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# Exercises

1. Estimate the highest possible frequency (in hertz), and the smallest possible wavelength, of a longitudinal wave in a thin Aluminium rod, due to the discrete atomic structure of this material. The mass density, Young's modulus, and atomic weight of Aluminium are , , and , respectively. (Assume, for the sake of simplicity, that Aluminium 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 and , where . The masses are connected by identical chemical bonds that are modeled as springs of spring constant . Show that the frequencies of the system's longitudinal modes of vibration either lie in the band 0 to or in the band to . 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 identical simple pendula of mass and length that are suspended from equal height points, evenly spaced a distance apart. Suppose that each pendulum bob is attached to its two immediate neighbors by means of light springs of unstretched length and spring constant . The following figure shows a small part of such an array. Let be the equilibrium position of the th bob, for , and let be its horizontal displacement. It is assumed that for all . Demonstrate that the equation of motion of the th pendulum bob is

Consider a general normal mode of the form

Show that the associated dispersion relation is

Suppose that the first and last pendulums in the array are attached to immovable walls, located a horizontal distance away, by means of light springs of unstretched length and spring constant . Find the normal modes of the system. Suppose, on the other hand, that the first and last pendulums are not attached to anything on their outer sides. Find the normal modes of the system.

4. Consider a periodic waveform of period , where for all , which is represented as a Fourier series,

where . Demonstrate that

where and , and

where

5. Demonstrate that the periodic square-wave

where for all , has the Fourier representation

Here, . 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

where for all , and , has the Fourier representation

Demonstrate that if then the most significant terms in the preceding series have frequencies (in hertz) that range from the fundamental frequency to a frequency of order .

Next: Traveling Waves Up: Longitudinal Standing Waves Previous: Fourier Analysis
Richard Fitzpatrick 2013-04-08