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Homework Help: Modeling a vibrating string

  1. Apr 28, 2010 #1
    1. The problem statement, all variables and given/known data

    hi im reading Guenther & Lee, "partial differential equations of mathematical physics and integral equations" on the
    first chapter second section i think, on "Small vibrations of an elastic string" they give this argument:
    1. consider a string of x length L, the string is assumed to be a continuum and tied to posts at [itex] x=0[/itex]
    and [itex] x=L[/itex]. a continuous density function, [itex] \rho [/itex], with its integral over any segment of the
    string gives the mass of the segment. the string is perfectly elastic, vibrations are very small.
    2. an axis perpendicular to the x axis is constructed at [itex] x=0[/itex] , the equilibrium position of the string
    is the horizontal segment [itex] 0 \leq x \leq L [/itex] . the position of a given point which was at x during equilibrium
    will be [itex] u(x,t) [/itex] at time t. if time is kept constant the function gives the shape of the string at that instant.
    3. the function [itex] \rho_0 (x) [/itex] denotes the density at equilibrium, and [itex] \rho (x,t) [/itex] at time t.
    as the string stretches the density will change; if we focus on an arbitrary interval between [itex] x=x_1 [/itex] &
    [itex] x=x_2 [/itex] along the string we find that the mass m in this interval satisfies:
    [tex] \int_{x_1}^{x_2} \rho_0 (x) dx = \int^{x_2}_{x_1} \rho (x,t) [ 1 + u^{2}_{x} (x,t) ]^{\frac{1}{2}} dx [/tex]

    this last expression has me stumped, it seems to me like he pulled it right from under his sleeve, why would the mass satisfy that
    expression? i think the squared term is a partial derivative with respect to x imho.


    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Apr 28, 2010
  2. jcsd
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