Question

(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.

(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.

$$ \begin{array}{l} u_{t t}=c^{2} u_{x x}, \quad 0< x < \infty, t>0 \\ u(x, 0)=A e^{-\alpha x} \quad \text { and } \quad u_{t}(x, 0)=0, \quad 0< x < \infty \\ u(0, t)=A \cos \omega t, \quad t>0 \\ \lim _{x \rightarrow \infty} u(x, t)=0, \quad \lim _{x \rightarrow \infty} u_{x}(x, t)=0 \end{array} $$

where \(c, A, \alpha\) and \(\omega\) are positive real constants. Solve \(u(x, t)\) using Laplace transform.

Answer 1

A & uxx - 9 Uzt 0 SUCIB) - CAD ) = ca doucro dau 8 dxa UWA = G expl&x)+ exp-a) + lim ucx, t) to > L [ Lion «xt] → - Agenda Ascal exx) X-so lim UCx, 2)>0 X100 G=0 个 G expc&a) + As ce ceva UGGB) = lo expl-8) + (8) UCO,t) =A@Cut) + LCucogt)) = LCA QSCwts] 060,8)= AC3) 894 wa → ☺

By ☺ and @ As core 98) A8 G 82_xaca = 82twa Adele 82_xaca As 82twa po UCX78) = A8 As metas Jexpca) + 82twa 8 xəcalex682) + Ascarea 82 ega [ CA * Content As every pexp(-82) + As ce que 82_oga ucat) As created exp(-8X) 89_xaca + LASER 89 29 urt) = Het-) /Acis cuter) – Ace Pels R(@Qt-x) - + Acole ad espact) Answer .