Let h() represent the initial displacement in [0, L], extended into (L, 0) as an odd

Chapter 10, Problem 20

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Let h() represent the initial displacement in [0, L], extended into (L, 0) as an odd function and extended elsewhere as a periodic function of period 2L. Assuming that h, h_, and h__ are continuous, show by direct differentiation that u( x, t) as given in equation (28) satisfies the wave equation (1) and also the initial conditions (9). Note also that since equation (20) clearly satisfies the boundary conditions (3), the same is true of equation (28). Comparing equation (28) with the solution of the corresponding problem for the infinite string ( 16), we see that they have the same form, provided that the initial data for the finite string, defined originally only on the interval 0 x L, are extended in the given manner over the entire x-axis. If this is done, the solution for the infinite string is also applicable to the finite one.