Physics 434
Autumn 2013
Module 4
Impulse Response of the Kundt’s Tube and Analysis by
Fourier Transformation
Introduction
Previously, you developed an automated system to characterize the response of the
Kundt’s tube to sound waves of known frequency. You generated a transmission
spectrum of the tube by scanning the frequency of sound sent to a speaker at one end of
the tube and measuring the average of the absolute value of the time varying amplitude of
the sound recorded for a known time interval using a microphone at the other end of the
tube. You noted a series of peaks in the spectrum occurring at frequencies (very roughly)
approximating the ideal fn = n (cs/2L), for a tube closed at both ends, where L is the
length of the tube, cs is the speed of sound and n is an integer.
Now, you will reconfigure the Kundt’s tube to measure its response to an impulse. You
will observe that the tube exhibits a “ring down” behavior, something akin to the sound a
bell makes when struck with a sharp blow. Fourier transformation of the ring down will
generate the same spectrum of the Kundt’s tube.
The impulse response experiment is a specific case of the stimulus response experiment.
Here, you will generate the impulse using the timer-counter facility of the DAQ board to
produce a logic pulse of software selectable time duration. Then, at the end of the pulse
you will acquire the sound-wave (pressure) response of the system using a microphone
connected to the ADC. The response for the data acquisition is given in Figure 1, for an
ideal tube, with perfect reflection of the pulse, and a slight decay. Because the
experiment is over so quickly, and so readily repeated, you will develop a facility in your
VI to add many experiments together to improve the signal to noise ratio.
Figure 1. Response from impulse in ideal tube
Wiring Instructions
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Autumn 2013
1. Connect the general-purpose counter #0 output (PFI 12) pin 89 on the breakout box)
and the digital ground (pin 90) to the speaker input of the audio amplifier box.
2. Also connect pin 68 to pin 73, TRIG1. (For the triggered mode.)
3. Connect the two analog inputs ACH0 (pin 68) and ACH8 (pin 34) to the microphone
amplifier output as before.
In addition connect your oscilloscope to the experiment. Connect channel 1 to the output
of the digital counter where it inputs to the speaker amplifier. This will allow you to view
the pulse output. Remember that it is a logic signal that goes between 0 and 3.5 - 5.0
volts. Connect channel 2 of the oscilloscope to the output of the microphone amplifier
so you can view the ring down. Arrange to trigger the sweep on the ascending portion of
the excitation pulse. Use the normal (not auto) position, so you trigger only on a pulse.
Set your time base for 20 msec/cm. By this means you should be able to view the output
on the oscilloscope simultaneously with the output on the computer. Use the test VI for
this exercise.
Initial study with the test VI
A VI that demonstrates the basic pulse generation and acquisition is available,
ImpulseTest.vi. You may use it as a starting point. Note the following:
1. Two independent while loops; one generates the pulses, the second collects and
displays the data
2. Front panel controls: pulse duration; No of data points, Scan rate, cycle delay;
This VI illustrates a small part of the capability for triggered, automatic data acquisition.
The front panel controls are set be default to reasonable values to allow you to see the
waveform (and compare it what you see on the oscilloscope). It moreover allows you to
probe the limits of the AT_E board and compute interface, which you should note in your
documentation:
1. What is highest scan rate supported?
2. Is there a limit on the number of samples? Note that it is more than the 4K size of
the ADC FIFO.
Enhancement to allow summing and frequency analysis
When you are satisfied that you understand the functioning of the test VI, modify or
replace it with a VI that (at least) allows the following:
1. The acquisition loop also adds the data to average out noise. A shift register is the
best way to do this.
2. There is a control to specify the number of pulses to average.
3. When the loop ends, the averaged waveform is sent to a FFT VI.
4. The FFT output of amplitude vs. frequency, up to the Nyquist frequency only, is
displayed in a labeled graph. That is the x-scale must be in frequency units.
(Slight) bonus: Clear the graph until data is available. It can be shown that the
frequency step is given by f = fs/N, where fs is the data acquisition rate and N is
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the number of data points in the ring down waveform. The frequency at the ith
point in the spectrum is i*f.
5. Writes out the final frequency spectrum to a spreadsheet file for analysis.
Experimental Procedure
The power spectrum generated as a graph on your VI may not meet your expectations.
This is because a Fourier transformation yields a function that is symmetric about the
Nyquist frequency. The first half of the spectrum corresponds to the spectrum you
measured in before.
Once the system is running and you have good data from the standard settings, try the
varying the following parameters of the experiment:
1. What is the effect of the pulse width on the experiment? Try settings like 0.1, 1.0,
and 10 msec. Report your observations in your documentation.
2. What is the effect of the acquisition rate? Try settings like 1000, 10000 and
100000 points per sec. Also note what the minimum buffer size and scan rate is
needed to describe the response up to 2000 Hz.
3. Try the effect of signal averaging on the appearance of the power spectrum. For
this experiment, reduce the gain of the audio amplifier until the signal is barely
detectable
Finally, run the analysis VI from Module 3 and compare some of the resonant
frequencies.
Submit your fully commented VI as before, saved with a sample output, and with the
documentation section containing answers to the questions above.
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Autumn 2013
Appendix
The fitting formula used here and in module 3 assumes that the response near each mode
corresponds to a damped, driven oscillator, for example a series LRC circuit.
The basic differential equation is:
 d 2 r d
2
 2 
 r  f (t )  g (t )
Q dt
 dt

where r is the resonant frequency and Q the dimensionless damping factor. For an LRC
circuit one has r2 = C/L, and Q  LC / R . The driving term is g(t), and the response
function f(t). Given a g(t), one solves this equation for f. The usual technique is to assume
a single frequency for g, g=exp(-it), for which the solution is
1
exp( it )
    r  i r / Q
The amplitude squared is then just the magnitude squared,
1
2
f 
2
2
2
  2   r   r / Q 
or, normalizing to 1 on resonance,
1
A( ) 
2
2
 2

2 r  

1  Q 

 r 
f (t ) 


2
2


 1  Q 2  r /    /  r 
A possible LabVIEW formula node is then
2 1 / 2
where x is the frequency (either  or f=/2), a0 the normalization factor, a1=r or
r/2, and a2 =Q.
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