HW4 - ECE215A/Materials206A Winter 2008 Prof. Brown/ECE...

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ECE215A/Materials206A Winter 2008 Prof. Brown/ECE Dept/UCSB 1 Homework 4 1. Monatomic linear lattice . Consider a longitudinal wave ( ) cos s uu t s K a ω =− which propagates in a monatomic linear lattice of atoms of mass M, spacing a , and nearest-neighbor interaction C. (a) Show that the total energy of the wave is given by () 2 2 11 1 22 , ss s E M du dt C u u + =+ where s runs over all atoms. (b) By substitution of , s u in this expression, show that the time-average total energy per atom is ( ) 2 1 42 2 1c o s , M uC K a u M u ωω +− = where in the last step we have used the longitudinal-wave dispersion relation. 2. Diatomic chain. Consider the normal modes of linear chain in which the force constants between nearest-neighbor atoms are alternately C and 10C. Let the masses be equal, and let the nearest- neighbor separation be 2 a . Find ω (k) at k = 0 and k = π /a. 3. Show that for long wavelengths the lattice-wave equation of motion reduces to the continuum elastic wave equation 2 , v tx = where v is the velocity of sound.
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HW4 - ECE215A/Materials206A Winter 2008 Prof. Brown/ECE...

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