HEARING THE SHAPE OF A DRUM: AMUSINGS
G.S.R Sarma, Gottingen, Germany
Notes on Drums
On the amusing mathematical drum question of Mark Kac mentioned earlier, I’d like to add some of my simple musings. In this context we way consider the following inputs from Physical observations as pertinent to the problem.
(a) For a given stroke, the larger the size of the vibrating membrane viz., face area A of the drum fewer its vibrations i.e., lower the pitch (frequency), , of the note.
(b) Also, higher the density of the membrane material, fewer the vibrations, lower V.
(c) Higher the tension T applied to the membrane, higher the pitch, .
The area A, tension T (force applied per unit length), and the material property, density (mass per unit area) of the membrane may be taken a prior as defining the simple drum. The only time scale based on , T, and which may be regarded as an intrinsic time for The membrane. The corresponding membrane frequency is then The spectrum of frequencies of the notes produced by its natural vibrations are therefore expressible as is a dimensionless constant for the – frequency mode.
The above mathematical relation is in tune with the physical observations we started with. More precisely, the tone emitted by the membrane decreases with the square root of its area and its material density but increases with the square root of the tension on the membrane. There is a well-known verse from a famous telugu poet, Vemana, with the rhetorical questions, ‘Kanchu mrogunatlu kanakabumroguna….’ (does gold sound like bronze….).without going into value judgments and the moral behind this verse, we can say from the above dimensional analysis that gold tone is definitely more subdued than that of bronze all other things beings the same since gold density more than twice that bronze. Vemana goes on to teach us that trivial person talks tall while a sound one speak soft.
The actual value of for the frequency spectrum would in general have to be fixed by experiments or a full solution of the boundary value problem, But the above information, confirming our qualitative physical observations and quantifying them up to the multiplicative , we have gained without actually setting up and analyzing the detailed underlying mathematical model. The latter is indeed the wave equation for the vibrations of the membrane subject to the Dirichlet boundary condition of zero displacement at the periphery. This problem occupied mathematicians through the centuries challenging even Poincare and was solved in early 20th century. Weyl analyzed the corresponding eigenvalue problem in a general form and showed among other things the important asymtotic property of the distribution of its discrete eigenvalues of the spatial part of the problem, namely, that where N is the number of eigenvalues of the problem less than .
Thus the frequency parameter in the temporal part of the eigenvalue problem. namely is also accordingly related to A as well as the of our dimensional argument. It is of course not to be expected that the high notes are simple multiples of the base note. Also, several different modes are usually mixed in the actual acoustic signal produced.
The above descriptive account illustrates the symbiosis between physics and mathematics.