Acoustic diffraction effects at the Hellenistic amphitheater of Epidaurus: Seat rows responsible for the marvelous acoustics

Abstract

The Hellenistic theater of Epidaurus, on the Peloponnese in Greece, attracts thousands of visitors every year who are all amazed by the fact that sound coming from the middle of the theater reaches the outer seats, apparently without too much loss of intensity. The theater, renowned for its extraordinary acoustics, is one of the best conserved of its kind in the world. It was used for musical and poetical contests and theatrical performances. The presented numerical study reveals that the seat rows of the theater, unexpectedly play an essential role in the acoustics--at least when the theater is not fully filled with spectators. The seats, which constitute a corrugated surface, serve as an acoustic filter that passes sound coming from the stage at the expense of surrounding acoustic noise. Whether a coincidence or not, the theater of Epidaurus was built with optimized shape and dimensions. Understanding and application of corrugated surfaces as filters rather than merely as diffuse scatterers of sound, may become imperative in the future design of modern theaters.

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Published online: 14 December 2004; | doi:10.1038/news041213-5
Mystery of 'chirping' pyramid decoded
Philip Ball
Acoustic analysis shows how temple transforms echoes
into sounds of nature.
El Castillo's strange echoes have
fascinated visitors for
generations
© Punchstock
A theory that the ancient Mayans
built their pyramids to act as giant
resonators to produce strange and
evocative echoes has been
supported by a team of Belgian
scientists.
Nico Declercq of Ghent University
and his colleagues have shown
how sound waves ricocheting
around the tiered steps of the El
Castillo pyramid, at the Mayan
ruin of Chichén Itzá near Cancún
in Mexico, create sounds that
mimic the chirp of a bird and the
patter of raindrops1.
The bird-call effect, which resembles the warble of the Mexican
quetzal bird, a sacred animal in Mayan culture, was first
recognized by California-based acoustic engineer David Lubman
in 1998. The 'chirp' can be triggered by a handclap made at the
base of the staircase.
Declercq was impressed when he heard the echo for himself at
an acoustics conference in Cancún in 2002. After the conference,
he, Lubman and other attendees took a trip to Chichén Itzá to
experience the chirp of El Castillo at first hand. "It really sounds
like a bird", says Declercq.
Sound structure
But did the pyramid's architects know exactly what they were
doing? Declercq's calculations show that, although there is
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16 November 2000
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23 February 2000
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evidence that they engineered the pyramid to produce surprising
sounds, they probably couldn't have predicted exactly what they
would resemble.
Lubman was at first convinced that the pyramid-builders did
create the bird-chirp effect intentionally. But that's not
necessarily so, Declercq and his colleagues argue. Their analysis
of the pyramid's acoustics show that the precise sound caused by
the echoes depends on the sound that excites them. Drums, for
example, might produce a different type of resonance.
The researchers hope that others will make more on-site
measurements of El Castillo's acoustics to see what effects other
sounds sources induce.
Indeed, Declercq heard one such variation during the 2002 trip.
As other visitors tramped up the steps of the 24-metre high
pyramid, he noticed a flurry of pulse-like echoes that seemed to
sound just like rain falling into a bucket of water.
Declercq wonders whether this, rather than the quetzal call,
could have been the aim of El Castillo's acoustic design. "It may
not be a coincidence," he says - the rain god played an important
part in Mayan culture.
ADVERTISMENT
But perhaps such meaningful interpretations are
fanciful. Declercq's team has shown that the height
and spacing of the pyramid's steps creates like an acoustic filter
that emphasizes some sound frequencies while suppressing
others. But more detailed calculations of the acoustics shows that
the echo is also influenced by other, more complex factors, such
as the mix of frequencies of the sound source.
Ultimately, then, it will be virtually impossible to prove that any
specific echo effect is intentional. "Either you believe it or you
don't," says Declercq. He himself is now sceptical of the quetzal
theory - not least because he has now heard similar effects
produced by staircases at other religious sites. At Kataragama in
Sri Lanka, for example, a handclap by a staircase leading down
to the Menik Ganga river produces an echo in response that
resembles the quacking of ducks.
Top
References
1. Declercq N. F., Degrieck J., Briers R. & Leroy O. J.
Acoust. Soc. Am., 116. 3328 - 3335 (2004). | Article |
Top
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A theoretical study of special acoustic effects caused by the
staircase of the El Castillo pyramid at the Maya ruins of
Chichen-Itza in Mexico
Nico F. Declercqa) and Joris Degrieck
Soete Laboratory, Department of Mechanical Construction and Production, Faculty of Engineering,
Ghent University, Sint Pietersnieuwstraat 41, 9000 Gent, Belgium
Rudy Briers
KATHO (RENO dept), Sint Jozefstraat 1, 8820 Torhout, Belgium
Oswald Leroy
Interdisciplinary Research Center, KULeuven Campus Kortrijk, E. Sabbelaan 53, 8500 Kortrijk, Belgium
共Received 21 February 2003; revised 18 February 2004; accepted 3 May 2004兲
It is known that a handclap in front of the stairs of the great pyramid of Chichen Itza produces a
chirp echo which sounds more or less like the sound of a Quetzal bird. The present work describes
precise diffraction simulations and attempts to answer the critical question what physical effects
cause the formation of the chirp echo. Comparison is made with experimental results obtained from
David Lubman. Numerical simulations show that the echo shows a strong dependence on the kind
of incident sound. Simulations are performed for a 共delta function like兲pulse and also for a real
handclap. The effect of reflections on the ground in front of the pyramid is also discussed. The
present work also explains why an observer seated on the lowest step of the pyramid hears the sound
of raindrops falling in a water filled bucket instead of footstep sounds when people, situated higher
up the pyramid, climb the stairs. © 2004 Acoustical Society of America.
关DOI: 10.1121/1.1764833兴
PACS numbers: 43.20.El, 43.20.Fn, 43.28.En 关DKW兴Pages: 3328–3335
I. INTRODUCTION
During the post meeting tour of the first PanAmerican/
Iberian meeting on Acoustics that was held in Cancun
共Mexico兲in 2002 共hereafter called ‘the post meeting tour’兲,
the participants were shown that there are plenty of interest-
ing sound effects that occur at Chichen-Itza. Chichen Itza is
a Maya ruin where, besides the famous ‘‘ball court,’’1there
is a pyramid 共El Castillo兲that produces a sound echo, in
response to a handclap, which sounds like the chirp of a
Quetzal bird. This effect has been one of the major subjects
during plenty of talks given by David Lubman2–5 and
others.6–8 Lubman has stressed the fact that the Quetzal bird
chirp is actually caused by Bragg scattering. However, there
has never been presented an actual simulation of the effect,
except for some heuristic simulations based on the ray
theory2–5 or a heuristic approach for the case of incidence at
45° measured from the normal to the surface.8–12 In what
follows, a full diffraction simulation is presented of the echo,
based on a 共time-兲delta function like handclap and also a
real handclap, based on the physical parameters of the stair-
case of the Pyriamid at Chichen-Itza and based on the
monofrequent single homogeneous plane wave diffraction
theory of Claeys et al.,9,10 which is a simplified case of the
inhomogeneous plane wave diffraction theory.11 The present
work describes the first simulations of a spherical sound
pulse, based on that monofrequent pure plane wave diffrac-
tion theory.9,10 Furthermore, it is for the first time that the
theory has been applied to audio frequencies.
Before presenting this development, it is of cultural im-
portance to stress the fact that some people believe that the
Quetzal bird chirp echo is caused by accident and others
believe that it is caused as a consequence of the Pyramid
builders’ purpose. Nevertheless, it is known that the Quetzal
bird has played a very important role in Mayan culture,
which is probably due to the fact that Mayans originally
lived for many centuries in the forest before getting involved
in the construction of cities and religious sites. However,
what is sure about this pyramid is that it certainly functioned
as a great solar calendar. For example a large serpent is built
on one side that causes special light effects around the time
of spring and fall equinox. This serpent is culturally con-
nected to the Quetzal bird 共as can be seen on a Mayan glyph
from the Dresden Codex兲, whence the generation of a
Quetzal bird echo might not be a real coincidence. It is also
known that an echo in Mayan culture represents a spirit.
However, it must also be notified that a Quetzal bird echo
also occurs at other Pre-Columbian sites and Ancient Mexi-
can ruins.12 Furthermore the first author encountered similar
effects as in Chichen Itza at two religious sites in Sri Lanka.
There, the short concrete staircase, that enables people to
take a bath in the Menik Ganga river at the religious site of
Katharagama, produces the low frequency sound of quacking
ducks in response to a handclap. Furthermore high frequency
echoes occur on the immense staircase leading to the reli-
gious site of Sri Pada 共Adam’s peak兲. Nevertheless, the ef-
a兲Electronic mail: NicoF.Declercq@UGent.be
3328 J. Acoust. Soc. Am. 116 (6), December 2004 0001-4966/2004/116(6)/3328/8/$20.00 © 2004 Acoustical Society of America

fects in Sri Lanka are probably a coincidence and are not a
result of purposely construction.
The last part of this paper is devoted to the less known
fact that an observer seated on the lowest stair step of the
great pyramid at Chichen Itza, hears pulses that sound like
raindrops falling in a water filled bucket, when other people
are climbing the pyramid higher up. This phenomenon 共here-
after called ‘‘raindrop effect’’兲, has been observed by the first
author and by a student fellow Ce
´cile Goffaux during the
post meeting tour. Since the ‘‘rain god’’ plays a very impor-
tant role in the Yucatan Mayan culture, this finding might be
an impetus for future cultural studies.
II. THEORETICAL DEVELOPMENT OF THE ECHO
SIMULATION
The staircase is seen as a periodically corrugated 共infi-
nite兲surface, being sawtooth shaped 共see Fig. 1兲. This is only
true within the interval of the physical staircase. This infinite
mathematical model is matched to reality by modeling a
handclap not by a truly spherical wave, but by a wave that
only contains propagation directions from the emitter di-
rectly to the staircase within the angular interval 关
␣
1,
␣
2兴
that assures impingement on the staircase and within the in-
terval 关
␣
3,
␣
4兴if, in addition, reflections on the ground are
considered as well. Hence, the handclap is only spherical if
observed on the staircase. Whatever sound patterns are emit-
ted to areas outside of the considered intervals is unimportant
for the present study. The vectors d,h,D, and Hare defined
in Fig. 1. For exand ezbeing unit vectors along the x, re-
spectively, zdirection, straightforward geometrical consider-
ations result in
␣
1⫽arccos共h⫹d兲•ez
冑
h2⫹d2,共1兲
␣
2⫽arccos 共h⫹d兲•ez
兩
h⫹d⫺
冑
D2⫹H2ex
兩
,共2兲
␣
3⫽arccos 共⫺h⫹d兲•ez
兩
⫺h⫹d⫺
冑
D2⫹H2ex
兩
,共3兲
␣
4⫽arccos共⫺h⫹d兲•ez
冑
h2⫹d2,共4兲
with
冉
dx
dz
冊
⫽
冉
cos
␰
sin
␰
⫺sin
␰
cos
␰
冊
冉
兩
d
兩
0
冊
,共5兲
冉
hx
hz
冊
⫽
冉
cos
␰
sin
␰
⫺sin
␰
cos
␰
冊
冉
0
兩
h
兩
冊
,共6兲
and
␰
⫽arctanH
D⫹
␲
2.共7兲
The diffraction theory of Claeys et al. that is applied here
can be found in the literature.9,11 Nevertheless, some charac-
teristics of that theory are outlined below. The theory is
based on the decomposition of the diffracted acoustic field
into pure plane waves, which is essentially only allowed
whenever the Lipmann conditions9,11 are fulfilled, stating
that the incident wave length must be of the same order of
magnitude as the corrugation period and that the corrugation
height must not exceed the incident wave length. If these
conditions do not hold, then errors will occur in the descrip-
tion of the sound field within the corrugation. Elsewhere the
errors will be small, except when the Lipmann conditions are
seriously violated of course. Basically, each of the reflected
and transmitted wave fields are decomposed into a series of
plane waves, each plane wave of order mhaving a wave
vector
Km⫽kx
mex⫹kz
mez,共8兲
with
kx
m⫽kx
inc⫹m2
␲
冑
2q,共9兲
and kz
mdetermined by kx
m, the material properties of the
considered medium and the dispersion relation k2⫽
␻
2/v2,
omega being the angular frequency and vbeing the plane
wave velocity. The sign of kz
mis chosen such, as to fulfill the
necessity of plane waves to propagate away from the inter-
face and, whenever kz
mis purely imaginary, the amplitude
must decay away from the interface. The continuity condi-
tions demand continuity of normal stress and normal particle
displacements on each spot of the pyramid’s staircase. It can
be found in Claeys et al.9,11 that this leads to a set of equa-
tions that is periodical in x, whence the discrete Fourier
transform can be applied, resulting in an equal number of
equations and unknown amplitudes of all diffracted orders. It
can also be found in Claeys et al.9,11 that this discrete infinite
set of equations and unknowns can be chopped to a square
linear matrix equation that can be solved by a computer.
III. NUMERICAL RESULTS AND DISCUSSION
The following parameters are chosen such as to match
the physical reality of the reported experiments13 at 10 m in
front of the pyramid 共see Fig. 1兲. The observer’s height is
FIG. 1. Depiction of the pyramid’s staircase with and observer in front of it.
3329J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004 Declercq
et al.
: Acoustic effects at Chichen-Itza

chosen h⫽1.80m, the observers distance d⫽10 m, the pyra-
mid’s dimensions D⫽23.84m, H⫽24.02m, q⫽0.263m. It
then follows from 共1兲–共4兲that
␣
1⫽35.01°,
␣
2⫽78.15°,
␣
3⫽82.22°, and
␣
4⫽55.42°. The material properties in the
humid Yucatan air have been taken as
␳
⫽1.1466 kg/m3for
the density and v⫽343m/s for the sound velocity. Those for
the limestone14 staircase have been taken as
␳
⫽2000 kg/m3
for the density, vl⫽4100m/s for the longitudinal wave ve-
locity and vs⫽2300m/s for the shear wave velocity. Damp-
ing has not been taken under consideration. For the param-
eters just given, the Lipmann conditions are given as
follows: For frequencies lower than 1844 Hz, the numerical
simulations will be perfect. For frequencies higher than 1844
Hz, there will be small errors in the description of the sound
field within the corrugation, but not elsewhere. For very high
frequencies, say more than 5000 Hz, errors may also occur in
the prediction of the sound field outside of the corrugation,
i.e., in the air and where the observer is situated. The errors
gradually grow for higher frequencies and are due to
‘‘shadow zones’’ and neglecting internal reflection within the
stairs.
A. Direct echo coming from a delta pulse
Within the angular interval 关
␣
1,
␣
2兴, the incident sound
is considered to be spherical and contains 500 frequencies
equally distributed between 500 and 3000 Hz. All incident
plane waves have the same amplitude regardless of their di-
rection and frequency. The former is necessary to produce
the spherical wave, the latter is needed to produce a delta
function like handclap. The spherical wave is modeled by
300 plane waves propagating along equally distributed
angles within the interval 关
␣
1,
␣
2兴. There is no serious vio-
lation of the Lipmann conditions. Only for frequencies above
1844 Hz can there be some errors in the sound field descrip-
tion within the corrugation, but that is not of significant im-
portance here because we are only interested in effects at the
observer’s position.
In Fig. 2, the calculated echo as a function of time is
given, corresponding to an incident spherical pulse. This sig-
nal looks very clean, i.e., there is not too much noise outside
of the echo, and is somewhat similar to the normalized plot
in Fig. 3 of the actual sound of a real Quetzal bird in the
forest. The latter signal was downloaded in*.wav format
from the website of David Lubman.13 The few delta function
like peaks in the middle of that latter plot are the result of
cracks that can be heard in the recorded sound file and are
probably due to wood creaks in the bird’s biotope. Figure 4
shows a normalized plot of the pyramid’s echo and is ob-
tained from a*.wav file that was also downloaded from Lub-
man’s website.13 This signal is far from clean. This is prima-
rily due to low frequency noise coming from the interaction
of wind with the microphone. Since it is almost impossible to
compare sound signals in time–space, it is necessary to study
sonograms or spectrograms of the obtained signals.15 A sono-
gram depicts the amplitude as a function of time ‘‘t’’ and as
a function of frequency ‘‘f.’’ It is obtained by a time limited
Fourier transform. Here, we used a gaussian window of
0.002 s width. The sonograms are plotted by means of a
gamma correction of 2. If the recorded sound is truly and
solely an echo that comes from diffraction on the staircase,
some patterns that will be mathematically described now,
may appear in the sonogram. From 共9兲an ⫺m’th order echo
may appear if the following relation holds:
kx
inc⫽m
␲
冑
2q.共10兲
FIG. 2. Normalized calculated direct echo coming from a delta pulse. FIG. 3. Normalized recorded signal produced by a Quetzal bird in the forest.
FIG. 4. Normalized recorded signal of the echo coming from the pyramid.
3330 J.Acoust. Soc. Am., Vol. 116, No. 6, December 2004 Declercq
et al.
: Acoustic effects at Chichen-Itza

