Harry E. Rauch

On the Poincaré Relation

This chapter discusses the problem of obtaining relations among the periods of the differentials of first kind on a compact Riemann surface. More precisely, if S is a compact Riemann surface of genus g together with a canonical homology basis γ 1 ,…, γ g , δ 1 ,…, δ g , then the vector space of holomorphic differentials on S is g dimensional and a basis can be found for this space φ 1 ,…, φ g so that ∫γj φ i = δ ij where δ ij is the Kronecker δ. This basis is uniquely determined and the matrix π = (π ij ) is formed, where π ij = ∫ δ j φ i , which is called the period matrix. It is well known that π is a symmetric matrix and that Im π is positive definite. The problem is to find the relations among the elements π ij of the matrix π.

CRUDE THEORETICAL MODELS OF JET‐DRIVEN WIND INSTRUMENTS AND THE EDGETONE

I f one makes the request, “please tell me how the trumpet or clarinet works, but do it quickly, simply, and without a lot of mathematics or other theory,” then a ready response is found in the classic work of Lord Rayleigh’ (

Theta functions with applications to Riemann surfaces

322k). If a little more detail is desired at the price of a little matkmatics then

Period Relations of Schottky Type on Riemann Surfaces

46 of the same work suffices. Rayleigh attributes his response to Helmholtz,2 (Appendix VIII), but Benade3 cites Wilhelm Weber, 1830, as an earlier source. A similar request for enlightenment about the operation of the flute or flue organ pipe (or even a blown bottle) brings a simple, but unfortunately long since discredited, response, again from Rayleigh (ibid.) and attributed to Helmholtz (ibid., pp. 92-93), and again, the writer surmises from a citation by Coltman: anticipated in 1830, this time by Sir William Herschel. The discredited theory, or model, will be sketched in the next section. If one pursues the matter in search of a valid model, one is soon lost in a swamp of technical papers and controversy. Authors on all sides express grave reservations about attempts a t simplified theories. One early investigator’ proposed a simplified theory of the edgetone, which proved inaccurate. However, the excellent work, theoretical and experimental, of several modern investigators, some of them mothated by the study of the closely related phenomenon of the edgetone, has revealed several basically simple mechanisms which, this writer feels, can be assembled into crudely simplified models of the desired type-models which, according to one’s taste, are tolerably consistent with reality, and plausible or a good target for critics to shoot at. The investigators whose works were drawn upon by the writer are Powell,” Cremer and k i n g 6 C ~ I t m a n , ~ and, later, for the edgetone, Karamcheti and sometime coworkers, Shields’ and Stegen.’ The resulting models like those of the reed-controlled instruments are small-signal, linear ones. A tone generator is linear if it can produce a pure (sinusoidal) tone. No real generator is linear, but, if tone amplitudes are sufficiently small, most real generators are sufficiently linear. In other words, the present models, if they have any validity at all, have more validity for softly playing instruments or softly sounding edgetones The writer takes this opportunity to call attention to some cogent observations of one of the referees. The first is that the present paper deals only with the feedback mechanism of the flute, but not with its resonator, particular tone quality, or the effect of shape, design, and material on the tone quality. The model here is a very limited one in this sense. The second observation is that the proposed criteria of plausibility of the model (the last section) may be difficult to verify. He proposes that it might be 54