1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted