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Can you hear the shape of a drum?

Q) Areas and perimeters again! My Year 9 students are bored and so am I. How can I make it interesting?

A) Mathematical research has been done that I think would make a great basis for exploring perimeters, areas and volume and would give practical exercises real purpose. The story begins about 200 years ago. Napoleon Bonaparte was fascinated by an experiment shown to him by scientist Ernst Chladni, which demonstrated "visible" sound waves caused by drawing a bow across the edge of a metal plate, which had been sprinkled with sand (pictures of Chladni patterns can be found on the internet). Napoleon offered a prize of 3,000 francs to anyone who could provide an explanation of how the shape of the patterns and the sound related to the shape of the metal surface.

The investigations continued and in 1966, mathematician Mark Kac published a famous paper in American Mathematical Monthly, entitled "Can one hear the shape of a drum?".

I contacted Professor Steve Zelditch at the John Hopkins University, who is a member of an American research team boasting a grant of $975,298 to study this topic. He says: "Hearing the shape of a drum is something 14-year-olds can understand and find interesting. It is known as an 'inverse problem'.

Trying to determine the shape of a drum from its sounds is very similar, mathematically, to figuring out what a star is made of by looking at the light that it emits. The 'hear the shape of a drum problem' is one of the simplest and most concrete inverse problems to state, but very difficult to solve - and it is still mainly unsolved. Everyone is familiar with drums, while few know much about stars.

"I've tried to illustrate the problem in a classroom, but I ran into the problem of finding ready-made drums of different shapes to bang on. You can easily find circular drums, but what about drums in the shape of an ellipse or a more complicated shape? My grant is about trying to understand how the shape of the drum affects not only the sounds it makes, but the shapes of its vibrations."

The application of the findings from the research is likely to be useful in medicine, earthquake research, in fact anything where you are using information that occurs externally - such as the emissions from stars - to gain information about an object's interior. In the same way, pupils can investigate how the shapes and dimensions of drums influence their sounds.

There is room also for extending the bright mathematician in your class, sending them on a fact-finding mission. Is it the perimeter, surface area or shape that influences the sound most?

"Hearing the shape of a drum" sent me touring the internet for background information and suitable materials to play with. You can see my efforts here.

I tried Correx and cling-film, but found Correx was too bendy and the cling film too delicate. Technology teachers may have some suggestions to get around this. The most successful surface I found was a black bin-liner attached with Sellotape. Tension can be achieved by carefully stretching the bin-liner.

I began by making a regular hexagonal drum, which involved interior and exterior angles, and using Pythagoras to lay out the base economically on the board. I made the height of the sides of each of the drums the same, at 12cm.

Next, I wanted to keep the surface area the same as the hexagon for the triangular and square drums. This provided a "live" staged problem. I then looked at keeping the perimeter of the shapes the same. I wanted to play some more, but ran out of time!

Professor Kate Okikiolu is also well known for working on hearing the shape of a drum. "The sound of a drum changes as its shape changes. Listening to the drum very carefully tells me how to change its shape to make it sound like a round drum and when it sounds like a round drum, it is indeed round!"

Animations of a vibrating circular drum can be seen at:l www.kettering.edudrussellDemosMembraneCircleCircle.html

* www.maa.orgmathlandmathland_4_14.html


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