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APPENDICES
C
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AppendixA. U
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sandFoγmulas, Vectoγ Relati側s
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417
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INTRODUCTIONTO
PLASMAPHYSICS
ANDCONTROLLED
FUSION
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Volume1
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[I ・ 13]
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De白 ning
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[l ・ 15]
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l
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o
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r
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f
u
lformso
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q
.[
1
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1
5
]
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λo = 69(T/n)112m,
λ0=7430(KT/n)112m,
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THEPLASMA PARAMETER 1
.
5
[
1
‑
1
9
]
[
l
‑
1
7
]
We a
r
e now i
nap
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CRITERIAFORPLASMAS 1
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6
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33
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i
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i
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47
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a
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7
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M
o
t
i
o
n
s
48
C
h
a
p
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r
Two
TheThirdAdiabaticInvariant， φ
perpendiculare
n
e
r
g
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e
sa
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edefinedby
W ＝ ~mv~
+~mv~
= ~mv~
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(
2
‑
8
1
)
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ucanbew
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i
t
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n
vu=[(2/m)(W 一 µ.B)]112
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‑
8
2
)
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s
t
a
n
t
,andonlyB v
a
r
i
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s
.Therefore,
内
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ｵ
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B
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.
B
ｵ
.
B
2F士五五＝ 2W11 一戸了
(
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‑
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3
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snotzeroo
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h
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S
i
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rmotion:
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r
B ＝ ー・ー＝
d
r dt
V,,r"
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q
R:W ・ VB
[
2
‑
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4
)
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t
i
1
1
ｵ
.(R,× B)·VB
lmv~ （BXVB ）・ Re
v
u‑ :q
R~B2
‑ 2q
R~B2
B
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i
g
. 2・ 16, wes
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et
h
a
tt
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to
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y
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a
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t
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r
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lmagnetic
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l
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ed
r
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r
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a
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e
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ti
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o
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o
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t
a
n
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l
u
c
t
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a
t
i
o
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so
fB occuronatimes
c
a
l
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h
o
r
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h
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r
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e
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h
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esomer
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e
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ee
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i
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i
nt
h
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ot
h
e
d
r
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ttimeo
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a
r
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i
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ee
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r
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h
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a
r
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c
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e
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r
e
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r
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h
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nt
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fthephase
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h
t
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h
ewavecanbee
x
c
i
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e
dbyt
h
econversiono
fp
a
r
t
i
c
l
ed
r
i
f
tenergy
49
2
.
8
.
3
Single-Paγticle
M
o
t
i
o
n
s
t
owavee
n
e
r
g
y
.
[
2
‑
8
5
)
Thef
r
a
c
t
i
o
n
a
lchangei
nv
u8
si
s
I d
1d
8
s 1d
v
1
1
一一一（ vu ＆）＝一
v
uO
sd
t "
＋ーー－
O
Sd
t v
ud
t
2・ 13.
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FromE
q
s
.[
2
‑
8
0
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2
‑
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5
)
,wes
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et
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,however. In
Thisi
snote
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18
sbetweentheturningp
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son8s ’ do notc
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i
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r
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so
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r
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n
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i
g
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‑
1
7
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.However,anye
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r
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ri
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s
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e
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l
i
g
i
b
l
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h
eturningp
o
i
n
t
s
,v
n
e
a
r
l
yz
e
r
o
.Consequently,wehaveproved
f =…
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u
d
s
v
1
1d
s= viιand
(
a
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e
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di百erentiate
w
i
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p
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m
e
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h
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hF
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iT
h
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e
e
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e
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i
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t
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o
nbetweenB andH,weassumeM t
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t
i
o
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a
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oBorH:
ps
i
56
l
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h
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t
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den、rative i
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a
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o
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fthe 戸articles. Ont
h
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e
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wew凶 to haveane
q
u
a
t
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o
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o
rf
l
u e
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e
r
wouldbei
m
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r
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e
.Consideradrop一 of creami
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saf
l
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e
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.Ast
h
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e
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t
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e
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i
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y
.
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l
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i
delementa
taf
i
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e
ds
p
o
ti
nt
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p
,however,r
e
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o
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a
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t
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h
et
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o
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a
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a
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(
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r
t
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faf
l
u
i
di
no
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e
‑
d
i
m
e
n
s
i
o
n
a
lxspace・ The
changeo
fG w
i
t
ht
i
m
ei
naframemovingw
i
t
ht
h
ef
l
u
i
di
st
h
esumo
ftwo
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e
r
m
s
:
ft
h
ei
o
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c
l
o
t
r
o
nf
r
e
q
u
e
n
c
yi
sd
e
n
o
t
e
db
yn
,andtheionplasmafrequency
3
‑
2
.I
i
sd
e
f
i
n
e
db
y
n
.= (ne2 ／ε。品，f)l/2
whereM i
st
h
ei
o
nm
a
s
s
,underw
h
a
tc
i
r
c
u
m
s
t
a
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ed
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t
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a
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on~ ； n~ ？
3
.
3 THEFLUID EQUATIONOFMOTION
d
G
(
x
,t
) aG aGd
x aG
aG
づ「 ＝ a1 ＋ 耳石 ＝ a1 ＋ 叫xa;
Maxwell ’S e
q
u
a
t
i
o
n
st
e
l
lu
swhatE andB a
r
ef
o
rag
i
v
e
ns
t
a
t
eo
ft
h
e
p
l
a
s
m
a
. To s
o
l
v
et
h
es
e
l
f
‑
c
o
n
s
i
s
t
e
n
t problem, we musta
l
s
o have an
og
i
v
e
nE andB
.I
nt
h
ef
l
u
i
d
e
q
u
a
t
i
o
ng
i
v
i
n
gt
h
eplasma’s responset
approximation,wec
o
n
s
i
d
e
rt
h
eplasmat
obecomposedo
ftwoo
rmore
inteψenetratiπE f
l
u
i
d
s
,onef
o
reachs
p
e
c
i
e
s
.Int
h
es
i
m
p
l
e
s
tc
a
s
e
,when
t
h
e
r
ei
so
n
l
yones
p
e
c
i
e
so
fi
o
n
,wes
h
a
l
lneedtwoe
q
u
a
t
i
o
n
so
fm
o
t
i
o
n
,
onef
o
rt
h
ep
o
s
i
t
i
v
e
l
ychargedi
o
nf
l
u
i
dandonef
o
rt
h
en
e
g
a
t
i
v
e
l
ycharged
e
l
e
c
t
r
o
nf
l
u
i
d
.Inap
a
r
t
i
a
l
l
yi
o
n
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z
e
dg
a
s
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h
a
l
la
l
s
oneedane
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u
a
t
i
o
n
f
o
rt
h
ef
l
u
i
do
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e
u
t
r
a
la
t
o
m
s
.Then
e
u
t
r
a
lf
l
u
i
dw
i
l
li
n
t
e
r
a
c
tw
i
t
ht
h
e
i
o
n
sande
l
e
c
t
r
o
n
so
n
l
ythroughc
o
l
l
i
s
i
o
n
s
.Thei
o
nande
l
e
c
t
r
o
nf
l
u
i
d
s
w
i
l
li
n
t
e
r
a
c
tw
i
t
heacho
t
h
e
reveni
nt
h
eabsenceo
fc
o
l
l
i
s
i
o
n
s
,because
o
ft
h
eEandB f
i
e
l
d
st
h
e
yg
e
n
e
r
a
t
e
.
(3訓］
!hefirst ほm ont
h
er
i
g
h
trepres側S t
h
echangeo
fG a
taf
i
x
e
dp
o
i
n
t
ms
p
a
c
e
,andt
h
esecondtermr
e
p
r
e
s
e
n
t
st
h
echangeo
fGast
h
eo
b
s
e
r
v
e
r
movesw
i
t
ht
h
ef
l
u
i
di
n
t
oar
e
g
i
o
ni
nwhichG i
sd
i
f
f
e
r
e
n
t
.I
nt
h
r
e
e
]g
e
n
e
r
a
l
i
z
e
st
o
d
i
m
e
n
s
i
o
n
s
,E
q
.(3・3 I
dG aG
d
t a
t
－一＝ー＋（u·V)G
[
3
‑
3
2
]
Thisi
sc
a
l
l
e
dt
h
ec
o
n
v
e
c
t
i
v
ed
e
r
i
v
a
t
i
v
eandi
ssometimesw
r
i
t
t
e
nDG/
D
t
.
Notet
h
a
t(
uｷV
)i
sas
c
a
l
a
rd
i
f
f
e
r
e
n
t
i
a
lo
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e
r
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t
o
r
.S
i
n
c
et
h
es
i
g
no
ft
h
i
s
termi
ssometimesas
o
u
r
c
eo
fc
o
n
f
u
s
i
o
n
,weg
i
v
etwos
i
m
p
l
ee
x
a
m
p
l
e
s
.
l
e
c
t
r
i
cwaterh
e
a
t
e
ri
nwhicht
h
eh
o
tw
a
t
e
r
Figure3・ 1 showsane
h
a
sr
i
s
e
nt
ot
h
etopandt
h
ec
o
l
dwaterh
a
ssunkt
ot
h
eb
o
t
t
o
m
.LetG
(
x
,t
)
bet
h
etemperatureT;VGi
sthenupward.Consideraf
l
u
i
delement
ft
h
eh
e
a
t
e
relementi
sturnedo
n
,t
h
ef
l
u
i
d
neart
h
eedgeo
ft
h
et
a
n
k
.I
elementi
sheated ぉ it moves,andwehavedT/dt>0
.I
f
,i
na
d
d
i
t
i
o
n
,a
i
x
e
d
paddlewheels
e
t
supaf
l
o
wp
a
t
t
e
r
na
sshown,t
h
etemperaturei
naf
f
l
u
i
delementi
sloweredbyt
h
econvec出n 。f c
o
l
dwa町 from t
h
ebot~om.
I
nt
h
i
sc
a
s
e
,we haveoT/ax>0 andUx>0
,s
ot
h
a
tuｷVT>O
.The
T
/
a
t
,i
sg
i
v
e
nbyab
a
l
a
n
c
e
temperaturechangei
nt
h
ef
i
x
e
de
l
e
m
e
n
t
,a
3
.
3
.
