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## Homework Statement

A string of length l has a zero initial velocity and a displacement y[itex]_{0}[/itex](x) as shown. (This initial displacement might be caused by stopping the string at the center and plucking half of it). Find the displacement as a function of x and t.

See the following link for the figure. It's the first one on the page.

http://web.physics.ucsb.edu/~physCS33/spring2011/hw1.pdf [Broken]

## Homework Equations

## The Attempt at a Solution

Now, to be honest with you all, I'm not really having trouble with the process. I understand how to find a form for y given the boundary conditions, and in fact I come up with,

y=Bn * sin(n[itex]\pi[/itex]x/l) * cos(n[itex]\pi[/itex]vt/l)

where Bn is equal to the fourier sine series,

Bn=2/l ∫ f(x)*sin(2[itex]\pi[/itex]nx/l)dx.

My problem is that my answer doesn't look like that in the book, or the answer on the page I linked above. They might be equivalent, and I'm just not seeing it?

I got y=2h/[itex]\pi^{2}[/itex] [itex]\sum[/itex]1/n[itex]^{2}[/itex] [2sin([itex]\pi[/itex]n/2) - sin([itex]\pi[/itex]n] sin(n[itex]\pi[/itex]x/l) cos (n[itex]\pi[/itex]vt/l).

I got Bn from setting f(x)=y[itex]_{0}[/itex](x) = 4hx/l for 0<x<l/4 and 2h-4hx/l for l/4<x<l/2.

Now my main question is -- way back to pre-algebra -- are my f(x) equations correct? I've done the fourier series integrals over and over again, and can't find any errors, and so I'm left to think that I must have concluded f(x) incorrectly.

Sorry for the pretty silly question, but I'm at a loss here!

Thanks so much.

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