Background
Primer on dispersion in waveguides -- find out the basic physics
Run the LabVIEW Virtual Instrument Mode Conditions.vi. This models a travelling wave of a certain frequency and particular direction as it travels down the acoustic waveguide. It gives a snapshot frozen in time of the wavefronts of the wave, together with three reflections from the walls. Under the 'wrong' conditions, the reflections are in no particular relation to the original wave, or to each other, and they tend each other out. Under the 'right' conditions, after two bounces, from opposing walls, the reflected wave drops onto the original wave in phase -- the wavefronts are 'in step' -- and add constructively. If the oscillations cancel each other, the wave does not propagate down the channel. If the reflections are constructive with each other, the wave does propagate down the tube.
Play with the angle, at a fixed frequency; play with the frequency, at a fixed angle.
- What determines the range of frequencies, for constructive interference at a fixed launch-angle (angle of incidence)?
- What determines the minimum and maximum angles possible, for a propagating solution, and fixed frequency?
- How do the minimum/maximum angle change, as the frequency is raised or lowered?
Set up the model to give a constructive-interference solution (a propagating wave).
- After two reflections, how many cycles does your reflected wave lag behind by? It could be 1, 2, 3 ...
- Measure the distance on the computer monitor from wavefront to wavefront at the wall of the guide (the effective wavelength in a waveguide) rather than perpendicular to the wavefronts (the free-space wavelength).
- Change the frequency slightly; then touch-up the angle of incidence to keep the wavefronts in phase -- the propagating wave at the new frequency. Now again measure the effective wavelength in the waveguide. Repeat until you have a dataset, and then plot it as wavenumber k on the x-axis, and angular frequency on the y-axis, as in Figure 1 of the Primer on Dispersion.
Hands On
The animated model above will have given you a pretty good idea of what you're looking for inside the waveguide -- now do it for real.
- Use the LabVIEW Virtual Instrument PolyDriver.vi which sends digital waveforms of your design to a digital-to-analog converter on the National Instruments card in a PCI slot in the computer. That converter will drive signals on AO0 and AO1 on the break-out panel next to the computer. You can use BNC-to-bananaplug cables to connect those outputs to the blue patch-box which connects to three speakers at the end of the acoustic waveguide. How you drive these speakers will determine what mode you can excite, in the waveguide, depending on frequency and launch-angle.
- Start by driving all three speakers identically -- patch one output into speaker A, and then use jumpers to put the other speakers B & C in parallel. Call this mode (+,+,+), where the ordered triple stands for the speakers (A,B,C) and the + signs mean that all have the same phase and amplitude.
- Set PolyDriver.vi to drive a sine wave oscillation; note the frequency and amplitude controls. Start at 3 kHz frequency and about 750 mV amplitude. Monitor this signal on Channel 1 on the digital oscilloscope, using it to trigger the scope.
- There is a travelling microphone on a tiny cart that can move up and down the acoustic waveguide, to sample the oscillations at different places. It is labelled Rover and it it connected to a tape measure that will let you push and pull it along inside the waveguide and measure its displacement in the z-direction. Use the microphone bias-voltage box to power that microphone and to connect its output to Channel 2 on the digital oscilloscope. (Unplug the battery when finished)
- With Channel 1 as a reference, you can see how the signal from Rover will help you find the effective wavelength inside the waveguide. As you drag Rover down the waveguide, you'll see the measured waveform slide in time, as the relative phase changes. When it has slid a whole cycle, and looks like the waveform you began with, then you have moved Rover through one whole waveguide wavelength.
- Make a series of measurements at decreasing frequencies, down at least to 500 Hz. Plot this as waveguide wavenumber k on the x-axis, and angular frequency on the y-axis.
- Now drive a different mode: use PolyDriver.vi to drive the second analog output channel AO1 out of phase by pi, or 180 degrees, relative to AO0 -- use the multiplier "1" to do this. Then wire the speakers to drive in this pattern: (+,0,). Repeat the experiment you made for (+,+,+). Can you still get to 500 Hz? Add your new data to the plot you made above.
- Arrange again, to provide a drive which is (+, , +), where the double minus sign means opposite phase and twice the strength. Repeat your measurements, again going to as low a frequency as you find feasible.
Analysis and Interpretation
- How do your observations relate to the animated-model 'numerical experiment' you made earlier? Note any differences that may arise from different dimensions of the waveguides.
- How do your observations compare to the analytic theory formulae in the Primer on Dispersion? Be quantitative -- overlay experiment with theory if possible. (Note that the lab computer includes StarOffice, which includes a spreadsheet function completely compatible with Microsoft Excel; it is also possible to plot formulae in Kaleidagraph, on the lab computers. Lastly, Kaleidagraph can fit to your experimental data curves of the form theory dictates.)