1 TheConvectiveDerivative
Thee
q
u
a
t
i
o
no
fmotionf
o
ras
i
n
g
l
ep
a
r
t
i
c
l
ei
s
d
t
(3・30]
~hi.s 爪 however, notaco同町1t formt
ou
s
e
.I
nE
q
.（臼9), t
h
et
i
m
e
3・ 1. D
e
r
i
v
et
h
euniform司plasma l
o
w
‑
f
r
e
q
u
e
n
c
yd
i
e
l
e
c
t
r
i
cc
o
n
s
t
a
n
t
,E
q
.[3羽］，
b
yr
e
c
o
n
c
i
l
i
n
gt
h
et
i
m
ed
e
r
i
v
a
t
i
v
eo
ft
h
ee
q
u
a
t
i
o
nVｷ
D= Vｷ
（εE) = 0w
i
t
ht
h
a
t
o
ft
h
evacuumP
o
i
s
s
o
ne
q
u
a
t
i
o
n[
3
‑
1
]
,w
i
t
ht
h
eh
e
l
po
fe
q
u
a
t
i
o
n
s[
3
‑
2
4
]a
nd[
2
‑
6
7
]
.
市立＝ q(E+vxB)
qη （ E+u × B)
[
3
‑
2
9
]
Assumef
i
r
s
tt
h
a
tt
h
e
r
ea
r
enoc
o
l
l
i
s
i
o
n
sandnothermalm
o
t
i
o
n
s
.Then
a
l
lt
h
ep
a
r
t
i
c
l
e
si
naf
l
u
i
delementmovet
o
g
e
t
h
e
r
,andt
h
eaveragev
e
l
o
c
i
t
y
uo
ft
h
ep
a
r
t
i
c
l
e
si
nt
h
eelementi
st
h
esamea
st
h
ei
n
d
i
v
i
d
u
a
lp
a
r
t
i
c
l
e
一」』ー
A
PROBLEMS
du
mn 一一 ＝
d
t
ELm
Dais
v
e
l
o
c
i
t
yv
.i
:
h
efluid 叩ation i
so
b
t
a
i
n
e
ds
i
m
p
l
ybym
u
l
t
i
p
l
y
i
n
gE
q
.(
3
‑
2
9
]
byt
h
ed
e
n
s
i
t
yn:
Thismeanst
h
a
tt
h
ee
l
e
c
t
r
i
cf
i
e
l
d
sduet
ot
h
ep
a
r
t
i
c
l
e
si
nt
h
eplasma
g
r
e
a
t
l
ya
l
t
e
rt
h
ef
i
e
l
d
sa
p
p
l
i
e
de
x
t
e
r
n
a
l
l
y
.A plasmaw
i
t
hl
a
r
g
eεshields
outa
l
t
e
r
n
a
t
i
n
gf
i
e
l
d
s
,j
u
s
ta
saplasmaw
i
t
hs
m
a
l
lλD s
h
i
e
l
d
soutdcf
i
e
l
d
s
.
C
h
a
p
t
e
r
T
h
r
e
e
9 出ゐ
58
-
m
一ー司－一ーー←
meaningt
h
a
tt
h
es
a
l
i
n
i
t
yi
n
c
r
e
a
s
e
sa
tanyg
i
v
e
np
o
i
n
t
.Ofc
o
u
r
s
e
,i
fi
t
S
/
d
ti
st
o
r
a
i
n
s
,t
h
es
a
l
i
n
i
t
yd
e
c
r
e
a
s
e
severywhere,andan
e
g
a
t
i
v
etermd
beaddedt
ot
h
emiddlep
a
r
to
fE
q
.[3司34].
Asaf
i
n
a
lexample,t
a
k
eG t
obet
h
ed
e
n
s
i
t
yo
fc
a
r
snearafreeway
e
n
t
r
a
n
c
ea
trushhour.Ad
r
i
v
e
rw
i
l
ls
e
et
h
ed
e
n
s
i
t
yaroundhimi
n
c
r
e
a
s
i
n
g
a
s he approaches t
h
e crowded f
r
e
e
w
a
y
. This i
st
h
ec
o
n
v
e
c
t
i
v
e term
(
uｷ
V)C.Att
h
esamet
i
m
e
,t
h
el
o
c
a
ls
t
r
e
e
t
smaybef
i
l
l
i
n
gw
i
t
hc
a
r
st
h
a
t
e
n
t
e
rfromd
r
i
v
e
w
a
y
s
,s
ot
h
a
tt
h
ed
e
n
s
i
t
yw
i
l
li
n
c
r
e
a
s
eeveni
ft
h
eo
b
s
e
r
v
e
r
doesn
o
tmove.Thisi
st
h
ea
G
/
a
tterm.Thetotalincreaseseenbythe
o
b
s
e
r
v
e
ri
st
h
esumo
ft
h
e
s
ee
f
f
e
c
t
s
.
I
nt
h
ec
a
s
eo
fap
l
a
s
m
a
,wet
a
k
eG t
obet
h
ef
l
u
i
dv
e
l
o
c
i
t
yuand
s
w
r
i
t
eE
q
.[3 ・30] a
FIGURE3
‑
1 Motiono
ff
l
u
i
de
l
e
m
e
n
t
si
na
h
o
tw
a
t
e
rh
e
a
t
e
r
.
r
a
u
a
t
+(u V)uJ
= qn(
E+uxB)
o
ft
h
e
s
ee狂ects,
mnj
dT
一一＝ーー－ u ｷVT
a
t d
t
。T
1
[
3
‑
3
5
1
wherea
u
/
a
ti
st
h
etimed
e
r
i
¥
'
a
t
i
v
ei
na 五xed f
r
a
m
e
.
[3 ・33]
I
ti
sq
u
i
t
ec
l
e
a
rt
h
a
ta
T
/
a
tcanbemadezero,atleastforashorttime.
Asasecondexamplewemayt
a
k
eG t
obet
h
es
a
l
i
n
i
t
ySo
ft
h
ewater
fxi
st
h
eupstreamd
i
r
e
c
t
i
o
n
,t
h
e
r
e
neart
h
emoutho
far
i
v
e
r(Fig. 与 2). I
TheS
t
r
e
s
sTensor 3
.
3
.
2
Whenthermalmotionsa
r
etakeni
n
t
oa
c
c
o
u
n
t
,ap
r
e
s
s
u
r
ef
o
r
c
eh
a
st
o
beaddedt
ot
h
er
i
g
h
t
‑
h
a
n
ds
i
d
eo
fE
q
.[
3
‑
3
5
]
.Thisf
o
r
c
ea
r
i
s
e
sfromt
h
e
y
OCEAN
z
FIGURE3・2
D
i
r
e
c
t
i
o
no
ft
h
es
a
l
i
n
i
t
yg
r
a
d
i
e
n
ta
tt
h
emoutho
far
i
v
e
r
.
ノ／／
x
O
r
i
g
i
no
ft
h
ee
l
e
m
e
n
t
so
ft
h
es
t
r
e
s
st
e
n
s
o
r
. FIGURE3 ・3
上
JaUゐ
[
3
‑
3
4
]
Emω
a
s
ax
A
a
s
at
一一＝ -u笠一一＞ O
‘
bFf
i
snormallyag
r
a
d
i
e
n
to
fSsucht
h
a
tas/ax く 0. Whent
h
et
i
d
ecomes
i
n
,t
h
ee
n
t
i
r
ei
n
t
e
r
f
a
c
ebetweens
a
l
tandf
r
e
s
hwatermovesupstream,
andUx>0
.Thus
C
h
a
p
t
e
r
T
h
r
e
e
ps
60
[
3
‑
3
9
]
Equation(3喝38] nowbecomes
U山 L
where6
.
n
vi
st
h
enumbero
fp
a
r
t
i
c
l
e
sperm3w
i
t
hv
e
l
o
c
i
t
yvx ・
もγe c
anc
a
n
c
e
ltwotermsbyp
a
r
t
i
a
ld
i
f
f
e
r
e
n
t
i
a
t
i
o
n
:
6
.
n
v= 6
.
v
xJ
Sf(v,,v
,
,v
,
)d円 dv,
a
u
x'
aη
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mnat 十間＇＂ at= -muχ a;;-.
Eachp
a
r
t
i
c
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r
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i
e
samomentumm1ら・ The d
e
n
s
i
t
ynandtemperature
K Ti
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ohavet
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ecube’s
n
t
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tx
0throughA
c
e
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t
e
r
.ThemomentumPA+earnedi
i
sthen
PA+=:
2
.6.nvmv!6
.
y6
.
z= 6
.
y6.z[m；；~nlxu-"x
8πa
ae- ＋ ζ （i山χ ） = 0
[3品］
｜円KT
PB ＋＝ ムy ムz ［間；； hJxり
ん＋一九÷＝ 6
.
y6
.
zkm （（η之lxu-C.x
au~＼
a
p
¥
a
t
+Uxa
;)= ‑
a
;
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r
au
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n
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;
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6
.
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y6
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z
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I
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c
t
m
n3.3.5
一一一一一一
(
3
‑
4
3
]
(
3
‑
4
4
]
What we have d
e
r
i
v
e
di
so
n
l
yas
p
e
c
i
a
lc
a
s
e
:t
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et
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a
n
s
f
e
ro
fx
momentumbymotioni
nt
h
exd
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r
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c
t
i
o
n
;andwehaveassumedt
h
a
tt
h
e
f
l
u
i
di
si
s
o
t
r
o
p
i
c
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ot
h
a
tt
h
esamer
e
s
u
l
th
o
l
d
si
ntheyandzd
i
r
e
c
t
i
o
n
s
.
Buti
ti
sa
l
s
op
o
s
s
i
b
l
et
ot
r
a
n
s
f
e
rymomentumbymotioni
nt
h
exd
i
r
e
c
t
i
o
n
,
f
o
ri
n
s
t
a
n
c
e
.Suppose,i
nF
i
g
.3
‑
3
,t
h
a
tuヲ is z
e
r
oi
nt
h
ecubea
tx= x
0
buti
sp
o
s
i
t
i
v
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i
d
e
s
.Thena
sp
a
r
t
i
c
l
e
smigratea
c
r
o
s
st
h
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a
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e
s
A andB,t
h
e
ybringi
nmorep
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s
i
t
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v
eymomentumthant
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e
yt
a
k
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t
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andt
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l
u
i
delementg
a
i
n
smomentumi
nt
h
eyd
i
r
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c
t
i
o
n
.Thiss
h
e
a
r
i
¥
"
e
nbyat
e
n
s
o
r
s
t
r
e
s
scannotberepresentedbyascalar 合 but mustbeg
This r
e
s
u
l
tw
i
l
l be j
u
s
t doubled by t
h
ec
o
n
t
r
i
b
u
t
i
o
no
f left‑movmg
l
s
omovei
nt
h
e
p
a
r
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i
c
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s
,s
i
n
c
et
h
e
yc
a
r
r
yn
e
g
a
t
i
v
exmomentumanda
o
p
p
o
s
i
t
ed
i
r
e
c
t
i
o
nr
e
l
a
t
i
v
et
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h
eg
r
a
d
i
e
n
to
f
The刷al changeo
f
oi
st
h
e
r
e
f
o
r
e
momentumo
ft
h
ef
l
u
i
delementa
tx
a
[
3
‑
4
2
]
Thisi
st
h
eu
s
u
a
lp
r
e
s
s
u
r
e
‑
g
r
a
d
i
e
n
tf
o
r
c
e
.Addingt
h
eelect附nagn附
eti
f
o
r
c
e
sandg
e
n
e
r
a
l
i
z
i
n
gt
ot
h
r
e
ed
i
m
e
n
s
i
o
n
s
,wehavet
h
ef
l
u
i
dequation
[
3
‑
3
7
]
~m(-6.x ）；.（♂）
a
x
I
mn ｛~Ux
=6
.
