SeeS – Sound 2006-09-11
Reference Material
Paper by Erich Rajch
Thunder drums
Thunder drums, cheap but quite complex “toys” allow to visualize some essential topics in wave
Physics. The three types of drums differ mainly in shape – narrow, 5 cm in diameter (fig1.a) or
large (15 cm in diameter), fig1b and fig.2, but similar in length (18 cm for the cylinder type and
21/38 cm in length for the oblique one). Apart from this all of them are open on one end and closed
on the other one by a rather rigid membrane with a 5 mm diameter spring attached, made of 0,4 mm
steel wire and 42 cm/ 88 cm long for the two types shown in
fig.1, respectively.
Fot. 1. Duża (59$) i mała (10$)„tuba
burzowa” (Educational Innovations)
Out of the three elements of the drum: 1) the tube, 2) the
membrane and the 3) spring, all of them perform a unique role in
generating the sinister thunder sound [hear also at xxx]. These
sound are low in frequency, pretty loud, growing and lowering in
intensity like a real, distant thunder. Below, we analyze these
sounds and recall some simple formula on Acoustics. This article
in the multimedia form [xx] supplies the reader with all sounds
and real movie records of the Fourier analysis.
The sounds were recorded by a cheap microphone connected to
a16-bit PC sound card and then analyzed in a separate computer (PC Pentium III with 750 MHz
clock) using the freeware “Oscilloscope xxxxxxx. The program allows to visualize the recorded
signal as well as perform the Fourier analysis of the component frequencies of the sound during its
execution.
Rys. 3. Widmo częstotliwości dźwięku
wytwarzanego poprzez pocieranie górnej
krawędzi kieliszka, zarejestrowane za
pomocą programu Oscilloscope.
(zakres częstotliwości na skali „wirtualnego
oscyloskopu” f = 0 – 2210,7 Hz)
As shown in fig.2 for the cognac glass (playing when rubbed on the upper border), this simple set
up allows to a very neat analysis of the sound components.
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SeeS – Sound 2006-09-11
Reference Material
Paper by Erich Rajch
Examples of the frequency analysis of the two tubes from fig. 1 are shown in, fig.3.
265 Hz
350 Hz
135 Hz
435 Hz
355 Hz
740 Hz
1140 Hz
1300 Hz
2200 Hz
1600 Hz
Rys. 4 b. Widmo dźwięku dużej „rury
burzowej”. Zakres częstotliwości na obu
rysunkach a i b taki sam f = 0 – 7046,4 Hz.
(kopia stop-klatki z „oscyloskopu
wirtualnego”)
Rys. 4 a. Widmo dźwięku mniejszej „rury
burzowej” (kopia stop-klatki z „oscyloskopu
wirtualnego”)
1. Open tubes
Open tubes, as recalled by the schematic drawing in see fig. 4 generate odd harmonics of the
fundamental frequency f= xx
a)
S
W
b)
S
S
W
W
Rys. 6. Obraz drgań podłużnych słupa powietrza
w piszczałce jednostronnie zamkniętej.
a) drgania podstawowe
b) drgania harmoniczne o częstotliwości 3 razy
większej od częstotliwości drgań podstawowych
c) drgania harmoniczne o częstotliwości 5 razy
większej od częstotliwości drgań podstawowych
c)
S
W
S
W
S
W
Fig.4 the fundamental pitch for this flute, if open at the end (and with all other registers closed) is 587 Hz,
and higher ones, for a stronger whistle, as analyzed by the present record system are 1174 and 1370 Hz. For
the
closed the extreme lower opening the pitch is xx Hz while higher frequencies are not so neatly multiplies of
the fundamental one, amounting to xx and xx Hz.
Sets of tubes covering one octave are commercially available by several suppliers (we recall one of
them in fig. 5). Sounds can be created by whistling on one end or also by simple stroking one and
by open hand [we recall the evening talk by xxx on GIREP 2003 seminar]. In the multimedia
version of this paper we recorded the first, second and third harmonics of the open or closed flute,
shown in fig 4.
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Reference Material
Paper by Erich Rajch
The fundamental frequency for an open tube 18 cm long would be xxx (we adopt the sound
velocity of xx m/s for the dry air at 20°C [CRC]). The spectrum of the narrow tube in fig.2 shows
wide peaks centered around 435 Hz, 1300 Hz and 2200 Hz, surprisingly following exact 1:3:5 ratios
for the fundamental and higher frequencies in fig. 5. Note, that the “effective” length of the open
tube is always higher than the exact, geometrical one – some changes of pressure due to the
standing wave take place also outside the tube [Berkeley]. The fundamental frequency would
correspond to the 19, xx cm length tube. Note however, that the maximum intensity of the sound
analyzed in fig. 2 falls at 350 Hz – a prominent, narrow peak. We will come back to this question
later on.
