natural

tragus position to an experimental group for which the tragus

had been bent out of the pinna aperture ?Lawrence and Sim-

mons, 1982?. However, an overview of the spatial distribu-

tion of sensitivity with sufficiently high resolution in angle,

frequency, and tragus position has not been provided. This

prohibits putting features of individual transfer functions ob-

tained for different directions into perspective and hampers

understanding the global effect of the tragus underlying these

properties. Furthermore, since prior data have been limited to

providing black-box system descriptions, they cannot yield

definitive insights into the physical mechanisms, which so

far have not been observed directly.

Here, a different methodological approach is introduced

to this problem, which addresses the above-mentioned short-

comings and hence is capable of yielding a comprehensive

picture of the role of the tragus: Based on x-ray computer

tomographic cross sections, an accurate high-resolution rep-

resentation of the shape of an outer ear can be generated.

Such a representation can be used for a numerical finite-

element simulation study of the diffraction effects taking

place around the ear surface. The far-field directional sensi-

tivity arising from the diffraction effects can also be pre-

dicted by projecting the wave field outward from the bound-

aries of the finite-element model’s computational domain.

This approach offers three key advantages over physical

measurements: First, directivity patterns can be estimated

with a high angular and frequency resolution, since the time

needed to obtain a sensitivity gain estimate is much shorter

than in any physical setup with moving parts. Second, the

digital representation of the pinna shape can be manipulated

in a well-defined way, drawing from the wide range of avail-

able computer graphics methods. Because the efficiency ad-

vantage of numerical evaluation applies to the manipulated

shapes as well, the effects of the manipulations can be stud-

ied using high-resolution directivity estimates to describe the

functional properties of the outer ear. Third, estimates of the

wave field in the volume surrounding the pinna surface,

where all diffraction effects take place, are generated as part

of the method and can be evaluated to obtain insights into the

physical mechanisms behind changes in the directivity pat-

tern. These advantages outweigh disadvantages given by the

approximations and simplifying assumptions made by the

numerical model. While these approximations represent

3701J. Acoust. Soc. Am. 116 (6), December 20040001-4966/2004/116(6)/3701/12/$20.00© 2004 Acoustical Society of America

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sources of error absent in physical measurement, it should be

noted that physical measurements have their own sources of

error.

In the results presented below, a numerical approach will

be used to

?i?

obtain a comprehensive spatial picture of the effect

the tragus has on the sensitivity of the outer ear;

understand the role of tragus position ?rotation?;

gauge the shape specificity of the spatial sensitivity

patterns; and

characterize the underlying wave-field diffraction ef-

fects.

?ii?

?iii?

?iv?

II. METHODS

A. Shape measurement

An outer-ear sample was taken from a big brown bat

?Eptesicus fuscus? carcass, which had been preserved by

cooling. The outer ear was severed from the head with the

concha and part of the ear canal left attached. It was mounted

?glued to a cylindrical holder? in a life-like manner and sub-

jected to x-ray microtomography in a Skyscan 1072 scanner.

X-ray prefiltering was applied to reduce beam-hardening ar-

tifacts in the reconstruction. Shadow images of 1024?1024

pixels with a pixel size of 19.35 ?m were collected over a

half-circle ?180°? of target rotation performed with 0.9° reso-

lution. Cross sections of the ear shape were reconstructed

usingFeldkamp’scone-beam

implemented in software provided by Skyscan. Cross-section

images ?gray scale, 8-bit resolution? were manually postpro-

cessed to remove noise and obvious artifacts which were

clearly separated from the ear cross section in the images. A

pixel classification into the two categories ‘‘tissue’’and ‘‘air’’

was performed by manual adjustment of a gray-value thresh-

old. Stacking of the resulting binary cross-section images

produced a three-dimensional voxel-array representation of

the shape. It was processed in two ways: for visualization of

the shape, a triangular mesh approximation of the boundary

surface between ear tissue and air was generated using the

marching cubes algorithm ?Lorensen and Cline, 1987?; these

meshes were postprocessed by low-pass filtering and deci-

mation ?Schroeder et al., 1992?. A rendering of the final

shape representation produced by this method is shown in

Fig. 1.

Visualization and rendering was performed using func-

tions provided by the VISUALIZATION TOOLKIT software li-

brary ?VTK, Schroeder et al., 2003?. A mesh of finite ele-

ments filling the air around the structure was generated by a

custom algorithm which evaluated all possible positions for

the placement of finite elements in a lattice of cubic elements

of uniform size. An element was placed at a given position

only if all eight corner voxels for this position had been

classified as air. The bias of this decision criterion against

placing air elements was adjusted at the voxel classification

level, where thresholding was preceded by a three-

dimensional smoothing operation using an isotropic Gauss-

ian kernel of adjustable bandwidth.

reconstructionalgorithm

B. Shape manipulation

The generated ear shape was subjected to two different

kinds of manipulation: tragus rotations and shape distortions.

Tragus rotation was performed in order to characterize the

influence which the relative position of tragus and ear aper-

ture exerts on the directivity. To define the operation, a

cuboid was placed manually in the voxel array to enclose the

portion of the tragus to be rotated. The cuboid was posi-

tioned to include the entire freestanding length of the tragus.

The classification of each source voxel ?tissue or air? inside

the cuboid volume was then transferred to a target voxel,

which was found by rounding the rotated position of the

source voxel to the resolution of the voxel grid. The classi-

fication of all source voxels was set to air, unless they were

the target voxel for another portion of the rotated structure. A

morphological closing operation was performed after the

transfer of the voxel classifications in order to fill any holes

produced by the rounding of the target positions in the voxel

grid and thereby ensure that the rotated structure was again a

homogeneous volume. Rotations were always about an axis

placed at the proximal end of the freestanding tragus portion.

