EXERCISES 4.2

4.2.1. 
(a) Using the equation
$$\rho_0(x)\frac{\partial^2 u}{\partial t^2} = T_0\,\frac{\partial^2 u}{\partial x^2} + Q(x,t)\rho_0(x),$$
 compute the sagged equilibrium position $u_E(x)$
if $Q(x, t) = -g$. The boundary conditions are $u(0) = 0$ and $u(L) = 0$.

(b) Show that $v(x, t) = u(x, t) - u_E(x)$ satisfies 
$$\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}
,\quad $c^2=\frac{T_0}{\rho_0(x)}.$$

4.2.2. 
Show that $c^2$ has the dimensions of velocity squared.

4.2.3. 
Consider a particle whose x-coordinate (in horizontal equilibrium) is designated
by $\alpha$. If its vertical and horizontal displacements are $u$ and $v$,
respectively, determine its position $x$ and $y$. Then show that
$$\frac{dy}{dx} = \frac{\partial u/\partial \alpha}{1+\partial v/\partial \alpha}.$$

4.2.4. 
Derive equations for horizontal and vertical displacements without ignoring
$v$. Assume that the string is perfectly flexible and that the tension is
determined by an experimental law.

4.2.5. 
Derive the partial differential equation for a vibrating string in the simplest
possible manner. You may assume the string has constant mass density
$\rho_0$, you may assume the tension $T_0$ is constant, and you may assume small
displacements (with small slopes).