EXERCISES 4.4

4.4.1. 
Consider vibrating strings of uniform density $\rho_0$ and tension $T_0$.

*(a) What are the natural frequencies of a vibrating string of length $L$ fixed
at both ends?

*(b) What are the natural frequencies of a vibrating string of length $H$,
which is fixed at $x = 0$ and "free" at the other end [i.e., $\partial u/\partial x(H,t)=0$]? 
Sketch a few modes of vibration as in Fig. 1, H4.4.

(c) Show that the modes of vibration for the odd harmonics (i.e., $n =
1, 3, 5, \ldots$) of part (a) are identical to modes of part (b) if $H = L/2$.
Verify that their natural frequencies are the same. Briefly explain using
symmetry arguments.

4.4.2.
In Sec. H4.2 it was shown that the displacement u of a nonuniform string
satisfies
$$\rho_0(x)\frac{\partial^2 u}{\partial t^2} = T_0\,\frac{\partial^2 u}{\partial x^2} + Q(x,t)\rho_0(x),$$
where $Q$ represents the vertical component of the body force per unit length.
If $Q = 0$, the partial differential equation is homogeneous. A slightly different
homogeneous equation occurs if $Q = \alpha u$.

(a) Show that if $\alpha < 0$, the body force is restoring (toward $u = 0$). Show
that if $\alpha > 0$, the body force tends to push the string further away
from its unperturbed position $u = 0$.

(b) Separate variables if po(x) and a(x) but To is constant for physical
reasons. Analyze the time-dependent ordinary differential equation.

*(c) Specialize part (b) to the constant coefficient case. Solve the initial
value problem if $\alpha < 0$:

$u(0, t) = 0, $ u(x,0) = 0$,

$(L, t) = 0$, $\frac{\partial u}{\partial t}(x, 0) = f(x)$.

What are the frequencies of vibration?

4.4.3. 
Consider a slightly damped vibrating string that satisfies
$$\rho_0(x)\frac{\partial^2 u}{\partial t^2} = T_0\,\frac{\partial^2 u}{\partial x^2} -\beta\frac{\partial u}{\partial t}.$$
(a) Briefly explain why $\beta > 0$.

*(b) Determine the solution (by separation of variables) that satisfies the
boundary conditions

$u(0, t) = 0$ and $u(L, t) = 0$

and the initial conditions

$u(x,0) = f(x)$ and  $\frac{partial u}{\partial t}(x,0) = g(x)$-

You can assume that this frictional coefficient $Q$ is relatively small
($\beta^2< 4\pi^2 \rho_0 T_0 / L^2$).

4.4.4. Redo Exercise 4.4.3(b) by the eigenfunction expansion method.

4.4.5. Redo Exercise 4.4.3(b) if $ 4\pi^2 \rho_0 T_0 / L^2 < \beta^2 < 16\pi^2\rho_0T_0/L^2$.

4.4.6. For (4.4.1)-(4.4.3), from (4.4.11) show that
$$u(x, t) = R(x - ct) + S(x + ct),$$
where $R4 and $S$ are some functions.

4.4.7. 
If a vibrating string satisfying the one-dimensional string equation with fixed ends is initially at rest, $g(x) = 0$,
with shape $f(x)$ given, then show that
$$u(x, t) = \frac12[F(x - ct) + F(x + ct)],$$
where $F(x)$ is the odd periodic extension of $f (x)$. 

Hints.\\
1. For all $x$, $F(x)= \sum_{n=1}^\infty A_n sin(n\pi x/L)$. \\
2. $\sin a \cos b = \frac12[\sin(a + b) + \sin(a - b)]$.

Comment: This result shows that the practical difficulty of summing an
infinite number of terms of a Fourier series may be avoided for the one-dimensional
wave equation.

4.4.8. 
If a vibrating string satisfying the one-dimensional string equation with fixed ends is  initially unperturbed, $f (x) =0$, with the initial velocity $g(x)$ given, then show that
$$
u(x, t) = \frac{1}{2c}\int_{x-ct}^{x+ct} G(u)du,
$$
where $G(x)$ is the odd periodic extension of $g(x)$. 

Hints:\\
1. For all $x$, $G(x)=\sum_{n=1}^\infty B_b\frac{n\pi c}{L}\sin(n\pi x/L)$.\\
2. $\sin a \sin b = \frac12[\cos(a - b) - \cos(a + b)]$.\\
See the comment after Exercise 7, H4.4.

4.4.9 
From $u_{tt}=c^2u_{xx}$, derive conservation of energy for a vibrating string,
$$\frac{dE}{dt} = c^2 \left. u_x(x,t)u_t(x,t)\right|_{x=0}^{x=L},$$
where the total energy $E$ is the sum of the kinetic energy, defined by
$\int_0^L \frac12(u_t)^2dx$, and the potential energy, defined by $\int_0^L \frac{c^2}{2}(u_x)^2dx$.

4.4.10. 
What happens to the total energy $E$ of a vibrating string (see Exercise 9, H4.4)\\
(a) If $u(0, t) = 0$ and $u(L, t) = 0$\\
(b) If $u_x(0,t) = 0$ and $u(L,t) = 0$\\
(c) If $u(0, t) = 0$ and $u_x(L, t) = -\gamma u(L, t)$ with $\gamma > 0$ \\
(d) If $\gamma < 0$ in part (c)

4.4.11. 
Show that the potential and kinetic energies (defined in Exercise 9, H4.4) are
equal for a traveling wave, $u = R(x-ct)$.

4.4.12. 
Using $$\frac{dE}{dt} = c^2 \left. u_x(x,t)u_t(x,t)\right|_{x=0}^{x=L},$$
 prove that the solution of the one-dimensional string equation with fixed ends is unique.

4.4.13. 

(a) Using 
$$\frac{dE}{dt} = c^2 \left. u_x(x,t)u_t(x,t)\right|_{x=0}^{x=L},$$ 
calculate the energy of one normal mode.

(b) Show that the total energy, when $u(x, t)$ is a superposition of product solutions representing the solution,
is the sum of the energeis contained in each mode.