In the original mounting of the ear, the tragus was placed

upright; this position—shown in Fig. 1—is referred to as

‘‘0°.’’Tragus rotation was in the direction outwards from the

pinna aperture. It was performed in steps of 15°, spanning a

total of 60° from the original position ?Fig. 2?.

Shape distortions were performed to assess the robust-

ness of the obtained results with regard to any source of

shape error or variability by means of a perturbation para-

FIG. 1. Tomographic shape reconstruction. Shown is a rendering of a

smoothed, triangular surface mesh ?119 287 triangles? approximating the

boundary between tissue and air.

FIG. 2. Shape renderings for tragus rotations performed on the ear shape

representation rendered in Fig. 1. The difference in rotation angle between

neighboring shapes is 15°; the rotation angle of the upright tragus ?left? is

defined as 0°.

3702 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

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digm: the voxel array representation was subjected to ran-

dom distortions by translating each voxel according to a ran-

dom displacement vector. The three Cartesian components of

the displacement vectors were drawn independently from a

zero-mean Gaussian distribution. A strong spatial smoothing

of the directions was performed in order to prevent discon-

tinuities in the displacement field which would have resulted

in breaking the continuous volume describing the ear tissue.

The smoothing operation was carried out using a three-

dimensional, isotropic Gaussian smoothing kernel with

15-mm standard deviation applied to each Cartesian vector

component. This is equivalent to computing a weighted vec-

tor average. The resulting spatially averaged displacement

vectors were normalized in length and then scaled with a set

of displacement magnitudes drawn independently from a

zero-mean Gaussian distribution. The standard deviation of

the displacement magnitude distribution was the parameter

used to control the amount of shape distortion. The field of

displacement magnitudes was spatially smoothed with a ker-

nel identical to the one used for the Cartesian components of

the displacement directions. Displacements were applied to

the voxel array in the same manner as the tragus rotations

?see Fig. 3?.

C. Generation of a simple model shape

In order to investigate to what extent the observed prop-

erties are specific to the bat ear shape, a simple model shape

was studied in parallel with the bat ear. This model shape is

based on the obliquely truncated horn introduced by Fletcher

and Thwaites ?1988? ?see also Fletcher, 1992? as an idealized

model for the mammalian pinna. In its original form the

model does not account for a tragus, however. Therefore, the

model was augmented here by addition of a frontal flap in-

tended to mimic the tragus. The overall dimensions of the

model shape were adjusted to match the natural ear shape

approximately ?Fig. 4?. The obliquely truncated horn is,

however, strictly bilateral symmetric, whereas the natural ear

shape strongly deviates from bilateral symmetry.

A detailed description of the augmented obliquely trun-

cated horn model, along with a discussion of other possible

features, can be found in Mu ¨ller and Hallam ?2004?. Here,

the obliquely truncated horn was subjected to the same tra-

gus rotations as the natural ear shape ?Fig. 2?. The shape

model was implemented using Boolean combinations of im-

plicit functions describing cones ?for the inner and outer sur-

face?, cutting planes ?for the outer-ear aperture and trunca-

tion of pinna and ear canal?, and cylinders for modeling an

ear-canal stump. Binary ?1-bit resolution? cross-section im-

ages were generated by evaluating the Boolean combinations

of the implicit functions and thresholding their function val-

ues. The binary cross-section images were then fed into the

same processing chain as the tomographic cross sections in

order to generate shape visualizations ?see Fig. 4? and finite-

element meshes.

D. Finite-element model

The finite-element simulation used cube elements ?all

edges of equal length, eight nodes per element? to approxi-

mate the air volume surrounding the outer ear. The air was

assigned a density of 1.205 kg/m3and a bulk modulus of

1.41767?105Pa; this corresponds to a sound speed of 343

m/s. The cube mesh was generated from the voxel represen-

tation of the shape as described in Sec. IIA. Element edge

length was set to eight voxels, corresponding to 154.8 ?m.

Thus, for the shortest considered wavelength ??4.6 mm at

75 kHz?, ?30 elements were placed within one wavelength.

A cuboidal computational domain around the ear shape was

meshed with finite elements, leaving a minimum distance of

several elements between the ear surface and the outer do-

main boundary on all faces but the bottom face, where the

foot plate of the ear shape was placed right on the domain

boundary. Where the outer boundary of the computational

domain was made up of elements representing air, it was

covered with two-dimensional square-shaped absorbing

boundary elements, which approximated reflection-free out-

ward propagation ?Enquist and Majda, 1977?. The ear sur-

face was assumed to be perfectly reflecting.

The finite-element model simulated an acoustical source

placed in the proximal ear-canal cross section, treating the

ear as a loudspeaker. This approach is valid to determine the

ear’s properties as a receiver by virtue of the reciprocity

principle ?Pierce, 1981?. It offers the methodological advan-

tage that a single simulation run gives predictions for the

field on all faces of the computational boundary, from which

FIG. 3. Example realizations of shapes distorted by random displacement

fields. Shown are the boundaries of the finite-element mesh, in which the

cubic shape of the individual elements is clearly visible. The parameter ?0,

0.12, 0.23, 0.46 mm? is the standard deviation of the displacement magni-

tude distribution.

FIG. 4. Comparison of the simple shape model ?obliquely truncated horn

augmented by a tragus-like flap? with the bat ear shape.

