Mathematics and Drums

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Mathematics and Drums

• W. Y. Chan and Shane Mizicko
• Math Club Presentation
• Johnson Hall 219
• November 7, 2007

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• The sounds we hear everyday arise from variations
  in air pressure.
• The rate at which an air pressure pattern
  occurring again is called frequency.

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Question 1

• Suppose that a new music note has an octave
  higher than the original, the ratio of the
  frequency of the new note to the original is
• 2
• 4
• 8
• The answer is (a).

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• Guinness World Record Biggest Drum, The Ireland
  Millennium Drum, designed by Brian Fleming and
  Paraic Breathnac, has a diameter of 15 ft. 6 in.
  and a depth of 6 ft. 3 in. It was first played at
  the St. Patrick's Festival in Dublin, Ireland's
  Millennium Festivals.

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• The relation of mathematics and music has been
  studied by a lot of scientists such as
  Pythagoreans, Mersenne, Galileo, Fourier, Ohm,
  and von Helmholtz.
• Fourier discovered that a repeated wave pattern
  can be broken up into a series of sine or cosine
  functions (cf. The Math Behind the Music by
  Harkeroad).
• Our discussion below is similar to the method of
  Fourier series, we will write a repeated wave in
  terms of Bessel functions.

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Friedrich W. Bessel (1784 1846)

• He was a German.
• When he was young, he was interested in
  navigation problem of finding the position of a
  ship at sea. Later, he became an astronomer and
  mathematician.
• The Bessel function was used to study planetary
  motion (cf. Introduction to Bessel Functions pp.
  122 124) (cf. http//www-history.mcs.st-andrews.
  ac.uk/Biographies/Bessel.html)

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Question 2

• Given two functions F(x,t) and G(x,t) below, if
  we multiply F(x,t) to G(x,t), determine the
  coefficient of t0 and t1 in terms of x.

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The coefficient of t0 is

• The coefficient of t1 is

• In fact, F(x,t) e(xt/2) and G(x,t)
  e(-x/(2t))

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• The coefficient of tn is

• Jm(x) is the Bessel function of the first kind
  with the order m 0, 1, , n, Also, if m -1,
  -2, , -n, , then

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• Base on the above formula, we see that Jn(x) is
  an even function when n is even, it is an odd
  function when n is odd.

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• The graphs of the functions J0(x), J1(x), and
  J-1(x) are

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• Bessel functions can be used to describe the
  vibrations of a uniformly stretched uniform
  circular membrane.

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Kettle Drum
(0,0)
(x, y)
r
a
a radius of the drum
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Vibration of a stretched circular membrane
Source James Stewart, Calculus Early
Transcendentals, 2005.
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• If there is no external force applying on the
  membrane and its motion is symmetrical about the
  origin, the displacement z of the membrane at the
  point (x, y) will satisfy a partial differential
  equation.

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• where z is the displacement of the membrane,
  c2 is a positive constant depending on tension
  and mass density of the membrane, t is time, and
  r is the distance between the point (x, y) to (0,
  0). The solution of this problem subjects to the
  homogeneous boundary condition (i.e. z is equal
  to zero), and nonzero initial condition.

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• Assume that c 1, the solution to this partial
  differential equation is given by

It is a complicated expression.
where an bn and ln are constants.