Question

)Consider the wave equation for a vibrating string of semi-infnite length with a fixed end at z = 0, t > 0 a(0,t) = 0, and initial conditions 0 < x < oo u(z,0) = 1-cos(nz), ut(x,0) = 0, Complete the table below with the values of u(0.5, t) at the specified time instants 0.5 0.5 x 0.5 0.5 0.5 2 0.5 0.75 t 0.25 u(x,t)

Answer 1

The solution is

(z,t) _ fir-ct) + f(x + ct) + 2c1rst g(s) ds

Where

f(x) = u(x,0) = 1-cos(m), g(x)-ut(z, 0) = 0, c= 1

And so $u(x,t)=[1-\cos(\pi(x-t))+1-\cos(\pi(x+t))]/2$

Which can be simplified to

u(x, t) 1-cos(πί) cos(nz)

So based on this the completed table is given below:

x	0.5	0.5	0.5	0.5	0.5
t	0.25	0.5	0.75	1	2
u(x,t)	1	1	1	1	1

$\blacksquare$

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