y6
.
z
隅ナ（ηVx)
[
3
‑
4
1
]
wehavef
i
n
a
l
l
y
Thust
h
en
e
tg
a
i
ni
nxmomentumfromright司moving p
a
r
t
i
c
l
e
si
s
n
[
3
‑
4
0
]
a
l
l
o
w
su
st
oc
a
n
c
e
lt
h
etermsn
e
a
r
e
s
tt
h
ee
q
u
a
ls
i
g
ni
nE
q
.(
3
‑
4
0
]
.Defining
t
h
ep
r
e
s
s
u
r
e
一宮
(
n
m
u
x
)6
.
x6
.
y6
.
z=
a
t
a
;
Thee
q
u
a
t
i
o
no
fmassc
o
n
s
e
r
v
a
t
i
o
n
*
Thesumover6
.
n
vr
e
s
u
l
t
si
nt
h
eaverageV
xovert
h
ed
i
s
t
r
i
b
u
t
i
o
n
.The
comesfromt
h
ef
a
c
tt
h
a
to
n
l
yh
a
l
ft
h
ep
a
r
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i
c
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e
si
nt
h
ecubea
t
f
a
c
t
o
rk
x
0‑6
.
xa
r
egoingto叩ard f
a
c
eA.S
i
m
i
l
a
r
l
y
,t
h
emomentumc
a
r
r
i
e
dout
throughf
a
c
eB i
s
二＇...
a
u
. a
m
n
u
x
‑
;
;
:
‑‑ (nKT)
一（
一一一←一一一一
閉山
I
E
~ (nmux)=‑m‑/;[n 応札＋之）］ =-m 子 rn(u;
＋~）
l
¥
m JJ
2
白山ゐ
6
.
n
vV
x6
.
y6
.
z
I
2
m
v
x
r=2KT
A
whereU
xi
st
h
ef
l
u
i
dv
e
l
o
c
i
t
yandV
x
ri
st
h
erandomthermalv
e
l
o
c
i
t
y
.For
aone‑d1mensionalMaxwelliand
i
s
t
r
i
b
u
t
i
o
n
,wehavefromE
q
.(
1
‑
7
]
nU5
randommotiono
fp
a
r
t
i
c
l
e
si
nandouto
faf
l
u
i
delementanddoesn
o
t
.
x6
.
y6
.
z
appeari
nt
h
eequationf
o
ras
i
n
g
l
ep
a
r
t
i
c
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e
.Letaf
l
u
i
delement6
becentereda
t(
x
0
,~6.y, ~6.z) (
F
i
g
.3・3). Fors
i
m
p
l
i
c
i
t
y
,wes
h
a
l
lc
o
n
s
i
d
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r
o
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l
yt
h
excomponento
fmotionthrought
h
ef
a
c
e
sA andB.Thenumber
o
fp
a
r
t
i
c
l
e
spersecondp
a
s
s
i
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gthrought
h
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a
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eA w
i
t
hv
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．加
62
C
h
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t
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fr
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t
a
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t
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s
u
l
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gf
o
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r
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e
na
s
‑mn(u‑u
o
)
/
r
.Thee
q
u
a
t
i
o
no
fmotion (
3
‑
4
4
]can beg
e
n
e
r
a
l
i
z
e
dt
o
i
n
c
l
u
d
ea
n
i
s
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r
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o
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sa
sf
o
l
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o
w
s
:
rau
1
明η （ u-uo)
LOL
」
T
mnJ で＋ (
uｷV)uI
= qn(
E+uX B)‑VｷP 一一一一一一－
、vntten
lρ0
0
¥
P OJ
¥
O o pI
P=IO
C
o
l
l
i
s
i
o
n
sbetweenchargedp
a
r
t
i
c
l
e
shaven
o
tbeeni
n
c
l
u
d
e
d
;t
h
e
s
ew
i
l
l
b
‑
:
:t
r
e
a
t
e
di
nChapter5
.
[
3
‑
4
5
]
V.pi
sj
u
s
tV
p
.I
nS
e
c
t
i
o
n1
.
3
,wenotedt
h
a
taplasmacouldhavetwo
t
e
m
p
e
r
a
t
u
r
e
sTム and T
1
1i
nt
h
epresenceo
famagnetic 自eld. Int
h
a
tc
a
s
e
,
t
h
e
r
ewould be two p
r
e
s
s
u
r
e
sP ょ ＝ nKTょ and P
1
1= n
K
T
1
1
. Thes
t
r
e
s
s
ComparisonwithOrdinaryHydrodynamics 3
.
3
.
4
Ordinaryf
l
u
i
d
sobeyt
h
eNavier‑Stokese
ｷ
q
u
a
t
i
o
n
tt ， EEE
OO
li、
t
P［~ +(uｷV)uJ
= ‑Vp+pv¥
7
2
u
[
3
‑
4
6
]
tf
－－
九州
’’’’
一一
114
aFeaEhEth
』6』
n
r
oho
hoo
t
e
n
s
o
ri
st
h
e
n
[
3
‑
4
7
]
[
3
‑
4
8
]
Thisi
st
h
esamea
st
h
eplasmae
q
u
a
t
i
o
n(
3
‑
4
7
)e
x
c
e
p
tf
o
rt
h
eabsence
o
fe
l
e
c
t
r
o
m
a
g
n
e
t
i
cf
o
r
c
e
sandc
o
l
l
i
s
i
o
n
sbetweens
p
e
c
i
e
s(
t
h
e
r
ebeing
o
n
l
yones
p
e
c
i
e
s
)
.Thev
i
s
c
o
s
i
t
ytermpvV2u
,whereνis t
h
ek
i
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e
m
a
t
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c
h
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v
i
s
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t
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o
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f
f
i
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n
t
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u
s
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o
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l
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i
o
n
a
lp
a
r
to
fVｷP‑V台 in t
o
fmagneticf
i
e
l
d
s
.Equation(
3
‑
4
8
)d
e
s
c
r
i
b
e
saf
l
u
i
di
nwhicht
h
e
r
ea
r
e
frequentc
o
l
l
i
s
i
o
n
sbetweenp
a
r
t
i
c
l
e
s
.Equation(
3
‑
47
]
,ont
h
eo
t
h
e
rhand,
wasd
e
r
i
v
e
dwithoutanye
x
p
l
i
c
i
tstatemento
ft
h
ec
o
l
l
i
s
i
o
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a
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i
n
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t
h
etwoe
q
u
a
t
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o
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sa
r
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t
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c
a
le
x
c
e
p
tf
o
rt
h
eE andB t
e
r
m
s
,canE
q
.
(
3
‑
47
]r
e
a
l
l
yd
e
s
c
r
i
b
eaplasmas
p
e
c
i
e
s
?Theansweri
saguardedy
e
s
,
andt
h
er
e
a
s
o
n
sf
o
rt
h
i
sw
i
l
lt
e
l
lu
st
h
el
i
m
i
t
a
t
i
o
n
so
ft
h
ef
l
u
i
dt
h
e
o
r
y
.
Int
h
ed
e
r
i
v
a
t
i
o
no
fE
q
.[
3
‑
4
7
)
,wed
i
da
c
t
u
a
l
l
yassumei
m
p
l
i
c
i
t
l
y
t
h
a
tt
h
e
r
ewerec
o
l
l
i
s
i
o
n
s
.Thisassumptioncamei
nE
q
.(
3
‑
3
9
]whenwe
took t
h
ev
e
l
o
c
i
t
yd
i
s
t
r
i
b
u
t
i
o
nt
o be M
a
x
w
e
l
l
i
a
n
. Such a d
i
s
t
r
i
b
u
t
i
o
n
g
e
n
e
r
a
l
l
ycomesabouta
st
h
er
e
s
u
l
to
ffrequentc
o
l
l
i
s
i
o
n
s
.However,t
h
i
s
assumptionwasusedonlyt
ot
a
k
et
h
eaverageo
fv ；，・ Any o
t
h
e
rdistribu・
t
i
o
nw
i
t
ht
h
esameaveragewouldg
i
v
eu
st
h
esameanswer.Thef
l
u
i
d
t
h
e
o
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y
,t
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e
r
e
f
o
r
e
,i
sn
o
tv
e
r
ys
e
n
s
i
t
i
v
et
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e
v
i
a
t
i
o
n
sfromt
h
eMaxwellian
d
i
s
t
r
i
b
u
t
i
o
n
,althought
h
e
r
ea
r
ei
n
s
t
a
n
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si
nwhicht
h
e
s
ed
e
v
i
a
t
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o
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r
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i
m
p
o
r
t
a
n
t
.K
i
n
e
t
i
ctheorymustthenbeu
s
e
d
.
Therei
sa
l
s
oane
m
p
i
r
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c
a
lo
b
s
e
r
v
a
t
i
o
nbyI
r
v
i
n
gLangmuirwhich
h
e
l
p
st
h
ef
l
u
i
dt
h
e
o
r
y
.Inworkingwitht
h
ee
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e
c
t
r
o
s
t
a
t
i
cprobeswhich
bearh
i
sname,Langmuird
i
s
c
o
v
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r
e
dt
h
a
tt
h
ee
l
e
c
t
r
o
nd
i
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i
b
u
t
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o
nf
u
n
c
ｭ
t
i
o
nwasf
a
rmoren
e
a
r
l
yMaxwellianthancouldbeaccountedf
o
rbyt
h
e
c
o
l
l
i
s
i
o
nrate・ This phenomenon,c
a
l
l
e
dLangmuir ’s p
a
r
a
d
o
x
,hasbeen
wheret
h
ec
o
o
r
d
i
n
a
t
eo
ft
h
et
h
i
r
d.
r
o
worcolumni
st
h
ed
i
r
e
c
t
i
o
no
fB
.
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ss
t
i
l
ld
i
a
g
o
n
a
landshowsi
s
o
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o
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.