If this is combined with the dispersion relation, the angle, as
a function of the frequency at which the echo may appear,
can be calculated. If a ray-consideration is then applied, the
time delay as a function of each angle, taking into account
the wave speed in air, can also be obtained. This ultimately
results in:
t共⫺m,f兲
⫽
兩
dz⫹hz
兩
vcosRe
冋
␲
2⫺arctan
冑
冉
2
␲
f
v
冊
2
⫺
冉
m
␲
冑
2q
冊
2
m
␲
冑
2q
册
.
共11兲
In Fig. 5 the curves that are represented by 共11兲are depicted
by means of a sonogram. In all sonograms that are presented
here, the vertical axis represents the frequency in the range
from 0 Hz 共bottom兲to 5000 Hz 共top兲. The horizontal axis
always spans a range of 0.2 s. However, the instant values on
the horizontal axis do not always range from 0 to 0.2 s. It is
only the difference between the right side of the horizontal
axis and the left side that is 0.2 s. This is of course due to the
fact that sound recordings contain no information about the
absolute values of the start of recording and the end of re-
cording. However, in order to compare the different sono-
grams that are presented here, we have taken into account
physical considerations like the presence of the handclap in
the recordings of Lubman13 or the knowledge of the time-
origin in our calculations, to draw a time–frequency window
on each of the presented sonograms that is absolutely the
same in each sonogram. This window will therefore function
as the reference window for the discussions below. The ab-
solute position of the window is chosen as to contain the
relevant information that is present in Fig. 6, which is the
sonogram that corresponds to the calculated echo of Fig. 2.
This sonogram shows almost the same structure as the one of
Fig. 7, which corresponds to the recorded Quetzal bird chirp
in the woods 共see also Fig. 3兲. The only important difference
is the frequency at which the patterns appear and their dura-
tion. The actual bird chirps at lower frequencies than the
calculated pyramid’s echo. The authors do not know how a
young Quetzal bird sounds like, but perhaps the resemblance
would then be better. If Fig. 5 is compared with Fig. 6, it is
noticed that even though the classical grating equation pre-
dicts the possibility of elevated amplitude lines in the sono-
gram, not all lines are associated with a relevant amplitude if
the continuity conditions are also taken into account 共see Fig.
6兲. However, the elevated amplitude patterns that do appear
correspond more or less to the lines of Fig. 5. Especially
there is a strong appearance of the m⫽⫺4orm⫽⫺5 back
reflected sound. The fact that it is not simple to decide which
order is actually determining the elevated amplitudes is prob-
ably due to the interference of several plane waves because
the incident sound is spherical. This is slightly in contrast
with the assumption of Lubman5that the Bragg-orders can
FIG. 5. Bragg diffraction lines on a sonogram. The sonogram shows infor-
mation as a function of time 共horizontal axis兲and frequency 共vertical axis兲.
The square window is a reference window that represents the same time–
frequency values in each sonogram in this report. FIG. 6. Sonogram of the calculated direct echo coming from a delta pulse.
The axes are equal to those of Fig. 5.
FIG. 7. Sonogram of the recorded Quetzal bird chirp in the forest. The axes
are equal to those of Fig. 5.
3331J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004 Declercq
et al.
: Acoustic effects at Chichen-Itza

be well seen in the sonogram of the recorded echo. In order
to examine this contradiction, we have calculated the sono-
gram that actually corresponds to the recorded pyramid’s
echo of Fig. 4. The result is shown in Fig. 8. Within the
reference time/frequency window, the same pattern can be
found more or less 共if you look through the noise兲as in Fig.
6. However, Fig. 8 shows that it is absolutely not for certain
that all patterns that are noticeable would correspond to the
lines of Fig. 5. There is even something more obscure, which
is the presence of ‘‘patterns’’ outside the reference window.
If these were simply coming from Bragg diffraction, they
would also appear in Fig. 6, where not only the mathematical
grating equation is taken into account, but also the continuity
conditions. Since they do not appear in Fig. 6 共or have an
amplitude which is too small to be noticed兲, it can already be
concluded that these patterns cannot simply be the result of
pure Bragg diffraction and that an extra effect must be in-
volved.
B. Direct echo coming from a handclap
The answer to the critical question as to what then actu-
ally causes these patterns can be revealed if one considers
Fig. 9. The latter figure depicts the sonogram of the handclap
taken from the recordings of Lubman13 and being isolated
from the echo of the same recording. A handclap is actually
far from a delta function, because not all frequencies have
the same amplitude. Actually, the handclap contains several
frequency bands. For this purpose we have also simulated
the echo resulting from a real handclap instead of a pulse.
The handclap itself 共as taken from Lubman13兲, which takes
0.02 s and must be followed by 0.18 s of silence in order to
get a realistic time window of 0.2 s, needs to be represented
by 4096 frequencies in between 5 and 25000 Hz. Because of
the amount of RAM memory needed and due to a limited
CPU speed, taking into account all these frequencies in our
diffraction procedure would result in a calculation time that
exceeds the lifetime of our high speed computer.
This, together with the fact that the higher the frequen-
cies, the more seriously Lipmann’s conditions are violated,
and a trade off between handclap reproducibility and calcu-
lation time, led to the decision to reduce the number of fre-
quencies to 1968 in between 400 and 10240 Hz. Taking into
account higher frequencies would have violated Lipmann’s
conditions and would have taken us too much time. Consid-
eration of only frequencies up to 5000 Hz led to an incident
handclap that didn’t sound right and led to an echo that did
not at all correspond with reality. The reason of the latter
effect is that a complicated handclap is much harder to deal
with than the pulse of last section. Whereas a frequency chop
for a pulse results in a new pulse that is quickly followed by
a period of silence within the 2 s time window of interest, a
frequency chop for a handclap results in unnegligible noise
following the handclap, which is too strong if only frequen-
cies up to 5000 Hz are considered. This noise, which is less
important if frequencies up to 10240 Hz are taken into ac-
count, is also diffracted and due to time shifts may even
overlap with neighboring time windows after diffraction.
Therefore the numerical echo 共as can be seen in Fig. 10兲,
corresponding with an incident numerical handclap with fre-
FIG. 8. Sonogram of the recorded echo coming from the pyramid. The
vertical axis is equal as in Fig. 5, the horizontal axis spans the same time
interval length. The reference window is situated at the same time/frequency
values as in Fig. 5.
FIG. 9. Sonogram of the recorded 共and mathematically isolated兲handclap.
Same comments on the axes as in Fig. 8.
FIG. 10. Calculated direct echo coming from a handclap.
3332 J.Acoust. Soc. Am., Vol. 116, No. 6, December 2004 Declercq
et al.
: Acoustic effects at Chichen-Itza

quencies higher than 10240 Hz neglected, is, contrary to
physical experiments, not limited in time. In Fig. 10, for
reasons of calculation time limitations, we have, just as in
the previous calculations, considered the results for all ap-
plied plane waves at all applied frequencies, but we have
only taken into account 1024 positions of time within the
interval of interest for reproducing the result. This means that
a time limited Fourier transform cannot extract frequencies
higher than the sampling frequency of 3034 Hz. However, if
we take a look at the sonogram in Fig. 11, which corresponds
with the numerical signal in Fig. 10 and is made just like all
previous sonograms, we can see 4 frequency bands instead of
only 2 in Fig. 6. Even more important is that they coincide
with the experimentally measured frequency bands of Fig. 8.
Therefore, even if, because of computer limitations, a true
temporal description cannot be obtained, still what the fre-
quencies are concerned the simulation reproduces the experi-
mental result obtained by Lubman.13 This proves that the
lower two frequency bands in the experiments are mainly
caused by the nature of the handclap and not as much by the
diffraction process itself. In other words the echo is a func-
tion of the kind of incident sound.
C. Direct and indirect echo coming from a handclap
In Sec. IIIA we discussed the echo coming from a pulse
and showed that the presence of 4 frequency bands in the
reflected sound instead of 2 was probably due to the kind of
incident sound. In Sec. IIIB this statement was proved by
simulating the echo coming from the handclap in the
experiments.13 Yet another important question that needs to
be resolved is the influence of the ground in front of the
stairs of the pyramid. Up until now we have neglected this
effect. We now consider the extreme condition where the
ground is a perfect reflector. Hence sound coming from the
handclap is not only propagating strait to the pyramid, but is
also reflected on the ground before propagating towards the
pyramid. Furthermore sound reflected from the pyramid may
be received after strait propagation from the stairs or may
again be reflected by the ground before being received.
Therefore, the received signal Gconsists of 4 parts:
共i兲G1: Sound traveled directly to the pyramid and being
received directly;
共ii兲G2: Sound traveled directly to the pyramid and being
received after being reflected by the ground;
共iii兲G3: Sound being reflected by the ground before hav-
ing traveled to the pyramid and being received di-
rectly;
共iv兲G4: Sound being reflected by the ground before hav-
ing traveled to the pyramid and being received after
being reflected by the ground.
We call the person in front of the pyramid ‘‘person’’ and his
mirror image 共see Fig. 1兲the ‘‘mirror person.’’ The ground is
replaced by a mathematical mirror plane. Mathematically G1
is emitted by the person and again received by the person. G2
is emitted by the person and received by the mirror person.
G3is emitted by the mirror person and received by the per-
son. G4is emitted and received by the mirror person. By
filling in the correct coordinates of the person 共d⫹h兲and the
mirror person 共d⫺h兲, simulation is again possible of each
signal. Then
G⫽G1⫹G2⫹G3⫹G4.共12兲
The result of G can be seen in Figs. 12 and 13. Again these
figures cannot really tell anything about the temporal distri-
bution of the frequencies, nevertheless it is seen that the
ground has no influence on the presence or absence of the 4
frequency bands. In the future it would be great if someone
would do some experiments at the pyramid by placing a
reflector or an absorber in front of the staircase in order to
see what effect it has on the received echo.
IV. EXPLANATION OF THE RAINDROP EFFECT
If people are climbing the pyramid, their shoes produce
sound pulses containing all frequencies. Even though such
pulses are more complicated, we model them here by means
of a superposition of normally incident pure plane waves.
Figure 14 shows the amplitude of the reflection coefficient of
the zero order and the ⫺1st order as a function of the fre-
quency. Since we are only interested in understanding the
raindrop effect, we focus in Fig. 15 on the frequency zone
FIG. 11. Spectrogram of the calculated direct echo coming from a handclap.
FIG. 12. Calculated direct and indirect echo coming from a handclap.
3333J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004 Declercq
et al.
: Acoustic effects at Chichen-Itza

where the ⫺1st order reflected sound undergoes a transition
from evanescent to bulk waves. That happens at a frequency
fgiven by
f⫽
v
冑
2q⫽919.57 Hz. 共13兲
In addition it can be verified with what has been explained
above that this transition zone fulfills the Lipmann condi-
tions whence the validity of the numerical calculations can-
not be cast doubt on.
On the right side of the transition frequency in Fig. 15,
the ⫺1st order reflected sound is as important, regarding its
amplitude, as the zero order reflected sound. Furthermore, in
Fig. 16, the propagation direction 共measured from the pyra-
mid’s surface兲of the ⫺1st order reflected sound is depicted
as a function of the frequency. On the right of and close to
the transition frequency, the ⫺1st order diffracted sound
travels almost parallel to the pyramid’s surface. Now, since
that sound is bulk in nature 共not evanescent兲and since it has
a considerable amplitude 共see Fig. 15兲, it is actually hearable
for the observer seated on the lowest stair step. The observed
frequency range is limited since 共see Fig. 16兲only a limited
bunch of frequencies produce sound that can reach the ob-
server’s ear, which is situated at small angles from the pyra-
mid’s surface. Frequencies between 920 and 1000 Hz indeed
sound like the main frequency that is present in the bunch of
frequencies generated by a raindrop falling in a bucket filled
with water.
V. CONCLUDING REMARKS
It is shown that the echo that is produced by the pyramid
consists of diffracted sound coming from the staircase. The
echo is formed by a process which is connected with Bragg
reflection, but more effects are as important as well, such as
the continuity conditions on the stairs and the frequency pat-
tern of the incident sound. Therefore we would be pleased if
someone could do some extended experiments in front of the
pyramid in order to measure the echo as a function of the
incident sound. We would not be surprised if the use of
drums or timber wood to produce sound pulses would result
in a better echo. The model also showed that the ground in
front of the pyramid has no influence on the reflected fre-
quency bands. Nevertheless it could not be shown what the
FIG. 14. The zero order reflection coefficient 共solid line兲and the ⫺1st order
reflection coefficient 共dotted line兲as a function of the frequency, for normal
incident sound on the pyramid. The left side of the dashed line corresponds
to evanescent ⫺1st order reflected waves, while the right side corresponds
to bulk ⫺1st order reflected waves.
FIG. 15. Close up of Fig. 14.
FIG. 16. The propagation angle of the ⫺1st order reflected sound as a
function of the frequency, measured from the pyramid’s surface.
FIG. 13. Spectrogram of the calculated direct and indirect echo coming
from a handclap.
3334 J.Acoust. Soc. Am., Vol. 116, No. 6, December 2004 Declercq
et al.
: Acoustic effects at Chichen-Itza

