Question

do question 3 with the info provided

f 0 Question 3 Given the graph above represents a string being plucked at point (g). The wave equation generated when the string is released after being plucked, is given by the wave equation in question 1, and that additionally: 1. u(0, t) 0 u(4, t) 2. u(x, 0) f(x) as in question 2 au 3. atlt-0 Solve wave equation subject to the restrictions above. [10] Question 2 a) In the General Fourier series bn sin L f(x) [an cos + 2 n 1 = Given that the solution to the second order equation d2y dx +p°y= 0 is y(x) Acos(px)+ B sin(px) Use the technique of separation of variables and assume u(x, t) = X(x)T(t) Show that the solution to the general wave equation: 1 au (A cos ) (C cos(At) D sin (At)) where cp = 2 B sin is u(x, t) +

Answer 1

Sloc)= O2DC 2 4-2 h7207T Page-1 let ula,t) X() TIt) be the Solution -D uxdT d2x T given eauation becomes Td2x C2 dt2 =-p 1Say) = d2T T dt2 d2x-2 dax2 2T dt2 dTtpTeo dt2 X t 0 dx2 TIt)= CCos Cpt +D Sincbt X)= ACas p+BSinpx becomes ulast) AGpx+BSin px) (ccascpt + D Sincpt) -e when c0, ulo,t) = O (A+o) (Cascpt + D Sincpt) AO when oc- u, l4141o (otB Sinup) arekt +D Sincht)

Sin4p o => B0, Sn4p Page-2 Sin4p Sinnn P=n, nel,23 SOr each value of n 12) becomes alt= Sin nnc AnCos nict 4 (ニ人 +8n Sin nTct hesre AnBC & Brn= BD pastially diSarentiata (3) w.9r.t t u ESin -nTC An Sinnict Uuuis 41 +hnc Bn Cay nTCt Cahen 七=0, àu.=0 Sina Hence 13) becomes ul04 AnSin nmz Cash nct When t =o Slx)- EAn Sinnn

h val wQ which is a halt Range Sine Sesries 4 n=근( gla) Sin pned oc 1 (4-x) Sinn de 2 -)/-l 6 SinnnI 4 + 4-Cor -16 Sin nTX 272 = 8 Sinn 8 Cos 16 Sinnil 2. 2 2 Sinnir 2. 32 2 16 2 - Hence greguised solution bocomos. Sin nT Sin nn Gor nnct 4 16 2 n! Ans Page-3)