# Ray Tracing and Beyond: Phase Space Methods in Plasma Wave by Tracy E.R., Brizard A.J., Richardson A.S., Kaufman A.N.

By Tracy E.R., Brizard A.J., Richardson A.S., Kaufman A.N.

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**Sample text**

X 2 ∂t 2 See comments in the previous section. 7 A plot of the dispersion curves for the electromagnetic wave in a uniform unmagnetized plasma. The dispersion surface D(k, ω) ≡ −ω2 + k 2 c2 + ωp2 = 0 has two branches. Each branch has a good projection to the k-axis, therefore the dispersion relations ω = ± (k) are globally defined. But the projection to the ω-axis is singular (two-to-one) at ω = ±ωp . Therefore, the dispersion relations k = κ± (ω) are not well-behaved at those points, and there are no real solutions for −ωp < ω < ωp .

The singular behavior when ω ∼ ωp reflects the fact that only frequencies greater than the plasma frequency propagate and care must be taken when applying this integral if the boundary conditions have frequencies near ωp , or lower. 57 That is, we assume ψ(x, t) has a carrier oscillation of the form exp[i(k0 x − ω0 t)], but with a varying amplitude. The amplitude function (also sometimes called the envelope) A(x, t) is assumed to vary on length and time scales that are long compared to a carrier wavelength and period, which means58 k0 |Ax | , |A| and |At | .

Almost all of the infinitude of rays emitted from S will miss the point O. There are two that make it to O: the straight-line path and the one that bounces off the mirror and satisfies the law of reflection along the way. 3. There is good reason to consider the path that bends around the edge of B as a ray, but it requires careful treatment at the edge where it encounters the obstacle. In fact, the light can diffract around the edge if it is sharp enough, so some light could reach O from S by this route.