Partial Differentials Notes_Part_19

Partial Differentials Notes_Part_19 - For example, on a...

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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 ...
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