Lab Experiments 55
KamalJeeth Instrumentation and Service Unit
Experiment-360
F
HELMHOLTZ’S
VOLUME RESONATOR
Jeethendra Kumar P K
KamalJeeth Instrumentation & Service Unit, 610, Tata Nagar, Bengaluru-560 092.INDIA.
Email: [email protected]; [email protected]
Abstract
Using a separating funnel of fixed neck length, resonance frequency is
determined for different volumes and the Helmholtz equation is verified. By
plotting a graph of 1/V versus f, the neck correction is obtained and verified with
the actual neck volume.
Introduction
Hermann von Helmholtz’s study of sound is one of the most important contributions to
Physics. He studied various combinations of vibrations, beats, of musical tones and their
inter-relationships, and the perception of sound by human ear. His study of resonators is the
most important among these. A coke bottle is a simple volume resonator, which makes a
unique sound when air is blown into it [1].
Study of Helmholtz resonators is an important experiment in physics because of its wide
application. Volume resonators are used as sound absorbers in noise reduction applications.
The resonators are placed inside walls of the sound producing device and sound is absorbed
at the resonance frequency. This has been used in ducting employed in air-conditioning and
in silencers fitted in automobile engines. These resonators are also used in musical
instruments, including guitar and violin. There has been extensive research on the Helmholtz
resonator, especially in sound absorption by changing the neck shape and size.
Herman von Helmholtz was a medical doctor by profession who made significant
contributions in several fields of physics. He first became a professor of physiology, then
professor of anatomy, and finally ended up as professor of physics at Berlin University,
Germany. While studying the functions of human eye, Helmholtz invented the
Ophthalmoscope, the eye testing equipment being used even now. Study of human ear, led
him to conceive the idea of resonator. The inner part of human ear consists of cochlea which
is a hollow winding pipe- like organ which is broad and open on the exterior side and gets
tapered and closed at the interior side. Helmholtz proposed that the cochlea acts as a long
chain of innumerable number of resonators with continuously diminishing volume from the
exterior to the interior [2, 3]. He went on to give a theory for the volume resonator, and stated
that the resonating volume of a resonator is inversely proportional to the square of the sound
frequency during the so called process of resonance, as governed by the equation
f2 α V, or
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KamalJeeth Instrumentation and Service Unit
f2V= constant
…1
which is known as the Helmholtz’s equation.
It is one of most fundamental observations by Helmholtz. Over the years, oversimplification of this experiment led to its deletion the from University syllabus in India.
Hence we thought it appropriate to revisit this classic experiment with a new design. In this
design a 300ml round glass vessel fitted with a cock for draining water and detachable neck
with speaker microphone is used to study Helmholtz resonance. A wide band AC voltmeter
or digital storage oscilloscope may be used to observe the resonance.
Theory of Helmholtz Resonator
Helmholtz resonator is a vessel with a narrow neck, as shown in Figure-1. If air is blown into
it, the air in the neck gets compressed and expanded similar to a spring and mass system.
Instead, one can make the air in the neck portion to move by the sound wave generated by
placing a tuning fork above the neck. Similarly, the air in the neck portion can be made to
vibrate by a speaker placed near it generating sound wave. The amplitude (loudness) of the
sound produced depends on the neck length, cross-sectional area of the neck and volume of
the resonator.
Neck area A
Neck dia
Neck
2r
Neck Length L
Brim
Res onator Volume V
Figure-1: (a) Helmholtz resonator (b) physical description of the neck
Helmholtz resonators do not work on the principle of formation of standing wave pattern.
Instead, they can be modeled as a simple mass loaded on a spring system. The air in the neck
acts as the mass, while the air pressure on both inside and out of the resonator acts as the
spring. The effective mass of the air in the neck is given by
m=ρAL
where
…1
ρ is the density of air
A is the neck area
L is the neck length
The air pressure acts as a spring in the system. The “spring constant” of the air pressure is
given by
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KamalJeeth Instrumentation and Service Unit
஺మ
k = ρc2 ௏
…3
where
c is the velocity speed of sound in air
V is the volume of the cavity
A is the area of the neck
The resonator can be excited using a frequency generator and speaker tuned to the resonant
frequency. This sound acts as a driving force, given by
F =PA
…4
,
where
P is the sound pressure.
