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Vibrating string displacement, partial differential problem

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leoflc
#1
Feb9-08, 08:46 PM
P: 56
1. The problem statement, all variables and given/known data
A damped vibrating string of length 1, that satisfies
u_tt = u_xx - ([tex]\beta[/tex])u_t

with the boundary conditions:
u(0,t)=0
u(1,t)=0

initial conditions:

u(x,0)=f(x)
u_t(x,t)=0

solve for u(x,t) if [tex]\beta[/tex]^2 < 4Pi^2


3. The attempt at a solution

if u(x,t)=F(x)G(t)

So by using partial differential equations, I got:

G''+[tex]\beta[/tex]G'+GP^2=0
and
F''+FP^2=0

I solved for F(x) with B.Cs and got:
F(x)=C*Sin(P*x), where C is a const.

when I tried to solve for G(t), I got a long equation with 2 constants in there.
If I try to solve for u(x,t) by using the F(x)G(t), I will have something with three unknow constants.

Am I on the right track?
I'm not sure how/when to apply I.Cs and [tex]\beta[/tex] to solve for u(x,t).


Thank you very much!
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