An Undergraduate Physics Project About Music: Week 1

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:

Week 1
Week 2
Week 3
Week 4

Over the next ten weeks I’m going to keep a rolling record of my work on my final year physics project as part of my undergraduate degree, I’ll document it all here so I can keep my work on track and aid in writing presentations and reports at the end of the year.

What is my project though? One of my first tasks was to nail down on what exactly I’m trying to do. What questions do I want to answer? The project was given to me as a research into tin whistles and how they work. Can I make a model that can explain the sound of a tin whistle? Can I use this model to infer how tin whistles will behave given various pieces of information such as their diameter and length? And lastly, what can I learn about how tin whistles are made that defines their sound output?

How should a tin whistle be modelled?

This week I researched several papers that would help in creating this model. The most useful papers compared tin whistles to transmission lines. I then went to explore what exactly a transmission line was and how it worked. This video proved to explain them very well:

Essentially a transmission line is designed to carry alternating electrical current at high enough frequencies that their wave nature makes them more difficult to manage. An ordinary cable would radiate the wave’s energy and the signal would be lost. In a transmission line two cables, separated by a dialectric carry the signal and slow it down using capacitance and inductance, which help to ensure the signal isn’t radiated away.

The reason this is useful is the equations for transmission lines which can be seen on it’s wikipedia page, model the change in voltage and current through the length of the cable. Voltage can be analogous to pressure and current to velocity, and so knowing this we can use the same equations to model a tin whistle. Where we have equations that tell us what the pressure and air velocity is along the length of the whistle.

I did find some information regarding modelling the holes in a tin whistle as branches in a transmission line, but this got quite complicated and my objective right now is to find the simplest model possible to try explain the workings of a tin whistle.

Studying notes

In reading about these equations I wondered how close a tin whistle was to producing a sine wave, or pure wave in a particular frequency. I opened up Audacity, a free sound editing program and created a tone in a given frequency (329.63Hz, E4) and played it. It didn’t sound anything like a tin whistle.

I then went hunting for reasons why musical instruments sound different to pure sine waves of the note they are supposed to be playing. Discussion with peers and my supervisor revealed that notes often contain many other frequencies that are created unique to the instrument. It’s the combination of all of these that create the actual sound. A friend had a music editing software on his computer and had some note samples of different instruments, so I loaded them up and did Fourier analysis to break apart their frequency make up. Fourier analysis, in simple terms, means breaking up a signal into each of the frequencies that contribute to it. We can plot these against the strength of each frequency and see how the sound is created. Here are some of the results:

Frequency spectrum of a flute
Frequency spectrum of piano
Frequency spectrum of brass instrument

It’s very evident from this that notes are very complicated mixes of frequencies and so if my model is to recreate the sound of a tin whistle I need to work out which harmonics past the fundamental (1st harmonic) are present in the sound. Harmonics are tones that are exact fractions of the fundamental frequency of vibration in the instrument, in the case of a tin whistle the vibration of the air in the whistle.

Next week

Now that I have a simple basis to work from and some understanding of the expected frequency content of a whistle I can attempt to solve the transmission line equations, in their most simple form and observe their behavior when simulated. I will need some method of solving differential equations within a defined boundary, known as a boundary value problem. Further reading will also be done to supplement my surrounding knowledge of acoustics and transmission line theory.