temporal effect is. It could elongate the echo or shorten it
depending on the reflective properties of the ground. It
would also be interesting to test the effect of the sound speed
in air on the produced echo. This speed can vary in the dry
season and wet season and can also vary with temperature. It
is also explained how an observer seated on the lowest stair
step may hear ‘‘raindrops’’ falling in a water filled bucket
when other people are climbing the upper stairs.
ACKNOWLEDGMENTS
The authors are thankful to ‘‘The Flemish Institute for
the Promotion of the Scientific and Technological Research
in Industry 共I.W.T.兲’’ for sponsoring this research. The ex-
perimental work of David Lubman and the spread of his data
through the internet is sincerely acknowledged. We hope that
this theoretical work will form an impetus for him to do even
more such interesting experiments in the near future. The
first author strongly acknowledges the inspiring interdiscipli-
nary contacts with fellow students, research colleagues and
seniors from all over the world, during the First Pan
American/Iberian Meeting on Acoustics in Cancun-Mexico
in December 2002. Furthermore we express our sincere
gratitude to the editor’s and reviewer’s comments and their
encouragement to perform, besides the calculations for the
direct pulse, also the simulations for the exact handclap and
also for the effect of the ground in front of the pyramid.
1Wayne Van Kirk, ‘‘The accidental 共acoustical兲tourist,’’ J. Acoust. Soc.
Am. 112共5兲, 2284 共2002兲.
2David Lubman, ‘‘Acoustical features of two Mayan monuments at
Chichen Itza: Accident or design,’’ J. Acoust. Soc. Am. 112共5兲, 2285
共2002兲.
3David Lubman, ‘‘Singing stairs,’’ Sci. News 共Washington, D. C.兲155,44
共1999兲.
4David Lubman, ‘‘Mayan acoustics: Of rainbows and resplendent
quetzals,’’ J. Acoust. Soc. Am. 106共4兲, 2228 共1999兲.
5David Lubman, ‘‘Archaeological acoustic study of chirped echo from the
Mayan pyramid at Chiche
´n Itza
´,’’ J. Acoust. Soc. Am. 104共3兲, 1763
共1998兲.
6Bijal P. Trivedi, ‘‘Was Maya Pyramid Designed to Chirp Like a Bird?’’
National Geographic Today, Dec. 6, 2002.
7Fernando J. Elizondo-Garza, ‘‘Quetzal or not Quetzal, that is the ques-
tion... on the stairs of the Castillo monument in Chichen Itza,’’J. Acoust.
Soc. Am. 112共5兲, 2285 共2002兲.
8Jorge Carrera and Sergio Beristain, ‘‘Theoretical interpretation of a case
study: Acoustic resonance in an archeological site,’’ J. Acoust. Soc. Am.
112共5兲, 2285 共2002兲.
9J.-M. Claeys and O. Leroy, ‘‘Diffraction of plane waves by periodic sur-
faces,’’ Rev. Cethedec 72, 183–193 共1982兲.
10J. M. Claeys, Oswald Leroy, Alain Jungman, and Laszlo Adler, ‘‘Diffrac-
tion of ultrasonic waves from periodically rough liquid–solid surface,’’ J.
Appl. Phys. 54共10兲, 5657–5662 共1983兲.
11 Nico F. Declercq, Joris Degrieck, Rudy Briers, and Oswald Leroy, ‘‘A
theoretical elucidation for the experimentally observed backward displace-
ment of waves reflected from an interface having superimposed periodic-
ity,’’ J. Acoust. Soc. Am. 112共5兲, 2414 共2002兲; Nico F. Declercq, Joris
Degrieck, Rudy Briers, and Oswald Leroy, ‘‘Theoretical verification of the
backward displacement of waves reflected from an interface having super-
imposed periodicity,’’ Appl. Phys. Lett. 82共15兲, 2533–2534 共2003兲; Nico
F. Declercq, Joris Degrieck, Rudy Briers, and Oswald Leroy, ‘‘Theory of
the backward beam displacement on periodically corrugated surfaces and
its relation to leaky Scholte-Stoneley waves,’’in press with J. Appl. Phys.
12Sergio Beristain, Cecilia Coss, Gabriela Aquino, and Jose Negrete, ‘‘Tonal
response on the stairway of the main pyramid at La Ciudela, Teotihuacan
archeological site,’’ J. Acoust. Soc. Am. 112共5兲, 2285 共2002兲; paper
3aAA4 in Proceedings of the first PanAmerican/Iberian meeting on acous-
tics, 2002.
13David Lubman: http://www.ocasa.org/MayanPyramid.htm 共site visited on
12/13/2002兲.
14Jacques R. Chamuel and Gary H. Brooke, ‘‘Transient Scholte wave trans-
mission along rough liquid–solid interfaces,’’ J. Acoust. Soc. Am. 83共4兲,
1336–1344 共1988兲.
15Leon Cohen, Time-Frequency Analysis 共Prentice-Hall, Englewood Cliffs,
NJ, 1995兲.
3335J. Acoust. Soc. Am., Vol. 116, No. 6, December 2004 Declercq
et al.
: Acoustic effects at Chichen-Itza

Acoustic diffraction effects at the Hellenistic amphitheater of
Epidaurus: Seat rows responsible for the marvelous
acoustics
Nico F. Declercqa兲and Cindy S. A. Dekeyser
Georgia Institute of Technology, George W. Woodruff School of Mechanical Engineering, 801 Ferst Drive,
Atlanta, Georgia 30332-0405
and Georgia Tech Lorraine, 2 rue Marconi, 57070 Metz, France
共Received 13 November 2006; revised 25 January 2007; accepted 25 January 2007兲
The Hellenistic theater of Epidaurus, on the Peloponnese in Greece, attracts thousands of visitors
every year who are all amazed by the fact that sound coming from the middle of the theater reaches
the outer seats, apparently without too much loss of intensity. The theater, renowned for its
extraordinary acoustics, is one of the best conserved of its kind in the world. It was used for musical
and poetical contests and theatrical performances. The presented numerical study reveals that the
seat rows of the theater, unexpectedly play an essential role in the acoustics—at least when the
theater is not fully filled with spectators. The seats, which constitute a corrugated surface, serve as
an acoustic filter that passes sound coming from the stage at the expense of surrounding acoustic
noise. Whether a coincidence or not, the theater of Epidaurus was built with optimized shape and
dimensions. Understanding and application of corrugated surfaces as filters rather than merely as
diffuse scatterers of sound, may become imperative in the future design of modern theaters. © 2007
Acoustical Society of America. 关DOI: 10.1121/1.2709842兴
PACS number共s兲: 43.55.Gx, 43.20.El, 43.20.Fn 关NX兴Pages: 2011–2022
I. INTRODUCTION
In the classical world, the “asclepieion” at Epidaurus
was the most celebrated and prosperous healing center;1in
its vicinity there was the amphitheater, designed by Polyclei-
tus the Younger in the fourth century B.C. and famous for its
beauty and symmetry. The original 34 seat rows were ex-
tended in Roman times by another 21 rows. The theater is
well preserved because it has been covered for centuries by
thick layers of earth. A recent picture of the theater is pre-
sented in Fig. 1.
Marcus Vitruvius Pollio 共first century B.C.兲describes in
his famous books “De Architectura”2the state of the art in
architecture and shows evidence that man was aware of the
physical existence of sound waves. He writes, “Therefore the
ancient architects following nature’s footsteps, traced the
voice as it rose, and carried out the ascent of the theater
seats. By the rules of mathematics and the method of music,
they sought to make the voices from the stage rise more
clearly and sweetly to the spectators’ ears. For just as organs
which have bronze plates or horn sounding boards are
brought to the clear sound of string instruments, so by the
arrangement of theaters in accordance with the science of
harmony, the ancients increased the power of the voice.”
This indicates that the construction of theaters was per-
formed according to experimental knowledge and experience
and that it was done such as to improve the transmission of
sound from the center of the theater 共the orchestra兲toward
the outer seats of the “cavea.” It has however always been
believed, even in the same chapter written by Vitruvius,2or
the work by Izenour,3that it was mainly the aspect of the
slope of the theater, as a result of the constructed seats, rather
than the seats themselves, that have been a key factor in the
resulting acoustic properties.
The current study was triggered by the marvels of Epi-
daurus and by recent advances in the explanation of a variety
of diffraction effects on corrugated surfaces.4–9
The theory of diffraction of sound is based on the con-
cepts of the Rayleigh decomposition of the reflected and
transmitted sound fields.10–12 The theory, earlier applied to
describe a number of diffraction effects for normal incident
ultrasound on corrugated surfaces,13 has been used success-
fully to understand the generation of ultrasonic surface
waves in the framework of nondestructive testing. The
theory was later expanded to include inhomogeneous waves
and enabled a description and understanding of the backward
displacement of bounded ultrasonic beams obliquely incident
on corrugated surfaces, a phenomenon which had been ob-
scure for 3 decades.9,14 Even more, it was later exposed that
predictions resulting from that theory were in perfect agree-
ment with new experiments.15
An expansion of the theory to pulsed spherical acoustic
waves revealed special acoustic effects at Chichen Itza in
Mexico.7,8 The advantage of the theory is its ability to make
quantitative simulations as they appear in reality. From those
simulations, it is then possible to detect and characterize pat-
terns and characteristics of the diffracted sound field such as
in the case of a short sound pulse incident on the staircase of
the El Castillo pyramid in Chichen Itza. The study indicated
that the effects were slightly more complicated than the ear-
lier considered principle of Bragg scattering. In the mean-
time, the fact that the Quetzal echo at Chichen Itza is influ-
a兲Author to whom correspondence should be addressed; electronic mail:
nico.declercq@me.gatech.edu
J. Acoust. Soc. Am. 121 共4兲, April 2007 © 2007 Acoustical Society of America 20110001-4966/2007/121共4兲/2011/12/$23.00

enced by the properties of the sound source as well as the
existence of the “raindrop effect,” have been experimentally
verified by Cruz et al.16 Bilsen17 later showed that if one is
only interested in the position of time delay lines on a sono-
gram and not in the entire amplitude pattern, that it is pos-
sible to apply a simpler model based on the gliding pitch
theory.
For a study of acoustic effects at Epidaurus however, we
are not interested in the response to a pulse. We are merely
interested in how, for each frequency, sound behaves after
interaction with the seats of the theater. Therefore the exten-
sive diffraction theory, as used earlier,7is the pre-eminent
tool.
Until now, there have appeared a number of “explana-
tions” for the excellent acoustics of Epidaurus, such as that
sound is driven by the wind because the wind is mostly
directed from the orchestra toward the cavea. The wind di-
rection has indeed some influence, but it is also known that
the acoustics of Epidaurus is very good when there is no
wind or when wind comes from other directions; wind even
has a general negative effect because it produces undesirable
noise. Another theory is the importance of the rhythm of
speech but there are also modern performances taking place
at Epidaurus where the typical rhythm of Hellenistic poems
and performances composed by Homerus, Aeschylus,
Sophocles, or Euripides is not there; still the acoustics seems
perfect.
The last theory is that special masks, worn by perform-
ers, may have had a focusing effect on the generated sound,
but that does not explain why speakers with weak voices are
also heard throughout the theater.
Izenour3points out that the acoustics is so good because
of the clear path between the speaker and the audience. The
current work proves numerically that the effect of diffraction
on the seat rows is probably an even more important effect
than the “clear path effect.”
In what follows, we describe the geometry of the theater.
Consequently we explain briefly how the numerical simula-
tions are performed. Then we present and explain the nu-
merical results. We end our paper with the most important
conclusions.
The material parameters at Epidaurus have been taken
as: 2000 kg / m3for the density of the theater’s limestone,
and a shear wave velocity of 2300 m/ s and longitudinal
wave velocity of 4100 m/ s.
For the air at Epidaurus, we have taken two cases: “sum-
mer,” corresponding to an air density of 1.172 kg/ m3and a
共longitudinal兲wave velocity of 348.04 m/ s; and “winter,”
corresponding to an air density of 1.247 kg/ m3and a 共lon-
gitudinal兲wave velocity of 337.50 m/ s.
II. GEOMETRY: MILLER PROJECTION
In this paper, we only focus on the geometrical proper-
ties of the theater that are important for the acoustics. The
theater is almost semicircular. This means that the acoustics,
for a sound source situated at the center of the theater, will
have a circular symmetry similar to the theater itself. A
Miller projection 共as in cartography兲, mathematically trans-
forming the semicircular theater into a rectangular theater
resulting in seat rows in the cavea becoming straight rows
having the same length as the outer seat row; and transform-
ing or “stretching” the central spot 共at the center of the or-
chestra兲into a straight line parallel with the transformed the-
ater and having the same length as the seat rows; makes the
sound source at the center of the orchestra become a line
source that generates cylindrical waves. For simplicity, we
do not take into account edge effects at the edges of the seat
rows. We may therefore disregard one Cartesian coordinate
and study the entire problem in a two-dimensional space.
All this is physically correct if we also perform a Miller
projection of the entire sound field. In other words the sound
amplitude must be multiplied by a function describing the
sound density variation along the theater slope due to the
Miller projection. An inverse function must then be applied
to the results if we want to transform the results back to the
circular theater. Conveniently it is therefore unnecessary to
consider this function because we do not want to show re-
sults that are valid for the transformed theater, but in the real
circular theater. Furthermore a source not exactly situated at
the center of the orchestra will deliver exact results along the
theater radius passing through the source but will yield
slightly deviating results for other positions in the theater’s
cavea.
III. GEOMETRY: SHAPE, SIZE, AND DISTANCES
A relict of the extension from 34 seat row to 55 in Ro-
man times is the presence of a “diazoma” in between both
constructions as can be seen in Fig. 1. This acoustic discon-
tinuity is neglected in our study because the upper seat rows
are built along the same straight line as the inner seat rows
and this neglect mathematically corresponds to adding a few
seat rows and makes the two-piece theater a single-piece
theater having 60 seat rows instead of 55.
It is necessary to define a number of vectors and angles
of importance. They are depicted in Figs. 2 and 3.
Tables I–III explain the variables depicted in Figs. 2 and
3 and indicate the numerical values for the theater at Epidau-
rus.
FIG. 1. Picture of the theater of Epidaurus 共picture taken by the authors兲.
2012 J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus

Straightforward geometrical considerations yield for the
direct distance between source and receiver:
RD
=冑共posL cos共
␣
兲+ posS兲2+共hL + posL sin共
␣
兲−hS兲2.
共1兲
Analogously we obtain for the distance between source and
receiver, taking into account the mirror effect caused by the
foreground:
RM = 冑共posS + posL cos共
␣
兲兲2+共hL + posL sin共
␣
兲+hS兲2.
共2兲
The numerical procedure developed for this paper is based
on consecutive consideration of diffraction in subsequent
spots of diffraction. With respect to these spots “P” of dif-
fraction, we define a number of valuable distances:
RSP = 冑共P−hSx− posSx兲2+共posSz+hSz兲2共3兲
which is the distance between the actual sound source and
the spot of diffraction;
RMP = 冑共P+hSx− posSx兲2+共posSz−hSz兲2共4兲
is the distance between the mirror source and the spot of
diffraction and
RPL = 冑共xL −P兲2+zL2,共5兲
being the distance between the diffraction spot and the lis-
tener where
xL = posL + hL sin共
␣
兲and zL = hL cos共
␣
兲.共6兲
A considered ray of sound is incident at the diffraction spot
at the angle of incidence
␪
in 共cf. Fig. 2兲. If we do not con-
sider any reflections on the foreground, then this angle is
equal to
␪
in,D= arccos
冉
posSz+hSz
RSP
冊
.共7兲
If we also consider a reflection on the foreground then the
angle of incidence is
␪
in,M= arccos
冉
posSz−hSz
RMP
冊
.共8兲
IV. ACOUSTIC SIMULATIONS
A. Sound field description
As noted earlier, the sound source is considered cylin-
drical. The generated sound field is thought of as a bunch of
rays spread over all directions and widening with increased
distance 共involving an amplitude inversely proportional to
the square root of the traveled distance兲from the source just
as a real cylindrical sound field. The phase of the sound
within each considered beam also behaves as the actual cy-
lindrical sound field. Normalization yields the summation of
the amplitudes of all beams to be equal to unity. These rays
interact with the theater. Because at considerable distances
from the sound source the sound field pattern in each of the
“rays” approximates a plane wave, the interaction of the rays
with the theater is modeled as a plane wave interaction, al-
lowing the use of earlier developed techniques based upon
Rayleigh’s theory of diffraction.9,10,13 The diffracted sound
fields are also thought of as a widening sound ray continuing
the same widening pattern and pace as the incident sound
ray. We apply Rayleigh’s decomposition, therefore the inci-
dent sound field 共displacement field兲is given by
Ninc =Ainc
␸
inc共ikx
incex+ikz
incez兲.共9兲
The reflected 共
␨
=r兲and transmitted longitudinal 共
␨
=d兲
sound fields are given by
TABLE I. Measured values describing the theater.a
Quantity Value Meaning
b0.746 m Width of the seats
␣
26.6° Slope of the theater
SDmax 22.63 m Distance between center of orchestra and lower seat row
ltheater 49.88 m Length of the seats
See Ref. 1.
TABLE II. Calculated values describing the theater.
Quantity Equal to Value Meaning
hbtan共
␣
兲0.367 m Height of the seats
⌳冑b2+h20.831 m Periodicity of the seat rows
nltheater
⌳60 Number of seat rows
FIG. 3. Vectors and angles used to describe the theater and the acoustics.
FIG. 2. Vectors and angles used to describe the theater and the acoustics.
J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus 2013

