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Dp R'(a) = R'(b) = O. a < P < b. 36) 44 CHAPTER 1. 36) has the form where C1 and C2 are arbitrary constants and N 1L!!. p) is the Bessel function of 'PO second kind. 36); namely, they provide the system of equations { + C2N~n (vIAa) = 0, C1 J~n (vIAb) + C2N~n (vIAb) = O. b) 'PO 'PO 'PO 'PO is equal to zero. In other words, Am,n = [JL~)J 2, where JL~) are the roots of the equation N~n(vIAa) 'PO We see that the radial function has the form Thus, the natural oscillations of our plate are described by the functions Um,n(P, cp) = [J7rn (JL~)p)N~n (JL~)a) ~ ~ = Rm,n(p)

Ul'(' zero. ion that a quantity of 11('<11 Q - (,()lIst ic; ('xtract('d it. CHAPTER 1. ELLIPTIC PROBLEMS 24 Solution. 17) (here k is the internal heat conduction coefficient). The eigenvalues and eigenfunctions of the problem are found by solving the auxiliary boundary value problem (Sturm-Liouville problem) { X" + A2 X = 0, 0 < x < a, X(O) = 0 = X'(a) = O. (2n+l)7r] 2 and Xn(x) -_ sm " [(2n+l)7r] _ We get A; = [-2-a-2-a- X , n - 0,1, ... We will seek the solution of the above problem in the form of an expansion in eigenfunctions ~ u(x, y) = ~ Yn(Y) sin [(2n+1)7f ] 2a x, where the functions Yn(Y) are subject to determination.

Hence. , (-p+ p24).. cose+ (2:32). p - p:l sm 2eSlIl(2

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