Consider a circular membrane r = a. For noncircularly symmetric eigenfunctions (m = 0)

Chapter 9, Problem 9.6.9

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Consider a circular membrane r = a. For noncircularly symmetric eigenfunctions (m = 0), (9.6.18) is valid with 0 = c1(1) 0 + c2(2) 0 , where (1) 0 and (2) 0 are two mutually orthogonal eigenfunctions corresponding to the same eigenvalue 0. Here, c1 and c2 are arbitrary constants. (a) Determine a homogeneous linear system of equations for c1 and c2 derived from the fact that 1 has two homogeneous solutions (1) 0 and (2) 0 . This will be the compatibility condition for (9.6.18) with (9.6.19). (b) Solve the linear system of part (a) to determine the perturbed frequencies and the corresponding natural modes of vibration.

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