N␵=兺
m
Am
␵
␸
m,␵共ikx
m,␵ex+ikz
m,␵ez兲,␵=r,d.共10兲
Finally, the transmitted shear sound field is written as
Ns=兺
m
Am
sPm,s
␸
m,s共11兲
with
␸
␶
= exp i共kx
␶
x+kz
␶
z兲共12兲
and
kx
m,sPx
m,s+kz
m,sPz
m,s=0. 共13兲
B. Mechanical continuity conditions
The sound fields described in Eqs. 共9兲and 共10兲must
correspond to incident and diffracted sound on the air-solid
interface formed by the seat rows.
In order to determine the unknown coefficients Am
r,Am
d,
Am
sPx
m,s, and Am
sPz
m,swe impose continuity of normal stress
and normal displacement everywhere along the interface be-
tween air and solid. The corrugated surface is given by a
function z=f共x兲. Periodicity of the corrugation yields
f共x+⌳兲=f共x兲共14兲
with ⌳the corrugation period. For further use, we define the
function g共x,z兲as follows:
g共x,z兲=f共x兲−z.共15兲
Along the interface we have g共x,z兲=0.
We do not consider viscous damping effects. The stress
tensor T
␶
共
␶
=1 in air,
␶
=2 the solid兲, is calculated as
Tij
␨
=兺
␩
␭
␶
␧
␩␩
␶
␦
i,j+2
␮
␶
␧i,j
␶
共16兲
in which ␭
␶
and
␮
␶
are Lamé’s constants.
The strain tensor ␧
␶
is calculated as
␧ij
␶
=1
2共
⳵
iNj
␶
+
⳵
jNi
␶
兲.共17兲
We also incorporate the dispersion relations for longitudinal
waves
k
␨
=冑
␳␻
2
␭
␶
+2
␮
␶
共18兲
with
␨
=“inc” or “m,r” and for shear waves
k
␨
=冑
␳␻
2
␮
␶
共19兲
with
␨
=s,2 for shear waves in the solid.
The dispersion relations 共18兲and 共19兲reveal the value of
kzcorresponding to each of the values for kxfor the different
diffraction orders. The sign of kzis chosen according to the
well-known “Sommerfeld conditions” stating that each of the
generated waves must propagate away from the interface and
demanding that whenever kzis purely imaginary 共evanescent
waves兲, its sign must be chosen such that the amplitude of
the wave under consideration diminishes away from the in-
terface.
Continuity of normal stress and normal displacement ev-
erywhere along the interface between air and solid yield
共Ninc +Nr兲·⵱g=共Nd+Ns兲·⵱galong g=0, 共20兲
兺
j
Tij
1共ⵜg兲j=兺
j
Tij
2共⵱g兲jalong g=0. 共21兲
Relations 共13兲,共20兲, and 共21兲result in four equations
that are periodical along the xaxis. A discrete Fourier trans-
form with repetition period ⌳is eminent and each of the
Fourier components on both sides of the equations are then
equal to one another.
Straightforward calculations ultimately result in four
continuity equations
Equation 1:
AincIinc,pi共−共k1兲2+kx
inckx
p兲+兺
m
Am
rIm,r,pi共−共k1兲2+kx
mkx
p兲
+兺
m
Am
dIm,d,pi共−共kd,2兲2+kx
mkx
p兲−兺
m
Am
sPx
m,sIm,s,p
⫻共kx
p−kx
m兲+兺
m
Am
sPz
m,sIm,s,p共kz
m,s兲=0. 共22兲
Equation 2:
−AincIinc,p
␳
1共kx
p−kx
inc兲−兺
m
Am
rIm,r,p
␳
1共kx
p−kx
m兲
+兺
m
Am
dIm,d,p
␳
2
冉
−kx
m+
冉
1+2共kx
m兲2−共kd,2兲2
共ks,2兲2
冊
kx
p
冊
+兺
m
Am
sPx
m,sIm,s,pi
␳
2
冉
1− kx
mkx
p
共kd,2兲2+
冉
1
共kd,2兲2
−1
共ks,2兲2
冊
共kx
m兲2
冊
+兺
m
Am
sPz
m,sIm,s,pi
␳
2共kz
m,s兲
冉冉
1
共kd,2兲2
−1
共ks,2兲2
冊
kx
m−
冉
1
共kd,2兲2−2
共ks,2兲2
冊
kx
p
冊
=0. 共23兲
TABLE III. Explanation of abbreviations used throughout the paper.
Abbreviation Explanation
hS height of the source
posS position of the source
hL height of the listener
posL position of the listener
RD direct distance
RSP distance between source and point of diffraction
RPL distance between point of diffraction and listener
RM direct distance for mirror source
RMP distance between mirror source and diffraction point
2014 J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus

Equation 3:
AincIinc,p
␳
1共kz
inc兲+兺
m
Am
rIm,r,p
␳
1共kz
m,r兲
+兺
m
Am
dIm,d,p共kz
m,d兲
␳
2
冉
−1+ 2
共ks,2兲2共kx
mkx
p兲
冊
+兺
m
Am
sPx
m,sIm,s,pi共kz
m,s兲
␳
2
冉
冉
1
共kd,2兲2−1
共ks,2兲2
冊
kx
m
−kx
p
共ks,2兲2
冊
+兺
m
Am
sPz
m,sIm,s,pi
␳
2
冉
冉
1
共kd,2兲2−1
共ks,2兲2
冊
⫻共kz
m,s兲2+1− kx
mkx
p
共ks,2兲2
冊
=0. 共24兲
Equation 4:
共Am
sPx
m,skx
m,s+Am
sPz
m,skz
m,s兲
␦
m,p=0. 共25兲
␦
m,pin Eq. 共25兲is Kronecker’s delta.
The grating equation 共similar to the one in optics兲takes
care of kx
mand kx
pas follows:
kx
␤
=kx
inc +
␤
2
␲
⌳,
␤
=m,p苸Z.共26兲
The Fourier transformation also leaves integrals within Eqs.
共22兲–共24兲:
Iinc,
␩
=1
kz
inc
冕
⌳
exp i关共kx
inc −kx
␩
兲x+kz
incf共x兲兴dx ,共27兲
Im,
␰
,
␩
=1
kz
m,
␰
冕
⌳
exp i关共kx
m−kx
␩
兲x+kz
m,
␰
f共x兲兴dx.共28兲
The integrals 共27兲and 共28兲can be solved numerically or
analytically.9They contain information about the surface and
are therefore called “surface integrals.”
C. The number of diffraction orders
Equations 共22兲–共25兲actually represent an infinite num-
ber of equations and unknown variables because the orders m
and pconstitute a discrete infinite interval of integer num-
bers Z. As discussed in earlier papers,5,7,9,13 finiteness of en-
ergy makes a limitation of the number of diffraction orders
authorized because only a few orders are really propagating;
the others are evanescent and with increasing value of mor
p, play a less important role in the energy transformation
upon diffraction.13 Therefore we only take into account two
forward and two backward “propagating” evanescent waves
for each of the considered frequencies. In other words, for
each frequency we consider the propagating bulk waves
共their number depends on the frequency兲and add two more
evanescent waves in each direction. The developed proce-
dure therefore automatically determines the number of waves
involved.
D. The formation of observed diffracted sound per
generated ray
By “observed diffracted sound” we mean sound that
reaches a given observer in the cavea of the theater. Because
we model the acoustics by means of cylindrically expanding
rays, it is clear that not all of these diffracted rays will ulti-
mately reach the observer.
It is known from textbooks on geometry that the dis-
tance from the diffracted ray to the observer is given by
冨
共xL + posL−p兲Re共kz
m兲
Re共kx
m兲−zL
冑
冉
Re共kz
m兲
Re共kx
m兲
冊
2
−1
冨
.共29兲
If this distance is smaller than a predetermined limit, the ray
is considered to reach the observer. If the distance is larger,
we further discard that ray. The limit is determined by the
width of the ray at the point of observance. We approximate
this width by its slightly larger value
limit = 2 max共
␪
in
+1,
␪
in
−1兲共RSP + RPL兲共30兲
with
␪
in
+1 and
␪
in
−1 the angle 共in rad兲between the considered
ray and the consecutive ray, respectively, the angle be-
tween the considered ray and the preceding ray.
Whenever we consider sound that is reflected in the or-
chestra on the foreground of the theater, we replace “RSP”
by “RMP” in Eq. 共30兲.
E. The integrated effect for all considered rays
The previous paragraph describes the interaction of one
ray at one single spot of the theater and it is determined
whether or not an observer will “detect” or “hear” the dif-
fracted rays. Calculation of the integrated effect consists of a
repetition of the previous procedures for each considered
generated ray and adding up all rays that are detected by the
observer. To approach physical reality, we model the gener-
ated cylindrical sound field by a bunch of rays that fulfill
specific incorporated requirements. The distribution of rays
is made such that the rays would be incident at spots Pon
the theater at equally spaced positions and such that there
would be three spots of incidence per wavelength, therefore
producing realistic simulations. Furthermore, we can apply
the above-mentioned procedure for each position of the lis-
tener “posL” and then plot the result as a function of the
position of the listeners on the theater.
F. The Lipmann and Wirgin criteria
The considered model, based on Rayleigh’s decomposi-
tion, is a simplified approach of more complicated models
such as the differential18,19 and integral equation
approach20–23 and Waterman’s theory;24–28 it is not valid for
any situation. There are certain requirements that need to be
fulfilled as studied by Lipmann29 and later also by Wirgin.12
Wirgin has shown12 that “contrary to prevailing opinion,
the Rayleigh theory is fully capable of describing the scat-
tering phenomena produced by a wide class of corrugated
J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus 2015

surfaces, including those whose roughness is rather large.”
Furthermore, Wirgin12 proves that the Rayleigh theory is
valid for ␭the largest wavelength involved in the diffraction
phenomenon, for ⌳the corrugation period and for hthe cor-
rugation height, whenever
h⬍0.34⌳共31兲
and
␭⬎1.53348h.共32兲
The Wirgin criteria are somewhat tighter than the older
Lipmann criteria.22,29 Nevertheless we may expect that the
Rayleigh theory for our purpose is reliable for frequencies
below 750 Hz 共in summer and in winter兲. Actually the theory
may even be valid to a large extent above 800 Hz. A limita-
tion to 750 Hz means that for a piano with 88 keys, our
model would simulate the acoustics at Epidaurus for the first
58 keys, this is almost 70% and is not too bad.
V. NUMERICAL RESULTS
In all our calculations we have considered an observer
whose ears are 80 cm above his seat. First consider a smooth
Epidaurus theater, i.e., a theater that consists of a smooth
slope without seat rows, making the sound rays undergo no
diffraction but simple reflections on the slope transmitting a
part of their energy into the limestone slope and reflecting
most of their energy. Calculations then reveal the observed
frequency spectrum for all positions posL on the slope. Con-
sider a sound source placed in the center at 22.63 m from the
first seat row and having a height of 2 m. This height is
reasonable since in the Hellenistic era the performers, who
were not very tall, wore “cothurns” or high theater sandals.
For simplicity, reflections on the foreground are not con-
sidered for the moment. Figure 4 shows the calculated re-
sults. The grayscale indicates the received sound intensity
whereas the horizontal axis gives the frequency and the ver-
tical axis corresponds to the height along the slope of the
theater 共posL兲.
Notice the appearance of distinct patterns due to the in-
terference between sound reaching the listener uninterrupted
and sound reaching the listener after being reflected upon the
slope of the theater. The “bands” of diminished intensity are
actually due to a phase canceling effect and are positions
where the audience will receive a much lower intensity than
at other positions in the cavea.
Figure 5 is similar to Fig. 4, except that here reflections
on the foreground are also taken into account, resulting in a
more complicated pattern, but with less distinct regions of
diminished intensity. Reflections on the foreground are there-
fore responsible for a better distribution of sound throughout
the theater.
Consider the situation at Epidaurus. Figure 6 shows the
results for Epidaurus with the seat rows installed 共at a peri-
odicity of 0,831 m兲and with reflections on the foreground.
The sound source is again situated as in Fig. 5.
FIG. 4. The received intensity in decibels for listeners situated at heights along the slope of a smooth theater, i.e., without seat rows, given along the vertical
axis and for frequencies given along the horizontal axis. The geometry corresponds to the geometry of Epidaurus and the sound source is situated at 22.63 m
from the first row of seats, i.e., at the center of the theater. Reflections on the foreground are neglected.
2016 J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus

FIG. 5. Similar to Fig. 4, but with incorporation of reflections on the foreground. Reflections on the foreground are responsible for a better distribution of
sound throughout the theatre.
FIG. 6. Calculated intensities, comparable with Fig. 5, but with the seat rows installed. The sound patterns are now influenced by the effect of diffraction. Note
that there is a relatively increased amplitude noticeable for high frequencies, whereas the overall sound intensity is lower than in the case without seat rows.
J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus 2017

Note that the intensities are slightly lower than for the
case without seat rows. In other words, the installation of
seat rows has a negative effect on the overall intensity of
sound throughout the theater.
As a matter of fact, the results are not really simple to
interpret because they show the cumulative effect caused by
the installed seat rows, caused by reflections on the fore-
ground and caused by the effect of the slope of the theater.
In order to highlight the particular effect of the seat
rows, which is the main purpose of this paper, it is necessary
to subtract Fig. 6 from Fig. 5. The result is shown in Fig. 7.
Because of the complexity of the diffraction phenom-
enon, the results are not really “smooth.” Still there are cer-
tain tendencies visible. First, the relative intensities are al-
most everywhere negative. This means that the presence of
stairs has a “damping effect” due to scattering in multiple
directions. Nevertheless, an overall drop of intensity is not
dramatic as the human ear is capable of adjusting its sensi-
tivity. What is more important is the fact that frequencies
beyond 530 Hz are less damped than frequencies between 50
and 530 Hz. Therefore there is a relative amplification of
high frequencies. There is also a dependency of the position
in the theater on the observable intensity, but this is mainly
caused by the slope and not really by the seat rows, as can be
seen in Fig. 5.
Note that reflections on the foreground are also very
important for the real theater of Epidaurus. This can be
clearly seen in Fig. 8, which is comparable to Fig. 7, except
that reflections on the foreground are neglected.
There are high intensity bands appearing from down un-
der to right up, which are merely due to interference effects
due to straight sound and zero order diffracted sound that
reaches the listener after diffraction. These bands correspond
to low physical intensities and are very distinct when the
theater contains no seats. In other words the existence of a
reflective foreground results in a better distribution of sound
throughout the theater and this redistribution is further im-
proved, in addition to the filtering effect in favor of high
frequencies, by the presence of seat rows.
Further results 共left out of the paper兲have revealed the
influence of the seat row periodicity on the acoustics. For
Aphrodisias, with a periodicity of 0.736 m, the relatively
amplified frequencies are higher than 600 Hz. For Pergamon,
with a periodicity of 1.657 m the relatively amplified fre-
quencies begin around 300 Hz. In other words, the periodic-
ity of the seat rows influences the band of amplified frequen-
cies: the smaller the periodicity, the higher the amplified
frequency band.
Additional results 共equally left out of the paper兲show
that patterns appearing at a certain height on the slope of the
theater shift to higher positions if the source is placed higher.
We have also studied the effect on the acoustics of the
distance between the source and the first seat row 共the “pro-
hedriai”兲. Apart from an increased overall intensity when the
source is positioned closer to the seat rows, we did not detect
any spectacular effects except that reflections on the fore-
ground become less important for a source closer to the seat
rows, therefore destroying the positive effect of a better dis-
FIG. 7. Comparison of Fig. 6 with Fig. 5, highlighting the effect of diffraction due to the installation of seat rows. At most positions and for most frequencies,
the intensity is diminished. However for frequencies beyond 530 Hz, one can see a relatively increased intensity. This is due to the filter effect causedbythe
seat rows.
2018 J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus

