PHYS_342_hmwrk2_early_quantum - PHYS342 FallSemester2011...

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1 PHYS 342 Fall Semester 2011 Homework No. 2 Due: Sept. 12, 2011 1. This problem is designed to lead you to the conclusion that mathematically speaking, the superposition of two waves moving in opposite directions is exactly equivalent to a standing wave. The displacement of the nth mode of oscillation in the y-direction of a stretched string tethered at both ends of length L with a mass per unit length μ subjected to a tensional force T can be written as a standing wave given by Note that each allowed mode of vibration is specified by the integer index n. The amplitude of each mode of vibration is specified by A n . The angular frequency of each mode is specified by n . a) According to Euler’s formula, you can always write It is important that you understand these identities, since we will make use of them often during the semester. By multiplying these two expressions together, show that you can rewrite y n (x,t) as  (,) s i n c o s nn n n nx y xt A t L T n L    s i n s i n 2 n n A t t LL      11 sin and cos ( ) 22 in x in x it n ee t e e Li
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2 Note the symmetry of this equation. One sine function has an argument that is given by the difference of two quantities; the other sine function has an argument that is the sum of the same two quantities. b) When discussing waves on a string, the spatial variation of the argument of a sine wave is usually written as where λ n defines the wavelength of the n th mode.
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PHYS_342_hmwrk2_early_quantum - PHYS342 FallSemester2011...

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