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ewavefrequencyv
s
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nt
h
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s
[
4
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6
1
]
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nt
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4
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t
[
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4
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c
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ex
.wecanusetheBoltzmannrelationforelectrons.Inlinearized
form,t
h
i
si
s
x
M~ ＝－eVφ1 +ev;1ﾗ Bo
[
4
‑
6
3
]
[
4
‑
6
2
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Plasm邸
？「
112
C
h
a
p
t
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r
F
o
u
r
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~－－：：三三千戸
FIGURE4‑24 S
c
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3
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‑
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‑
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a
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r
,wehave
Eqs 目
1 －~） ＝一明（ 1 －~）
M(
川河） = m w~ +MD.~ = e叫十卦
2
4
6
8
10
B(kG)
FIGURE4
‑
2
5 M
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[
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1
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r
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4.11 THELOWERHYBRID FREQUENCY
2~2
w2＝~手－＝
[
4
‑
7
l
a
]
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o
w
‑
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n
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i
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113
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114
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4
.
1
2 ELECTROMAGNETIC WAVES WITH Bo=0
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nt
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o
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i
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o
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f
o
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)
:
。v••
m －」ニ＝
[
4
‑
7
2
]
。t
c2VxB1=E
1
c2Vx(VxB
1
)=V x 亘 1 ＝一丞 l
[4・74]
Againassumingp
l
a
n
e
swavesv
a
r
y
i
n
ga
sexp[i(kx ー wt)], wehave
c
2
[
k
(
kｷ
B
i
)‑k
2
B
i
]
w2B1= ‑c2kx(
kﾗBi)=‑
[
4
‑
7
5
]
S
i
n
c
ekｷBi=‑iVｷBi=0byanothero
fMaxwell ’S e
q
u
a
t
i
o
n
s
,t
h
er
e
s
u
l
t
i
s
w2=fl
k
2c2
[4・ 76]
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st
h
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e
l
o
c
i
t
yw/ko
fl
i
g
h
tw
a
v
e
s
.
,E
q
.[4・72] i
sunchanged,butwemustadd
I
naplasmaw
i
t
hBo=0
oE
q
. [4・73] t
oaccountf
o
rc
u
r
r
e
n
t
sdue t
o first・order
atermji ／εo t
chargedp
a
r
t
i
c
l
em
o
t
i
o
n
s
:
[
4
‑
8
3
]
e
E
1
V,i ＝で一一一
imw
TL
(
4
‑
7
7
]
E,
ｷ
(
4
‑
7
8
]
w
h
i
l
et
h
ec
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r
lo
fE
q
.[
4
‑
7
2
]i
s
Vx(VﾗE
i
)=V(V.E
i
)‑V2Ei=‑
vxBi
(
4
‑
7
9
]
E
l
i
m
i
n
a
t
i
n
gV xBiandassuminganexp[
i(
kｷ
r 一 wt)] d
ependence,we
have
2
-k(k·Ei）吋 2Ei ＝と吉ji ＋与Ei
C
ε oC
(4・801
Byt
r
a
n
s
v
e
r
s
ewaveswemeankｷE
i=0
,s
ot
h
i
sbecomes
(
w
2‑c
2
k
2
)
E
i= -iwji ／ε0
(4・ 81]
2
'2
[
4
‑
8
5
]
Thisi
st
h
ed
i
s
p
e
r
s
i
o
nr
e
l
a
t
i
o
nf
o
re
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t
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g
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e
t
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c!
L
'
a
v
e
sp
ropagating
i
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i
t
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i
e
l
d
.Wes
e
et
h
a
tt
h
evacuumr
e
l
a
t
i
o
n
[
4
‑
7
6
]i
smodifiedbyaterm r
e
m
i
n
i
s
c
e
n
to
fplasmao
s
c
i
l
l
a
t
i
o
n
s
.The
phasev
e
l
o
c
i
t
yo
fal
i
g
h
twavei
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r
e
a
t
e
rthant
h
ev
e
l
o
c
i
t
yo
f
l
i
g
h
t
:
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!
+
>
Pし
at
2
ωρ ＋ ck
一一
ε。
2＝
C
‘
ω
一一
1a
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o
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i
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i
g
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‑
4
5
)
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q
.[
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]
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componento
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ε。（w2 ‑c2k2)E1= -iwη 0e(v;, ‑v
,
.
)
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4
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1
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m w w,
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1
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l － ~f1E1
I
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1
2
2
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9
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w ‑c/
。；
2
ε 0M e·Br,
2
c
w
2
c
11 ＝ 了τ石五五百＝ 1+(pµ，。／ B'f,)c2
c
c
(
4
‑
1
2
4
)
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o
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k
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/
k
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4
‑
1
2
8
]
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v
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oE
q
s
.[4‑120)and[4‑121).Thus,t
h
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l
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t
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5
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di
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g
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w
/
k
)
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/B
0
I
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h
i
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h
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c
c
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d
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n
gt
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q
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sj
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l
u
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o
c
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y!
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x
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0
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h
ef
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h
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l
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i
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e
so
s
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ω ザ the particles 叫ere s
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regardeda
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‑
2
8
)
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quation [4・ 124)
low‑frequency p
e
r
p
e
n
d
i
c
u
l
a
r motions (
E
q
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s
i
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p
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c
t
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medium:
αJ
vxE,=‑Bi
(
4
‑
1
2
3
)
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h
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r
t
h
e
rassumptionw2 < n;; hydromagnetic
waveshavef
r
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q
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139
(
4
‑
1
2
5
]
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2
6
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εR ＝ ε／ε。＝ 1+(c2/v~ ）
B0
x
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/
[4圃 127]
Notet
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ｭ FIGURE4‑46
a
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igure4
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9shows measurements o
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Therefore,t
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n
1
143
W
a
v
e
si
n
k
[
4
‑
1
3
2
]
－－＝一－ v,,
π。
α3
s
ot
h
a
tE
q
.[
4
‑
1
3
1
]becomes
U同＝一一一－
v.
Bn ＋ ーτ
,
.
Mw
日＂
w<
[
4
‑
1
3
3
]
v.
り
問一
M
可Vith
M
t
h
i
sbecomes
v；，（ ト A ）＝ 一旦~v目
(
4
‑
1
3
4
]
α3
Mπ 。一一＝
a
t
t
h
ea
b
b
r
e
v
i
a
t
i
o
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AR
一ぷ
OV;1
もγith
A
F
i
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o
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i
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rlow‑frequencye
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s
sB
o
.Againwemayt
a
k
eBo=B0zandE
1=£1 圭， but wenowl
e
t
k=kチ（ Fig. 4
‑
5
0
)
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eVptermi
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h
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o
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s
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k2γiKT;
i
e
4.19 MAGNETOSONIC WAVES
e
n
o
(
E
1+Vi1
× Bo）一 γ；KT;Vn,
(4・ 129]
Combiningt
h
i
sw
i
t
hE
q
.[
4
‑
1
3
0
]
,wehave
i
n
,I i
f
)
,
¥
ourc
h
o
i
c
eo
fE
1andk
,t
h
i
sbecomes
MW
U悶＝ τ与一 （Eχ ＋ v;,Bo)
[
4
‑
1
3
1
]
W
＼
αJ
I
(
4
‑
1
3
5
]
••(I 一山）＝日
1‑A
Mw
(
4
‑
1
3
0
]
ルfα3
ie
白 k y、KTi 札、
＝一一 （－ v,.B0） 十一ーでアー
JI;,氏。
w
M
no
,
Vix ＝てτ－ E. ＋一一よ l 一一一）（ 1-A ）「 U悶
Thisi
st
h
eo
n
l
ycomponento
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;1wes
h
a
l
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et
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yn
o
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h
ewavee
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u
a
t
i
o
n[
4
‑
8
1
]i
s
仁omponent
ε0(w2 ‑c
2
k
2
)
E
.= -iwη0e(v;, ‑v
,
.
)
z
[4・ 136]
Too
b
t
a
i
nU口’ we needo
n
l
yto maket
h
ea
p
p
r
o
p
r
i
a
t
echangesi
nE
q
.
[
4
‑
1
3
5
]andt
a
k
et
h
el
i
m
i
to
fs
m
a
l
le
l
e
c
t
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o
nm
a
s
s
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ot
h
a
tw2 《 w~ and
w2 < k2v;h,:
i
e w2{
. k2γ，KT，＼ 【
ik 2γ，KT， 【
1 一一苛一一一一 u：，一歩一ーーーτ 一一一一－！：， 守
. mww;¥ ω ゐ間 I ‑
wBo e
‑
v.
.＝一一一一一一百ー I
(
4
‑
1
3
7
]
y
P
u
t
t
i
n
gt
h
el
a
s
tt
h
r
e
ee
q
u
a
t
i
o
n
st
o
g
e
t
h
e
rwehave
k
9
x
FIGURE4
‑
5
0 Geometry o
f am
a
g
n
e
t
o
s
o
n
i
c wave
p
r
o
p
a
g
a
t
i
n
ga
tr
i
g
h
ta
n
g
l
e
st
oB0・
9
o
ri
e
I
1 A
¥
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'
k
'
)
E
,= -ituηoel-E.I
• • )
L
M
t
u \1-A ー（fl.~ ／ w")/
+ik2ぜ γ，KT, ~＿l
一一一
wB0 eM ‑
J
(
4
‑
1
3
8
]
Pl邸明白
写事竺
145
s
-
市山一
M
Y一
¥
-
K一
H,
g
＋一
M
βqv A ノ
-
K一
、
e
Plasm出
T一
9
o of
γl KT,¥
I o
。剤-KT\
ザ－ c
"
k
"
(1＋七ァγ ）＋オ（ w"‑k" 竺ー）＝ O
b
’民
n
:
l
V
l
γ一
一一
VA
2
'"
9
一一
k2c2γ，KT,
2
(w -ck ）＝ ーポ ω （ト A ）＋ 一γ てア
Wavesi
n
2S
U
zR
ω
9h
Bo= 0ork
I
I
B
o
:
け
z-
F、
rι
v
γ，
e、
u
o
t
u
(
t
a
t
nU
t
e
f
e
u
a
Aυ
。！
伺H
n;,
Wes
h
a
l
la
g
a
i
nassumew2 <
s
ot
h
a
t1‑ Acanben
e
g
l
e
c
t
e
dr
e
l
a
t
i
v
e
t
o!l~／w2. Witht
h
eh
e
l
po
ft
h
edefinit問1s o
f!
l
pandvA
,wehave
叫
Fouγ
ri
144
Cha争teγ
；
(
A
c
o
u
s
t
i
cw
a
v
e
s
) [
4
‑
1
4
5
]
[
4
‑
1
3
9
]
w2= n~
k ム Bo:
I
(
E
l
e
c
t
r
o
s
t
a
t
t
c1
0
0
4
‑
1
4
6
]
c
l
o
t
r
o
nw
a
v
e
s
) [
+k
2
v
'
f
C)
S
i
n
c
e
。；/!l~ = c
2/v~
o
r
[4・ 140]
E
q
.[
4
‑
1
3
9
]becomes
W
て l ＋~）＝ c2k2(1 ＋叫 γぷT,) =c2
＼凡1VA
VA
}
＼凡／
[
4
‑
1
4
1
]
Electroπ 即日 ves
(
L
o
w
e
rh
y
b
r
i
d
2
,
.