tribution of sound throughout the theater. Still diffraction of
sound on the seat rows makes the effect less dramatic.
We have also studied the influence of the slope of the
theater on the acoustics. This effect is very important for a
smooth theater without reflections on the foreground. The
effects are still noticeable in the case of installed seat rows
and reflections on the foreground, but it is less outspoken.
The slope does not really influence the frequency values
where the amplified frequency band appears.
The previous results were all for summer. Another as-
pect that we have studied is the influence of the season on
the acoustics of Epidaurus. The season has an influence on
both the sound velocity in air and the density of air. The
differences in the limestone are negligible. We found that
there was no significant difference between the acoustics in
summer and the acoustics in winter.
VI. THE PHYSICAL ORIGIN OF THE HIGH PASS
FILTER EFFECT
For low frequencies, the seat rows do not really diffract
sound, which means that there is no big difference compared
with a smooth slope. For higher frequencies, diffraction
plays a role and higher order reflected sound is generated,
causing sound to be distributed in different directions upon
reflection into the air and upon transmission into the bulk of
the theater’s slope. Figures 9 and 10 show the intensity of the
isolated reflected diffraction orders “-1” and “-2,” respec-
tively.
There is a “vertical raster” added to the figure that indi-
cates the transition from evanescent waves to propagating
bulk waves; for frequencies below the raster, sound is eva-
nescent and is stuck to the slope of the theater, for frequen-
cies passing the raster, sound is really propagating in space
and is observable. Note that there are negative first-order
diffracted waves observable at frequencies above 200 Hz,
but that their intensity is really small 共−15 dB and much
less兲. The second negative order diffracted waves appear be-
yond around 450 Hz and their intensity is higher 共−10 dB
and higher兲. These facts result in the following analysis: For
low frequencies there is no significant effect caused by the
seat rows. For higher frequencies the reflected sound is dis-
torted by the diffraction effect resulting in a distribution of
the sound energy in many directions and actually causing a
drop in the measured sound intensities for the audience. For
frequencies in between 100 and 500 Hz, there is a physical
influence of the negative first-order diffracted waves on the
acoustics of the theater. The amplitude of these first-order
waves is very small and therefore it mainly causes a distor-
tion of the sound field and diminishes the observed intensi-
ties. For frequencies beyond 500 Hz, the negative second-
order waves become important and they do have a significant
intensity. These negative second-order waves actually consist
of backscattered sound; for a given listener somewhere on
the cavea of the theater, they consist of sound that has passed
the listener and is backreflected toward this person. Because
the accompanied intensity is considerable, it results in an
FIG. 8. Comparable to Fig. 7, but in the case of the absence of a reflective foreground. Frequencies above 530 Hz are still favored by the filtering effect, but
there appears a position dependent intensity which is caused by the slope, just as in Fig. 4, and is only partly annihilated by the seat rows. This is mainly due
to the interaction of uninterrupted sound beams and zero-order diffracted 共i.e., undiffracted兲sound beams reflected from the cavea.
J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus 2019

increased observed intensity. Contrary to lower frequencies,
here the diffraction effect plays a constructive effect.
Besides negative diffraction orders, there are of course
also positive diffraction orders involved at Epidaurus. These
orders, however, are evanescent throughout the entire consid-
ered frequency interval and are therefore never observable by
the audience.
VII. CONSEQUENCES OF NUMERICAL RESULTS FOR
EPIDAURUS
We have shown in Sec. VI that the most important effect
caused by the seat rows at Epidaurus is the effect of relative
amplification of a frequency band above 530 Hz. In this sec-
tion, we discuss the consequences of this effect for the
acoustics of the theater.
Izenour3already pointed out that background noise is
very important for the acoustics of a theater. Background
noise is extremely important in a modern motorized society,
still at the old Epidaurus there were many visitors that must
have caused noise too. Furthermore there is also wind,
typically30,31 up to 500 Hz, rustling trees, etc. Most of the
noise produced in and around the theater was probably low
frequency noise and even if high frequency noise was pro-
duced to some extent, it would have been filtered out by the
fact that low frequency noise always spans much further in
open air than high frequency noise. The presented calcula-
tions indicate that a high frequency band is favored at the
expense of lower frequencies. This is true for sound pro-
duced at the location of the speaker. Sound coming from
other directions will be influenced differently. Nevertheless,
we have shown that the position of the sound source and its
height has no significant influence on the properties of the
amplified frequency band. This means that the conclusions
hold for noise coming from any direction.
Still, a reduction of the lower frequencies does not only
filter out low frequency noise, but it also filters out the fun-
damental tones of the human voice 共85– 155 Hz for men,
165– 255 Hz for women兲. This is not dramatic as the human
nerve system and brain are able to reconstruct this funda-
mental tone, by means of the available high frequency infor-
mation; this is the phenomenon of virtual pitch in the case of
a missing fundamental tone.32–36 As a matter of fact, virtual
pitch is the basic effect behind the creation of the illusion of
bass in small radios, miniature woofers, and in telephones.37
In other words the seat rows of the theater filter out low
frequency noise which has a positive influence on the clarity
of a speaker throughout the theater, despite the fact that the
lower tones of the human voice are filtered out as well.
VIII. COMPARISON WITH OTHER CLASSICAL
THEATERS
Table IV shows the physical parameters of different an-
cient theaters.
FIG. 9. The diffraction spectrum as a function of frequency and position along the “slope” of the cavea, of the -1 order diffracted sound waves. The raster at
approximately 200 Hz indicates the transition between evanescent sound and propagating sound. Note that the amplitudes at frequencies beyond 200 Hz are
very small: −15 dB and much less.
2020 J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus

Note that most theaters, apart from Pergamon, have a
seat row periodicity that is comparable to Epidaurus. The
slope values are more scattered. The discussion of our ob-
tained numerical results shows that the periodicity is the key
factor for the filtering effect of the stairs. Within that scope,
it is not surprising that most theaters copy Epidaurus’ seat
rows.
Still, the fact that the acoustics of Epidaurus is much
more renowned than the acoustics of the other theaters is
probably because of the fact that Epidaurus has been re-
nowned from the very beginning 共historical reason兲and that
it has been preserved so well 共conservational reason兲.
Dionysus is the theater whose dimensions best resemble
the dimensions of Epidaurus, but is in a much worse condi-
tion and therefore it will never be really possible to compare
the acoustics of both theaters experimentally.
IX. CONCLUDING REMARKS
It is shown that reflections on the foreground of the the-
ater result in a better distribution of sound throughout the
cavea so that all positions become acoustically similar to one
another. The installation of seat rows on a smooth cavea
generates diffraction effects that change the acoustic proper-
ties of the theater.
The intensity observed by the audience will be lower
than in the case of a smooth cavea. This is not dramatic
because the human ear is capable of adapting its sensitivity.
It is more important that the damping effect is frequency
dependent: the seat rows act like a filter. For frequencies
beyond a certain threshold, second-order diffracted sound
plays an important role and causes sound to be backscattered
from the cavea to the audience making the audience receive
sound from the front, but also backscattered sound from be-
hind. This has a positive outcome on the reception of sound.
FIG. 10. Comparable to Fig. 9, but for the -2 order diffracted sound. The raster is now situated at approximately 450 Hz. The amplitude for frequencies
beyond 450 Hz are −10 dB or higher. It is this -2 order diffracted sound that is responsible for the filter effect and for favoring frequencies beyond 500 Hz
for the audience.
TABLE IV. Properties of classical theaters.
Theater Dated Location Style ⌳共m兲
␣
共deg兲
Epidaurusa300 B.C. Greece Hellenistic 0.831 26.6
Aphrodisiasb300 B.C. Turkey Hellenistic 0.736 31.1
Aspendosb161–180
A.D.
Turkey Roman 0.788 33.1
Dionysusc
共Athens兲
400–300
B.C.
Greece Greek 0.829 23.5
Ostia Anticab19–12
B.C.
Italy Roman 0.762 22.1
Pergamonb197–159
B.C.
Turkey Roman 1.657 62.7
Pompeiid
共Odium兲
80 B.C. Italy Roman 0.805 32.3
Prieneb330 B.C. Turkey Hellenistic 0.772 31.2
Sideb? Turkey Greek 0.749 34.1
aSee Ref. 1.
bSee Ref. 38.
cSee Ref. 39.
dSee Ref. 40.
J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus 2021

For frequencies below the threshold 共mostly noise兲, the ef-
fect of backscattering is less important and is to a great ex-
tent filtered out of the observed sound. The threshold fre-
quency of the filtering effect is mainly determined by the
periodicity of the seat rows in the cavea of the theater. For
Epidaurus this threshold is around 500 Hz, which is usually
the upper limit for wind noise.30,31
The slope of the cavea does not really influence the fre-
quency values where the amplified frequency band appears
and there is no significant difference between the acoustics in
summer and the acoustics in winter.
1A. Von Gerkan and W. Muller-Wiener, Das Theater von Epidaurus (The
Theater of Epidaurus) 共Kohlhammer, Stuttgart, 1961兲共in German兲.
2M. Vitruvii Pollionis, De Architectura (On architecture), book V, Public
places, Chap. 3 共1st Century BC兲.
3G. C. Izenour, Theater Design, 2nd ed. 共Yale University Press, New Ha-
ven, CT, 1996兲.
4N. F. Declercq, J. Degrieck, and O. Leroy, “Diffraction of complex har-
monic plane waves and the stimulation of transient leaky Rayleigh
waves,” J. Appl. Phys. 98, 113521 共2005兲.
5N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, “Diffraction of ho-
mogeneous and inhomogeneous plane waves on a doubly corrugated
liquid/solid interface,” Ultrasonics 43, 605–618 共2005兲.
6N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, “Theory of the
backward beam displacement on periodically corrugated surfaces and its
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of special acoustic effects caused by the staircase of the El Castillo pyra-
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8P. Ball, “Mystery of ‘chirping’ pyramid decoded,” News@nature.com, 14
December 2004; doi:10.1038/news041213-5.
9N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, “Theory of the
backward beam displacement on periodically corrugated surfaces and its
relation to leaky Scholte-Stoneley waves,” J. Appl. Phys. 96, 6869– 6877
共2004兲.
10Lord Rayleigh, Theory of Sound 共Dover, New York, 1945兲.
11H. W. Marsh, “In defense of Rayleigh’s scattering from corrugated sur-
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16Private communication between N. F. Declercq and Jorge Antonio Cruz
Calleja 共Department of Acoustics, Escuela Superior de Ingeniería
Mecánica y Eléctrica UC, Avenida Santa Ana No. 1000 México D. F. Del.
Coyoacan. C. P. 04430. San Francisco Culhuacan, e-mail:
jorgeacruzc@hotmail.com兲.
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2022 J. Acoust. Soc. Am., Vol. 121, No. 4, April 2007 Declercq and Dekeyser: Acoustics at Hellenistic amphitheater of Epidaurus

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Published online: 23 March 2007; | doi:10.1038/new s070319-16
Why the Greeks could hear plays from the back
row
An ancient theatre filters out low-frequency background
noise.
Philip Ball
The wonderful acoustics for which
the ancient Greek theatre of
Epidaurus is renowned may come
from exploiting complex acoustic
physics, new research shows.
The theatre, discovered under a
layer of earth on the Peloponnese
peninsula in 1881 and excavated,
has the classic semicircular shape
of a Greek amphitheatre, with 34
rows of stone seats (to which the
Romans added a further 21).
Its acoustics are extraordinary: a
performer standing on the open-air stage can be heard in the
back rows almost 60 metres away. Architects and
archaeologists have long speculated about what makes the
sound transmit so well.
Now Nico Declercq and Cindy Dekeyser of the Georgia Institute
of Technology in Atlanta say that the key is the arrangement of
the stepped rows of seats. They calculate that this structure is
perfectly shaped to act as an acoustic filter, suppressing low-
frequency sound — the major component of background noise
— while passing on the high frequencies of performers' voices
1
.
It's not clear whether this property comes from chance or
design, Declercq says. But either way, he thinks that the Greeks
and Romans appreciated that the acoustics at Epidaurus were
something special, and copied them elsewhere.
Sound steps
In the first century BC the Roman authority on architecture,
Vitruvius, implied that his predecessors knew very well how to
Modern actors can be hear d clearly
60 metres away on a wind less day.
Nico Declercq
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The theatre at Epidaurus
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The Journal of the Acous tical Society of
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design a theatre to emphasize the human voice. "By the rules of
mathematics and the method of music," he wrote, "they sought
to make the voices from the stage rise more clearly and
sweetly
to the spectators' ears... by the arrangement of theatres in
accordance with the science of harmony, the ancients increased
the power of the voice."
Later writers have speculated that the excellent acoustics of
Epidaurus, built in the fourth century BC, might be due to the
prevailing direction of the wind (which blows mainly from the
stage to the audience), or might be a general effect of Greek
theatre owing to the speech rhythms or the use of masks acting
as loudspeakers. But none of this explains why a modern
performer at Epidaurus, which is still sometimes used for
performances, can be heard so well even on a windless day.
Declercq and Dekeyser suspected
that the answer might be connected
to the way sound reflects off
corrugated surfaces. It has been
known for several years now that
these can filter sound waves to
emphasize certain frequencies, just as microscopic corrugations
on a butterfly wing reflect particular wavelengths of light. The
sound-suppressing pads of ridged foam that can plastered on
the walls of noisy rooms also take advantage of this effect.
Declercq has shown previously that the stepped surface of a
Mayan ziggurat in Mexico can make handclaps or footsteps
sound like bird chirps or rainfall (see 'Mystery of 'chirping'
pyramid decoded'). Now he and Dekeyser have calculated how
the rows of stone benches at Epidaurus affect sound bouncing
off them, and find that frequencies lower than 500 hertz are
more damped than higher ones.
Murmur murmur
"Most of the noise produced in and around the theatre was
probably low-frequency noise," the researchers say: rustling
trees and murmuring theatre-goers, for instance. So filtering
out the low frequencies improves the audibility of the
performers' voices, which are rich in higher frequencies, at the
expense of the noise. "The cut-off frequency is right where you
would want it if you wanted to remove noise coming from
sources that were there in ancient times," says Declercq.
Declercq cautions that the presence of a seated audience would
alter the effect, however, in ways that are hard to gauge. "For
human beings the calculations would be very difficult because
the human body is not homogeneous and has a very
complicated shape," he says.
Filtering out the low frequencies means that these are less
audible in the spoken voice as well as in background noise. But
that needn't be a problem, because the human auditory system
can 'put back' some of the missing low frequencies in high-
frequency sound.
"There is a
neurological
phenomenon
called virtual
pitch that
enables the
human brain to
reconstruct a
sound source
even in the
absence of the
lower tones,"
The acoustic cu t-off
frequency is righ t where you
would want it
Nico Declercq, G eorgia Institute of
Technology, Atlant a
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...
3/24/2007
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-
16.html