3703 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

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the directional sensitivity of the ear shape can be inferred for

all directions as described below. If instead a planar wave

impinging from a particular direction were to be simulated, a

separate simulation would have to be run for each direction.

In order to test the importance of the type of simulated

source on the measured system properties, two different

sources were tested and the results compared: a point source,

where a time-varying boundary condition on pressure was

placed on one node near the center of the proximal ear-canal

cross section only, and a piston source, where all nodes lying

in the cross section were subjected to the same time-varying

boundary condition. Taking advantage of the linearity as-

sumption, the signals emitted by the simulated sources were

given lattice spectra, which allowed for testing system be-

havior for an entire set of frequencies of interest in a single

simulation. The frequency components present in the signals

were spaced 500 Hz apart from 25 to 75 kHz; this covers

most of the frequency band of the emission of the big brown

bat as reported by Hartley and Suthers ?1989? ?25 to 90

kHz?.

Assembly of the master stiffness matrix from the finite-

element mesh and computation of a solution via a precondi-

tioned conjugate gradient method was performed using the

COUPLED FIELD SIMULATION ?CFS??? program developed by

the Department of Sensor Technology at the University of

Erlangen-Nu ¨rnberg, Germany.

E. Directivity estimation and visualization

The wave fields generated by diffraction from shapes

with as little symmetry as the shapes considered here will in

general show a joint dependence on all three spatial coordi-

nates and frequency. However, in the far field, the dependen-

cies on distance and angle separate and the wave-field am-

plitude P can be expressed as

P??,?,r,??

→

r?8a2/?

D??,?,??1

re?jkr,

?1?

where D is the directivity, which depends on direction ?two

angles, azimuth ? and elevation ?? and frequency ? only

?Urick, 1983?. The second term ?(1/r)e?jkr? is the expres-

sion for the wave field of a point source; it depends only on

distance r and wave number k, which is proportional to fre-

quency. Hence, the directivity D is a complete far-field sys-

tem description which contains all the information specific to

a particular source. In the following, a normalized version of

D is presented in order to treat the influence of direction

separately from overall gain. The lower boundary for the far

field given in Eq. ?1? is r?8(a2/?), where a is the charac-

teristic dimension of the source ?here the ear? and ? is the

wavelength. The height of the ear shape is ?18 mm; hence,

it may be assumed that a?9 mm. The smallest wavelength

considered here is ?4.6 mm ??75 kHz?. Consequently, the

far-field boundary is at r?14.4cm and—at the chosen reso-

lution of the finite element mesh—?1000 elements would

have to placed along each dimension to reach it. In three

dimensions, the size of the finite-element problem would be-

come a concern in terms of computation resources and accu-

mulation of numerical errors, if the computation were to be

extended significantly beyond the far-field boundary.

To avoid these problems, the wave fields were forward-

projected from the boundary faces of the computational do-

main, which were themselves still inside the near field.

Sound propagation into a space free of any diffracting ob-

jects ?and sources? can be described by a Kirchhoff integral

formulation ?Jackson, 1999?, which was used here to per-

form the forward projection of the wave field ?Ramahi,

1997?: In the frequency domain, the wave field ??x? at po-

sition x outside the boundary surface S can be expressed as

4??

S

R

??x???

1

ejkR

n•????jk?1?

j

kR?

R

R??ds,

?2?

where R is the vector between the surface element ds and

the position x, n the surface normal, ? in the argument of the

integral the field value on S, and k the wave number. The

product n•?????/?n is the derivative of the field ? with

respect to the surface normal n. In order to obtain a predic-

tion of the directivity, the positions x were chosen on a hemi-

sphere which was placed over the ear. The radius of the

hemisphere was chosen large enough to satisfy the condition

given in Eq. ?1?. Because all positions on the surface of the

hemisphere differ only in direction and not in distance, the

normalized field amplitudes ??x? can be taken as estimates

of the directivity D.

The transformation from the time-domain finite-element

results to the frequency-domain representation used in Eq.

?2? as well as for the directivities was performed using the

Goertzel algorithm ?Mitra, 2001?. Since directivity is a sta-

tionary system characterization, data points containing tran-

sients generated at the start of the time-domain simulation

were discarded prior to this transformation.

The directivity results were visualized using Lambert’s

azimuthal equal-area projection ?Weisstein, 2002?. The

equal-area property of this projection is desirable for the vi-

sualization of directivities, because the relative cross-

sectional area of lobes in the beam patterns can be compared

in the map. At the same time, the projection keeps distortions

of lobe shape close to the theoretical minimum, even if a

large portion of the directivity ?up to a hemisphere? is shown

in a single map.

The combination of directivity and spherical spreading

losses, like the geometrical attenuation included in Eq. ?1?,

results in the formation of three-dimensional ‘‘sound

beams:’’ in the directions where the directivity has a ?local?

maximum, sensitivity falls below a given threshold at a

larger distance than for the directions of the adjacent ?local?

minima. These sound beams were visualized by rendering of

iso-sensitivity surfaces. Since directivity is also a function of

frequency, there is a different isosurface for each frequency.

Isosurfaces for a set of linearly spaced frequencies were ren-

dered with different gray levels and overlaid in one image to

allow for an easy comparison of the sound beam patterns.

The isosurfaces were generated by evaluating the prod-

uct of directivity and spherical spreading given in Eq. ?1?.

Absorption as a source of spreading loss was not considered

for the sake of simplicity, since only the effect of the

frequency-dependent effect of directivity was to be visual-

3704 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

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ized in a three-dimensional way. To this end, it suffices that

the directivity is multiplied with a term which is a monotone

decreasing function of distance and independent of direction.

