A first pulse of wavenumber at leaves a density ripple in a Maxwellian electron…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Kinetic theory, collisions and transport · Kinetic theory, collisions and transport

Plasma Physics, Ch The Vlasov equation and Landau damping

Problem

A first pulse of wavenumber

k_1=100\ \mathrm{m^{-1}}

t=0

leaves a density ripple

\propto\hat f_0(v)\cos(k_1x)

in a Maxwellian electron plasma with

v_{te}=1.0\times10^{6}\ \mathrm{m\,s^{-1}}

. A second pulse of wavenumber

k_2=150\ \mathrm{m^{-1}}

follows at

\tau=1.0\ \mu\mathrm{s}

. Treat the electrons as streaming freely between and after the pulses, and take as given that just after the second pulse the part of the perturbation that can echo is

C\hat f_0(v)\cos[(k_1-k_2)x-k_1v\tau]

, with

C

a constant. Use

\int\hat f_0(v)\cos(\alpha v)\,\mathrm{d}v=e^{-\alpha^2v_{te}^2/2}

. ((a)) Show that the first pulse's own density falls by a factor

e

within a time

\sqrt2/(k_1v_{te})

, and evaluate it. ((b)) Let the echo term stream freely after

\tau

. Show that its density is

\propto\cos[(k_1-k_2)x]\exp[-(k_2-k_1)^2v_{te}^2(t-t_{\rm echo})^2/2]

with

t_{\rm echo}=k_2\tau/(k_2-k_1)

. ((c)) Evaluate

t_{\rm echo}

and the time on either side of it within which the echo is above

1/e

of its peak, and say why no echo follows the second pulse if

k_2<k_1

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