Declercq says.
"This effect
causes small
loudspeakers to
produce
apparently better
sound quality
than you'd
expect."
Although many
modern theatres
improve
audibility with
loudspeakers,
Declercq says
that the filtering
idea might still be relevant: "In certain situations such as sports
stadiums or open-air theatres, I believe the right choice of the
seat row periodicity or of the steps underneath the chairs may
be important."
Visit our newsblog to read and post comments about this
story.
Top
References
1. Declercq N. F. & Dekeyser C. S.. A. J. Acoust. Soc. Am.,
in press (2007).
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SIT on the steps of Mexico's El Castillo pyramid in Chichen Itza and you may
hear a confusing sound. As other visitors climb the colossal staircase their
footsteps begin to sound like raindrops falling into a bucket of water as they near
the top. Were the Mayan temple builders trying to communicate with their gods?
The discovery of the raindrop "music" in another pyramid suggests that at least
some of Mexico's pyramids were deliberately built for this purpose. Some of the
structures consist of a combination of steps and platforms, while others, like El
Castillo, resemble the more even-stepped Egyptian pyramids.
Researchers were familiar with the raindrop sounds made by footsteps on El
Castillo - a hollow pyramid on the Yucatán Peninsula. But why the steps should
sound like this and whether the effect was intentional remained unclear.
To investigate further, Jorge Cruz of the Professional School of Mechanical and
Electrical Engineering in Mexico City and Nico Declercq of the Georgia Institute
of Technology compared the frequency of sounds made by people walking up El
Castillo with those made at the solid, uneven-stepped Moon Pyramid at
Teotihuacan in central Mexico.
At each pyramid, they measured the sounds they heard near the base of the
pyramid when a student was climbing higher up. Remarkably similar raindrop
noises, of similar frequency, were recorded at both pyramids, suggesting that
rather than being caused by El Castillo being hollow, the noise is probably
caused by sound waves travelling through the steps hitting a corrugated surface,
and being diffracted, causing the particular raindrop sound waves to propagate
down along the stairs (Acta Acustica united with Acustica, DOI:
10.3813/AAA.918216).
El Castillo is widely believed to have been devoted to the feathered serpent god
Kukulcan, but Cruz thinks it may also have been a temple to the rain god Chaac.
Indeed, a mask of Chaac is found at the top of El Castillo and also in the Moon
Pyramid. "The Mexican pyramids, with some imagination, can be considered
musical instruments dating back to the Mayan civilisation," says Cruz, although
he adds that there is no direct evidence that the Mayans actually played them.
Francisco Estrada-Belli, an archaeologist at Boston University, Massachusetts,
says: "Most if not all Maya pyramids were conceived as sacred mountains, which
were the places where the clouds gathered and created rain." However, while the
acoustics may have emphasised the metaphor of water, "the fact that there were
echoes around them does not mean that they were musical instruments", he
says - adding that Mayan texts do not mention such a use.
Elizabeth Graham of University College London points out that the pyramids
have been restored. "The authors need to provide a good reason for why they
think the restored building surfaces are enough like ancient building surfaces,"
she says.
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Mayans 'played' pyramids to make music for rain god
16 September 2009 by Linda Geddes
Magazine issue 2726. Subscribe and get 4 free issues.
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Have your say
Sacred Sounds
Thu Sep 17 18:31:18 BST 2009 by Jeremy
It seems unlikely that designing the pyramids to produce this sound could
be done without a lot of trial and error construction, and I doubt if the
Mayans had enough experience building new and improved pyramids over
many years to do this.
However, their written language consisted of heiroglyphics and they had
musical instruments, so who knows how they recorded sound or what they
considered as sacred sounds. Western cathedrals were designed with
acoustic effects in mind, although these were mostly an artefact of sound
bouncing off the stone interiors that resembled Pagan ritual sites deep in
forests. It's more the other way around: music was composed for the
cathedrals' acoustics.
I recall an image recovered by an archaeologist of a Mayan playing a flute
that had various-sized round bubbles floating in the air above the flutist.
Intuitively I thought the bubbles represented musical notes. But who knows.
We may never know what sounds they made or how they used their
pyramids
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ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 95 (2009) 849 –856 DOI 10.3813/AAA.918216
The Acoustic Raindrop Effect at Mexican
Pyramids: The Architects’ Homage to the Rain
God Chac?
JorgeAntonio Cruz Calleja
Escuela Superior de Ingeniería Mecánica yEléctrica UC, Department of Acoustics, Avenida Santa Ana No. 1000,
México D.F.Del. Coyoacan. C.P.04430. San Francisco Culhuacan, Mexico. jorgeacruzc@hotmail.com
Nico F. Declercq
Georgia Institute of Technology,George W. WoodruffSchool of Mechanical Engineering, 801 Ferst Drive,At-
lanta, GA 30332-0405, USA and Georgia Tech Lorraine, Laboratory for Ultrasonic Nondestructive Evaluation,
2rue Marconi, 57070 Metz, France. nico.declercq@me.gatech.edu
Summary
Mesoamerican pyramids have been in the center of attention ever since their discovery by westerners because
of their architectural beauty,their physical connection to ancient Indian cultures, their relationship to astronomy
and religion or simply because of their monumental size and attractive decor for tourist pictures. An acoustic
effect first encountered by Declercq (reported in J. Acoust. Soc. Am. 116(6),3328-3335, 2004)isthe raindrop
effect. When visitors climb the colossal staircase of Maya pyramids, their footsteps are transformed into sound
having distinct frequencies similar to raindrops falling in abucket filled with water.The current paper reports in
situ experiments followed by numerical simulations of the raindrop effect together with aphysical explanation.
In addition to numerical simulations, arule of thumb formula is extracted from the calculations that enable
the prediction of the acoustic raindrop frequencyatany other pyramid in Mexico. If the raindrop effect is a
phenomenon that wasintentionally incorporated in the construction of the Maya pyramids, such as the pyramid in
Chichen Itza, then it wasmost probably related to the rain god Chac for which there is ubiquitous archaeological
evidence decorated on the pyramid itself.
PACS no. 43.55.Gx, 43.20.El, 43.20.Fn
1. Concise introduction to the acoustics and
the cultural-historical background
The purpose of this section is to sketch the background of
the studied phenomenon in terms of its general importance
in the framework of history and culture.
There are manypyramids in Mexico. Theyare all step
pyramids. Some have distinct different ‘floors’ (big steps
or levels), likethe ones at Teotihuacan, others look more
likeEgyptian pyramids without distinctive ‘floors’, such
as the El Castillo pyramid at Chichen Itza (see right side
of Figure 1).The El Castillo pyramid is believedtohave
served as atemple to the god Kukulkan (the Maya name
for Quetzal Coatl), or the feathered serpent, see for ex-
ample Figure 2. It can be found in the Dresden Codex
[1, 2, 3, 4, 5, 6, 7] (kept by the ‘Sächsische Landesbib-
liothek’, the state and university library in Dresden, Ger-
many) that the Quetzal Coatl wasconnected to the Re-
splendent Quetzal, nowadays Mexico’snational bird [8].
Received20January 2009,
accepted 6February 2009.
Special acoustic effects, where the pyramid produces an
echo in response to ahandclap, that resembles the chirping
Resplendent Quetzal bird, have been studied by anumber
of scientists, including Lubman [9, 10, 11, 12, 13], De-
clercq [14, 15, 16], VanKirk [17], Bilsen [18], Trivedi
[19] ,Elizondo-Garza [20] and Beristain [21]. The dif-
ferent studies showthat there is most likely an acoustic
connection between the pyramid’sstaircase and the Re-
splendent Quetzal chirp. As Lubman stated, it is as if the
staircase forms arecording of the Quetzal chirp. However
there are also well-known archeological connections. In-
deed the pyramid contains sculptures of plumed serpents
running down the sides of the northern staircase.
There are manyhistorical and astronomical reports
[22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31] that focus on
aspects not of particular interest to acoustics, that show
that the pyramid undoubtedly served as acalendar system
connecting astronomical events and bodies to the temporal
cycles on Earth.
Observation of the particular shape of pyramids likethe
El Castillo pyramid also reveals adistinctive mismatch be-
tween the staircases and the outermost beauty of the geo-
metrical pyramid that theycover. This is because the in-
©S.Hirzel Ve rlag ·EAA 849

ACTA ACUSTICA UNITED WITH ACUSTICA Calleja, Declercq: Acoustic raindrop effect at Mexican pyramids
Vol. 95 (2009)
Figure 1. Left: the Coo waterfall in
Belgium. Each of the 4staircases of the
El Castillo pyramid look likewaterfalls
originating from the top of the pyra-
mid. Right: The pyramid of Chichen
Itza. Center: the mask of Chac enlarged
(pictures taken by Declercq).
clination of the staircases differs from the inclination of
the pyramid’swalls themselves. What could have been the
reason for constructing staircases at adifferent inclination
than the walls of the pyramid? One explanation is that this
difference in inclination is made to produce the specific
features of the descending serpent during the solar eclipse
[32]. Howeverthe effect is only visible on one side of the
pyramid whereas there are atotal of four of such staircases
on the pyramid. An unprejudiced person may perhaps see
aresemblance between the pyramid’sstaircases and water-
falls. It is as if the four staircases are solidified waterfalls
coming from 4gates at the top of the pyramid as can be
seen in Figure 1. Within that framework, it is interesting to
knowthat, according to Roman Piña [33] Chichen Itza is
Mayan for ‘at the mouth of the well of the water magician’
(‘itz’means ‘magician’ and ‘ha’means ‘water’).
Indeed water has been, and still is, very important for
the growth and continuation of civilization on the dry
peninsula of Yucatan [34, 35, 36, 37, 38, 39, 40, 41, 42].
There is also archeological evidence for the connection of
the El Castillo pyramid to the rain god (orwater god)Chac
[43]. In fact, aclose look at the pyramid uncovers the pres-
ence of amask of Chac on top of the pyramid (onall four
sides). This is also shown in Figure 1. There are also other
curved elements at twofaces of the temple related to the
same Deity [44].
Furthermore, Chac wasthe unification of 4separate
gods based in the four cardinal directions [45]: ‘Chac Xib
Chac’ (orRed Chac of the East), ‘Sac Xib Chac’ (orWhite
North Chac), ‘Ek Xib Chac’ (orBlack West Chac)and
‘Kan Xib Chac’ (orYellowSouth Chac). These 4separate
gods also correspond to the 4staircases of the pyramid.
Even more, according to Thompson [46] the God Chac is
depicted 134 times in the Dresden Codex.
Declercq et al. [14] have reported what is nowwidely
known as the ‘raindrop effect’, aphenomenon which De-
clercq had discovered, together with acolleague during
avisit to Chichen Itza on the occasion of the first Pan-
American meeting of the Acoustical Society of America
in 2002. If the raindrop would have been an intentional
acoustic effect incorporated by the builders of the pyramid,
then it is most likely the acoustic fingerprint (orrecording)
of Chac, the rain god. The effect involves footsteps, pro-
duced by people climbing the immense staircase of the El
Castillo pyramid, to be transformed into the sound of rain-
drops falling in abucket filled with water,atleast to an
observer seated on the lower stairs of the pyramid. With-
out anymeasurement, butwith the memory of this par-
ticular sound in his mind, Declercq et al. [14] mentioned
the effect in apaper about the Quetzal echo and produced
apreliminary simulation. The simulation showed that the
effect wasprobably due to the diffraction of sound, caus-
ing sound of aparticular frequencytopropagate along the
stairs down towards the observer.
The current paper investigates this raindrop effect more
thoroughly and compares in-situ recordings with newnu-
merical simulations.
2. Acoustic experiments
The awareness of the raindrop effect originates from De-
cember 2002, when Declercq and acolleague were sitting
on the lower step of the El Castillo pyramid in Chichen
Itza. Theyheard the sound of raindrops falling in abucket
of water,and not the sound of footsteps, while people were
climbing the stairs higher up. We have performed experi-
ments to study the effect quantitatively and found asimilar
effect if one is sitting higher up and people are climbing
the pyramid lower down. The newexperiments also reveal
that the raindrop effect is actually only detectible very near
the surface of the staircase and is best perceivedinbetween
the steps.
In what follows, we present the recorded measurements
of the ‘raindrop effect’ caused by twoMexican pyramids:
the Moon Pyramid of Teotihuacan and also the El Castillo
Pyramid at Chichen Itza. Both pyramids were chosen be-
cause of their significant staircase reaching from the base
to the top of the pyramid. In each of the experiments we
measured the sound in between twoofthe lower pyramid
850

Calleja, Declercq: Acoustic raindrop effect at Mexican pyramids ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 95 (2009)
Figure 2. God Chac in Dresden Codex.
steps while astudent wasclimbing the pyramid higher up.
All measurements were made during the night in order to
avoid interference of sound caused by chirping birds or
other visitors.
2.1. Measurements at Teotihuacan
In [14] avalue ‘q’isintroduced to characterize the steps of
the pyramid and is related to the step periodicity Λ(crucial
for the diffraction effect)asq=Λ/
√
2. ForTeotihuacan,
Figure 3. Analyses of the Teotihuacan raindrop effect record:
sonogram.
Figure 4. Analyses of the Chichen Itza raindrop effect record:
sonogram.
we have measured the steps and found amean value for
the step parameter q=0.298 m.
The reported signals were recorded at asample rate
of 44.1 kHz. We have applied asonogram analysis using
Hanning windowing of 8192 samples wide. Asonogram
shows the evolution of the frequencyspectrum as afunc-
tion of time.
Figure 3shows the sonogram of the in situ recorded
raindrop pulse at Teotihuacan. The sonogram shows that
the main amplitude is situated at 271.86 Hz for all times
depicted. The shape of the flanks of that peak is depend-
ing on time and so is the overall amplitude. The peak at
271.86 Hz means that the main frequencypresent in the
Raindrop effect is actually 271.86 Hz.
2.2. Measurements at Chichen Itza
ForChichen Itza [14], the mean value for qis 0.263 m.
We have followed the same experimental procedure as in
Teotihuacan.
Figure 4shows the sonogram of the in situ recorded
raindrop pulse at Chichen Itza. We only showthe time in-
terval that highlights the peak as we figured out that the
851

ACTA ACUSTICA UNITED WITH ACUSTICA Calleja, Declercq: Acoustic raindrop effect at Mexican pyramids
Vol. 95 (2009)
time evolution as giveninFigure 3for Teotihuacan plays
arole, butdoes not change the position of the amplitude
peak. The sonogram shows that the peak amplitude is sit-
uated at 304.69 Hz.
3. Numerical simulations of acoustic
phenomena
We have used the same material parameters as in Declercq
et al. [14]. Therefore the material properties in the hu-
mid Yucatan air have been taken as ρ=1.1466 kg/m3
for the density and v=343 m/s for the sound velocity.
Those for the limestone [47] staircase have been taken as
ρ=2000 kg/m3for the density, vl=4100 m/s for the lon-
gitudinal wave velocity and vs=2300 m/s for the shear
wave velocity.Visco-elastic damping effects have not been
taken under consideration.
Simulation of the interaction of sound with the staircase
is performed using aplane wave expansion technique, i.e.
Rayleigh’stheory of diffraction [48, 49, 50, 51]. The inci-
dent sound field (displacement field)isgiven by
Ninc =Aincϕinc ikinc
xex+ikinc
zez.(1)
The reflected (ζ=r)and transmitted longitudinal (ζ=d)
sound fields are givenby
N
ζ=
m
A
ζ
m
ϕ
m,ζ ikm,ζ
xex+ikm,ζ
zez,ζ=r, d. (2)
Finally,the transmitted shear sound field is written as
Ns=
m
As
mPm,sϕm,s ,(3)
with
ϕτ=exp ikτ
xx+kτ
zz (4)
and
km,s
xPm,s
x+km,s
zPm,s
z=0.(5)
In order for the sound fields described in (1)–(2) to be
the incident and diffracted sound on the air-solid interface
formed by the staircase, it is necessary to determine the un-
known coefficients Ar
m,Ad
m,As
mPm,s
xand As
mPm,s
z.For this
reason we impose continuity of normal stress and normal
displacement along the interface. The corrugated surface
is givenbyafunction z=f(x). Periodicity of thecorru-
gation yields
f(x+Λ)=f(x),(6)
with Λthe corrugation period. Forfurther use, we define
the function g(x, z)as
g(x, z)=f(x)−z. (7)
Along the interface we have g(x, z)=0.
The stress tensor Tτ(τ=1inair, τ=2the solid), is
calculated as
Tζ
ij =
η
λτετ
ηηδi,j +2µτετ
i,j,(8)
in which λτand µτare Lamé’sconstants.
The strain tensor ετis calculated as
ετ
i,j =1
2∂iNτ
j+∂jNτ
i.(9)
We also incorporate the dispersion relations for longitudi-
nal waves,
kζ=ρω2
λτ+2µτ,(10)
with =ζ=“inc” or “m, r”and for shear waves
kζ=ρω2
µτ,(11)
with ζ=s, 2for shear wavesinthe solid.
The dispersion relations (10) and (11) reveal the value
of kzcorresponding to each of the values for kxfor the
different diffraction orders. The sign of kzis chosen ac-
cording to the well-known ‘Sommerfeld conditions’ stat-
ing that each of the generated wavesmust propagate away
from the interface and demanding that whenever kzis
purely imaginary (evanescent waves),its sign must be
chosen such that the amplitude of the wave under consid-
eration diminishes away from the interface.
Continuity of normal stress and normal displacement
everywhere along the interface between air and solid yield
Ninc +Nrg=Nd+Nsgalong g=0,(12)

j
T1
ij (g)j=
j
T2
ij (g)jalong g=0.(13)
Relations (5),(12)and (13) result in 4equations that are
periodical along the x-axis. Adiscrete Fourier transform
with repetition period is eminent and each of the Fourier
components on both sides of the equations are then equal
to one another.
Straightforward calculations ultimately result in 4con-
tinuity equations
AincIinc,p i−(k1)2+kinc
xkp
x
+
m
Ar
mIm,r,pi−(k1)2+km
xkp
x
+
m
Ad
mIm,d,pi−(kd,2)2+km
xkp
x
−
m
As
mPm,s
xIm,s,pkP
x−km
x(14)
+
m
As
mPm,s
zIm,s,pkm,s
z=0,
852