III. RESULTS

A. Robustness

1. Influence of the excitation source

A comparison of directivity results which were not nor-

malized for each frequency ?not shown? indicated a strong

effect of the chosen excitation source ?piston and point

source?. In the normalized directivities ?Fig. 5?, however,

only minor differences were found and the overall shape of

the directivities was the same for both sources.

Therefore, the choice of the source affected primarily

the overall sensitivity as a function of frequency, but it had

only a negligible effect on how this sensitivity depended on

direction for any given frequency.

2. Influence of random shape distortions

The influence of shape distortion on directivity estimates

was tested over a range of values for the standard deviation

of the magnitude of the random displacements. The tested

values resulted in distortions ranging from very slight to ma-

jor ?Fig. 3?. Directivity patterns estimated for example real-

izations ?Fig. 6? were found to be consistent with the notion

that—in general—larger shape distortions can be expected to

result in larger changes in the directivity patterns. Small

shape distortions ?standard deviation?0.1 mm, i.e., ?2/3 of

a finite-element edge length; see Fig. 6? did not alter the

pattern of the major lobes in the directivity; the location,

shape, and relative weight of these features were largely pre-

served. Larger shape distortions did change the overall lay-

out of the directivity pattern, although some common fea-

tures were identifiable even for the largest shape distortion

shown here ?standard deviation 0.46 mm; see Fig. 3 for a

rendering of the tested realization?. From the examples ex-

amined, it was not apparent that the degree to which shape

distortions changed the directivity pattern was dependent on

frequency. Hence, it cannot be said that within the studied

frequency band ?25–75 kHz? low-frequency directivities

were more or less susceptible to shape distortion than high

frequencies.

B. Tragus rotation and directivities

Rotation of the tragus had a clear, systematic effect on

the directivity pattern in the medium frequency range stud-

ied. Between ?35 and ?50 kHz, the strength—and to some

extent the location and shape—of an extensive, asymmetric

?one-sided? sidelobe was dependent on tragus rotation. In the

upright tragus position ?0°?, the sidelobe had the highest rela-

tive sensitivity; it was gradually attenuated as the tragus was

rotated out of the pinna aperture ?see Figs. 7 and 8?.

While tragus rotation did result in a large number of

small changes to the directivity, the effect on the asymmetric

sidelobe stands out by virtue of its magnitude and systematic

nature. For frequencies below 35 kHz ?e.g., 30 kHz in Fig.

7?, no sidelobe is formed, and the effect of tragus rotation on

the directivity pattern is minor. For frequencies above 50

kHz, sidelobes are common, but none of them was found to

show a systematic dependence on tragus rotation. The results

for 53 kHz in Fig. 7 are an example of a strong sidelobe in

this frequency band; like the broad main lobe at 30 kHz, it is

largely unaffected by the tragus rotation. At a higher fre-

quency resolution ?Fig. 8?, it is evident that the orientation of

the asymmetric sidelobe depends on frequency in a system-

atic fashion: the sidelobe shifts to higher elevations as fre-

quency increases.

C. Wave field correlates

In the wave fields simulated inside the computational

domain of the finite-element model effects were found which

showed a systematic dependence on tragus rotation. This de-

pendence paralleled the effects seen in the directivities. The

effects were particularly obvious in the phase component of

monochromatic wave fields. In the frequency band of the

asymmetric sidelobe, rotation of the tragus caused systematic

FIG. 5. Comparison of directivity pattern estimates for two different sources

of excitation ?point and piston source, columns?. The directivities are nor-

malized individually for each of the frequencies ?rows? shown. The graphs

show Lambert’s azimuthal equal-area projections of the directivity.

3705 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

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shifts in wave field phase in two regions: on the outer side of

the tragus, in the area where the tragus obstructed the direct

line of sight from the source in the ear canal, and inside and

upwards from the tragus ?Fig. 9?.

Taking the wave field phase values at 60° tragus rota-

tion, where the sidelobe is strongly attenuated, as a reference,

it can be seen that phase angles in the former region tend to

be larger than the reference, whereas those in the latter tend

to be smaller ?Fig. 9?. The extent of the phase advance region

increases as the tragus is rotated upwards. Likewise, the re-

gion of phase lag changes shape and the lag magnitude in-

creases ?Fig. 9?. In a quantitative comparison of phase values

along a transect which cuts through both regions, it was

found that a strict ordering of phase magnitudes according to

tragus rotation exists ?Fig. 10?. In the phase advance region,

upward rotations of the tragus always resulted in larger phase

values, whereas in the phase lag region the reverse was the

case.

D. Comparison with simple model shape

In general, the directivity patterns predicted for the aug-

mented truncated horn were found to be quite different from

those predicted for the bat outer ear. However, despite the

overall dissimilarity, an asymmetric sidelobe was found to be

FIG. 6. Directivity estimates for ex-

ample realizations of distorted shapes.

The graphs show Lambert’s azimuthal

equal-area projections of the normal-

ized directivity. The corresponding

shape realizations for 0, 0.12, 0.23,

and 0.46 mm are shown in Fig. 3.

3706J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

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present at 34.5 kHz which showed the same dependence on

tragus/flap rotation as the sidelobe seen in the natural ear

shape ?Fig. 11?.

It was strongest in the upright position of the flap and

systematically decreased in sensitivity gain as the flap was

rotated downward out of the aperture.

