Download 5B: ACOUSTIC RESONANCES

5B: ACOUSTIC RESONANCES
A. INTRODUCTION:
Often a system has natural frequencies of vibration. In the case of the column of air in the
PVC tube we are studying, these natural frequencies are called the harmonic frequencies,
f1, f2,É. A vibrating source, such as the speaker, is said to drive the column of air. As
the frequency, f, of the driver (speaker) is slowly varied, the amplitude of the driven
system (air column) gets larger and larger until it reaches a peak at one of its own resonant
frequencies, f1, f2,É. Therefore, as we sweep the speaker frequency past one of the
harmonic frequencies of the air column, we hear the increase in amplitude of the vibrating
air column as an increase in loudness of the radiated sound from the tube. At resonance the
driving system passes energy very efficiently on to the driven system and the amplitude of
the driven system reaches a maximum called Amax. (See Rossing, The Science of Sound,
Ch. 4.) A graph of the amplitude as a function of frequency is shown here.
Output
signal
voltage
maximum signal
Df =linewidth
Amax
=resonant
amplitude
0.71 Amax
background level
f1
f =frequency
Note that the resonant amplitude peaks at frequencyf1 and that the peak has a width, Df
called the linewidth. The linewidth is usually measured at an amplitude of 71% of Amax.
For a system which loses energy rapidly through damping or friction, the maximum
amplitude, Amax, is small and the linewidth large, and the resonance is said to be broad.
Similarly, for a resonating system which loses energy very slowly, the maximum
amplitude is very large and the linewidth is very small. The resonance is said to be sharp.
P108 Lab 5BÑ1
The quality, Q, characterizes the sharpness of one of the resonances. For example, for the
fundamental frequency the Q value is:
Q=
f1
.
Df
(1.0)
A high Q circuit is one with a sharp resonance. Once set oscillating it loses energy very
slowly. A tuning fork is an example of an object with a very high Q. To drive an object
with high Q, the driving frequency must be very close to the resonant frequency, f1.
B. MEASUREMENT OF THE Q OF A RESONANCE:
1. High Q System:
It is convenient to measure the Q of one of the harmonics of your long PVC tube with one
end closed. Tape over one of your tube and glue a glass plate to it.
a.
Set up the scope with channel 1 looking at the signal from the signal generator (2002)
and with channel 2 looking at the signal from the microphone. Set the scope to
trigger on channel 1. With a small sine wave signal (about 1 volt) from the signal
generator, attach the small loudspeaker to the generator. Also, attach the frequency
counter to the generator.
b.
Find a convenient harmonic of your open tube with a frequency above 350 Hz. (The
small speakers we are using have a resonant frequency of their own at about 260 Hz.
ItÕs nice to avoid this frequency region for this reason.) Choose a sensitivity
(VOLTS/DIV) for the microphone input into CH2 so that the full signal can be seen
on the scope face at resonance.
P108 Lab 5BÑ2
c. Now, starting below the resonance frequency and scanning over the resonance in 10 or
12 steps, record the amplitude of the microphone signal in millivolts. As a first try
you might start about 50 Hz below resonance and step over the resonance in 10 Hz
steps. You may want to fill in a few extra points where the amplitude changes
rapidly.
Driving Frequency
d.
Amplitude (mv)
Comments
On a separate sheet of graph paper make a plot of amplitude (on the verticle scale)
against frequency (on the horizontal scale). Using the example plots above, connect
the points of your plot, calculate the width of your response curve at 71% of Amax,
Df, and the Q for this resonance. Notice that Amax is usually measured as the
amplitude above the background level. (See the diagram on the first page.)
i.
ii.
iii.
iv.
f:
Df:
Q = Dff :
Your comments:
P108 Lab 5BÑ3
2. Lower Q System:
A resonating system with low Q is one which loses energy rapidly when not being driven.
