HEARING THE SHAPE OF A DRUM: MUSINGS
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. Usually, an equivalent physical problem including practical observation lease to the solution of a more general mathematical problems, at least as special case, in its own turn leading to further developments both in mathematics and physics.
As an interesting aside, I may mention that I have thought of this problem for the first time ever, during my reminiscing mood for the previous contribution on SMS. After I gathered the above pieces of information, I got a bit curious to know if something simple like this is already in the books. It was almost a sure bet I could feel. Lo and behold, I see in science & Music by James Jeans (Dover, 1968), on page 64, the following Mersenne's Laws stated.
I. When a string and its tension remain unaltered, but the length is varied, the period of vibration is proportional to the length. (The law of Pythagoras).
II. When a string and its length remain unaltered, but the tension is varied the frequency is proportional to the square root of the tension.
III. For different strings of the same length and tension. the period of vibration is proportional to the square root of the weight of the strings.
The apparent difference between the one-dimensional strings case and the two-dimensional membrane case is not a momentous one and we recognize that the membrane equivalents of the Mersenne's Laws following from the formula we have derived above, noting that square root of the area is a length in fact , a real string and real membrane are both cylinders and are neither a zero-thickness line nor a zero-thickness surface. Also it may be noted that the 'mathematical membrane' problem is not restricted to two dimensions and is generally formulated for an arbitrary number of dimensions . Nor are 'Mathematical drums' necessarily like anything we would normally recognize as such [cf. C. Gordon & D. Webb, you can't hear the shape of a drum, American Scientist, Vol. 84, No. 1. pp. 46-55 (1996)]. John Milnor (Fields Medalist 1962) proved in 1964 the existence of two noncongruent isospectral sixteen dimensional tori for the Laplace Beltrami operator, thus settling Kac's generally query.
SMS made unique and valuable contribution to this class of boundary value problem as well. Work on the generic 'membrane problem' by T. Carlman in Sweden was used as the starting point for his famous work with Pleijel in Princeton. Hermann Weyl pointed out that the work of SMS was independent, more general and shed new light on the whole problems. SMS also developed in this context a generalization of the Epstein zeta functions which he employed in the above work.
After Weyl's pioneering advance on the Inverse Spectral Geometry problem by connecting the area to the asymptotic eigenvalue spectrum of the 'Mathematical membrane' problem, work of SMS, Pleijel, Kac, Mckean Jr, Singer, Patodi, went further to identify also the perimeter and curvature of the boundary from the asymptoc spectrum. Since the membrane eigenvalue problem is formatted for the Spatial part involving the Laplace operator and the boundary domain, the focus here is only on the geometrical aspects: area, perimeter and curvature of the boundary domain. The subsumed features of the full problem related to the density of the membrane and the tension on it are brought to light through the dimensional considerations above. Thus we see that Mersenne's Law II and III are also sunstained by the same formula that checks with Weyl's asymptotic results, namely, Mersenne's Law I. For latest development e.g. on nonuniform density distribution etc., see H.W.P Gottlied (2004) [Inverse Problems, Vol.20 (1), pp. 155-161] and the interesting topical surveys cited therein.
After so many words on theoretical drums it may be refreshing to hear a practical application. An example of the Pythagoras -Weyl formula bearing on real drums seems appropriate.
Figures 1 and 2 show a drum-combo for playing North Indian style Raga Music. The maestro (Figure 1) tunes the set of drums to a set specified pitches. Swaras, to render various musical compositions. This is quite a difficult percussion art form known as Tabla Tarang (Tabla 'waves'). It is still practiced on the Indian subcontinent and has also spread abroad. Figure 2 illustrates a seven piece set where in the tone goes from the lowest pitch on the largest faced drum to the highest pitch on the smallest faced drum tuned appropriately). Of course, there are further finer details of the design affecting the musical quality like the wooden resonator box, the layered structure of the membrane and the black central patch. This patch made of a special mix brings out the tangy tone. the strokes on the patch, Karine are particularly delightful punctuations for the audience to look forward to on the ancient Indian instrument, Mridangam, a two-sided drum, during a South Indian style classical recital of music or dance. We see also that a Mridangam, as an accompaniment for vocal music, has to be larger in size for male artists due to their lower Shrithi (pitch) levels. As a lay connoisseur of music, that’s as far as I should let my fancy play on this example. I'd only add that, thanks to Kac, [Amer. Math. Vol 73 (1966), 1-23]. we are reassured that a circle, such as the face of a Tabla or Mridangam is uniquely associated with its eigenvalue spectrum, i.e., if you hear the frequency spectrum of a circular membrane then it had better be one.
Mersenne on Primes
Back to the man Marin Mersenne who is indeed a very interesting personage to get to know. From the literature we learn that he was a French mendicant-order monk by confession but remarkably multifaceted talent by profession: mathematician, natural scientist, philosopher, theologian and publicist. He formulated the above mentioned laws on stringed instruments from several empirical observations and recorded them in his treatise Harmonie Universelle (1636). He was a well-heeled traveler and a good friend of Descartes. Desaurges. Fermat, Galileo, and Pacal to name a few contemporary celebrities. Mersenne was mentioned along with Fermat and Pascal by Hilbert in the Introduction for his epoch-making lecture on mathematical problems' before the International Congress of Mathematicians. Paris (1900). These Hilbert problems have been engaging mathematicians even into 21st century.
Mersenne was veritably a mobile Journal himself, publicizing the work of contemporary savant along with his own work and discoveries. He was into Number Theory of the times and is known perhaps for his conjecture-formula for prime, namely, N(p)=
where p is a prime. He asserted in the Preface to his work Cogitata Physica- Mathematica (1644), Thoughts on mathematical physics' we might now say, the formula is good for p=2,3,5,7,13,17,19,31,67,127, and 257 and that N(n) is otherwise a composite number for all n<257.>not true for p=67 and 257. Actually, it turns out however, it is not true for P=67 and 257 but on the other hand hold for p=61,89,and 107. So, his prediction from the 'formula' was not perfect. Primes are, after all, not so' predictable' and it took centuries of effort to check this one out.
A personal anecdote here. I had the privilege of using in 1956 the then widely acclaimed experimental main frame computer ILLIAC (IIlinois Automatic Computer) II at the university of IIIinois, Urbana on a rather simple problem of experimental data correlation on super conductivity, a la training cannons on sparrows. This famous computer was used in 1963 to produce the 21st and then largest Mersenne prime number N(p) for the high exponent p=11,213 with N(p) having 3,376 digits taking 135 minutes on ILLIAC II. It was such a great sensation in the 'Prime World' and the mathematics and Computer Science Department in particular that the U.S. Post Office (located, by a curious coincidence, also in the Altgeld hall housing the Department) was persuaded to place the postage meter stamp: 
is prime on all outgoing envelopes from that post office as shown in Figure 3 below.
The latest record set up on 15th May 2004 is for the 41st Mersenne prime with the now reigning exponent p=24,036,58, yielding 723,573,314,471,465 digits for N(p) which took for its confirmation 14 days on a 2.4 GHz Pentium 4 PC. There are websites on these discoveries of Mersenne primes, historical anecdotes and open problems. See e.g. the interesting site www.homes.uni-bielfeld.de/achim/mersenne.html maintained by Dr. Achim Flammenkamp.
