A taut string of length 2l , fastened at both ends, is disturbed from its position of equilibrium by imparting to each of its points on an initial velocity of magnitude k(2lx-x^2). Find the displacement function y(x,t)
1
Expert's answer
2022-01-12T08:00:29-0500
The displacement y(x,t) of any point 'x ' of the string at any time 't ' is given by
∂t2∂2y=a2∂x2∂2y−−−(1)
We have to solve equation (1) satisfying the following boundary conditions.
y(0,t)=0,fort≥0−−−−(2)
y(2l,t)=0,fort≥0−−−−(3)
y(x,0)=0,for0≤x≤2l−−−−(4)
δtδy(x,0)=k(2lx−x2),for0≤x≤2l−−−−(5)
The suitable solution of Eq (1), consistent with the vibration of the string,is
y(x,t)=(Acospx+Bsinpx)
(Ccospat+Dsinpat)−−−−(6)
Using boundary conditions (2) in (6), we have
A(Ccospat+Dsinpat)=0forallt≥0
A=0
Using boundary conditions (3) in (6), we have
Bsin2lp(Ccospat+Dsinpat)=0
forallt≥0
Either B=0 or sin2lp=0
If we assume that B=0, we get a trivial solution.
sin2lp=0
2lp=nπ
p=2lnπ
Where n=0,1,2,...∞
Using boundary conditions (4) in (6), we have
Bsinpx.C=0
for0≤x≤2l
As B=0, we get C=0
Using these values of A,p,C in (6), the solution reduces to
y(x,t)=ksin2lnπsin2lnπat
−−−−(7)
where n=0,1,2,3...∞
The most general solution of Eq.(1) is
y(x,t)=∑n=1∞λnsin2lnπcos2lnπat−−−−(8)
Differentiating both sides of (8) partially with respect to t, we have
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