Unformatted text preview: For example, on a unit square R = 0 < x, y < 1 , an accidental degeneracy occurs whenever m2 + n2 = k 2 + l2 (12.163) for distinct pairs of positive integers (m, n) = (k, l). The simplest possibility arises whenever m = n, in which case we can merely reverse the order, setting k = n, l = m. In Figure 12.12 we plot three sample nodal curves sin 4 x sin y + sin x sin 4 y = 0, corresponding to three different linear combinations of the eigenfunctions with m = l = 4, n = k = 1. The associated vibrational frequency is, in all cases, 4,1 = c 17 , where c is the wave speed. Remark : Classifying accidental degeneracies takes us into the realm of number theory, [8, 31]. In the case of a square, equation (12.163) is asking us to locate all integer points in R 2 that lie on a common circle. Remark : An interesting question is whether a circular disk has accidental degeneracies, which would occur if two different Bessel roots were to coincide. However, the Bourget hypothesis, [142; p. 484], implies that m,n = k,l whenever (m, n) = (k, l). Thus, a disk has no such degeneracies, and all its nodal curves are concentric circles and diameters. 1/22/12 473 c 2012 Peter J. Olver ...
View
Full Document
 Fall '10
 Olver
 Differential Equations, Geometry, Equations, Partial Differential Equations, Integers, Elementary mathematics, accidental degeneracies, nodal curves sin

Click to edit the document details