.
.
= W t = H,W,
OS口 llations)
w
!+k2c2
cRα）；
7す ＝ c
k
k ム Bo, E1
I
IBo: ーす＝ 1 一号
2' 2
2v
s 十 UA
[4・ 142]
十 UA
2
,
2
R
白J
2 2
h
ωωω
Thisi
st
h
ed
i
s
p
e
r
s
i
o
nr
e
l
a
t
i
o
nf
o
rt
h
emagnetosoη ic wave propagating
perpendiculart
oB
0
.I
ti
sana
c
o
u
s
t
i
cwavei
nwhicht
h
ecompressions
andr
a
r
e
f
a
c
t
i
o
n
sa
1
eproducednotbymotionsalongE
,butbyEﾗB
d
r
i
f
t
sa
c
r
o
s
sE
.Int
h
el
i
m
i
tB0• O,vA • 0, t
h
emagnetosonicwavet
u
r
n
s
o
na
c
o
u
s
t
i
cwave. In t
h
el
i
m
i
tKT • 0, v , • 0, t
h
e
i
n
t
oan ordin且ry i
p
r
e
s
s
u
r
eg
r
a
d
i
e
n
tf
o
r
c
e
sv
a
n
i
s
h
,andt
h
ewavebecomesamodi自ed Alfven
wave.Thephasev
e
l
o
c
i
t
yo
ft
h
emagnetosonicmodei
salmosta
l
w
a
y
s
;f
o
rt
h
i
sr
e
a
s
o
n
,i
ti
so
f
t
e
nc
a
l
l
e
dsimplythe “ fast ” hydro”
l
a
r
g
e
rthanvA
magnetICwave.
k 上 Bo:
2
2
2
U
p
p
e
rh
y
b
r
i
d
2 (
= W p +W , = W h o
s
c
i
l
l
a
t
i
o
n
s
)
1
2
'2
Wp/W
7-1 一戸<;;;:M
c
2
k
2
α）
;
:
;
2
I αJ
1+(w,/w)
w
Ioπ 山aves
t.\wa、 e)
[
4
‑
1
5
0
]
(R w
a
v
e
)
(
w
h
i
s
t
l
e
rm
o
d
e
)
[
4
‑
1
5
1
]
(Lw
a
v
e
)
[
4
‑
1
5
2
]
(
e
l
e
c
t
r
o
m
a
g
n
e
t
i
c
)
Bo=0
:
None
W
2=(?.VA
'
22
(
A
l
f
v
e
nwa 、 e)
[
4
‑
1
5
3
]
。。
k 上 B ，》：
(
e
l
e
c
t
r
o
s
t
a
t
i
c
)
W
[
4
‑
1
4
9
]
一 Wh
一-r= 1 一一一」L一一一
4
.
2
0 SUMMARY OFELEMENTARYPLASMAWAVES
(
P
l
a
s
m
aoscilla <ions)
2
'2
C /
?
. ̲
kI
I
B
o
:
k
l
l
B
o
:
Bo= 0ork
I
I
B
o
: w2=w ！＋ ~k2v
(0w
a
v
e
)
2
＂＇白)n
k.LB0.E1 よ Bo ：ーす＝ 1--i 寸一ーす
Electron 叩aves
[4・ 148]
αJαJ
ーす一－－；；－
C
(
L
i
g
h
twa 、 es)
2
9. 9
2
w
[
4
‑
1
4
7
]
(
e
l
e
c
t
r
o
m
a
g
n
e
t
i
c
)
w2=
Bo=0 ・
wherev
,i
st
h
ea
c
o
u
s
t
i
cs
p
e
e
d
.F
i
n
a
l
l
y
,wehave
2
竺ー＝ ,2:1!.£土主主
k
2 " c2+v
4
‑
1
5
4
]
(
M
a
g
n
e
t
o
s
o
n
i
cw
a
v
e
) [
Thiss
e
to
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i
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r
s
i
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t
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ys
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o
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yt
h
ep
r
i
n
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i
p
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i
r
e
c
t
i
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r
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1
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V ﾗE =‑
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207
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g
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KT;n~ ，
V
;
o=Vo ；＝一τ：－－
y
en0no
[
6
‑
5
6
]
KT,n
b
,
V
,
o=Vo，＝一一τァ一一 y
[
6
‑
5
7
]
eno
η。
Fromoure
x
p
e
r
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n
c
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l
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r
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rVo，・ We s
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h
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athermodynamice
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.
5
)
:
t
i
o
n4.10 ）目 They w
6
.
8 RESISTIVEDRIFTWAVES
n
1
/
n
o= eφ 1/KT,
As
i
m
p
l
eexampleo
fau
n
i
v
e
r
s
a
li
n
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t
a
b
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l
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s
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a
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e
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c
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r
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r
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t
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t
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i
f
twaveshaveas
m
a
l
lbutf
i
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i
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e
componento
fk alongB
0
. The c
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t
a
n
td
e
n
s
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r
f
a
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s
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r
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f
o
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,
[
6
‑
5
8
]
Atp
o
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n
tA i
nF
i
g
.6
‑
1
4t
h
ed
e
n
s
i
t
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sl
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r
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e
rthani
nequilibrium ， ηI i
s
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m
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,n1andφi a
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s
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e
g
a
t
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.Thed
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np
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1
一..－－..・ー
FIGURE6
‑
1
2
216
I
fgi
sac
o
n
s
t
a
n
t
,Vow
i
l
lbea
l
s
o
;and(v0 ・ V)vo v
a
n
i
s
h
e
s
.Takingt
h
ec
r
o
s
s
producto
fE
q
.(
6
‑
3
6
]w
i
t
hB
0
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i
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e
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t
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o
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i
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M gﾗBo
, Notet
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a
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,however,i
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nv
0
.Forp
e
r
t
u
r
b
a
t
i
o
n
so
ft
h
eformexp[
i(ky ー wt)], wehave
1
Jf(w‑k
v
0
)
v
1= i
e
(
E
1+v
1X Bο）
g•
vo=-;B下＝一石； Y
[
6
‑
3
7
]
Thee
l
e
c
t
r
o
n
shaveano
p
p
o
s
i
t
ed
r
i
f
twhichcanben
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g
l
e
c
t
e
di
nt
h
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i
m
i
t
m/M • 0. Therei
snodiamagneticd
r
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f
tbecauseK T =0
,andnoEaxBo
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r
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f
tbecauseE0= 0
.
I
far
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p
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l
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h
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a
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ogrow(
F
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g
.6
‑
1
1
)
.
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.6
‑
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1
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1ﾗBo
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o
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th
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p
p
l
egrows
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e
s
u
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to
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s
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egrowthr
a
t
e
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h
eu
s
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a
ll
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e
a
r
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a
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o
rwavespropagatingi
ntheyd
i
r
e
c
t
i
o
n
:k= k タ The perturbed
i
o
nequationo
fmotioni
s
M(n0 ＋ ηillLa
主（vo
+v
1
)+(
v
o+v1）・ V(vo +vi)I
t
I
= e （η。＋ ni
)
[
E
1+(
v
o+v1 )× Bo] +M(no+n1
)
g
n
;ｻ(w‑ kvo)2
Af(n0+ni)(v0 ・ V)vo = e （目。＋ ηilvo X Bo+M(no+n1 ） 冨［6・39]
S
u
b
t
r
a
c
t
i
n
gt
h
i
sfromE
q
.[
6
‑
3
8
]andn
e
g
l
e
c
t
i
n
gsecond‑ordert
e
r
m
s
,we
have
[6・40]
[6・42]
t
h
es
o
l
u
t
i
o
ni
s
E,
vx ＝一－
Bo
w ‑kvoE,
v = -z 一一一一一ーー
η!l,
Bo
[6 ” 43]
Thel
a
t
t
e
rq
u
a
n
t
i
t
yi
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o
l
a
r
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r
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f
ti
nt
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o
nframe.Thec
o
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s
ｭ
pondingq
u
a
n
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re
l
e
c
t
r
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n
sv
a
n
i
s
h
e
si
nt
h
el
i
m
i
tm/1\lf • 0. Fort
h
e
e
l
e
c
t
r
o
n
s
,wet
h
e
r
e
f
o
r
ehave
v"=E,/B0
v
,
,=0
[
6
‑
4
4
]
Theperturbede
q
u
a
t
i
o
no
fc
o
n
t
i
n
u
i
t
yf
o
ri
o
n
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s
a
n
,
a
t
Wenowm
u
l
t
i
p
l
yE
q
.(6・36] by1+（π 1/n0) too
b
t
a
i
n
Mno［守＋何· V)v1]= eno(E1＋い Bo)
Thisi
st
h
esamea
sE
q
.(
4
‑
9
6
]e
x
c
e
p
tt
h
a
tw i
sr
e
p
l
a
c
e
dbyw ‑k
v
0
,and
e
l
e
c
t
r
o
nq
u
a
n
t
i
t
i
e
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r
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e
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e
dbyi
o
nq
u
a
n
t
i
t
i
e
s
.Thes
o
l
u
t
i
o
n
,t
h
e
r
e
f
o
r
e
,
i
sg
i
v
e
nbyE
q
.(
4
‑
9
8
]w
i
t
ht
h
ea
p
p
r
o
p
r
i
a
t
ec
h
a
n
g
e
s
.ForE,= 0and
一一＋
[6司38]
[
6
‑
4
1
]
V (nova)+(vυ·V)n1
+(v1
・ V)n0
十 n1V· vυ
+noV V
1+V ｷ(
n
1
v
1
)= 0
[
6
‑
4
5
]
Thezeroth” order termv
a
n
i
s
h
e
ss
i
n
c
ev
0i
sperpendiculartoVn0,and
t
h
en1V ｷv
0termv
a
n
i
s
h
e
si
fv
0i
sc
o
n
s
t
a
n
t
.Thef
i
r
s
t
‑
o
r
d
e
re
q
u
a
t
i
o
ni
s
,
t
h
e
r
e
f
o
r
e
,
‑iwn1+ikv0n1+v悶η~
+ikn0v;,= 0
[
6
‑
4
6
]
wherenb=c
i
n
0
/a
x
.Theelectronsfollowasimplerequation,sincev
,
0=O
andu町＝ 0
:
-zwn1 十 V.xnb = Q
[
6
‑
4
7
]
Notet
h
a
twehaveusedt
h
eplasmaapproximationandhaveassumed
n
;1= n
,Iｷ T
hisi
sp
o
s
s
i
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l
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n
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t
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h
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q
u
a
t
i
o
n
s[
6
‑
4
3
]and(
6
‑
4
6
]y
i
e
l
d
E旬ー
w ‑kvn E.屯
(w‑k
v
o
)
n
1+z 」川 ＋ ikn0 －一一」ーよ＝ 0
Bo
!
l
, Bo
[
6
‑
4
8
]
E
q
u
a
t
i
o
n
s[
6
‑
4
4
]and(
6
‑
4
7
]y
i
e
l
d
~.，《
FIGURE6‑11
wn1+i 云ム πo=U
Do
ι.
iwn,
云.z. ＝－，よ
Doπ 。
[
6
‑
4
9
]
217
E
q
u
i
l
i
b
r
i
u
m
andS
t
a
b
i
l
i
t
y
寸－
DENSE
FIGURE 6・ 13
LESSDENSE
Geometryo
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h
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i
c
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i
f
tw
a
v
e
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‑
1
4
h
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e
n
s
i
t
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f
o
f our p
r
e
v
i
o
u
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s
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m
p
t
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r
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c
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r
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e
n
s
i
t
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r
t
i
c
l
e
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6
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0
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g
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e
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e
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i
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.[
6
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0
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s
[
6
‑
5
9
]and[6 ” 58], wecanw
ik，φI
.
