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Wave problem

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Find the solution \(\boldsymbol{u}(\boldsymbol{x}, \boldsymbol{t})\) for the wave problem on a string of length \(\boldsymbol{L}=\pi\) with \(c^{2}=1\) and conditions given by:

\(\left\{\begin{array}{l}u(0, t)=0, u(\pi, t)=0, \quad t>0 \\ u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=\sin x, 0<x<\pi\end{array}\right.\)

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Answer #1

L. 224 3 x dT dt 2 2 X x at the solution of wome-dimensional www.e.equation be du 2x 2 = 22 at 2 2 be written as 46 14t) X(Xand d² df2 p2T 1 - 0 台 Case L i when kro: Let k=pr (ray); pto and c 2 I Given) Then dy p² X 0. dx2 Therefore their respectiveDate : 1 Page No. In this case dy 2 e Pl_ept - 0 P 20 - 21:07 p = 0 which is contrary to ous assumption that pto. . then we dDate: 1 Page No. Therefore equation @ becomes: uſset) =( 6 cos px + Cro scapx)CG cos put + G12 sempt) Using the boundary condNow use the brindery condition u(40)20 and to in uluol : on ४ ť An sin nx net To find An and Bni Since EAn sin nx = 0 Anzo. nDate : Page No. sin x. shnu a { nBn sin nx sinxx To T 02 sinx. smnx EnBn sin nx Incl Bnzo ...Fon the given boudery conditio n

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