IV. DISCUSSION AND CONCLUSIONS

The directivity pattern of the bat pinna shape studied

here arises from sound diffraction at a rather complicated

shape devoid of any symmetry which could be used to sim-

plify treatment. Furthermore, since the wavelengths and the

size of the outer ear and its structural features—such as the

tragus—are similar to each other, the approximations of geo-

metrical acoustics are not valid. The latter insight is not

taken into account by ideas which try to explain the acous-

tical properties of the pinna–tragus combination with distinct

‘‘secondary reflections’’ as alluded to in Lawrence and Sim-

mons ?1982? and Wotton et al. ?1995?. Instead, an impinging

wave is diffracted by the entire shape and the interactions

between different parts of the shape ?e.g., tragus and pinna?

are a continuous process rather than one which could be

readily broken up into separate, time-discrete effects. In such

a situation, a numerical approach offers the only access to

the mechanisms which cause the functionally significant fea-

tures of the directivity pattern.

A. Robustness and validity of the results

Because no information on the motion pattern of the

tympanic membrane in the big brown bat is available, an

arbitrary choice for the acoustic excitation source to be used

FIG. 7. The effect of tragus rotation ?rows? on a large

asymmetric sidelobe of the directivity pattern. Shown is

an example of a directivity pattern with sidelobe for 45

kHz ?center column, approximate sidelobe location

marked by an arrow?. Examples of directivities for fre-

quencies below ?30 kHz? and above ?53 kHz? where the

effect is absent are shown for comparison. The tragus is

pointing approximately to the left in the graphs; the

corresponding tragus rotations are shown in Fig. 2. The

graphs show Lambert’s azimuthal equal-area projec-

tions of the normalized directivity.

3707 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

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in the simulation had to be made. However, the comparison

of the results for two very different source choices tested

?piston and point source? indicates that this is not a relevant

issue for the prediction of directivities: while the nature of

the source does affect the overall transfer function of the ear,

i.e., the spectral weighting across frequency bands, it does

have only a minor effect on normalized directivities, i.e., the

spatial weighting within a given frequency band. These ex-

perimental results are in agreement with theoretical predic-

tions ?Fletcher, 1992?: if a source is placed at one end of a

pipe which has a diameter smaller than the wavelength, the

wavefront at the other end of the pipe will be planar to a very

good approximation, irrespective of the source type. The pre-

served portion of the ear canal represents a pipe for which

the scale condition ?wavelength larger than pipe diameter?

holds for all frequencies considered here. Therefore, both

theoretical considerations and experimental results corrobo-

rate the notion that directivity patterns can be predicted with-

out knowing the motion pattern of the tympanic membrane.

Inaccessibility of the overall frequency transfer function does

not affect the objective of the work reported here, namely

understanding the spatial sensitivity distribution generated

by the pinna. The directivities of the sonar emission and the

pinna are the only two points along the sonar communication

channel where the spatial sensitivity of the system can be

controlled. Understanding the spatial characteristics at these

two stages is hence of prime importance. In contrast to this,

the overall frequency transfer function is not a spatial prop-

erty and may be controlled at any internal stage of the chan-

nel.

Because only a single ear was used for this study, some

sources of variability in ear shape such as individual and

gender differences, geographical variation, and postmortem

changes cannot be considered. All these factors have the po-

tential for introducing a considerable amount of variability,

an assessment of which would most likely require the study

of a fairly large sample. However, the shape distortion ex-

periments ?see Sec. IIB? address this issue by taking a boot-

strap approach which creates variability around the sample in

order to assess the robustness of the results. Two insights can

be gained from the outcome of these experiments: first, the

change in directivity as the structure is perturbed is gradual

and the basic shape of the directivity is lost only as pertur-

bations get fairly large. This demonstrates that the results are

robust against small errors introduced by sample mounting

and the shape estimation procedure. Second, larger shape

distortions can alter the directivity pattern thoroughly, even if

the basic pinna–tragus combination is left intact. This find-

ing offers an experimental corroboration of the fundamental

insight that diffraction effects integrated over the entire

shape are responsible for the functional ?directivity? proper-

ties. The specific combination of properties seen in the ear

sample, i.e., an asymmetric sidelobe, which occurs over a

fairly wide frequency band, has a frequency-dependent axis

and a sensitivity controlled by tragus rotation, likely to de-

pend on several aspects of this particular shape. However,

the results obtained for the augmented obliquely truncated

horn model ?see Sec. IIID? demonstrate that some individual

aspects of the overall behavior can be reproduced by a model

FIG. 8. Three-dimensional iso-sensitivity surfaces for different tragus rota-

tions and frequencies. For each tragus rotation, isosurfaces for frequencies

from 25 to 75 kHz in steps of 500 Hz are rendered. Dark-gray values

correspond to low frequencies, brighter grays to higher frequencies. The

asymmetric sidelobe affected by the tragus rotation is marked by an arrow.

The ear shapes are shown as a reference for the orientation of the isosurfaces

with respect to the ear. They are 70? magnified with respect to the direc-

tivities and displaced outside the isosurfaces to provide an unobstructed

view.

3708 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

Page 9

which shares very little structural detail with the original ear

shape.

Besides the validity issues discussed above, it should be

noted that the results presented here solely pertain to the

shape of the pinna. In doing so, two simplifying assumptions

were made: first, it was assumed that the air–pinna boundary

is perfectly reflecting. This simplification is motivated by the

large difference between the acoustic impedances of air

??413 rayl under the simulated air properties? and tissue ?on

the order of magnitude of 106rayl). Second, the shape of the

pinna was considered in isolation rather than together with

the head. This was necessitated by the limited size of the

tomographic imaging volume. While it can be expected that

the major shaping influence on the directivity pattern is ex-

erted by the structures which immediately surround the

opening of the ear canal, it is possible that parts of the head

which are close to the pinna have a modifying influence.

