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phys documents (dragged) 29

# phys documents (dragged) 29 - Chapter 5 Waves 23 3 Ez and...

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Chapter 5: Waves 23 3. E z and B z are zero everywhere: the Transversal electromagnetic mode (TEM). Than holds: k = ± ω ε μ and v f = v g , just as if here were no waveguide. Further k IR , so there exists no cut-off frequency. In a rectangular, 3 dimensional resonating cavity with edges a , b and c the possible wave numbers are given by: k x = n 1 π a , k y = n 2 π b , k z = n 3 π c This results in the possible frequencies f = vk/ 2 π in the cavity: f = v 2 n 2 x a 2 + n 2 y b 2 + n 2 z c 2 For a cubic cavity, with a = b = c , the possible number of oscillating modes N L for longitudinal waves is given by: N L = 4 π a 3 f 3 3 v 3 Because transversal waves have two possible polarizations holds for them: N T = 2 N L . 5.6 Non-linear wave equations The Van der Pol equation is given by: d 2 x dt 2 - εω 0 (1 - β x 2 ) dx dt + ω 2 0 x = 0 β x 2 can be ignored for very small values of the amplitude. Substitution of x e i ω t gives: ω = 1 2 ω 0 ( i ε ± 2 1 - 1 2 ε 2 ) . The lowest-order instabilities grow as 1 2 εω 0 . While x is growing, the 2nd term becomes larger and diminishes the growth. Oscillations on a time scale ω - 1 0 can exist. If x is expanded as x = x (0) + ε x (1) + ε 2 x (2) + · · · and this is substituted one obtains, besides periodic,
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