Bo
. eφl
KT,
~
~
[
6
‑
6
1
]
Thuswehave
(
6
‑
5
9
]
w
Wes
h
a
l
lassumevIχdoes notv
a
r
yw
i
t
hxandt
h
a
tk
,i
smuchl
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s
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,
;t
h
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s
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h
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l
u
i
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c
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l
l
a
t
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c
o
m
p
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s
s
i
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c
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.Consider
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6
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一一一」←一一 一
245
K
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マ「
246
Cha1やter
S四肌
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7
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however, h
a
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F
i
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‑
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247
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v
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n FIGURE7
‑
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8
v= v~ c
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‑
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9
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7
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J
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‑
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7 Customaryp
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248
C
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e
r
g
y
.Theaverageenergychangei
sthen
Chapteγ
('1Wk)= ~m （ηuvi
S回開
W
,
k
+2u司 1V1)
2
5
3
K
i
n
e
t
i
cTheoη
[
7
‑
8
1
]
FromE
q
.(
7
‑
7
7
]
,wehave
20
E二
一U
R
一，
一m
2e一ω
一一
r、、
，
n
21
η
U
u
-9
[7・ 82]
t
h
ef
a
c
t
o
r~ r
e
p
r
e
s
e
n
t
i
n
g(
c
o
s
2(kx ー wt)). S
i
m
i
l
a
r
l
y
,fromE
q
.(
7
‑
7
9
]
,we
have
wko
2 2
Fマハ h叫
2叫（百 1V1)=nu~二寸
m (w-ku ）ー
[
7
‑
8
3
]
u
‑
l
v
1I u u
+
l
v
1I
C
o
n
s
e
q
u
e
n
t
l
y
,
2£~
「
mL(w‑ku"L
The q
u
a
d
r
a
t
i
c r
e
l
a
t
i
o
n b
e
t
w
e
e
n FIGURE7
‑
2
3
k
i
n
e
t
i
ce
n
e
r
g
yandv
e
l
o
c
i
t
yc
a
u
s
e
sa
s
y
m
m
e
t
r
i
cv
e
l
o
c
i
t
yp
e
r
t
u
r
b
a
t
i
o
nt
o
g
i
v
er
i
s
et
o an i
n
c
r
e
a
s
e
da
v
e
r
a
g
e
e
n
e
r
g
y
.
2ku 1
(w‑ku)J
('1W.)= mnu 寸←」アτl ！＋一一一一 1
4
2 2
nueEo w +ku
4 m (w‑ku)3
[7・ 84]
Thisr
e
s
u
l
tshowst
h
a
t(
'
1Wk)dependsont
h
eframeo
ft
h
eo
b
s
e
r
v
e
r
andt
h
a
ti
tdoesnotchanges
e
c
u
l
a
r
l
ywitht
i
m
e
.Considert
h
ep
i
c
t
u
r
eo
f
af
r
i
c
t
i
o
n
l
e
s
sb
l
o
c
ks
l
i
d
i
n
goveraw
a
s
h
b
o
a
r
d
‑
l
i
k
es
u
r
f
a
c
e(
F
i
g
.7
‑
2
2
)
.I
n
ki
sp
r
o
p
o
r
t
i
o
n
a
lto ー （ku )
‑
2
,a
ss
e
e
nby
t
h
eframeo
ft
h
ewashboard,1
1w
nE
q
.(7司84]. I
ti
si
n
t
u
i
t
i
v
e
l
yc
l
e
a
rt
h
a
t(
1
)(
Af竹） i
sn
e
g
a
t
i
v
e
,
t
a
k
i
n
gw = 0i
s
i
n
c
et
h
eb
l
o
c
kspendsmoret
i
m
ea
tt
h
epeaksthana
tt
h
ev
a
l
l
e
y
s
,and
(
2
)t
h
eb
l
o
c
kdoes n
o
tg
a
i
no
rl
o
s
eenergyon t
h
ea
v
e
r
a
g
e
, oncet
h
e
o
s
c
i
l
l
a
t
i
o
ni
ss
t
a
r
t
e
d
.Nowi
fonegoesi
n
t
oaframei
nwhicht
h
ewashboard
av
e
l
o
c
i
t
yu
n
a
f
f
e
c
t
e
dbyt
h
emotion
i
smovingw
i
t
has
t
e
a
d
yv
e
l
o
c
i
t
yw/k(
o
ft
h
eb
l
o
c
k
,s
i
n
c
ewehaveassumedt
h
a
tnui
sn
e
g
l
i
g
i
b
l
ys
m
a
l
lcompared
w
i
t
ht
h
ed
e
n
s
i
t
yo
ft
h
ewholep
l
a
s
m
a
)
.i
ti
ss
t
i
l
lt
r
u
et
h
a
tt
h
eb
l
o
c
kdoes
notg
a
i
norl
o
s
eenergyont
h
ea
v
e
r
a
g
e
,oncet
h
eo
s
c
i
l
l
a
t
i
o
ni
ss
t
a
r
t
e
d
.
h
ev
e
l
o
c
i
t
yw
/k
,andhence
ButE
q
.(
7
‑
8
4
]t
e
l
l
su
st
h
a
t(
AWk)dependsont
ont
h
eframeo
ft
h
eo
b
s
e
r
v
e
r
.I
np
a
r
t
i
c
u
l
a
r
,i
tshowst
h
a
tabeamh
a
s
口－
4初ラお＼
~
v
r
e
s
e
n
c
eo
ft
h
ewavethani
ni
t
sabsencei
fw ‑ ku く O
l
e
s
senergyi 日 the p
o
ru>vφ， and i
th
a
smoreenergyi
fw ‑ku>0o
ru く Uφ・ The r
e
a
s
o
n
f
o
rt
h
i
scanbet
r
a
c
e
dbackt
ot
h
ephaser
e
l
a
t
i
o
nbetweenn1andv1 ・ As
F
i
g
.7
‑
2
3s
h
o
w
s
,Wki
sap
a
r
a
b
o
l
i
cf
u
n
c
t
i
o
no
fv
.Asvo
s
c
i
l
l
a
t
e
sbetween
u ー Ivi
iandu+Ivi
i,Wkwillattainanaveragevaluelargerthanthe
h
a
tt
h
ep
a
r
t
i
c
l
espendsane
q
u
a
lamount
e
q
u
i
l
i
b
r
i
u
mvalue れも 0, providedt
o
ft
i
m
ei
neachh
a
l
fo
ft
h
eo
s
c
i
l
l
a
t
i
o
n
.Thise
f
f
e
c
ti
st
h
emeaningo
ft
h
e
f
i
r
s
ttermi
nE
q
.(
7
‑
8
1
]
,whichi
sp
o
s
i
t
i
v
ed
e
f
i
n
i
t
e
.Thesecondtermi
n
t
h
a
te
q
u
a
t
i
o
ni
sac
o
r
r
e
c
t
i
o
nduet
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h
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a
c
tt
h
a
tt
h
ep
a
r
t
i
c
l
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i
s
t
r
i
b
u
t
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t
st
i
m
ee
q
u
a
l
l
y
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nF
i
g
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‑
2
1
,ones
e
e
st
h
a
tbothe
l
e
c
t
r
o
na
ande
l
e
c
t
r
o
nbspendmoret
i
m
ea
tt
h
et
o
po
ft
h
ep
o
t
e
n
t
i
a
lh
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l
lt
h
a
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t
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a
c
h
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st
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a
tp
o
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n
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t
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rap
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o
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r
ｭ
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h
eb
o
t
t
o
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,bute
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e
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t
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g
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t
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v
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h
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e
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ot
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o
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r
e
.
a
f
t
e
rap
e
r
i
o
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t
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o
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h
er
i
g
h
t
)
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ot
h
a
tv1i
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1Wk)t
ochanges
i
g
na
tu= Uφ・
Thise
f
f
e
c
tc
a
u
s
e
s(
~
乞必必0:00必必必必必タ初？？タ務労労労労協労わ円円形労労労労労a 一一F
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f
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i
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i
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lConditions 7
.
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.
2
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s
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l
twehavej
u
s
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e
r
i
v
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d
,however,s
t
i
l
lh
a
snothingt
odow
i
t
h
l
i
n
e
a
rLandaudamping.Dampingr
e
q
u
i
r
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sacontinuousi
n
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r
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a
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h
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n
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c
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la
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.
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‑
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よ
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o
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h
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t(
untrappedp
a
r
t
i
c
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e
si
sc
o
n
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t
a
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e
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F
i
g
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t
o
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ee
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e
c
t
r
o
n
sa
tA
andD havegainede
n
e
r
g
y
,w
h
i
l
et
h
o
s
ea
tB andC havel
o
s
te
n
e
r
g
y
.