This is particularly likely for the lower frequencies where the

distance is smaller in terms of the number of wavelengths

needed to span it.

B. Directivities and the role of the tragus

The directivity D(?,?,?) is a complete description of

the spatial sensitivity of an antenna in the far field. However,

it is a function of three variables, two angles ??,?? and fre-

quency ???, which poses a challenge for visualization. Head-

related transfer functions are one method for making the di-

rectivity accessible: for selected combinations of angle

values ?? and ??, a one-dimensional function of sensitivity

gain versus frequency is plotted. Since the spatial variables

??,?? only appear as labels on a set of one-dimensional func-

tions, head-related transfer functions are a poor choice for

making the spatial dimension of the directivity accessible. In

the spatial view presented here, a two-dimensional function

of the spatial variables is plotted for a set of selected fre-

quency values. If the directivity functions are plotted as a

two-dimensional map ?as done in Figs. 5, 6, 7, and 11?, this

necessitates one graph for each frequency. From such maps,

the shape of the directivity for a single frequency can be

readily assessed, if an adequate map projection is chosen.

The choice of the projection is important, since the spatial

relations on the surface of a sphere can never be accurately

represented in a plane. Therefore, a carefully considered

compromise has to be made. Plots using azimuth and eleva-

tion as Cartesian coordinates ?as presented in Wotton and

Jenison ?1997?? heavily distort spatial relationships and

hence are difficult to read. Irrespective of the chosen projec-

tion, comparing the directivities obtained for different, fixed

FIG. 9. Difference in wave field phase for a monochro-

matic field of 45 kHz. The reference is 60° tragus rota-

tion. Depicted is the difference between the phase val-

ues of the fields obtained for the other rotations ?45° to

0°? and the reference; the corresponding tragus rota-

tions are shown in Fig. 2. Positive values indicate a

smaller reference.

FIG. 10. Wave field phase values along a linear transect

through computational domain. The wave field is

monochromatic ?excitation frequency 45 kHz?. Top

graph: phase values ?coded by gray values? for upright

tragus position ?0°? and location of the transect ?dashed

line?. Bottom graph: phase values along the transect for

all tragus rotations tested. Forward phase shifts with

smaller tragus rotation angles are enclosed by the el-

lipse, phase lags by the square. The interruptions in all

curves but the one for 0° at positions in the vicinity of

7.5 mm are due to the transect cutting through the tra-

gus for these shapes. The corresponding tragus rotations

are shown in Fig. 2.

3709 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

Page 10

frequencies requires superimposing two or more maps on

each other mentally, since a graphical superposition would

become cluttered quickly. This encumbers obtaining an un-

derstanding of the directivity as a function of all three inde-

pendent variables. Three-dimensional renderings of sensitiv-

ity gain isosurfaces for different frequencies can be

superimposed on each other in a single graph ?see Fig. 8?.

While occlusions of one surface by another hide some detail,

these renderings provide an intuitive overall view of sensi-

tivity gain as a joint function of spatial variables and fre-

quency.

The numerical simulation and visualization approach

taken here has produced a concise description of the impact

the tragus has on the spatial sensitivity of the outer ear in the

big brown bat: it causes the formation of an asymmetric side-

lobe limited to a certain frequency band ?here ?35 to ?50

kHz? and the direction the sidelobe is pointing in shifts

smoothly upwards with frequency ?see Fig. 8?. These find-

ings are at least in qualitative agreement with transfer func-

tion data presented in Wotton et al. ?1995?, where a spectral

notch was observed which shifted upwards in frequency

?from ?45 to ?55 kHz? as a function of elevation. An ex-

ample of directivity gains from the present numerical results

arranged as a function of one directional angle ?defined as

elevation? and frequency shows a similar behavior ?Fig. 12?.

A precise match between these data and the measure-

ment results of Wotton et al. ?1995? is not to be expected,

because of individual differences between animals, estima-

tion errors associated with both physical measurements and

numerical predictions, and—probably most importantly—the

arbitrary nature of the choices made in positioning the azi-

FIG. 11. In the directivity patterns predicted for the augmented obliquely

truncated horn ?see Fig. 4?, an asymmetric sidelobe can be seen for a fre-

quency of 34.5 kHz ?approximate location indicated by triangle?, which

exhibits a similar dependence on tragus rotation as found for the natural bat

ear shape ?see Fig. 7?. The graphs show Lambert’s azimuthal equal-area

projections of the normalized directivity; the flap modeling the tragus is

pointing to the bottom of the graphs.

FIG. 12. Example of directivity gains as a function of one angle ?elevation?

and frequency. Directivity gains are normalized for each elevation value.

The foot plate of the shape ?see Fig. 1? is aligned with the azimuthal plane

and zero elevation is at the equator.

3710J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

Page 11

muthal plane and the origins of azimuth and elevation. The

three-dimensional view presented here puts the results on

moving spectral notches into a spatial perspective: occur-

rence of a spectral notch at a certain frequency in the head-

related transfer function for a certain direction means that

this direction coincided with the direction of the spatial notch

in the directivity for that particular frequency. Moreover, a

full view of the directivity can be used as a basis for assess-

ing the role of the outer ear in a different class of biosonar

estimation tasks. Head-related transfer functions are useful

system descriptions in estimation tasks which can be per-

formed by spectral template matching. Elevation estimation

for targets with transfer functions which do not obscure the

spectral templates is an example of such a task. However,

this estimation approach cannot be extended to biosonar es-

timation tasks which involve extended targets with many re-

flecting facets ?Mu ¨ller and Kuc, 2000?. For these tasks, the

spatial view presented here is more useful because it reveals

the spatial weighting imposed on a random spatial distribu-

tion of scatters by the outer ear.

