An Undergraduate Physics Project About Music: Week 4
This is part of an on going series of articles documenting my final year undergraduate physics project. The objective of the project is to learn about the physics of tin whistles and create a model that can predict the frequencies for given configurations of the whistle.
Contents:
Modelling Holes
The first priority this week was completing the modelling of open and closed holes in my simulation. This was completed and the results are quite interesting to play with. I setup the system such that partially open holes can be modelling as well. At the center of the hole the pressure plot changes suddenly and does not return to it’s original solution. Here’s some examples:
Much more can be investigated into this and I’m excited to compare it to real data taken from a recording of a tin whistle
Rewriting of code
I spent a large amount of time this week rewriting my code as it’s become a mess of experiments and tests. The refactoring has made the code faster and easier to modify such that when I start plotting different variables and experimenting with the behaviour of the model I do not need to break large chunks of my code.
In the interest of not losing work I’ve also now put my code on a GitHub repository, both so I can keep track of my edits, but also so others can benefit from this work. It can be found here.
Questions to be answered
I wondered this week how the hole size and placement effects the change in note when the hole is opened. Furthermore it may be worth investigating how the length and radius of the whistle effects the note. I want to extend my code to find the fundamental frequency for a whistle of given dimension and remember it, then change the dimensions slightly and try find the fundamental again.
Plotting this would give me some idea of the relationship between the whistle dimensions and notes. This could be another sine curve like it was last time, but either way it may be useful in figuring out the dimensions of a whistle that I would like to target a specific note. One of the final goals of this project was to design a whistle in an unusual key and I feel this information would be helpful in that task.
Failings of the Model
Last week I mentioned that the model provided no information on the amplitude drop in higher harmonics and that this was an issue because it prevented me from finding the more complex composition of the sound.
To confirm that this was true I allowed my code to find frequencies over a massive range and print the maximum pressure each time it found a solution. The results from this confirmed that the pressure never drops. Solutions at extremely high frequencies (past human hearing) had the same pressure amplitude as at lower frequencies. This is disappointing but it does appear an intrinsic part of the model.
Since I’m attempting a solution for lots of different frequencies, I have to start with an initial value for pressure and flow rate at each attempt, and all I’m changing is the period (frequency) of the standing wave, of course the amplitude will never change.
It means however that the amplitude is dependent on my initial flow rate, which I’m guessing. It’s currently taken as one for convenience. The resulting pressure and flow rate amplitude is thus never physically valid.
Another option is to rearrange the equations so as to single out either flow rate or pressure, and then solve it analytically for a general solution. Imposing boundary conditions will single out frequencies that satisfy the conditions. But this will not provide us with the information either. This model is interested in a standing wave inside a tube. And in that case the pressure amplitude does not matter as any amplitude will still satisfy the conditions at a specific frequency.
I see no way around this without using an entirely different model. My immediate thought that I would like to try after analysing recordings of real tin whistles is to try and find a pattern to the ratio of amplitudes between different harmonics. Using this information we may be able to recombine multiple harmonics coming out of the model using amplitude ratios found in real tin whistles in an attempt to artificially recreate the sound.
I think the issue arises because in a real tin whistle blowing harder on the fipple causes more turbulent air that creates smaller vortices that excite the air to a higher frequency. Considering this, the flow rate effects frequency, in my current model, it does not. I think therefore the only way to approach this is to analyse how turbulent air is excited at the fipple depending on the flow rate of air.
Next Week
Next week I would like to further explore this analysis of the fipple and it’s interaction with oncoming air. I believe it will reveal more about how over-blowing works, and how multiple waves are created inside the whistle. I assume the air is excited in a wide range of frequencies, but most are lost through destructive interference. However I still don’t know why blowing harder would increase the dominant harmonic in the note.
I would also like to employ some stability analysis on the general solutions to the coupled wave equations. If I could vary coefficients and plot phase diagrams I may be able to get a feel for the entire space of solutions for the equations, which would enable me to predict it’s behaviour in a wide range of circumstances.
Lastly I must get on to recording real tin whistles and taking dimensions from physical samples so I can verify the model is predicting frequencies accurately. The spectral analysis of recordings may also help in understanding the composition of notes.