# Small oscillations of heavy string

I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points.

First of all I wrote down this Lagrangian:

$$S=\int dt ds \left[\frac{\rho}{2}(\dot{x}^2+\dot{y}^2)-\rho g y(s,t)+\frac{\lambda(s,t)}{2}\left(\left(\frac{\partial x}{\partial s}\right)^2+\left(\frac{\partial y}{\partial s}\right)^2-1\right)\right]$$

This Lagrangian describes heavy string with fixed ends in gravitational field. Where $\rho$ is density, $g$ is gravitational acceleration, $s$ is natural parameter.

So I have 3 equations from Euler-Lagrange equations.

$$\rho\ddot{x}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial x}{\partial s }\right)=0$$ $$\rho\ddot{y}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial y}{\partial s }\right)+\rho g=0$$ $$\left(\frac{\partial x}{\partial s }\right)^2+\left(\frac{\partial y}{\partial s }\right)^2=1$$

After that I've found stationary solution ($\frac{\partial x}{\partial t}=\frac{\partial y}{\partial t}=\frac{\partial \lambda}{\partial t}=0$). (I just put $\ddot{x}=\ddot{y}=0$)

$$y_0(x)=-\frac{C_1}{\rho g}\cosh\left(\frac{\rho g x}{C_1}+C_2\right)$$

Where $C_1,C_2$ is integration constants (depends on positions of ends of string). And $\cosh(x)$ is hyperbolic cosine.

To study small oscillations I've tried to use pertrubation theory.

So, I put $$y(s,t)=y_0(s)+\bar{y}(s,t)$$ $$x(s,t)=x_0(s)+\bar{x}(s,t)$$ $$\lambda(s,t)=\lambda_0(s)+\bar{\lambda}(s,t)$$

But after that I get difficult differential equations, which I can't solve.

Maybe someone know the more simplier aproach to solve this problem or know how to solve it in this way?

• I did the same thing (7-8 years ago) to find the galloping modes of an overhead power line (catenary + vibration), but I do not remember how I did it. If I come up with something I will try to post. – John Alexiou Oct 14 '13 at 19:47
• @ja72 It would be great! :) – Oiale Oct 15 '13 at 15:03
• I don't know if this will help, but I would parametrize the perturbation by a movement perpendicular to the static string. The second thing I would look into is if the movement could be treated as a wave in a media with a varying speed of travel. I base this on the idea that the tension is greater near the ends and so the speed of the wave is greater there as well. – John M. Cavallo Oct 16 '13 at 0:00
• I guess there are longitudonal and transverse waves. Is your focus on both, or one of them? – John Alexiou Oct 16 '13 at 15:10
• @ja72 Now I'm focusing on transverse waves. – Oiale Oct 16 '13 at 17:18