C. Wave field correlates

Digital shape manipulations allow well-localized and

quantifiable changes to the outer-ear shape to be made. This

was used in the tragus rotation experiment and revealed that

the strength of the asymmetric sidelobe was strictly decreas-

ing over five successive steps of outward tragus rotation.

These findings not only establish a convincing correlation

between the tragus and sidelobe formation, they also provide

a starting point for the search of further correlates closer to

the cause of these effects. Correlates matching the monotone

relationship between tragus position and sidelobe strength

for the entire sequence of rotation values are likely to be

nonrandom and linked to the mechanism by which tragus

rotation controls sidelobe strength. The identified changes in

the phase of the simulated wave field meet this criterion: the

extent of the phase-shift regions as well as the magnitude of

the phase shifts showed a monotone dependence on tragus

rotation.

Wave field properties in the immediate vicinity of the

diffracting structure are interesting, because the causes of

sidelobe formation must be localized where diffraction takes

place. A hypothesis for a mechanism of sidelobe control

must link the changes in shape to changes in the wave field

in the vicinity of the structure, and these wave field changes

to the far-field directivity. As to the latter step, it may be

hypothesized that the two observed phase shifts in the out-

going wavefront—forward on the outside and backwards in-

side and above the tragus—cause destructive interference in

the direction of the notch, which separates main- and side-

lobe, and constructive interference in the direction of the

sidelobe maximum. As to the link between shape- and wave-

field-changes, the phase advance on the outside of the tragus

may be attributed to diffraction around the tragus. If a wave

has to travel further to a certain point in space, its phase will

be advanced at that point. As the tragus is rotated outward

and out of the direction of sound propagation, the region

accessible only through diffraction around the tragus shrinks,

as was observed for the phase advance region. The connec-

tion between the observed phase lag and the properties of the

shape is not as readily made as for the phase advance. It may

depend on less well-localized diffraction effects and hence

requires further analysis.

D. Insights from simple shape models

Simplified shape models have the potential to facilitate

the analysis of the physical mechanism linking tragus rota-

tion and sidelobe formation. In order to qualify as a compre-

hensive model, they should reproduce all salient properties

of the sidelobe seen in the natural pinna shape, in particular

its dependence on tragus rotation and on frequency. The re-

sults presented here for the augmented obliquely truncated

horn model show only the reproduction of one property, a

sidelobe strength which depends on tragus rotation. They did

not show the gradual dependence of sidelobe orientation on

frequency. Instead, the directivity pattern changed pro-

foundly, even for slight changes in frequency. Therefore, at

present, this particular model shape could only be used to

study one aspect of the effect. These results do not prove,

however, that the general shape model is incapable of repro-

ducing the sidelobe behavior of the bat outer ear in a more

comprehensive fashion. This may still be possible for a dif-

ferent set of shape parameters. Further results on the aug-

mented obliquely truncated horn model published elsewhere

?Mu ¨ller and Hallam, 2004? demonstrate that interactions be-

tween the tragus and shape features of the pinna ?e.g., a

surface ripple? take place and can have a profound effect on

the directivity. Because this expands the parameter space be-

yond what has been presented here, finding a suitable com-

bination of parameter values would certainly require a major

search effort, which could be undertaken using a numerical

searching technique, for instance a genetic algorithm.

E. Tragus rotation as a control mechanism

Tragus rotations were performed here as an experimen-

tal method to firmly establish that the tragus is the cause for

the large asymmetric sidelobe. However, many bat species,

including the big brown bat, show intricate patterns of ear

movement ?Valentine et al., 2002?. If muscular actuation of

the ear allowed some of these species to control tragus rota-

tion relative to the pinna, these animals would also have

control over certain aspects of their ear directivities. Since

only the relative position of tragus and pinna matters, this

does not necessarily require that the tragus itself is being

moved. A fixed tragus and a mobile pinna or a bit of mobility

on both sides would be alternative ways to achieve the same

effect. If the pinna were to be moved, however, this would

most likely imply a coupling between the overall orientation

of the directivity function ?controlled by where the pinna is

pointing? and its shape. A—hypothetical—bat with a combi-

nation of the suitable ear mobility and a tragus with a similar

functional significance as in the big brown bat would have

control over the extent to which a sidelobe is present in its

directivity. Such a bat could extend or retract the sidelobe

whenever it facilitates achieving its sensing objectives. A

comparison of echoes received with and without the sidelobe

being present in the directivity pattern would allow the clas-

3711 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus

Page 12

sification of echo features as being caused by targets in the

direction of the main lobe or by targets in the direction of the

sidelobe. This classification could be performed even if

main- and sidelobe were both populated with a large number

of scattering facets which would prohibit the application of

conventional direction estimation techniques. In such a case,

the sidelobe could be used like the side view or rear-view

mirror of a car, to which the driver can attend at will, when-

ever deemed necessary.

ACKNOWLEDGMENTS

The ear sample was kindly provided by the Department

of Animal Physiology, University of Tu ¨bingen, Germany.

Joris J. J. Dirckx and Stefan Gea of the Department of Phys-

ics, University of Antwerp ?RUCA? made their microtomog-

raphy facilities available and provided essential help with the

data collection. Manfred Kaltenbacher and Alexander Stre-

icher of the Department of Sensor Technology, University of

Erlangen-Nu ¨rnberg shared the CFS?? source code and pro-

vided crucial advice on the finite element method. This work

was supported by the European Union ?IST Program, LPS

Initiative, CIRCE Project IST-2001-35144?.

