Exercises

1. Estimate the highest possible frequency (in hertz), and the smallest possible wavelength, of a longitudinal wave in a thin aluminum rod, due to the discrete atomic structure of this material. The mass density, Young's modulus, and atomic weight of aluminum are $2.7\times 10^{\,3}\,{\rm kg\,m}^{-3}$, $7\times 10^{10}\,{\rm N\,m}^{-2}$, and $27$, respectively. (Assume, for the sake of simplicity, that aluminum has a simple cubic lattice.)

2. A simple model of an ionic crystal consists of a linear array of a great many equally-spaced atoms of alternating masses $M$ and $m$, where $m<M$. The masses are connected by identical chemical bonds that are modeled as springs of spring constant $K$.
   1. Show that the frequencies of the system's longitudinal modes of vibration either lie in the band 0 to $(2\,K/M)^{1/2}$ or in the band $(2\,K/m)^{1/2}$ to $[2\,K\,(1/M+1/m)]^{1/2}$.

   2. Show that, in the long-wavelength limit, modes whose frequencies lie in the lower band are such that neighboring atomics move in the same direction, whereas modes whose frequencies lie in the upper band are such that neighboring atoms move in opposite directions. The lower band is known as the acoustic branch, whereas the upper band is known as the optical branch.

3. Consider a linear array of $N$ identical simple pendula of mass $m$ and length $l$ that are suspended from equal-height points, evenly-spaced a distance $a$ apart. Suppose that each pendulum bob is attached to its two immediate neighbors by means of light springs of unstretched length $a$ and spring constant $K$. Figure 5.8 shows a small part of such an array. Let $x_i=i\,a$ be the equilibrium position of the $i$th bob, for $i=1,N$, and let $\psi_i(t)$ be its horizontal displacement. It is assumed that $\vert\psi_i\vert/a\ll 1$ for all $i$.
   1. Demonstrate that the equation of motion of the $i$th pendulum bob is
      $$\ddot{\psi}_i = - \frac{g}{l}\,\psi_i + \frac{K}{m}\,(\psi_{i-1}-2\,\psi_i+\psi_{i+1}).$$

   2. Consider a general normal mode of the form
      $$\psi_i(t) = [A\,\sin (k\,x_i)+ B\,\cos(k\,x_i)]\,\cos(\omega\,t-\phi).$$
      Show that the associated dispersion relation is
      $$\omega^{\,2}= \frac{g}{l} + \frac{4\,K}{m}\,\sin^2(k\,a/2).$$

   3. Suppose that the first and last pendulums in the array are attached to immovable walls, located a horizontal distance $a$ away, by means of light springs of unstretched length $a$ and spring constant $K$. Find the normal modes of the system.

   4. Suppose that the first and last pendulums are not attached to anything on their outer sides. Find the normal modes of the system.

   Figure 5.8: Figure for Exercise 3.

   

4. Consider a periodic waveform $y(t)$ of period $\tau$, where $y(t+\tau)=y(t)$ for all $t$, which is represented as a Fourier series,
   $$y(t) = C_0 + \sum_{n>0}\left[C_n\,\cos(n\,\omega\,t) + S_n\,\sin(n\,\omega\,t)\right],$$
   where $\omega=2\pi/\tau$.
   1. Demonstrate that
      $$y(-t) = C_0 + \sum_{n>0}\left[C_n'\,\cos(n\,\omega\,t) + S_n'\,\sin(n\,\omega\,t)\right],$$
      where $C_n'=C_n$ and $S_n'=-S_n$.
   2. Show that
      $$y(t+T) = C_0 + \sum_{n>0}\left[C_n''\,\cos(n\,\omega\,t) + S_n''\,\sin(n\,\omega\,t)\right],$$
      where

      $$C_n'' = C_n\,\cos(n\,\omega\,T) + S_n\,\sin(n\,\omega\,T),$$

      $$S_n'' = S_n\,\cos(n\,\omega\,T) - C_n\,\sin(n\,\omega\,T).$$

5. Demonstrate that the periodic square-wave
   $$y(t)=A\left\{\begin{array}{ccc} -1&\mbox{\hspace{1cm}}&0\leq t/\tau\leq 1/2\\ [0.5ex] +1 &&1/2<t/\tau \leq 1 \end{array}\right.,$$
   where $y(t+\tau)=y(t)$ for all $t$, has the Fourier representation
   $$y(t) = -\frac{4\,A}{\pi}\left[\frac{\sin(\omega\,t)}{1}+ \frac{\sin(3\,\omega\,t)}{3}+ \frac{\sin(5\,\omega\,t)}{5}+\cdots\right].$$
   Here, $\omega=2\pi/\tau$. Plot the reconstructed waveform, retaining the first 4, 8, 16, and 32 terms in the Fourier series.

6. Show that the periodically repeated pulse waveform
   $$y(t)=A\left\{\begin{array}{ccl} 1&\mbox{\hspace{1cm}}&\vert t-T/2\vert\leq \tau/2\\ [0.5ex] 0 &&\mbox{otherwise} \end{array}\right.,$$
   where $y(t+T)=y(t)$ for all $t$, and $\tau < T$, has the Fourier representation
   $$y(t) =A\,\frac{\tau}{T} +\frac{2\,A}{\pi}\sum_{n=1,\infty} (-1)^{\,n}\,\frac{\sin (n\,\pi\,\tau/T)}{n}\,\cos(n\,2\pi\,t/T).$$
   Demonstrate that if $\tau\ll T$ then the most significant terms in the preceding series have frequencies (in hertz) that range from the fundamental frequency $1/T$ to a frequency of order $1/\tau$.