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Mathematical brass

Prikkel, J. (2008) Mathematical brass. Bachelor's Thesis, Mathematics.

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Abstract

Nowadays, you just can’t imagine a society without music. Everybody knows the background music in shopping areas and on television. Also, many people of many different ages play instruments for fun or to earn their living. These musicians use different kinds of instruments, we can divide those into three important groups: strings, wind and percussion instruments. During this research we’ll try to find out several things about brass instruments. In particular we’ll try to answer the following questions: • What physical process describes the production of sound waves by brass instruments? • Which simple representation can we use to model brass instruments and what are the differences with real instruments? • What are the solutions of the wave equation analytically and via a Finite Element Method? How do these solutions differ from reality? When a musician plays his instrument, for example the trombone, people nearby the musician can hear the sound. This is because the trombonist creates sound waves with his instrument. Those waves travel through the air and everybody nearby can hear the sound. At this point a question arises: what physical process creates those sound waves and what’s the main function of the instrument during this process? This is the first question we try to answer with this paper. This question will be answered in chapter 2, by examining the most important parts of brass instruments. A simple representation of a wind instrument is just a tube. From high school physics we already know that some waves are able to travel through such a straight, cylindrical or conical tube. The selection of those waves strongly depends on the length of the tube. Musicians influence the pitch of a tone by changing the length of the tube, this is obvious when you see a trombonist playing. In this case the frequency directly depends on the length of the tube. We also know from high school that a certain tube has a fundamental tone and several overtones or harmonics. Wind musicians can use this physical aspect by tensing their lips in different ways. By using some theory about those fundamental tones and corresponding harmonics we can calculate the frequencies of these tones. But, of course wind instruments differ from straight cylindrical or conical tubes. Most instruments obviously have bends and a bell. In chapter 3 we’ll find out which tube gives the best representation of brass instruments. In chapters 4 and 5 we’ll use this representation to do some experiments with a Finite Element Method. In chapter 3 we’ll also study the wave equation. The wave equation is a partial differential equation which has solutions describing a wave. By analytically solving this equation with several special boundary conditions the solution for a sound wave through a straight cylindrical tube can be found. This is the same equation we solved in the last chapters by using a Finite Element Method.

Item Type: Thesis (Bachelor's Thesis)
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 07:28
Last Modified: 15 Feb 2018 07:28
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/8494

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