Chapter 5: Waves233.EzandBzare zero everywhere: the Transversal electromagnetic mode (TEM). Than holds:k=±ω√εμandvf=vg, just as if here were no waveguide. Furtherk∈IR, so there exists no cut-offfrequency.In a rectangular, 3 dimensional resonating cavity with edgesa,bandcthe possible wave numbers are givenby:kx=n1πa,ky=n2πb,kz=n3πcThis results in the possible frequenciesf=vk/2πin the cavity:f=v2n2xa2+n2yb2+n2zc2For a cubic cavity, witha=b=c, the possible number of oscillating modesNLfor longitudinal waves isgiven by:NL=4πa3f33v3Because transversal waves have two possible polarizations holds for them:NT= 2NL.5.6Non-linear wave equationsTheVan der Polequation is given by:d2xdt2-εω0(1-βx2)dxdt+ω20x= 0βx2can be ignored for very small values of the amplitude. Substitution ofx∼eiωtgives:ω=12ω0(iε±21-12ε2). The lowest-order instabilities grow as12εω0. Whilexis growing, the 2nd term becomes largerand diminishes the growth. Oscillations on a time scale∼ω-10can exist. Ifxis expanded asx=x(0)+εx(1)+ε2x(2)+· · ·and this is substituted one obtains, besides periodic,
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