Calleja, Declercq: Acoustic raindrop effect at Mexican pyramids ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 95 (2009)
−AincIinc,p ρ1kp−kinc
x
−
m
Ar
mIm,r,pρ1kp
x−km
x(15)
+
m
Ad
mIm,d,pρ2−km
x+1+2(km
x)2−(kd,2)2
(ks,2)2kp
x
+
m
As
mPm,s
xIm,s,piρ21−km
xkp
x
(kd,2)2
+1
(kd,2)2−1
(ks,2)2(km
x)2
+
m
As
mPm,s
zIm,s,pρ2(km,s
z) 1
(kd,2)2−1
(ks,2)2km
x
−1
(kd,2)2−2
(ks,2)2kp
x=0,
AincIinc,p ρ1kinc
z
+
m
Ar
mIm,r,pρ1km,r
z(16)
+
m
Ad
mIm,d,pkm,d
zρ2−1+2
(ks,2)2km
xkp
x
+
m
As
mPm,s
xIm,s,pikm,s
zρ2
· 1
(kd,2)2−1
(ks,2)2km
x−kp
x
(ks,2)2
+
m
As
mPm,s
zIm,s,piρ2 1
(kd,2)2−1
(ks,2)2(km,s
z)2
+1−km
xkp
x
(ks,2)2=0,
As
mPm,s
xkm,s
x+As
mPm,s
zkm,s
zδm,p =0.(17)
δm,p in (17) is Kronecker’sdelta.
The grating equation (similar to the one in optics)de-
termines km
xand kp
xas
kβ
x=kinc
x+β2π
Λ,β=m, p ∈Z.(18)
The Fourier transformation also leavesintegrals within the
equations (14)–(16):
Iinc,η =1
kinc
zΛ
exp ikinc
x−kη
xx+kinc
zf(x)dx, (19)
Im,ζ,η =1
km,ξ
zΛ
exp ikm
x−kη
xx+km,ξ
zf(x)dx. (20)
The integrals (19) and (20) can be solved numerically or
analytically.
In other words, an incident plane wavesisconsidered
that interacts with the interface. The formulation described
above then delivers the amplitudes of the diffraction orders
in reflection and transmission. The experiments are per-
formed in the air,therefore we only focus on the numer-
ical results for the reflection amplitudes Ar
mas afunction
of the frequency.
In [14] Declercq et al. have numerically estimated the
frequencyofthe raindrop effect, applying the described
procedure, for sound diffracted by means of sound perpen-
dicularly incident on the staircase. ForChichen-Itza they
found aresult of 920 Hz. Giventhe measurements reported
in the current paper,this has clearly been an overestima-
tion of the exact frequency. At that time there were no
measurements to compare with except for amemory of
what Declercq had experienced when he visited Chichen
Itza. It wasmentioned [14] that the actual footstep must be
farmore complicated than anormal incident sound wave,
still for simplicity only normal incident waveswere con-
sidered. Furthermore the numerical results were only con-
sidered valuable in that range of frequencies where evanes-
cent wavesturn into bulk wavesbecause it wasbelieved
that the raindrop effect wasactually caused by skimming
bulk wavesinair.
Newnumerical simulations, having the newexperimen-
tal results in mind, showthat we should not focus on skim-
ming bulk waves, butonevanescent waves(surface waves)
and that we should not limit the simulation to perpen-
dicularly incident sound butextend it to oblique incident
sound. Indeed, the newexperimental observations have
shown that the raindrop can only be observed very near or
even in between the stairs; therefore it is likely that evanes-
cent wavesare involved. Furthermore afootstep on astair
is actually directing sound perpendicular to the stair itself
and not perpendicular to the staircase. Therefore oblique
incident sound wavesmust be considered in order to study
the effect. Forobliquely incident sound, we found that the
threshold frequencycorresponding to the transition from
evanescent wavestobulk waves, is lower than for per-
pendicularly incident sound butthis frequencyisstill too
high compared with the experiments. If howeverwedo
not limit our study to (skimming)bulk wavesand if we
also consider evanescent waves, it is interesting to study
the amplitude of these evanescent waves. Forthat purpose
Figure 5shows the numerical result for the diffracted zero
order sound wavesand also the diffracted first order sound
waves. As expected the zero order diffracted wavesshow
an amplitude drop at the threshold frequencywhere the
first order diffracted wavesturn from evanescent waves
into bulk waves. Asimilar effect wasalso seen in the sim-
ulations for normal incidence reported in [14]. The first
order diffracted wavesare evanescent for lowfrequencies
and within that evanescence regime theyshowconsider-
able amplitude values for certain frequencies. The ampli-
tude peak for the first order diffracted wavesatChichen
Itza wasfound to be 308.70 Hz and 272.55 Hz at Teoti-
huacan if we consider an angle of incidence of 85 degrees
measured from the direction perpendicular to the staircase.
As amatter of fact, the peak positions were almost un-
changed for angles around the almost grazing angle of in-
cidence of 85 degrees and were significantly increased for
smaller angles (80degrees or less). If we compare these
results with the experimental values, we findrather good
agreement. It actually means that it is precisely that por-
tion of the incident sound generated at around 85 degrees
853

ACTA ACUSTICA UNITED WITH ACUSTICA Calleja, Declercq: Acoustic raindrop effect at Mexican pyramids
Vol. 95 (2009)
Figure 5. Estimated frequencyfor raindrop effect in Chichen Itza
and Teotihuacan pyramids.
that causes evanescent surface wavestobeoriginated that
pass the pass-band staircase and result in ‘transmitted’ fre-
quencies corresponding to the raindrop frequencyand are
detected by the observer.
The reported numerical simulations are based on the
grating equation, butalso on the continuity of normal
stress and normal displacements on the corrugated surface.
Therefore if we want to predict the raindrop frequencyfor
anygiven pyramid, we must perform the entire calcula-
tion and then findthe peak value of the amplitude of first
order sound. There is howeverarule of thumb possible
to estimate the raindrop frequency, if we ignore disper-
sion effects of evanescent waves. In Figure 5, we clearly
see that the shift of the peak is comparable to the shift
of the threshold frequencywhere ‘evanescence’ turns into
‘skimming bulk’. We know[52, 53, 54, 16] that the thresh-
old frequencyisdetermined by the dispersion equation of
sound in air and by the grating equation. Forfirst order
diffracted sound, this results in (1)
ftreshold1−sin θinc =vair
√2q,(21)
with ftreshold the threshold frequency, for θinc the angle of
incidence and for vair the sound velocity in air.The formula
is extendable to anyskimming wave (not just askimming
bulk sound wave in air at the threshold frequency) gener-
ated at its corresponding frequency. If we suppose that the
evanescent wave that is responsible for the raindrop effect,
has avelocity that does not depend on the step periodicity
of the staircase, i.e. if we ignore dispersion effects, then
we find
fq =fq.(22)
Indeed, if we takethe example of Chichen Itza, then
q=0.263mand the simulated raindrop frequencyisf=
308.70 Hz. If we enter q=0.298m(corresponding to the
Teotihuacan pyramid), we findaraindrop frequencyfor
Teotihuacan of f=272.44 Hz, which corresponds almost
perfectly to the raindrop frequencyof272.55 Hz found in
the exact simulations. Equation (22) is likely to predict the
raindrop frequencyaccurately,atleast when dispersion ef-
fects for evanescent wavesare ignored. From our experi-
ence with the plane wave expansion technique we know
that dispersion can be ignore in those situations where
the plane wave expansion technique is valid, i.e. when the
wave length of sound in air is of the same order of magni-
tude as the step periodicity and the step height. The transi-
tion from evanescent to bulk wavesnaturally occurs in the
regime when the wave length of the first order diffracted
waves(bulk or evanescent)haveawavelength comparable
to the step periodicity.Therefore it is right to assume that
formula (22) is valid for anypyramid in Mexico because
the second condition of validity of the plane wave expan-
sion technique, is always fulfilled since the step heights is
always almost equal to the step periodicity.The raindrop
frequencycan therefore be predicted in confidence by for-
mula (23) which may be useful to other acousticians or
archeologists.
fraindrop =81.1881
q.(23)
Formula (23) is accurate for limestone staircases where
the step height is of the same order of magnitude as the
step periodicity.
The velocity corresponding to the raindrop effect can be
calculated by means of the grating equation as
vraindrop =sin θinc
vair
+1
qf√2−1(24)
If we enter the properties for Chichen Itza, we findaveloc-
ity for the raindrop effect, measured along the staircase, of
122 m/s. Since formula (23) predicts the results very well
for the Teotihuacan pyramid as well, we may assume that
the raindrop effect on pyramids under similar conditions
(steps made of limestone, air around humidity and tem-
perature as in Chichen Itza)propagates along the staircase
at the same velocity.
4. The raindrop effect is not caused by the
hollowness of the pyramid
It is known that some pyramids are hollow, including the
El Castillo pyramid at Chichen Itza. Hollowness of the
pyramid may also cause adistinct sound, at much lower
frequencies than the raindrop effect, when hit by heavy
tools, similar to hitting abrick wall with aheavy hammer.
First of all, the effect described here is not due to hollow-
ness because we have encountered the effect both at hol-
lowand solid pyramids. In addition, if hollowness wasthe
cause, it would be heard in all directions; contrary to atyp-
ical diffraction effect likethe raindrop effect, which causes
the phenomena only to be heard at skimming angles. Fur-
thermore the sound would not be ‘trapped’ by the stair-
case, as with the evanescence of surface waves, butwould
also propagate away from it, which is clearly not the case.
Hence it is clear that acoustic effects caused by the hol-
lowness of the pyramid, if theyexist, are totally different
from what is described here.
854

Calleja, Declercq: Acoustic raindrop effect at Mexican pyramids ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 95 (2009)
5. Conclusions
After an introduction to the pyramid of Chichen Itza and
after recalling the existence of aQuetzal echo at the pyra-
mid, we studied the raindrop effect. We explained howthis
raindrop effect may have been related to the rain god Chac
because the latter is depicted several time on the pyramid.
Then we presented in situ experiments and indicated the
measured frequencies of the raindrop effect at Chichen Itza
and also Teotihuacan. Consequent numerical simulations
showed that the effect is due to evanescent wavesalong the
staircase propagating at avelocity of 122 m/s. Astonish-
ing agreement between the experimentally found frequen-
cies and the numerically obtained frequencies led to the
formulation of arule of thumb useful to predict the rain-
drop frequencyfor anysimilar Mesoamerican pyramid as
long as dispersion effects do not play asignificant role for
the evanescent waves, which is most likely the case for all
Mexican pyramids since the height of the steps is always
of the same order of magnitude as the periodicity.
As afinal conclusion we would liketohighlight
that only specificarcheological evidence may ever prove
whether acoustics played arole in Maya culture [46, 55,
56, 57, 58, 59, 60, 61, 62, 63, 64]. Up until nowthere
are only so called archeological ‘indications’ found, men-
tioned in this paper,together with acoustic effects that are
currently under investigation. The Mexican pyramids, with
some imagination, can be considered musical instruments
dating back to the Mayan civilization, butwehavenoevi-
dence that the Mayans have ever played them. ..
At other locations, mainly at Epidaurus in Greece, there
is real evidence of ancient architectural structures for
which it is much more likely that theyhavebeen con-
structed mainly according to acoustic principles and for
the reason of acoustic performances [48, 65].
Acknowledgement
The authors are grateful to the following students who as-
sisted in the recording of the studied sounds: Angelica
Villanueva Almaraz, Alfredo Cruz Calleja, Israel Reyes
San Juan, and the acoustics carrier students. The au-
thors are also grateful to the INAH (Instituto Nacional de
Antropología eHistoria)Mexico.
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856

VOLUME 9, ISSUE 1 JANUARY 2013
Physical Acoustics
Acoustics
Today
A publication of
the Acoustical Society
of America
Diffraction by Periodic Structures
Microbubbles as Ultrasound
Contrast Agents
Electronically-phased Acoustic
Array
Standards Report
and more
v9i1_cvrs_8.5x11_Layout 1 3/29/13 9:20 AM Page 2

effect was due to a new type of surface
acoustic wave.
As scientific tools such as modern
calculus and numerical methods that are
applied today were not developed before
the end of the 17th century, one must
wonder if certain ancient phenomena
could possibly be of even higher interest
than modern day technology and expe-
riences. It was pointed out by others4–11
that Chichen Itza was known for the
existence of a transformed echo at the
base of the El Castillo pyramid
(Kukulkan pyramid) that sounded pretty
much like the chirp of a Quetzal Coatl.
Indications based on prediction of Bragg scattering angles as
a function of angle of incidence and frequency gave some
insight but lacked actual correspondence with the experi-
ments. As will be pointed out further Declercq et al.12 made
quantitative simulations taking into account mechanical cou-
pling and interference at the origin of the diffracted waves and
determined reflected features very much in agreement with
experiments. An additional discovery found that not only was
the staircase itself responsible for the specific feature of the
echo, but also the signature of the incident sound.
Declercq,12 while visiting Chichen Itza with fellow stu-
dent Goffaux in 2003, discovered the raindrop acoustic effect
in addition to the well-known chirp and this phenomenon
was further studied experimentally and theoretically by Cruz
and Declercq.13
Marcus Vitruvius Pollio, on the other hand, noted in the
first century BC that the Greeks built their theatres following
nature’s footsteps: “they traced the voice as it rose, and car-
ried out the ascent of the theater seats. By the rules of math-
ematics and the method of music, they sought to make the
voices from the stage rise more clearly and sweetly to the
spectators’ ears. For just as organs which have bronze plates
or horn sounding boards are brought to the clear sound of
string instruments, so by the arrangement of theaters in
accordance with the science of harmony, the ancients
increased the power of the voice.” New research revealed
some interesting features as is described below.14
It turned out that diffraction of sound with incorpora-
tion of mechanical coupling and acoustic interference
enables the understanding of both phenomena.
A recent re-emergence of interest in the field of diffrac-
tion of sound is connected with the development of phonon-
Introduction
Although physically very complex,
under a relatively wide range of
conditions the diffraction of
sound by a corrugated structure can be
described and understood under the
principle of a plane wave expansion. A
plane wave expansion was first consid-
ered by Lord Rayleigh and was later
applied by the team of Oswald Leroy
while incorporating the mandatory
mechanical coupling conditions between
the two media separated by the corruga-
tion. In essence the plane wave expan-
sion (i.e., expansion into diffraction
orders) is based on a Fourier series expansion with incorpo-
ration of propagation properties given by the dispersion rela-
tion and based on the acoustic wave equation. The funda-
mental connection between the periodic structure and the
decomposition into different diffraction orders is the grating
equation as also used in optics. Hence, it is not surprising that
research teams in the 1980’s and 1990’s that worked in this
area were also very experienced with Raman-Nath diffraction
in acousto-optics. Besides curiosity in the area of physics,
most researchers during the past century paid attention to dif-
fraction of sound by corrugated structures because of the abil-
ity to transform bulk acoustic waves into surface acoustic
waves. During that time special attention was paid to diffrac-
tion spectra and the appearance of anomalies. It was indeed
found that certain anomalies (called Wood anomalies) coin-
cided with the generation of Scholte-Stoneley surface acoustic
waves. The latter are known to be difficult to generate, hence
it was hoped corrugated surfaces would be more efficient.
Scholte Stoneley waves propagate over large distances and are
therefore of high interest for shallow water acoustic commu-
nication and also for high speed nondestructive testing of
large surfaces and structures. Encouraged by the physics of
sound being akin to that of light, Mack A. Breazeale’s team1
developed experiments to verify the existence of acoustic
counterparts of diffraction effects in optics. One such effect
was the Goos Hänchen effect2resulting in a backward dis-
placed optical beam. The experiments lead to the observation
of a similar effect in acoustics under the occurrence of dif-
fraction, but could not be explained by the assumptions made
at that time. It turned out later3that the use of inhomogeneous
waves in combination with the plane wave expansion tech-
nique would be the key to explaining the effect and that the
8Acoustics Today, January 2013
ON THE FASCINATING PHENOMENON OF
DIFFRACTION BY PERIODIC STRUCTURES
Nico Felicien Declercq
Georgia Institute of Technology
Unité Mixte Internationale (UMI) Georgia Tech –
French Centre National de la Recherche Scientifique (CNRS) 2958
George W. Woodruff School of Mechanical Engineering, Georgia Tech Lorraine
Metz, France 57070
Marcus Vitruvius Pollio
(first century BC):
“so by the arrangement of
theaters in accordance with
the science of harmony, the
ancients increased the power
of the voice.”
v9i1_p5_ECHOES fall 04 final 3/28/13 10:16 AM Page 8