The equation for spring mass oscillation can be written as
ௗమ ఋ
m ௗ௧ మ + ݇ߜ = ‫ܲܣ‬
…5
߱௢ = ට௠
…6
where ߜ is the displacement of the slug of air contained in the neck. Solving Equation-5, we
get the resonance frequency as
௞
Substituting for m and k in Equations- 1 and 3, we get
f=
௖
ଶగ
ට
஺
௏௅
…7
This shows that the resonance frequency is inversely proportional to the square root of the
volume as predicted by Helmholtz. A graph of 1/V versus f is a straight line. This equation is
known as Helmholtz equation. Unknown volume of hidden resonator can, therefore, be
determined by knowing its resonance frequency and the neck dimensions.
The assumption of spring mass oscillation in the resonator is not perfectly valid in actual
practice because the boundary conditions at the two ends from where the sound wave enters
into the resonator are different. Hence the theoretical frequency does not match with the
value give by Equaution-7. For this reason a correction needs to be applied to the neck
length. This correction factor can be estimated by performing the experiment. If L' is the
corrected neck length, the equation for frequency becomes
f=
௖
ଶగ
ට
஺
௏௅ᇲ
…8
If a graph is drawn taking f on Y-axis and √(1/V) along X-axis, the straight line graph will
have slope (mexpt) given by
mexpt =
௖
ଶగ
ට
஺
௅ᇲ
…9
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KamalJeeth Instrumentation and Service Unit
and the theoretical curve will have slope (mthet) given by
mthet = ଶగ ට ௅
௖
where,
஺
…10
mexpt is slope of the experimental curve and
mthet is slope of the theoretical curve.
Dividing Equation- 9 by Equation- 10 we get
௠೐ೣ೛೟
௠೟೓೐೟
௅
= ට௅ᇲ or
L' = ൬
௠೟೓೐೟
௠೐ೣ೛೟
ଶ
൰ ‫ܮ‬
...11
In this experiment employing a fixed neck resonator, its volume is varied by filling it with
water and the corresponding resonance frequency is determined.
Apparatus used
The experimental set-up consists of a 300ml separating funnel, a speaker-microphone system,
digital storage oscilloscope and a function generator. The experimental set-up used in the
experiment is shown in Figure-2.
Experimental procedure
1. Using digital calipers the neck length and inner diameter of the neck are determined
and the actual neck volume is calculated.
Neck length (L) = 3.5cm = 3.5x10-2m
Inner diameter of the neck = 2.4cm
Inner radius of the neck (r) = 1.2cm
Neck area (A) =ߨ‫ ݎ‬ଶ = 4.52cm2= 0.452X10-3 m2
Neck volume = πr2L =1.583x10-5m3
2. Room temperature (T) is noted from the digital thermometer
T =27.4°C
3. Velocity of sound (c) is calculated at room temperature.
c = 331+0.607x27.4 =347.7 m/s
4. The separating funnel is now placed on the circular clamp fitted to a retort stand. The
speaker–microphone system is now placed above the neck and microphone is
positioned at the centre of the neck opening.
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KamalJeeth Instrumentation and Service Unit
5. The speaker is connected to the sine wave output of the function generator and its
amplitude is set to its maximum value and frequency is set to X100Hz band.
6. The microphone is connected to the digital storage oscilloscope (DSO).
7. The function generator is switched on and sound is heard in the speaker.
8. Water is filled up to the brim of the separating funnel. Hence, the volume of the
resonator is zero.
Volume of water drained
(V) X10-6m3
ඨ
૚
ࢂ
Table-1
Resonance frequency (f) Hz
Expt
Thet
Before correction
After Correction
25
200.0
868
1257.7
50
141.4
640
888.7
75
115.3
537
724.7
100
100.0
469
628.5
125
89.4
424
561.9
150
81.8
389
514.1
175
75.5
359
474.5
200
70.7
335
444.3
225
66.3
316
416.7
250
63.2
299
397.2
275
60.0
285
377.1
Empty funnel
276
Variation of resonance frequency with volume
877
620
506
438
392
358
331
310
292
277
264
-
9. Now 25 ml water is drained out through the funnel using a graduated beaker which
creates 25ml space in the separating funnel or the resonator volume now is 25ml.
