Exercises

1. Consider a uniformly-beaded string with $N$ beads that is similar to that pictured in Figure 4.1, except that each end of the string is attached to a massless ring that slides (in the $y$-direction) on a frictionless rod.
   1. Demonstrate that the normal modes of the system take the form
      $$y_{n,i}(t) = A_n\,\cos\left[\frac{n\,(i-1/2)}{N}\,\pi\right]\,\cos(\omega_n\,t-\phi_n),$$
      where
      $$\omega_n = 2\,\omega_0\,\sin\left(\frac{n}{N}\,\frac{\pi}{2}\right),$$
      $\omega_0$ is as defined in Section 4.2, $A_n$ and $\phi_n$ are constants, the integer $i=1,N$ indexes the beads, and the mode number $n$ indexes the modes.

   2. How many unique normal modes does the system possess, and what are their mode numbers?

   3. Show that the lowest frequency mode has an infinite wavelength and zero frequency. Explain this peculiar result.

2. Consider a uniformly-beaded string with $N$ beads that is similar to that pictured in Figure 4.1, except that the left end of the string is fixed, and the right end is attached to a massless ring which slides (in the $y$-direction) on a frictionless rod. Find the normal modes and normal frequencies of the system.

   Figure 4.12: Figure for Exercise 3.

   

3. Figure 4.12 shows the left and right extremities of a linear LC network consisting of $N$ identical inductors of inductance $L$, and $N+1$ identical capacitors of capacitance $C$. Let the instantaneous current flowing through the $i$th inductor be $I_i(t)$, for $i=1,N$. Demonstrate from Kirchhoff's circuital laws that the currents evolve in time according to the coupled equations
   $$\ddot{I}_i = \omega_0^{\,2}\,(I_{i-1}-2\,I_i+I_{i+1}),$$
   for $i=1,N$, where $\omega_0=1/\sqrt{L\,C}$, and $I_0=I_{N+1}=0$. Find the normal frequencies of the system.

4. Suppose that the outermost two capacitors in the circuit considered in the previous exercise are short-circuited. Find the new normal frequencies of the system.

5. A uniform string of length $l$, tension $T$, and mass per unit length $\rho$, is stretched between two immovable walls. Suppose that the string is initially in its equilibrium state. At $t=0$ it is struck by a hammer in such a manner as to impart an impulsive velocity $u_0$ to a small segment of length $a<l$ centered on the mid-point. Find an expression for the subsequent motion of the string. Plot the motion as a function of time in a similar fashion to Figure 4.11, assuming that $a=l/10$.

6. A uniform string of length $l$, tension $T$, and mass per unit length $\rho$, is stretched between two massless rings, attached to its ends, that slide (in the $y$-direction) along frictionless rods.
   1. Demonstrate that the most general solution to the wave equation takes the form
      $$y(x,t) = Y_0 + V_0\,t + \sum_{n>0} A_n\,\cos\left(n\,\pi\,\frac{x}{l}\right)\cos\left(n\,\pi\,\frac{v\,t}{l}-\phi_n\right),$$
      where $v=\sqrt{T/\rho}$, and $Y_0$, $V_0$, $A_n$, and $\phi_n$ are arbitrary constants.

   2. Show that
      $$\frac{2}{l}\int_0^l\cos\left(n\,\pi\,\frac{x}{l}\right)\cos\left(n'\,\pi\frac{x}{l}\right)dx=\delta_{n,n'},$$
      where $n$ and $n'$ are integers (that are not both zero).

   3. Use the previous result to demonstrate that the arbitrary constants in the previous solution can be determined from the initial conditions as follows:

      $$Y_0 =\frac{1}{l}\int_0^l y_0(x)\,dx,$$

      $$V_0 =\frac{1}{l}\int_0^l v_0(x)\,dx,$$

      $$A_n = (C_n^{\,2} + S_n^{\,2})^{1/2},$$

      $$\phi_n =\tan^{-1}(S_n/C_n),$$

      where $y_0(x)\equiv y(x,0)$, $v_0(x)\equiv \partial y(x,0)/\partial t$, and

      $$C_n = \frac{2}{l}\int_0^l y_0(x)\,\cos\left(n\,\pi\,\frac{x}{l}\right) dx,$$

      $$S_n = \frac{2}{l}\int_0^l v_0(x)\,\cos\left(n\,\pi\,\frac{x}{l}\right) dx.$$

   4. Suppose that the string is initially in its equilibrium state. At $t=0$ it is struck by a hammer in such a manner as to impart an impulsive velocity $u_0$ to a small segment of length $a<l$ centered on the mid-point. Find an expression for the subsequent motion of the string.

7. The linear LC circuit considered in Exercise 3 can be thought of as a discrete model of a uniform lossless transmission line (e.g., a co-axial cable). In this interpretation, $I_i(t)$ represents $I(x_i,t)$, where $x_i=i\,\delta x$. Moreover, $C={\cal C}\,\delta x$, and $L={\cal L}\,\delta x$, where ${\cal C}$ and ${\cal L}$ are the capacitance per unit length and the inductance per unit length of the line, respectively.
   1. Show that, in the limit $\delta x\rightarrow 0$, the evolution equation for the coupled currents given in Exercise 3 reduces to the wave equation
      $$\frac{\partial^{\,2} I}{\partial t^{\,2}} = v^{\,2}\,\frac{\partial^{\,2} I}{\partial x^{\,2}},$$
      where $I=I(x,t)$, $x$ measures distance along the line, and $v=1/\sqrt{{\cal L}\,{\cal C}}$.

   2. If $V_i(t)$ is the potential difference (measured from the top to the bottom) across the $i+1$th capacitor (from the left) in the circuit shown in Exercise 3, and $V(x,t)$ is the corresponding voltage in the transmission line, show that the discrete circuit equations relating the $I_i(t)$ and $V_i(t)$ reduce to

      $$\frac{\partial V}{\partial t} =-\frac{1}{{\cal C}}\,\frac{\partial I}{\partial x},$$

      $$\frac{\partial I}{\partial t} = - \frac{1}{{\cal L}}\,\frac{\partial V}{\partial x},$$

      in the transmission-line limit.
   3. Demonstrate that the voltage in a transmission line satisfies the wave equation
      $$\frac{\partial^{\,2} V}{\partial t^{\,2}} = v^{\,2}\,\frac{\partial^{\,2} V}{\partial x^{\,2}}.$$

8. Consider a uniform string of length $l$, tension $T$, and mass per unit length $\rho$ that is stretched between two immovable walls.
   1. Show that the total energy of the string, which is the sum of its kinetic and potential energies, is
      $$E = \frac{1}{2}\int_0^l\left[\rho\left(\frac{\partial y}{\partial t}\right)^2 + T\left(\frac{\partial y}{\partial x}\right)^2\right] dx,$$
      where $y(x,t)$ is the string's (relatively small) transverse displacement.

   2. The general motion of the string can be represented as a linear superposition of the normal modes; that is,
      $$y(x,t) = \sum_{n=1,\infty}A_n\,\sin\left(n\,\pi\,\frac{x}{l}\right)\cos\left(n\,\pi\,\frac{v\,t}{l}-\phi_n\right),$$
      where $v=\sqrt{T/\rho}$. Demonstrate that
      $$E = \sum_{n=1,\infty} E_n,$$
      where
      $$E_n = \frac{1}{4}\,m\,\omega_n^{\,2}\,A_n^{\,2}$$
      is the energy of the $n$th normal mode. Here, $m=\rho\,l$ is the mass of the string, and $\omega_n = n\,\pi\,v/l$ the angular frequency of the $n$th normal mode.