Suppose that the circular membrane of Example 2 has

Chapter , Problem 1

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Suppose that the circular membrane of Example 2 has initial position u.r; 0/ D f .r/ and initial velocity ut.r; 0/ D 0. Derive by separation of variables the solution u.r; t / D X1 nD1 cnJ0 nr c cos nat c ; where f ng 1 1 are the positive roots of J0.x/ D 0 and cn D 2 c2 J1. n/2 Z c 0 rf .r/J0 nr c dr: 2.

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