Hence resonator volume, V, is given by
V =25ml
The frequency of the function generator is increased from 100Hz hearing sound and
watching the DSO. At the resonance frequency the sound will have the maximum
amplitude and the DSO shows the maximum amplitude. This is the resonance point.
The resonance frequency is noted from the DSO
f = 868Hz; for V=25ml =25x10-6m3
The theoretical resonance frequency, f, is calculated using Equation-7 as
f=
௖
஺
ට
=
ଶగ ௏௅
ଷସ଻.଻
ଶగ
ට
଴.ସହଶ௫ଵ଴షయ
ଶହ௫ଵ଴షల ଡ଼ ଷ.ହ୶ଵ଴షమ
= 1257.7 Hz
The values obtained are tabulated in Table-1.
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KamalJeeth Instrumentation and Service Unit
10. Trial is repeated by draining out 25ml water; making the volume of the resonator
cavity to 50ml and the resonance frequency for this volume is determined and
recorded in Table-1.
11. The trial is repeated until all the water is drained out and the resonance frequency of
empty funnel is determined.
Figure-2: Helmholtz resonator experimental set-up
12. A graph is plotted taking 1/V along X-axis and f along Y-axis which gives a straight
line. It is seen that the theoretical curve passes through the origin, whereas the
experimental curve does not pass through the origin. The slopes of two straight lines
are determined as
Slope of the theoretical curve (mthet.) = 6.285
Slope of the experimental curve (mexpt) = 4.36
௠
ଶ
଺.ଶ଼ହ ଶ
L' = ൬௠ ೟೓೐೟ ൰ ‫= ܮ‬ቀ ସ.ଷ଺ ቁ ܺ3.5ܿ݉ = 7.27ܿ݉
೐ೣ೛೟
Hence the corrected neck length is 7.27cm.
Using this value of L′, the resonance frequency is recalculated using Equation-7 and
presented in Table-1. A plot of the corrected theoretical frequency vs. the
experimental curve, given in Figure-4, shows good match between the theory and
the experiment after the neck correction.
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f (Hz)
Expt
Thet.
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
0
25
50
75
100
125
150
175
200
225
√(1/V)
Figure-3: Frequency variation with ඥ૚/ࢂ
1000
900
800
700
f(Hz)
600
500
400
300
200
100
0
0
50
100
150
200
250
(1/V)
Figure-4: Frequency variation of the theoretical curve with neck correction and the
experimental curve
13. For empty resonator the observed resonance frequency is 276 Hz which corresponds
to
ට = 58 or = 3364 or
௏
௏
ଵ
ଵ
V =297 ml
This is the volume of the resonator which can be verified by filling the separating
funnel with water completely, and its exact volume is determined.
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KamalJeeth Instrumentation and Service Unit
Exact volume of resonator 300ml
Results
The results obtained are tabulated in Table-2
Parameters
Volume of resonator (ml)
Corrected neck length (cm)
Actual neck length (cm)
Neck volume (ml)
Table-2
Experimental
297
Actual
300
7.2
3.5
32.57
Experimental results
15.83
Conclusions
After making the neck correction, the theoretical and experimental curves nearly match with
each other with very small deviation (less than 7 %), thus verifying the Helmholtz equation.
The accuracy of the volume determination of separating funnel is excellent. The resonator
behaves better when its volume is more than 150ml, being half the actual volume (300ml) of
the funnel. Because of this, an attempt has been made to determine the volume by filling the
resonator with marbles, sand and other liquids. The volume determined by filling small
marbles was not accurate may because of the air gap between the marbles.
References
[1] Kelly Patton, Studying the effects of filling a Helmholtz resonator with spheres,
http://www3.wooster.edu/physics/jris/Files/Patton_Web_article.pdf
[2] S P Basavaraju, A detailed text book of Engineering Physics Practicals, 1999,
Page-15.
[3] D Sudhakar Rao and Chitra G M, Helmhotz volume resonator, LE Vol-7, N0-4,
Dec.–2007, P-296.
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