Friction or resistance can cause such an energy loss. Also, a system can lose energy
quickly if it radiates large amounts of its energy as sound. Instrument sounding boards are
examples of low Q systems with broad resonances. For a vibrating column of air in a tube,
energy can be dissipated through contact with the air outside. One might expect that the Q
of a tube would then depend on the ratio of diameter to length of the tube and hence on the
areas at each end of the tube compared to the amount of vibrating air in the tube. Similarly,
a tube with a horn installed at one end might radiate sound more quickly to the outside air,
thus changing the Q value of the tube. This mechanism is often used in the design of brass
instruments.
The easiest of these system to test is a tube with a larger ratio of end area to air inside, that
is a shorter tube where you expect faster dissipation of energy and hence smaller Q value.
Pick a short PVC tube with a length of approximately 0.15 m.
a.
Calculate the expected fundamental frequency, f1, and record amplitudes around this
frequency as you did above. Record your values in a similar table:
Driving Frequency
b.
Amplitude (mv)
Comments
On a similar sheet of graph paper as above made a plot of these new data. Calculate
the width of this response curve at 71% of Amax, Df, and the Q for this resonance.
i.
ii.
iii.
f:
D f:
Q=
f
Df
:
P108 Lab 5BÑ4
C.
YOUR COMMENTS ON THE COMPARISON OF QÕS FOR THE
TWO TUBES.
P108 Lab 5BÑ5
D. ELECTRIC RESONANCES
1.
Introduction
In the previous section we saw that acoustic resonances occur when an air column was
driven by an external vibrating source with a frequency at or near the natural frequencies of
the air column. It is possible to have resonances in electrical circuits as well. An example
of an oscillatory circuit is shown in the following diagram.
L
C
vin
R
vout
A capacitor C, an inductance L, and a resistor R are in series with the function generator.
The resonance frequency of this circuit is
f0 =
1
2p LC
As the frequency of the sine wave generator is varied at constant input voltage vin, the
output voltage vout shows a maximum at the resonance frequency f0, which is analog to
maximum amplitude we saw before for the resonating air column. The maximum output
voltage and the sharpness of the resonance, Q, are determined by the total resistance and
the inductance in the circuit. Note that the inductor has some internal resistance.
2. Measurement of the Quality Factor Q
a.
Use the provided RLC circuit with L = 0.10 H and C = 0.01mF . Connect the 2002
Function Generator to the input of the circuit. Also connect the scope to show vin and
vout. Observe the resonance without recording any data and determine roughly the
range of frequencies required to go completely across the resonance.
P108 Lab 5BÑ6
b.
Now decide upon a set of 15 to 20 frequencies over this range for your
measurements.Set vin to 0.4 V peak-to-peak and measure vout at each chosen
frequency. Be sure to reset vin to the same amplitude each time, if necessary, and
verify that it remains sinusoidal.
vin in volts
frequency in Hz
c.
vout in volts
viyt/vub
On a similar sheet of graph paper as above made a plot of these new data. Calculate
the width of this response curve at 71% of Amax, Df, and the Q for this resonance.
i.
ii.
iii.
f:
D f:
Q=
f
Df
:
The observed fact that the output voltage can be much greater than the input voltage shows
why an LC resonance circuit is so useful in radio communication circuits (transmitters,
receivers). In this experimental setup, the resistor was added to make the resonance
broader. Otherwise the quality factor Q would have been too high to make the
measurements readily. In communication resonance circuits the resistance is kept to a
minimum to provide a very sharp resonance and thus a good selectivity of stations on
nearby frequencies.
P108 Lab 5BÑ7
E. COMPARISON OF ACOUSTICAL AND
ELECTRICAL RESONATING SYSTEMS
You have now measured the resonance behavior of an air column and an electrical RLC
circuit. In both cases you used a Function Generator to excite the system at various
frequencies. You displayed input and output voltages on an oscilloscope and measured thir
peak-to-peak amplitudes. The traces on the scope looked the same, except that the air
column and the electric circuit resonated at different frequencies.
If you now put the air column or the electric circuit into a black box with two sockets, one
for the input to drive the system and one for the output to measure the amplitude of the
resonating system, you would not be able to distinguish between the two systems. Both
systems show the same resonant behavior.
However, there are differences between a resonating air column and an electric RLC
circuit. What is the major difference? Does a resonating RLC circuit have higher harmonics
like the air column?
P108 Lab 5BÑ8