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Math Help - Modelling the shape of a vibrating guitar string

  1. #1
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    Modelling the shape of a vibrating guitar string

    Hello, I am Marckutt and I teach at an underprivelleged school outside Buenos Aires. We are trying to model the shape of a guitar string as it vibrates. More specifically, I need to be able to determine the displacement of any point along the string from the restpoint as the string oscillates. This requires incorporating two independent variables - the point along the string and time. What type of model could I use to model this scenario?
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  2. #2
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    Well for a start lets say displacement is a function of two variables the point x horizontally along the sting and t time, the time a note is held?

    \displaystyle d(x,t) = \dots \sin (\dots) +\dots

    What other information do you have?
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  3. #3
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    Start with the wave equation:

    \[\frac{\partial^2 y}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}\]<br />

    where v is the speed of propagation.

    We assume the at x=0 and x=L, the displacement y=0.

    The solution to the equation is separable, which means

    y(x,t)=X(x)T(t)

    Solving the wave equation with this general solution, be the method of separation of variables, we find that

    X(x)=sin(\frac{n\pi x}{L})

    and

    T(t)=sin(\frac{n\pi vt}{L})

    where n is an integer corresponding to the frequencies of vibrations (modes)

    Thus the displacement is

    y(x,t)=sin(\frac{n\pi x}{L})sin(\frac{n\pi vt}{L})

    The most general solution is an superposition of the solutions for all values of n

    y(x,t)=\sum_{n=1}^{\infty}B_{n}sin(\frac{n\pi x}{L})sin(\frac{n\pi vt}{L})
    Last edited by dats13; April 6th 2011 at 02:32 PM. Reason: Typo
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