For the wide tube, see fig. 3b, the frequency spectrum is much more complex and the
maxima at about 135, 355, 740, 1140 and 1600 Hz are not so well visible as the harmonic
progression for the narrow tube. Furthermore, the ratios between frequencies diverge from the
1:3:5:7:9 relation – higher harmonics are more distant than predicted for a simple tube. Similarly to
the narrow tube the most prominent maximum, at 265 Hz does not belong to the “main” series of
observed frequencies, being however in accordance with the theoretical value of 278 Hz for the tube
of the mean between two borders, i.e. 30 cm length tube. But this latter agreement can be somewhat
fortuitous, the narrow peak at and not caused by other elements of the tube, i.e. the membrane and
the spring.
2. Vibrating membranes
In the case of a cord, higher frequencies are integer multiplies of the fundamental one – but this is
not the case of membranes. Solution of the wave equation for the circle membrane are described by
Bessel functions multiplied by angular component (see for example xxx). For the radial
symmetrical vibrations the higher harmonics are multiples of the fundamental one but with noninteger factors, say 2.29, 3.6, xx and so on. Therefore, the full spectrum of the membrane, of a drum
for example, does not form a “harmonic” series and is perceived by our ear as the “thunder”,
“clasp”, “noise”.
In the case of elliptic membrane the theoretical spectrum of higher frequency components
can be continuous – the fact well visible in a wide maximum between 2 - 4 kHz in fig. 3b.
Unfortunately, we did not measure the fundamental frequencies of the membrane alone – this would
require demounting of the tubes. Note only, that due to the ratio between the radii the fundamental
frequencies would scale as 3:1 (we recall 435 Hz and 135 Hz for the first peak in the “series” for the
narrow and large tubes, respectively).
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Reference Material
Paper by Erich Rajch
3. Distant thunder
One of the characteristics of a thunder storm is that more distant lightings produce thunders with
lower pitch. This is due to the different attenuation of high and low sound frequencies in
atmosphere. Say, the frequency of 200 Hz becomes attenuated by a factor of 1.5 over 1 km distance
while the 2000 Hz by a factor of 2.5. As shown in a sequence of Fourier spectra over 1.5 s time in
fig. 10 the “thunder drums” show a similar feature – high frequencies, at around 3000 Hz decay
much quicker than the low ones (around 300 Hz). This is due to the higher energy stored at higher
frequencies, according to the formula where is the sound frequency and A its amplitude. The latter
formula explains also, why higher energy is required to generate higher harmonics, like those in the
present flute [x].
1)
2)
3)
4)
5)
6)
Rys. 11. Porównanie szybkości zaniku drgań o różnych częstościach w dużej „tubie burzowej” wyższe częstotliwości zanikają szybciej. Skala częstości f = 0 – 7046,4 Hz: Rysunki 1-6 to widma
fal zmierzone w odstępach czasu 0,2 sekundy
4. Sinister growing
Another specifics of a thunder storm is its irregular characteristics – the distant sounds grow and
fall, being reflected from clouds, ground, mountains where present. In the drums this effect is
reproduced by the spring – coupled to the membrane. The performance of the system springmembrane can be compared to two weakly coupled pendula, passing vibrations from one to another.
In the drums the spring can accumulate a substantial amount of the energy, as transverse vibrations,
not generating the sound for a while, and passing this energy to the acoustic mode, when reflecting
from the membrane, see fig. 12.
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Reference Material
Paper by Erich Rajch
Rys. 12. „Przelewanie się” częstotliwości dla dużej „tuby burzowej”. Dla uwidocznienia efektu w
sekwencji zmniejszono zakres częstotliwości f = 0 – 2706,4 Hz oraz zwiększono wzmocnienie.
The effect of growing and lowering of “thunder” is particularly well audible in the wide tube – due
to possible several modes of the transverse vibrations, see once more fig.12.
5. Metallic pitch
Apart from all above characteristics the drum tubes allow “to play” also other sounds. The
metallic pitch becomes easily audible when the spring is stroked by a hard object, like the border of
a table [big4.] In order to check in quantitative way if this effect corresponds to a longitudinal
acoustic wave inside the spring wire we calculate the pitch for the fundamental frequency in a 10 m
long wire (this is an approximate length of the wire for the narrow tube). Assuming the wave
velocity in the steel of 5800 m we obtain 290 Hz – close to the narrow peak at 350 Hz in the
spectrum in fig. 2. Obviously, numerous other, higher frequencies could be associated to the
longitudinal standing waves in the spring wire.
6. A very wise assembly
It is quite clear that all elements of the “Thunder drum” have been chosen and coupled in a wise
manner:
a. the main progression of frequencies is determined by the tube length (this is well
visible for the narrow tube)
b. the membrane, in particular the elliptic one, assures the richness of non-harmonic
overtones, resembling the thunder noise
c. coupling of membrane modes to the vibrations of the spring allows quick energy
transfer from one to another system, causing the effect of a real thunder propagating
form a long distance
d. the very material of the spring is responsible for additional frequencies – however
also them in the low of a few hundreds of Hz, like the fundamental modes of the
membrane and the tube.
As an effect: o portable thunder, rich of Physics!
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