Diffraction by Periodic Structures 9
ic crystals.15 It is known that such crystals exhibit very inter-
esting filtering properties of interest in electronics and in
seismology. Negative refraction for instance is found within
this framework. Recently potential applications have been
envisaged in which transformation of external bulk waves
into phononic crystal propagation modes are considered.
Although more complex than a corrugated surface, certain
physical phenomena such as conversion into Scholte-
Stoneley waves have been found already. Therefore one might
expect continued interest in the diffraction phenomena at
least for one more decade.
In what follows, a brief overview of a number of selected
phenomena studied in the recent past with concise explana-
tion and historical context will be presented.
Ultrasonic diffraction on periodic micro-structures
When ultrasound impinges a periodically corrugated
material, it either scatters as it would on a regular surface, or
it diffracts like light diffracts on a compact disk. Figure 1
shows an optical image of the cross section of a corrugated
brass sample.
When frequencies are used resulting in wavelengths of the
order of magnitude of the periodicity of the structure, it would
diffract. However, as in Fig. 2, when a scan is made of such a
sample at frequencies high enough to avoid diffraction, results
are obtained showing good images of the corrugation.
If incident sound having a wavelength of the same order
of magnitude as the corrugation periodicity is used, inter-
esting diffraction effects occur. One such effect is the
appearance of Wood anomalies in diffraction spectra of
sound impinging the corrugated surface at normal inci-
dence. In the 1980’s and the 1990’s such anomaly frequen-
cies were used to generate Scholte-Stoneley waves on solids
immersed in water.
In Fig. 3 experimental zero order reflection spectra are
depicted, obtained using normally incident longitudinal
waves impinging, from the waterside, a solid-water periodi-
cally corrugated surface.
In this framework one can seek optimization of surfaces
to enhance surface wave stimulation.18 Other anomalies also
exist and they are, of course, not limited to normal incident
sound. As a natural consequence, corrugated surfaces can be
used to polarize ultrasonic waves.19,20 They can also be used as
sophisticated filters for complex frequencies (transient sig-
nals)21,22 or to direct sound in particular directions in 3-
dimensional space23 when using 2-dimensional (doubly) cor-
rugated surfaces.
Before we move on to a very interesting physical phenom-
enon it is important to mention that diffraction effects can also
be exploited in air-coupled applications. A recent example is
the measurement of the thickness of cylinders in a dry envi-
ronment. Indeed Bragg scattering of sound on periodically
stacked cylinders reveals, with high accuracy,24 the cylinder
diameter and is a rather easy technique to apply in industries
such as steal-cord fabrication or even the pasta industry.
Perhaps a much more exciting phenomenon from a
physical acoustics point of view is the backward displacement
of beams when reflected off of a periodic structure. Breazeale
Fig. 1. Optical side view of a corrugated brass sample;
Λ
= 515 µm, h = 238 µm.16
Fig. 2. An ultrasonic image of the sample presented in Fig. 1, based on a C-scan
technique in which (in this example) the difference between the maximum ampli-
tude of the plateau reflection and that of the valley reflection is represented by a
color.16
Fig. 3. Experimental normal reflection spectrum into water for a brass – water
interface,
Λ
= 25 µm =, h = 66 µm. These results have been extracted from17 as a
representative example of the research undertaken on this subject in the 1980’s. The
labels STn are added to show where anomalies appear that are related to the gener-
ation of Scholte-Stoneley waves.
and Torbett1performed experiments in the 1970’s to deter-
mine whether there was an acoustic analogue of the optical
Goos-Hänchen effect.
The Goos-Hänchen effect2predicts that light incident
near the critical angle on a dielectric interface from an opti-
cally denser medium has a reflected beam that is laterally
shifted from the position predicted by geometrical optics.
The incident light beam transfers a portion of its energy
into the optically rarer medium and excites an electromag-
netic field that travels longitudinally for a certain distance
along the interface. This energy is leaked back into the
denser medium and interferes with the specularly reflected
beam. This interference results in a reflected beam which
exhibits a lateral displacement that appears as a forward
beam shift. More complex structures such as multilayered
media and periodically corrugated configurations of the
v9i1_p5_ECHOES fall 04 final 3/28/13 10:16 AM Page 9

10 Acoustics Today, January 2013
optical grating type guide electromagnetic fields of the
leaky-wave variety as well. The lateral displacement of a
light beam reflected from a leaky-wave structure when a
Gaussian light beam is incident upon it was studied by
Tamir and Bertoni.25 The Tamir and Bertoni25 theory pre-
dicts that at a certain critical angle, a reflected beam shift
may occur either in the forward or in the backward direc-
tion with respect to the incident beam. The early experi-
ments of Schoch26,27 using the acoustic analog of the Goos-
Hänchen effect for an ultrasonic beam reflected from a liq-
uid-solid interface showed a forward lateral displacement of
the reflected ultrasound beam. Later, Breazeale and
Torbett,1using a Schlieren photographic technique,
observed a backward beam shift of a 6 MHz ultrasonic
beam of 10 mm width, reflected from a superimposed peri-
odic grating, confirming the backward beam displacement
predicted by the theory of Tamir and Bertoni.25 Although
the backward shift was observed in acoustics, it was unclear
which acoustic phenomenon caused it and above all there
was no simulation method available at that time to model it.
Much later Declercq et al. developed the necessary theoret-
ical model based on a combination of the inhomogeneous
wave theory and the plane wave expansion theory of dif-
fraction to study the backward beam displacement.3,28 The
most important conclusions from that work were, first that
the backward beam displacement is caused by a backward
propagating Scholte-Stoneley wave generated by the inci-
dent beam. Such waves are known for their non-leaky fea-
ture when propagating on smooth surfaces, therefore they
are used for long-distance nondestructive testing and for
acoustic communication through sound propagating on the
seafloor. What had not been known was that these waves
become slightly leaky when propagating on corrugated sur-
faces. The developed theory described the leaky Scholte-
Stoneley waves and enabled simulation of the backward dis-
placement. Second, the theory also predicted that a bound-
ed beam can still be displaced forward or backward at
angles of incidence corresponding to leaky Rayleigh wave
generation even though this had not been observed by
Breazeale and Torbett.1The clue was that the beam must be
sufficiently narrow in order to obtain the effect. Indeed,
later experiments have confirmed the theory.29 Backward
beam displacement was considered an exotic phenomenon
that only exists under very specific laboratory conditions.
To verify these assumptions further research was performed
and it turned out that practically every angle of incidence
causes a backward displacement as long as the incident
wave is an ultrasonic pulse. Herbison et al.30 showed
through an angular frequency spectrum analysis that prac-
tically every ultrasonic pulse contains frequencies that dis-
place backward no matter what angle of incidence is used. It
was also shown that the backward displacement does not
just occur on the liquid side, as in the original experiments
of Breazeale and Torbett,1but also on the solid side.31 The
experiments of Herbison et al.30 were not based on the
Schlieren technique enabling visualization of sound, but
were based on quantitative measurements using a specially
designed ultrasonic polar scan system. The technique
allowed us to give the necessary evidence of the physical
cause of the backward displacement, namely leaky Scholte-
Stoneley waves. (see Fig. 4)
Influence of diffraction on the acoustic performance of
theaters and auditoria
Periodic structures, currently very popular as phononic
crystals, functioning as acoustic prisms and frequency selective
mirrors hold promise for applications ranging from seismic
wave deflection to accurate passive filters used in electronics.
The Hellenistic theatre of Epidaurus, on the Peloponnese in
Greece, which is well known for its extraordinary acoustic qual-
ities, attracted Declercq’s attention in the framework of his
investigations in this field. The theater, renown for its extraor-
dinary acoustics, is one of the best conserved of its kind in the
world. It was used for music and poetry contests and theatrical
performances.
Many assumptions existed concerning the reasons why the
acoustics of this theatre were so extraordinary, yet not a single
assumption was satisfactory. Declercq and Dekeyser14 proposed
that the acoustic quality of the theatre is due to the seat rows
forming a corrugated structure. This research clearly showed
and explained a filtering effect whereby the seat rows enhanced
the acoustic quality of the theatre.14 This study demonstrated
Fig.4. Angular spectrograms30 confirming the results obtained by Breazeale and
Torbett.1For a
θ
iof 22.5º, a spectrogram from the region of the specularly reflected
beam (a) shows backward shifted frequencies in the range 5.98-6.12 MHz. The com-
plementary spectrogram from a scan to detect the backward surface wave (b) shows it
occurring at 6.05 MHz. Propagating bulk modes also detected in (b) and shown the-
oretically by a dotted line. At other angles of incidence similar spectrum analysis
shows that there is practically always some portion of sound backward displaced.
v9i1_p5_ECHOES fall 04 final 3/28/13 10:16 AM Page 10

Diffraction by Periodic Structures 11
that an appropriate periodicity, in combination with properly
selected building materials, has a significant influence on the
theater’s acoustics, and provides critically important design
guidelines to improve the acoustical performance of modern
outdoor theatres and sports stadiums. Investigations on the
constructive influence of periodic seat rows, and later also on
corrugated ceilings32 as band pass filters on the acoustics of
rooms and theatres, resulted in a new architectural paradigm
that not only changed the way we look at ancient buildings but
also how we design new buildings. (see Fig. 5)
Explanation of acoustic marvels at the Pyramid of
Chichen Itza in Mexico
As pointed out earlier, perpendicular pulse-echo ultra-
sound experiments on corrugated surfaces were used in the
1990’s to determine the corrugation periodicity or to generate
surface waves at certain frequencies. More complicated peri-
odic structures, such as photonic and phononic crystals, were
developed in the late 1990’s and into the 21st century. The first
element of added complexity appeared in the 1990’s in experi-
ments on oblique pulse-echo ultrasound. These experiments
actually illuminated a striking feature of the El Castillo pyra-
mid in Chichen Itza, on Mexico’s Yucatan peninsula. There,
acoustic waves generated by a handclap were back-reflected by
the immense ziggurat causing them to sound not like a hand-
clap, but like a chirping Quetzal. This phenomenon has long
intrigued not only tourists and archeologists, but also acousti-
cians.4–11 Declercq et al.12 were the first to deliver a full explana-
tion of this phenomenon based on a theoretical model and
subsequent numerical simulations. This study received wide
attention because of the widespread interest in this phenome-
non in other branches of science and culture and because of
the possibility that Mayans actually built pyramids not only as
tremendous calendars but also as recordings of the chirp of the
holy Quetzal Coatl (Kukulkan).
Another phenomenon, the so called raindrop effect, ear-
lier believed to be caused by the partial hollowness of the
pyramid, was explained by Cruz and Declercq13 as a natural
effect caused by the staircase. (see Fig. 6)
Conclusions and continued research
Without being exhaustive we have tried to give a brief
overview of fields in which diffraction of sound by a period-
ic structure is of importance and we have to some extent
explained our share in this exciting research area. As the fab-
rication of new micron and nano materials has become com-
mon practice, it is believed that more problems and applica-
tions involving sound diffraction will need study the coming
years. New results and new physical phenomena related to
diffraction of sound are currently under investigation and
can be expected as publications in the near future. AT
Acknowledgments
Research mentioned in this document covers collabora-
tive work between Nico F. Declercq and (in alphabetical
order) Sarah Benchabane, Mack A. Breazeale, Rudy Briers,
Jorge Antonio Cruz Calleja, Joris Degrieck, Cindy S. A.
Dekeyser, Katrien Dewijngaert, Roger D. Hasse, Sarah
Herbison, Vincent Laude, Oswald Leroy, Jingfei Liu, Michael
S. McPherson, Rayisa P. Moiseyenko, Bart Sarens, Alem A.
Teklu, Katelijn Vanderhaeghe, John M. Vander Weide,
Patricia Verleysen and has been sponsored in one way or
another (travel grants, equipment grants, awards, project
grants) by the following funding agencies: The French
‘Agence National de la Recherche’ (ANR), the Belgian agency
for innovation by science and technology (IWT), the Belgian
Fund for Scientific Research (FWO), The North Atlantic
Treaty Organization (NATO), The French Conseil Regional
de Lorraine (CRL), Georgia Tech, the National Center for
Physical Acoustics (NCPA), the French ‘Centre National de la
Fig. 5. The theater of Epidaurus in Greece well-known for its wonderful acoustic properties.
v9i1_p5_ECHOES fall 04 final 3/28/13 10:16 AM Page 11

First PanAmerican/Iberian Meeting on Acoustics, 2002.
12 N. F. Declercq, J. Degrieck, R. Briers, and O. Leroy, “A theoretical
study of special acoustic effects caused by the staircase of the El
Castillo pyramid at the Maya ruins of Chichen-Itza in Mexico,” J.
the Acoust. Soc. Am. 116(6), 3328–3335 (2004); Reported by
Nature News, 14 December 2004; doi:10.1038/news041213-5.
13 J. A. Cruz Calleja and N. F. Declercq, “The acoustic raindrop effect
at Mexican pyramids: The architects’ homage to the rain god Chac?,”
Acta Acustica united with Acustica 95(5), 849–856, (2009); reported
by Linda Geddes “Mayans ‘played’ pyramids to make music for rain
god,” New Scientist, Issue 2726, 2 (16 September 2009).
14 N. F. Declercq and C. S. A. Dekeyser, “Acoustic diffraction effects
at the Hellenistic amphitheater of Epidaurus: Seat rows responsible
for the marvelous acoustics,” J. Acoust. Soc. Am. 121(4),
2011–2022 (2007); Reported by Nature News: Published online: 23
March 2007; | doi:10.1038/news070319-16.
15 R. P. Moiseyenko, S. Herbison, N. F. Declercq, S. Benchabane, and
V. Laude, “Phononic crystal diffraction gratings,” J. Appl. Phys.
111(3), 034907 (2012).
16 J. Liu and N. F. Declercq, “Ultrasonic geometrical characterization
of periodically corrugated surfaces,” Ultrasonics, DOI
10.1016/j.ultras.2012.12.006 (2012).
17 J. M. Claeys, O. Leroy, A. Jungman, and L. Adler, “Diffraction of
ultrasonic waves from periodically rough liquid-solid surface,” J.
Appl. Phys. 54(10), 5657–5662 (1983).
18 N. F. Declercq and B. Sarens, “Increased efficiency of surface wave
stimulation on the inaccessible side of a thick isotropic plate with
superimposed periodicity,” IEEE Trans. on Ultrasonics,
Ferroelectrics, and Frequency Control 54(7), 1409–1422 (2007).
19 N. F. Declercq, R. Briers, and O. Leroy, “The use of polarized
bounded beams to determine the groove direction of a surface cor-
rugation at normal incidence, the generation of surface waves and
the insonification at Bragg-angles,” Ultrasonics 40, 1–8. 345-348
(2002).
12 Acoustics Today, January 2013
Recherche Scientific’ (CNRS), The European Union, The
Acoustical Society of America (ASA), and The International
Commission for Acoustics (ICA).
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Acustica united with Acustica 97(4), 599–606 (2011).
Nico Declercq is an associate pro-
fessor at the Woodruff School of
Mechanical Engineering of the
Georgia Institute of Technology in
Atlanta. At Georgia Tech he has a
laboratory in France at Georgia
Tech Lorraine as part of the
French-American international
unit “UMI Georgia Tech – CNRS
2958.” Declercq teaches acoustics
and engineering classes at Georgia
Tech Lorraine and Georgia Tech
Atlanta. He is president of the
2015 International Congress on
Ultrasonics to be held in Metz,
France. Declercq is a European
Union Promodoc Ambassador. He received the International
Commission for Acoustics Early Career Award in 2007, the
International Dennis Gabor Award in 2006, and a Sigma Xi
Young Faculty Award in 2006. Prior to becoming a Georgia
Tech faculty member, Declercq was
subsequently a voluntary re-
searcher at the laboratory of
Oswald Leroy at the Kortrijk cam-
pus of the Catholic University of
Leuven in Belgium, Ph.D.
researcher at Ghent University, and
a postdoctoral fellow of the Belgian
National Science Foundation. His
first collaborations in the United
States were with Mack A. Breazeale
at the National Center for Physical
Acoustics, Oxford, Mississippi. He
received a Master’s degree in
(astro)physics from the Catholic
University of Leuven in 2000 and a
Ph.D. in Engineering Physics from Ghent University in 2005.
Declercq is married with 3 children and, although he is a glob-
al citizen, his family roots are entirely located in the south of
West Flanders in Belgium.
Nico Declercq and his daughter at Chichen Itza
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