By J C A Wevers

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D is not parallel with E if the polarizability P of a material is not equal in all directions. There are at least 3 directions, the principal axes, in which they are parallel. This results in 3 refractive indices ni which can be used to construct Fresnel’s ellipsoid. g. at trigonal, hexagonal and tetragonal crystals there is one optical axis in the direction of n1 . Incident light rays can now be split up in two parts: the ordinary wave is linear polarized ⊥ the plane through the transmission direction and the optical axis.

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 by: kx = , ky = , kz = This results in the possible frequencies f = vk/2π in the cavity: a b c f= v 2 n2y n2x n2z + + a2 b2 c2 For a cubic cavity, with a = b = c, the possible number of oscillating modes NL for longitudinal waves is given by: 4πa3 f 3 NL = 3v 3 Because transversal waves have two possible polarizations holds for them: NT = 2NL . 6 Non-linear wave equations The Van der Pol equation is given by: dx d2 x − εω0 (1 − βx2 ) + ω02 x = 0 dt2 dt βx2 can be ignored for very small values of the amplitude.

The distribution as a function of the absolute value of the velocity is given by: dN 4N mv 2 = 3 √ v 2 exp − dv 2kT α π The general form of the energy distribution function then becomes: c(s) P (E)dE = kT where c(s) is a normalization constant, given by: 1. Even s: s = 2l: c(s) = 1 (l − 1)! 2. Odd s: s = 2l + 1: c(s) = √ 2l π(2l − 1)!! 3 31 Pressure on a wall The number of molecules that collides with a wall with surface A within a time τ is given by: ∞ π 2π 3 nAvτ cos(θ)P (v, θ, ϕ)dvdθdϕ d N= 0 0 0 From this follows for the particle flux on the wall: Φ = 14 n v .