450fall09hw10_Sol

# 450fall09hw10_Sol - ECE 450 Fall 2009 Homework 10 Solutions...

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ECE 450 Fall 2009 Homework 10 Solutions Due: Tuesday, Nov 3 rd , 2009 1. The medium is dispersive, i.e., different frequency components will travel at different speeds, which is determined by the dispersion relation given as gk ω = . a. Given the relationships for v p and v g , express both them as a function of frequency ω what the dispersion relation indicates. There are a couple of ways to approach the problem, all of which lead to the same results. i. Given gk = , 2 2 p gk g k gg v kk ωω = =⇒= = = () 2 2 2 2 g gk gk gk g g v k =→ = ∂∂ =⇒ = == ii. Æ Given gk = and the fact that p k v = , 2 p p p g v g gv v = =♠ Æ Proceed with using k directly on the dispersion relation to determine v g . 1

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() 1 1 g 2 2 · 2 2 Using , 2 g gk gk g kk gk k g g v k ω ∂∂ TT == = = ** Relationship: v g = 1/2 v p and as a side note, notice how both decreases as frequency increases. b. So the rock falling into the calm, deep ocean creates a spectrum waves with periods longer than T min and shorter than T max , i.e., im T in ax m , ( T i exemplifies the period of an individual frequency component). Given the fact that the period T is inverse proportional to frequency (i.e., 1 T ), you can equate x min ma T and n max mi T ax --
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450fall09hw10_Sol - ECE 450 Fall 2009 Homework 10 Solutions...

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