Now,i
ff
0
(
v
)wasi
n
i
t
i
a
l
l
yuniformi
ns
p
a
c
e
,t
h
e
r
ewereo
r
i
g
i
n
a
l
l
ymore
tC
,andmorea
tD thana
tB.T
h
e
r
e
f
o
r
e
,t
h
e
r
ei
s
e
l
e
c
t
r
o
n
sa
tA thana
an
e
tg
a
i
no
fenergybyt
h
ee
l
e
c
t
r
o
n
s
,andhencean
e
tl
o
s
so
fwave
e
n
e
r
g
y
.Thisi
sl
i
n
e
a
rLandaudamping,andi
ti
sc
r
i
t
i
c
a
l
l
ydependenton
t
h
eassumedi
n
i
t
i
a
lc
o
n
d
i
t
i
o
n
s
.A
f
t
e
ralongt
i
m
e
,t
h
ee
l
e
c
t
r
o
n
sa
r
es
o
smearedo
u
ti
nphaset
h
a
tt
h
ei
n
i
t
i
a
ld
i
s
t
r
i
b
u
t
i
o
ni
sf
o
r
g
o
t
t
e
n
,andt
h
e
r
e
i
snof
u
r
t
h
e
raverageenergyg
a
i
n
,a
swefoundi
nt
h
ep
r
e
v
i
o
u
ss
e
c
t
i
o
n
.
h
o
s
ew
i
t
hu く Uφ，
I
nt
h
i
sp
i
c
t
u
r
e
,botht
h
ee
l
e
c
t
r
o
n
sw
i
t
hv>vφand t
whenaveragedoveraw
a
v
e
l
e
n
g
t
h
,g
a
i
nenergya
tt
h
eexpenseo
ft
h
e
w
a
v
e
.Thisapparentc
o
n
t
r
a
d
i
c
t
i
o
nw
i
t
ht
h
ei
d
e
adevelopedi
nt
h
ep
i
c
t
u
r
e
o
ft
h
es
u
r
f
e
rw
i
l
lber
e
s
o
l
v
e
ds
h
o
r
t
l
y
.
2
5
5
K
i
n
e
t
i
cT
h
e
o
r
y
v
v
ゆ
>)
J
Q
'
l
:
x
0
-eφ1
x
P
h
a
s
e
‑
s
p
a
c
et
r
a
j
e
c
t
o
r
i
e
s(
t
o
p
)f
o
re
l
e
c
t
r
o
n
smovingi
nap
o
t
e
n
t
i
a
lwave(
b
o
t
t
o
m
)
. FIGURE7
‑
2
4
Thee
n
t
i
r
ep
a
t
t
e
r
nmovest
ot
h
er
i
g
h
t
.Thea
r
r
o
w
sr
e
f
e
rt
ot
h
ed
i
r
e
c
t
i
o
no
f
e
l
e
c
t
r
o
nm
o
t
i
o
nr
e
l
a
t
i
v
et
ot
h
ewavep
a
t
t
e
r
n
.Thee
q
u
i
l
i
b
r
i
u
md
i
s
t
r
i
b
u
t
i
o
n/0（υ ）
i
sp
l
o
t
t
e
di
nap
l
a
n
ep
e
r
p
e
n
d
i
c
u
l
a
rt
ot
h
ep
a
p
e
r
.
256
7.6 A PHYSICALDERIVATIONOFLANDAU DAMPING
v1
/a
x
.Thuswetake
ordert
omatcht
h
esamef
a
c
t
o
ri
na
Cha戸ler
Seveη
257
K
i
n
e
t
i
cTheoη
Wea
r
enowi
nap
o
s
i
t
i
o
nt
od
e
r
i
v
et
h
eLandaudampingr
a
t
ew
i
t
h
o
u
t
n
t
e
g
r
a
t
i
o
n
.Asb
e
f
o
r
e
,wed
i
v
i
d
et
h
eplasmaupi
n
t
o
r
e
c
o
u
r
s
etocontouri
beamso
fv
e
l
o
c
i
t
yuanddensity
and examinet
h
e
i
rmotioni
nawave
eE1k
1
n
1= nu 一孟一戸て五子
~,
E = ｣1s
i
n(kχ ー wt)
[
7
‑
8
5
]
FromE
q
.[
7
‑
7
7
]
,t
ev
e
l
o
c
i
t
yo
feachbeami
s
eE1c
o
s(kx‑wt)
v
,
＝一一一一一一一一一一一
．市
w ‑ku
[
7
‑
8
6
]
Thiss
o
l
u
t
i
o
ns
a
t
i
s
f
i
e
st
h
ee
q
u
a
t
i
o
no
fmotion [
7
‑
7
6
]
, buti
tdoesn
o
t
tt=O
.I
ti
sc
l
e
a
rt
h
a
tt
h
i
si
n
i
t
i
a
l
s
a
t
i
s
f
yt
h
ei
n
i
t
i
a
lc
o
n
d
i
t
i
o
nv
1=0a
e
r
yl
a
r
g
ei
nt
h
e
c
o
n
d
i
t
i
o
nmustbeimposed; o
t
h
e
r
w
i
s
e
,v1wouldbev
h
eplasmawouldbei
nas
p
e
c
i
a
l
l
yprep呪red
v
i
c
i
n
i
t
yo
fu= w/k, andt
s
t
a
t
ei
n
i
t
i
a
l
l
y
.Wecanf
i
xupE
q
.[
7
‑
8
6
]t
os
a
t
i
s
f
yt
h
ei
n
i
t
i
a
lc
o
n
d
i
t
i
o
nby
u
n
c
t
i
o
no
fkx‑k
u
t
.Thec
omposites
o
l
u
t
i
o
nwould
addingan 昌rbitrary f
h
eoperatoront
h
el
e
f
t
‑
h
a
n
ds
i
d
eo
fE
q
.
s
t
i
l
ls
a
t
i
s
f
yE
q
.(7・ 76] becauset
[7・76], whena
p
p
l
i
e
dt
of(kx‑k
u
t
)
,g
i
v
e
sz
e
r
o
.O
b
v
i
o
u
s
l
y
,t
og
e
tv
1= O
a
tt= 0
,t
h
ef
u
n
c
t
i
o
nf(k涜－ kut) mustbet
a
k
e
nt
obe ‑cos(
k
x‑k
u
t
)
.
Thusweh
a
v
e
,i
n
s
t
e
a
do
fEq冒［ 7-86],
‑eE1c
o
s(
k
x wt)‑ c
o
s(
k
x‑k
u
t
)
V
1＝一一一一
'
m
w ‑Rn
[7・ 87]
N
e
x
t
,wemusts
o
l
v
et
h
ee
q
u
a
t
i
o
no
fc
o
n
t
i
n
u
i
t
y[7・ 78] f
o
rn1
,a
g
a
i
ns
u
b
j
e
c
t
tt=0
.S
i
n
c
ewea
r
enowmuchc
l
e
v
e
r
e
r
t
ot
h
ei
n
i
t
i
a
lc
o
n
d
i
t
i
o
nn
1=0a
t
h
a
nb
e
f
o
r
e
,wemayt
r
yas
o
l
u
t
i
o
no
ft
h
eform
n1 ＝冗 1[cos (kx ー wt)
‑c
o
s(
k
x‑k
u
t
)
]
[
7
‑
8
8
]
克1
sm(kx
ー wt)= -n ， 一一－
'm
2
(w‑ku)"
[7・ 90]
Thisc
l
e
a
r
l
yv
a
n
i
s
h
e
sa
tt= 0
,andonecane
a
s
i
l
yv
e
r
i
f
yt
h
a
ti
ts
a
t
i
s
f
i
e
s
E
q
.(
7
‑
7
8
]
.
Thesee
x
p
r
e
s
s
i
o
n
sf
o
rv1andη1 a
l
l
o
wu
snowt
oc
a
l
c
u
l
a
t
et
h
ework
donebyt
h
ewaveoneachbeam.Thef
o
r
c
ea
c
t
i
n
gonau
n
i
tvolumeo
f
eachbeami
s
Fu= ‑eE1
s
i
n(kx 一 wt)(n,, +n1)
[
7
‑
9
1
]
andt
h
e
r
e
f
o
r
ei
t
senergychangesa
tt
h
er
a
t
e
di-γ
一一＝ F
,
,
(
u+v
1
)=
d
t
eE1s
i
n(
k
x wt ）（九日＋ nuV1 +n1u＋ η1V1)
①②③③
[
7
‑
9
2
]
now t
a
k
et
h
es
p
a
t
i
a
l average o
v
e
ra wavelength 目 The f
i
r
s
t term
,
,
ui
sc
o
n
s
t
a
n
t
.The f
o
u
r
t
htermcan be n
e
g
l
e
c
t
e
d
v
a
n
i
s
h
e
s because n
becausei
ti
ssecondo
r
d
e
r
,buti
nanyc
a
s
ei
tcanbeshownt
ohavez
e
r
o
a
v
e
r
a
g
e
.Theterms ( and ( can bee
v
a
l
u
a
t
e
du
s
i
n
gE
q
s
.[
7
‑
8
7
]and
[
7
‑
9
0
]andt
h
ei
d
e
n
t
i
t
i
e
s
1へfe
k
x‑wt)c
o
s(
k
x‑k
u
t
)
)= － ~sin (
w
t‑k
u
t
)
(
s
i
n(
[
7
‑
9
3
]
(
s
i
n(
k
x wt)s
i
n(
k
x‑k
u
t
)
)＝ ~cos (
w
t‑k
u
t
)
Ther
e
s
u
l
ti
se
a
s
i
l
yseent
obe
(
o
/
i
)
,
,＝ ~n，， 戸ιず日
I
n
s
e
r
t
i
n
gt
h
i
si
n
t
oE
q
.[
7
‑
7
8
]andusingE
q
.(7 ・87] f
o
rv
1
,wef
i
n
d
eE1ks
i
n(
k
x‑wt)‑ s
i
n(kx‑kut)
×[ cos (kx 一 wt) ‑c
o
s(
k
x‑ kut ） ー （w ‑k
u
)
ts
i
n(
k
x‑k
u
t
)
]
[7・ 89]
A
p
p
a
r
e
n
t
l
y
,wewerenotc
l
e
v
e
renough,s
i
n
c
et
h
es
i
n(
k
x wt) f
a
c
t
o
r
u
t
)
,w
hichcame
doesn
o
tc
a
n
c
e
l
.Tog
e
tatermo
ft
h
eforms
i
n(kx‑k
,wecanaddatermo
ft
h
eformAts
i
n(
k
x‑k
u
t
)
fromt
h
eaddedtermi
nv1
t
on
1
.Thistermo
b
v
i
o
u
s
l
yv
a
n
i
s
h
e
sa
tt= 0
,andi
tw
i
l
lg
i
Y
et
h
es
i
n(
k
x‑
k
u
t
)t
ermwhent
h
eoperatoront
h
el
e
f
t
‑
h
a
n
ds
i
d
eo
fE
q
.(7 司 78] o
p
e
r
a
t
e
s
ont
h
etf
a
c
t
o
r
.Whent
h
eo
p
e
r
a
t
o
ro
p
e
r
a
t
e
sont
h
es
i
n(kx‑kut)f
a
c
t
o
r
,
i
ty
i
e
l
d
sz
e
r
o
.Thec
o
e
f
f
i
c
i
e
n
tA mustbep
r
o
p
o
r
t
i
o
n
a
lt
o(w ku)‑1 i
n
+ku
i
nw
-kut ） 一 （w
‑k
u
)
tc
o
s(wt‑kut)l
I
(w-ku ）ー」
[7・ 94]
Notet
h
a
tt
h
eo
n
l
ytermst
h
a
ts
u
r
v
i
v
et
h
ea
v
e
r
a
g
i
n
gp
r
o
c
e
s
scomefrom
t
h
ei
n
i
t
i
a
lc
o
n
d
i
t
i
o
n
s
.
