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Unformatted text preview: Chapter 5: Waves 23 3. Ez and Bz are zero everywhere: the Transversal electromagnetic mode (TEM). Than holds: k = √ ±ω εµ and vf = vg , just as if here were no waveguide. Further k ∈ I , so there exists no cut-off R frequency. In a rectangular, 3 dimensional resonating cavity with edges a, b and c the possible wave numbers are given n1 π n2 π n3 π , ky = , kz = This results in the possible frequencies f = vk/2π in the cavity: by: kx = a b c f= v 2 n2 n2 n2 y x z + 2+ 2 a2 b c For a cubic cavity, with a = b = c, the possible number of oscillating modes N L for longitudinal waves is given by: 4 π a3 f 3 NL = 3v 3 Because transversal waves have two possible polarizations holds for them: N T = 2NL . 5.6 Non-linear wave equations
The Van der Pol equation is given by: dx d2 x 2 + ω0 x = 0 − εω0 (1 − β x2 ) dt2 dt 2 1− β x2 can be ignored for very small values of the amplitude. Substitution of x ∼ e iωt gives: ω =
12 2 ε ). 1 2 ω0 (iε 1 2 εω0 . The lowest-order instabilities grow as While x is growing, the 2nd term becomes larger ± − and diminishes the growth. Oscillations on a time scale ∼ ω 0 1 can exist. If x is expanded as x = x (0) + (1) 2 (2) εx + ε x + · · · and this is substituted one obtains, besides periodic, secular terms ∼ εt. If it is assumed that there exist timescales τn , 0 ≤ τ ≤ N with ∂τn /∂ t = εn and if the secular terms are put 0 one obtains: d dt 1 2 dx dt 2 2 + 1 ω0 x2 2 = εω0 (1 − β x2 ) dx dt 2 This is an energy equation. Energy is conserved if the left-hand side is 0. If x 2 > 1/β , the right-hand side changes sign and an increase in energy changes into a decrease of energy. This mechanism limits the growth of oscillations. The Korteweg-De Vries equation is given by: ∂u ∂u ∂u ∂3u + − au + b2 3 = 0 ∂t ∂x ∂x ∂x
non−lin dispersive This equation is for example a model for ion-acoustic waves in a plasma. For this equation, soliton solutions of the following form exist: −d u(x − ct) = cosh2 (e(x − ct)) with c = 1 + 1 ad and e2 = ad/(12b2 ). 3 ...
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
- Spring '10