- How have you treated experimental error? How much should be expected of your experimental data? Within this expectation, does it agree with theory? Why, or why not? Is there systematic or random error that you've neglected?
- What is the nature of the different modes driven by (+,+,+), (+,0,) and (+, , +)? Why do these different drives give different modes? Can you construct a dot-product between these ordered triples that helps to explain?
Going back to experiment with revised/refined ideas
- If you have new questions, don't hesitate to go back to the experiment to answer them
- There are microphones set into the aluminum channel that can be moved transversely across the waveguide. Does this give information about the differences between different drive-modes?
- Or if you prefer, there is an array of 8 microphones on a different travelling cart, Rover 2, which can be fed into 8 channels of analog-to-digital conversion on the National Instruments multifunction card. The LabVIEW virtual instrument Rover2.vi can help you [[under development]]
- What is the nature of the different modes driven by (+,+,+), (+,0,) and (+, , +)? Why do these different drives give different modes? Can you construct a dot-product between these ordered triples that helps to explain?
The primer will have introduced you to the difference between phase velocity and group velocity, if you haven't already seen it in lectures in physics or engineering. Click here for an animation which shows a pulse travelling by at the group velocity, with the carrier wave travelling inside the envelope at the phase velocity.
How can you measure phase velocity, using Rover or the other microphones? You'll need to track a wave-crest, and follow it; see what time it takes to travel a measured distance.
Measure phase velocities for different modes, and for different drive-frequencies. Use the dispersion relations you found in Section I, above, and determine from them the phase speed which follows, for your mode and frequency. How does your measured value compare to the value you deduce or infer from your dispersion relation measurements? How does your measured phase velocity compare to predictions from analytic theory, starting with the dispersion formulae in the Primer.
Track your error estimates in all cases, and see whether your measurements give comparable answers, within estimated error.
How can you measure group velocity, using Rover or the other microphones? Hint: PolyDriver.vi lets you drive not just a sine wave, but a pulse of a certain carrier frequency. And you have microphones at different distances. How fast does the pulse travel, for different carrier frequencies? Be careful to use at least 20 cycles in your pulse -- if you go shorter, you'll have larger and larger freqency content in your shorter and shorter pulses, so you will have big error bars in actual frequency, and weird effects which are part of the next section. The bandwidth of your pulses, shown in PolyDriver.vi, will be part of your error-bars on the frequency.
Start at frequencies well above the cutoff, where the dispersion-curve is nearly linear, and then work down toward the cutoff frequency. What happens when you try to propagate a pulse that has a center-frequency below the cut-off frequency? How do you explain this? Can you quantify the effect, by measuring the amplitude of your pulse as it propagates down the waveguide, for different frequencies? Can you connect this to results from the simple analytic theory in the Primer?
Compare your measured group velocities, for different centre-frequencies and different modes, to the value which you can determine from your measured dispersion relations (how is that found?). Then compare also to values found from analytic theory, for the dimensions of the lab waveguide. As always, track your error estimates, which will serve as a prediction for how closely you might expect experimental results to match theory, or the values expected from other experimental results.
CoolEdit 2000 is software that does a very nice job with the SoundBlaster Card installed -- the input jack is in the back of the computer, and a stereo patch cord leads out, to make it easy for you to connect without fussing behind the computer. CoolEdit 2000 can let you look at the time/amplitude waveform, or let you visualize the spectrum evolving in time, in a time-frequency analysis. When opening a new file, to record into, use a digitization rate well above the Nyquist criterion, which is twice the highest frequency in the pulse spectrum; I use 44kS/s.
You know, from the dispersion relation, that the energy in different frequencies will travel at different group velocities. If you make a short pulse (only a few cycles, or even less if you wish to make a half-cycle pulse or an impulse), it will all at once contain a substantial range of frequencies, each travelling at it's own speed. So the pulse will unravel, or stretch out, as it propagates. Look at the shape of your pulse on the oscilloscope: as it was generated; as it appeared at different points along the waveguide using the fixed microphones.
Record the waveform from different microphones into CoolEdit 2000, and then use the menu to convert from waveform to spectrogram. Explain the shapes you see, in terms of the dispersion relations you measured above, or the analytic formulae. What is the lowest group velocity you see? You may want to make a series of rough measurements from the spectrogram and from them reconstruct the dispersion relations for each mode.
The experiment is ready, but the writeup is not -- ask your T.A. or Professor Marjoribanks for suggestions. Or do you need any? Hint: CoolEdit 2000 can let you record a short pulse launched in a mode, near the band-edge cutoff, and then time-reverse it to reverse the sign of the frequency chirp, and play it back into the driving speakers. This pulse will compress, as it travels, and reach optimum at the distance at which it was itself recorded.
Last revised: 12 March 2003 -- rsm