Coles, R., Guppy, A., Anderson, M., and Schlegel, P. ?1989?. ‘‘Frequency

sensitivity and directional hearing in the gleaning bat, Plecotus auritus

?Linnaeus 1758?,’’ J. Comp. Physiol., A 165, 269–280.

Enquist, and Majda, ?1977?. ‘‘Absorbing boundary conditions for the nu-

merical simulation of waves,’’ Math. Comput. 31, 629–651.

Firzlaff, U., and Schuller, G. ?2003?. ‘‘Spectral directionality of the external

ear of the lesser spear-nosed bat, Phyllostomus discolor,’’ Hear. Res. 181,

27–39.

Fletcher, N. H. ?1992?. Acoustic Systems in Biology ?Oxford University

Press, Oxford?.

Fletcher, N. H., and Thwaites, S. ?1988?. ‘‘Obliquely truncated simple horns:

Idealized models for vertebrate pinnae,’’ Acustica 65, 194–204.

Fuzessery, Z. ?1996?. ‘‘Monaural and binaural spectral cues created by the

external ears of the pallid bat,’’ Hear. Res. 95, 1–17.

Grinnell, A. D., and Schnitzler, H.-U. ?1977?. ‘‘Directional sensitivity of

echolocation in the horseshoe bat, Rhinolophus ferrumequinum. II. Behav-

ioral directionality of hearing,’’ J. Comp. Physiol. ?A? 116, 63–76.

Hartley, D. J., and Suthers, R. A. ?1989?. ‘‘The sound emission pattern of the

echolocating bat, Eptesicus fuscus,’’ J. Acoust. Soc. Am. 85, 1348–1351.

Jackson, D. E. ?1999?. Classical Electrodynamics, 3rd ed. ?Wiley, New

York?.

Jen, P.-S., and Chen, D. ?1988?. ‘‘Directionality of sound pressure transfor-

mation at the pinna of echolocating bats,’’ Hear. Res. 34, 101–118.

Lawrence, B. D., and Simmons, J. A. ?1982?. ‘‘Echolocation in bats: The

external ear and perception of the vertical positions of targets,’’ Science

218, 481–483.

Lorensen, W. E., and Cline, H. E. ?1987?. ‘‘Marching cubes: A high resolu-

tion 3D surface construction algorithm,’’ in Computer Graphics ?Proceed-

ings of SIGGRAPH ’87?, Vol. 21, pp. 163–169.

Mitra, S. K. ?2001?. Digital Signal Processing: A Computer-based Ap-

proach, McGraw-Hill Series in Electrical and Computer Engineering, 2nd

ed. ?McGraw-Hill/Irwin, Boston?.

Mu ¨ller, R., and Hallam, J. C. T. ?2004?. ‘‘From bat pinnae to sonar antennae:

Augmented obliquely truncated horns as a novel parametric shape model,’’

in Proceedings of the Eighth International Conference on the Simulation

of Adaptive Behavior, edited by S. Schaal, A. J. Ijspeert, A. Billard, and S.

Vijayakumar ?to appear?.

Mu ¨ller, R., and Kuc, R. ?2000?. ‘‘Foliage echoes: A probe into the ecological

acoustics of bat echolocation,’’ J. Acoust. Soc. Am. 108, 836–845.

Obrist, M., Fenton, M., Elger, J., and Schlegel, P. ?1993?. ‘‘What ears do for

bats: A comparative study of pinna sound pressure transformation in chi-

roptera,’’ J. Exp. Biol. 180, 119–152.

Pierce, A. D. ?1981?. Acoustics ?McGraw-Hill, New York?.

Ramahi, O. M. ?1997?. ‘‘Near- and far-field calculations in fdtd simulations

using Kirchhoff surface integral representation,’’ IEEE Trans. Antennas

Propag. 45, 753–759.

Schroeder, W., Martin, K., and Lorensen, B. ?2003?. The VISUALIZATION

TOOLKIT—An Object-oriented Approach to 3D Graphics, 3rd ed. ?Kitware,

Inc.?.

Schroeder, W., Zarge, J., and Lorensen, W. ?1992?. ‘‘Decimation of triangle

meshes,’’ in ‘‘Computer Graphics,’’ Vol. 26, pp. 65–70.

Urick, R. J. ?1983?. Principles of Underwater Sound, 3rd ed. ?McGraw-Hill,

New York?.

Valentine, D. E., Sinha, S. R., and Moss, C. F. ?2002?. ‘‘Orienting responses

and vocalizations produced by microstimulation in the superior colliculus

of the echolocating bat, Eptesicus fuscus,’’ J. Comp. Physiol., A 188, 89–

108.

Weisstein, E. W. ?2002?. CRC Concise Encyclopedia of Mathematics, 2nd

ed. ?CRC Press, Boca Raton?.

Wotton, J. M., and Jenison, R. L. ?1997?. ‘‘A backpropagation network

model of the monaural localization information available in the bat

echolocation system,’’ J. Acoust. Soc. Am. 101, 2964–2972.

Wotton, J. M., Haresign, T., and Simmons, J. A. ?1995?. ‘‘Spatially depen-

dent acoustic cues generated by the external ear of the big brown bat,

Eptesicus fuscus,’’ J. Acoust. Soc. Am. 98, 1423–1445.

3712 J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004Rolf Mu ¨ller: Role of the tragus