onsider the superposition of N equal-amplitude simple harmonic oscillator with t
ID: 3308595 • Letter: O
Question
onsider the superposition of N equal-amplitude simple harmonic oscillator with the same frequency, but random phases where n are random numbers, where n = 1,2, , N. What is the average value of the of the amplitude ofx? Express your answer in terms of N and a. Hints: In calculating the average amplitude it is best to start with the amplitude squared: A2 = XX * where * 1s the complex conjugate of x. In addition, note that el@n-Pm) are random locations on the unit circle in the complex plane soe«on-an) fr n # m.Explanation / Answer
x = aei(wt + p1) + aei(wt + p2) + ............... + aei(wt + pN) , where pi is the angle phi and i = 1,2,.......
<x> = x2 = xx* = aa* [ ei(wt + p1) + ..................... + ei(wt + pN)] [ e-i(wt + p1) + .................. + e-i(wt + pN)]
<x> = |a|2 [ N + ei(p1 - p2) + ei(p2 - p3) + ........................... + ei(pn-1 - pn)] , this is because since we multiply the above two brackets e-i(wt + pN) ei(wt + pN) will leads to e0 = 1 and since we multiply each of the like terms with the like terms in the next bracket the result will be N which is the first and since ei(pn - pm) where pn is not equal to pm , therefore, it yields the above result.
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