An Undergraduate Physics Project About Music: Week 2
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:
The task for this week was to begin solving the equations describing waves inside a tube, directly converted from the transmission line theory I researched last week into parameters suitable for an analysis of air pressure waves.
Boundary Value Problems
I have two differential equations that relate the way pressure and volume flow rate (speed of the air) change over distance. That distance being the distance from some reference point on the tin whistle. Or in a more intuitive sense, the position along the whistle. I know two things about the solution currently. I know that for a tube with two open ends (a whistle) the pressure will be 0 at either end, and the flow rate will be non zero. That is air is flowing in and out at either end.
Usually when this sort of problem is known as a boundary value problem, since we know the answer at the two boundaries of the area we want to solve in. Matlab, the system I’m using for running code, has built in solvers for these kinds of problems. I tried using them and immediately found they weren’t particularly helpful.
Shooting method
The problem is that one of the parameters in my model is the frequency of the pressure wave inside the whistle. I’m trying to find what frequency will solve the problem. And so if I guess a frequency, I may not find any solution that matches my given boundary conditions.
The solution to this is the shooting method. We start with the boundary conditions on one end of the whistle, and progressively figure out the solution step by step across the length of the tube. We can do this because we know the rate of change. We have the starting position, so we can see where it’s likely to go next given the rate of change. We continue doing this across the length of the whistle. If the pressure at the end matches what we want, we know the frequency is correct. If not we need to try another frequency. This carries on until we find the right frequency.
This worked fantastically well and I quickly got plots like this:
These graphs show the pressure inside the tin whistle at all points along it’s length. The frequency of the second wave is just about exactly double the first. Which is exactly what we’d expect out of a standing wave like this. Furthermore, if I change the boundary conditions to close one end, we get plots like these:
This time the second harmonic is 3 times that of the first. We’re only getting odd harmonics this time, which is again exactly as standing wave theory predicts.
Next Week
Although I’ve had great progress this week there is still lots to do. There is a correction factor that needs to be taken into account at the end of the whistle. I’m also curious about the pressure at the end of the whistle as we vary frequency during our shooting method. How does that change, and will it tell me anything about the system. I also would like to start modelling the holes along the whistle and affecting the note by closing and opening various holes.