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Unformatted text preview: Physics 762 Homework #2 Spring ‘11
Dr. Drake 1. Show that the Z function satisﬁes the following diﬀerential equation:
dZ (ξ )
= −2(1 + ξZ (ξ )).
dξ (1) 2. Calculate the kinetic dispersion relation for ion acoustic waves in a plasma with
a Maxwellian distribution of ions. The electrons are also Maxwellian but with
a mean velocity Ue = U z . Take the wavevector parallel to Ue . Show that the
general dispersion relation is given by
(k, ω ) = 1 − 2
ω − kU
) − De Z (
kvte 3. Show that the real frequency ωr for sound waves when Te Ti and U (2)
vte is given by
ωr = k 2 c2
1 + k 2 /kDe (3) where c2 = Te /mi . Keep corrections to obtain the growth rate of the instability,
√ ωr kDe ωr − kU
Te ωr −ωr /k2 vti
γ=− π 2 2 (
Ti kvti (4) For Ti = 0 show that instability occurs for
U 2 (1 + k 2 /kDe ) > c2 .
s Hint: you will ﬁnd that ω k vti and ω − kU vte . (5) ...
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This note was uploaded on 12/29/2011 for the course PHYSICS 762 taught by Professor Drake during the Fall '08 term at Maryland.
- Fall '08