(1-12) Acoustics of a Greek amphitheater
It is shown that a surface of porous
material or auditorium seats yields negative reflection at the grazing angle1).
It is also mentioned2) that the reflection from the front seats at
an amphitheater is less because of their steep angle (26.3 degrees) and that
its path deference from the direct sound from the stage is larger than at the
These discussions were done when the
surface is flat. Here it is discussed on the effect of the curvature of a
1) Sound reflection at the grazing angle
Reflection coefficient was defined for a
surface which is not rigid and it was measured with a large panel of the
surface. It was used to get the reflection of a limited dimension with the
surface to be convolved to the rigid panel of the dimension.
Thus defined reflection
coefficients of two different porous surfaces are show in Fig.1. They are for a
punch carpet layer and a urethane foam layer.
Fig.1 Reflection coefficients of two porous layers
It has a positive surface reflection at the
normal incidence but it shows a large negative surface reflection at 80 degrees
which is close to the grazing angle.
An apparent surface was given as
shown in Fig.2 for uneven auditorium seats. A reflection coefficient was
measured at the ear position for an audience. An incident angle is shown by θi. They are for hard and absorptive seats in Fig.2.
Fig.2 Apparent surface of auditorium seats
and a receiving point for the reflection coefficient measurement
Thus measured reflection
coefficients of auditorium seats at the grazing angle for world famous concert
halls are shown in Fig.3. They are shown in the time and frequency domains.
Fig.4 shows the sketches of their auditorium seats.
The first negative reflection
coefficients are gathered in Fig.5 with a few additions of Japanese
Fig.3 Reflection coefficients of auditorium
seats at six world famous concert halls in the time and frequency domains
Fig.4 Sketches of auditorium
seats of the six concert halls
Fig.5 Incident angles and first surface
reflections of auditorium seats at world famous concert halls and a few
While, a Greek amphitheatre has a
slope with 26.3 degrees, as shown in Fig.6. The angle refers to around 65
degrees in Fig.5, where the reflection coefficient is smaller in negative.
Fig.6 Steep slope of audience seats at a Greek amphitheater
In this way, a Greek amphitheatre has commonly
the audience slope of 26.3 degrees and they get the direct sound from the stage
2) Reflection from a rigid concaved surface
To have the reflection of a concaved
surface it was divided into small rectangles where each of them gets plane wave
incidence and it was summed up being calculated by the Fraunhofer equation. The
reflection from each rectangle has a pair of positive and negative rectangular
waves from the far field term and a trapezoid wave from the near field term as
shown in the above of Fig.7.
In the below of Fig.7, it is
shown how they are summed for a concave and convex surfaces. For a concave
surface, the reflection from each rectangle is in phase with the one of the
rectangle at the specular reflection and they compose a pair of large and
For a convex surface, the
positive pair remains but the negative pair is cancelled with the positive
pairs of the surrounding rectangles. So it tends to remain only a positive
reflection from the rectangle of the specular reflection and it is affected by
In fig.8, two examples are shown
compared with experimental results. The successive multiple reflections are not
calculated but they simulate the specular reflection well.
Fig.9 shows the calculated
reflection of a concaved surface when its curvature is changed. Specular
reflection is very large with the addition in phase and its spectrum is large
too. In this condition, the amplitude and spectral level are largest at the
curvature Rc 70cm.
Fig.7 Reflections from a divided rectangle
in the above and the addition of reflections of the surrounding rectangles to
the specular reflection in the below
The reflection of
a convex surface is given in the left and that of a concave surface in the
Fig.8 Comparison of a calculated and
measured result for a concave surface
Fig.9 Reflection of a concaved surface with
the change of its curvature
Thus, the negative reflection from the
front seats becomes small and delayed because of the steep slope, and the
received level is amplified by the curvature at the audience.
possible to have an impulse response at an audience with the practical dimensions
of an amphitheatre. It should be convolved with the transient response of our
hearing system and integrated in the time window after absolutization.
3) Another possibility which makes amphitheatre
It has been discussed only for the direct
sound from the stage but it needs a bit of reverberation.
The impulse response for the
curvature Rc 70cm in Fig.9 shows a large specular reflection with ± amplitude and a successive transient wave
after them. It is given from the area surrounding the specular reflection
rectangles with delayed reflections. They are imagined to add a bit of
Further discussion should be done
after the transient response of our hearing system is convolved, comparing with
the reflection at a usual auditorium seats.
4) Rectangular reflectors surrounding the stage
There are wooden rectangular panels at the
three sides on the stage to support performers as can be seen in the later
photos. To show the acoustical difference, I spoke or sang outside or inside
the enclosure, they recognized it very clearly.
It was clearly noticed too from
the stage when audience talks back to me. It is reciprocal.
It is apparent that the
rectangular enclosure gives a good support to the performers on the stage. Rich
normal modes are created by the enclosure. How would it be if a ceiling is
added to get richer normal modes or the dimensions of reflectors are changed referring
to the musical notes? Interesting questions are never finished.
An impulse response at an audience from a
sound source on the stage of rectangular enclosure, which is the direct sound,
can be calculated. The sound reflected by the front audiences can be calculated
by the convolution of the reflection coefficient and the reflection of the
concaved surface. Then the impulse response at an audience is obtained by the
addition of them.
The evaluation of the sound field
can be done there and after the transient response of our hearing system is
It is not difficult to get such slopes in
the green country NZ. It is very lucky to be able to enjoy music and/or plays
in the green space under the stars. We held music gatherings ten times once in
a year in March celebrating autumn harvests. If this kind of event would be
expanded, it would be wonderful. A few photo shots are given below.
1）Y. Sakurai and K. Nagata, “Practical estimation of sound reflection
of a panel with a reflection coefficient” , J. Acoust. Soc. Jpn (E), 3, 1,
2) Y.Sakurai, H.Morimoto and K.Ishida,” The
reflection of sound at grazing angles by auditorium seats”, p209-227, Applied
3) Y.Sakurai,” The early reflection of the
impulse response in an auditorium”, p127-138, J.Acoust.Soc.Jpn.(E), 8, 4(1987).
4) Y.Sakurai, ”Sound reflection of a curved
rigid panel”, p63-70, J. Acoust. Soc. Jpn(E), 2, 3(1981).
5) Y.Sakurai and H.Morimoto,” The transient
response of human hearing system”, J.Acoust.Soc.Jpn.(E), 10, 4(1989).
Photos from handmade music gatherings