# Modeling a vibrating string

1. Apr 28, 2010

### Mechdude

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 $x=0$
and $x=L$. a continuous density function, $\rho$, 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 $x=0$ , the equilibrium position of the string
is the horizontal segment $0 \leq x \leq L$ . the position of a given point which was at x during equilibrium
will be $u(x,t)$ at time t. if time is kept constant the function gives the shape of the string at that instant.
3. the function $\rho_0 (x)$ denotes the density at equilibrium, and $\rho (x,t)$ at time t.
as the string stretches the density will change; if we focus on an arbitrary interval between $x=x_1$ &
$x=x_2$ along the string we find that the mass m in this interval satisfies:
$$\int_{x_1}^{x_2} \rho_0 (x) dx = \int^{x_2}_{x_1} \rho (x,t) [ 1 + u^{2}_{x} (x,t) ]^{\frac{1}{2}} dx$$

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