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Unformatted text preview: Assignment 4 Find the following for the wave LI) 3 5’ W (I )4 + 3 +)
a) wave vector is) wavelength 0) frequency d) period c) phase velocity 1) amplitude Censider a series of square wave pulses shown below. re) Fourier analysis says that this pulse can expanded as:
m B ‘ = z n
P(+):_Ck + Z(Qawwi+ ”an‘i) 50?
'Z V! 3 i a) Show that An = 4/n 1: where n is odd, An =0 for n is even
Bn = O and A0 = 0 b) Plot the ﬁrst term, ﬁrst 2 terms and ﬁrst 3 terms in the sum. c) Hence, a pulse of light which can convey information, is composed of
many frequencies. Estimate the range of frequencies Av required to make
a one femtosecond laser pulse using the Heisenberg Uncertainty Principle
Av At > 2n. Superposition Principle a) Show that if (H and (Rare solutions of the 3 dimensional wave equation
that their sum also is a solution. 13) This may seem trivial but show that the superposition principle does not
hold for the following differential equatiou. Damped harmonic oscillator m 6‘ 2‘M 2 '— i< K "" Y _0{_‘£_) 37? 0” a) Consider a solution x = A e“, Solve for 2.. (Result will be complex) This 
approach is much simpler than using )4: = A cos (at + B sin not. 1)) Write down the general solution for the case of weak damping km >> if. c) What is the solution for the case the mass is initially at rest at distance x0?
Plot this solution. Show that the group velocity vg is related to the phase velocity v by the
following equation. Note that for the case of normal dispersion v3 < c. \J ;: C ‘3 n+wﬁl£ Jo ...
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 Winter '10
 A

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