Thet
o
t
a
lworkdoneont
h
ep
a
r
t
i
c
l
e
si
sfoundbysummingovera
l
l
t
h
ebeams:
~（おι ＝ f や（れぬ＝ η。 jぞ（説 du
258
C
h
a
p
t
e
r
S
e
v
e
n
I
n
s
e
r
t
i
n
gE
q
.(7・94] andusingt
h
ed
e
f
i
n
i
t
i
o
no
fω炉 we thenf
i
n
df
o
rt
h
e
r
a
t
eo
fchangeo
fk
i
n
e
t
i
cenergy
Comparingt
h
i
sw
i
t
hE
q
.[
7
‑
1
0
1
]
,wef
i
n
d
r
1e2E~ ε叩
（事）＝ 一－
εoEi w;I I
~ 州wt ‑kut)du
d
tI
2
L
Jfo(u ） ー一一一一ー－
w‑ku
（ ~Wk ）＝了一一」」了＝ニデ＝（ WE)
生
‑k
u
i
tc
o
s(
w
t‑k
u
t
)kndnl
(w-ku γ.. ~~つ
u
一向
tL
ai
「＂＇
w-Ru
dん sin
(w-k吋，
」∞
l
一一…
L
"
' du w ‑ku
‑J
(WE ）＝ εo(E2)/2 ＝ εoEi/4
[
7
‑
9
9
]
Thei
n
t
e
g
r
a
t
e
dp
a
r
tv
a
n
i
s
h
e
sf
o
rw
e
l
l
‑
b
e
h
a
v
e
df
u
n
c
t
i
o
n
sf
0
(
u
)
,andwe
have
Thesecondp
a
r
ti
st
h
ek
i
n
e
t
i
cenergyo
fo
s
c
i
l
l
a
t
i
o
no
ft
h
ep
a
r
t
i
c
l
e
s
.I
f
wea
g
a
i
nd
i
v
i
d
et
h
eplasmaupi
n
t
obeams,E
q
.[
7
‑
8
4
]g
i
v
e
st
h
eenergy
perbeam:
u 一k
ー，ilJ
一ω
対一一
ι
一u
reE』
E
EEL
21
E一，
I
一ω
pu－
一d吐
一一
九一四川
司i
品
A
m
-u
dWw
w 2 「＂＇ •
f
s
i
n(w‑k
u
)
!
l
寸ケ＝ Ww 了的｜ fb(u>I 一一一つ一一 I du
αE
[
7
‑
1
0
0
]
lP2 i
(
'
' f0(u ） 「
2ku 1
（~Wk ）＝÷一一一｜ 一一一一一τ11 ＋一一一一｜
4 m L＂＇（，曲 － ku γL
曲 － kuJ
\
W
-flu
」
r1・ 107]
昆J
τr t 帝国 L
曲 － llu
J
r1・ 108]
Thus
dWw T " 27TW 合／ω＼ー
ω ト，／ω＼
d
t =V
V
.
,
W
p
kk
J
O
¥
"
k
J= V
V
w
1
T
W
J
;
'
i
f
O
¥
"
k
}
[
7
‑
1
0
1
]
[
7
‑
1
0
9
]
S
i
n
c
eIm(
w
)i
st
h
egrowthr
a
t
eo
fE1,andWwi
sp
r
o
p
o
r
t
i
o
n
a
lt
oEi,we
musthave
dWw/dt= 2[Im(w))Ww
[
7
‑
1
0
2
]
u
[7・ 110]
Hence
UsingE
q
.(
7
‑
7
9
]f
o
rπ1, wehave
1＝~こす J」＝三二［∞主包4
Eom τ （w -ku ） 品
εom L
"
'(w‑ku)'
L
よ＂＇
δ!
｛ペ
k hm ｷ
r
s
i
n(w 二ku)tl
u‑
‑
;J＝一
I
. I
Thesecondtermi
nt
h
eb
r
a
c
k
e
t
scanben
e
g
l
e
c
t
e
di
nt
h
el
i
m
i
tw/k > v,h,
whichwes
h
a
l
lt
a
k
ei
nordert
ocomparew
i
t
hourp
r
e
v
i
o
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[
8
‑
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3
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297
Nonlinear
E
f
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298
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7
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‑
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8
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5
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q
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b
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.
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8
‑
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7
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2
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4
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6
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.[
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‑
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9
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8
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]
,and[
8
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8
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o
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i
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844 ］：・
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2
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FNL ＝一寸 V 一一τ一一一 ε 。
αJo
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o
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FNL ー
－－ ~j__
wil a
x笠~
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[
8
‑
6
4
]
Thisf
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c
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o
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l
o
wtowardt
h
ei
n
t
e
n
s
i
t
yminima.Theresult司
i
n
gd
e
n
s
i
t
yr
i
p
p
l
et
r
a
p
swavesi
ni
t
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o
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g
h
s
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h
u
senhancingt
h
e
modulationo
ft
h
ee
n
v
e
l
o
p
e
.
。ω
q ＝ 一羽刊
ω
[8 ・ 123)
Modulationali
n
s
t
a
b
i
l
i
t
yoccurswhenp
q>0
;t
h
a
ti
s
,whenδwand d
v
,
/
d
k
haveopposites
i
g
n
.Figure8
‑
2
7i
l
l
u
s
t
r
a
t
e
swhyt
h
i
si
ss
o
.InF
i
g
.8‑27A,
ar
i
p
p
l
ei
nt
h
e wave envelope h
a
s developed a
sar
e
s
u
l
t of random
sn
e
g
a
t
i
v
e
.Thent
h
ephasev
e
l
o
c
i
t
yw/k
,whch
f
l
u
c
t
u
a
t
i
o
n
s
.Suppose8wi
m
a
l
l
e
ri
nt
h
eregionofhigh
i
sproportionalt
ow,becomessomewhats
i
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i
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y
.Thisc
a
u
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e
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ewavec
r
e
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t
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eupont
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f
tofF
i
g
.8‑27B
st
h
e
r
e
f
o
r
el
a
r
g
e
andt
ospreadoutont
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er
i
g
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t
.Thel
o
c
a
lv
a
l
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fki
fd
v
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/
d
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o
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e
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o
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ont
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tands
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er
i
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t
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w
i
l
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a
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e
ront
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f
tthant
h
er
i
g
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t
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ot
h
ewaveenergyw
i
l
lp
i
l
eup
i
n
t
o as
m
a
l
l
e
rs
p
a
c
e
. Thus, t
h
er
i
p
p
l
ei
nt
h
e envelope w
i
l
l become
f8wandd
v
,
/d
kwereofthesame
narrowerandl
a
r
g
e
r
,a
si
nF
i
g
.8‑27C.I
s
i
g
n
,t
h
i
smodulationali
n
s
t
a
b
i
l
i
t
ywouldnothappen.
‑
‑
338
Chα：pteγ
E
i
g
h
t
339
Nonlinea γ
E
f
f
e
c
t
s
(A)
一一一一事F
ー－ー・ー
一一一－－－－
V中
/
//
(
B
)
Ane
n
v
e
l
o
p
es
o
l
i
t
o
n
. FIGURE 8
‑
2
8
ーーー一ーー－一ーーー
vE『
ー－－”ー
av
e
l
o
c
i
t
yV hast
h
emoreg
e
n
e
r
a
lform(
F
i
g
.8‑28)
ψ（x, t
)= （子） 1
1
2s
e
c
h[（~） 1
1
2(
x‑x0 日）］
B
o
)
]
(
C
)
xexpi
(At+‑ijx fit+
[
8
‑
1
2
5
)
、、
、、
wherex
0and 0
0arethei
n
i
t
i
a
lp
o
s
i
t
i
o
nand p
h
a
s
e
.I
ti
sseent
h
a
tt
h
e
magnitude of V a
l
s
oc
o
n
t
r
o
l
st
h
e numberofwavelengths i
n
s
i
d
et
h
e
envelopea
tanygivent
i
m
e
.
FIGURE 8
‑
2
7 M
o
d
u
l
a
t
i
o
n
a
li
n
s
t
a
b
i
l
i
t
yo
c
c
u
r
s when t
h
en
o
n
l
i
n
e
a
r
f
r
e
q
u
e
n
c
ys
h
i
f
tandt
h
egroupv
e
l
o
c
i
t
yd
i
s
p
e
r
s
i
o
nh
a
v
e
o
p
p
o
s
i
t
es
i
g
n
s
.
8・ 19. Showb
yd
i
r
e
c
ts
u
b
s
t
i
t
u
t
i
o
nt
h
a
tE
q
.(
8
‑
1
2
4
]i
sas
o
l
u
t
i
o
no
fE
q
.(
8
‑
1
2
2
]
.
r
emodulationally
Although planewaves
o
l
u
t
i
o
n
st
oE
q
. [8‑123] a
,t
h
e
r
e can bes
o
l
i
t
a
r
ys
t
r
u
c
t
u
r
e
sc
a
l
l
e
de
n
v
e
l
o
p
e
u
n
s
t
a
b
l
ewhenpq>0
s
o
l
i
t
o
n
swhicha
r
estable ・ These a
r
egeneratedfromt
h
eb
a
s
i
cs
o
l
u
t
i
o
n
叫X, t)= （子） 112sech[（~） 112xJ
e;A,
[8・ 124]
whereA i
s an a
r
b
i
t
r
a
r
y constantwhich t
i
e
s together t
h
e amplitude,
w
i
d
t
h
,andfrequencyo
ft
h
ep
a
c
k
e
t
.Atanyg
i
v
e
nt
i
m
e
,t
h
edisturbance
8
‑
1
0
0
]
)(thought
h
ehyperbolics
e
c
a
n
t
resemblesasimples
o
l
i
t
o
n(
E
q
.[
)o
s
c
i
l
l
a
t
e
i
snotsquaredh
e
r
e
)
,butt
h
ee
x
p
o
n
e
n
t
i
a
lf
a
c
t
o
rmakesw(x,t
betweenp
o
s
i
t
i
v
eandn
e
g
a
t
i
v
ev
a
l
u
e
s
.Anenvelopes
o
l
i
t
o
nmovingw
i
t
h
8・20.
V
e
r
i
f
yE
q
.(
8
‑
1
2
5
]b
yshowingt
h
a
tif 凶 （x, t
)i
sas
o
l
u
t
i
o
no
fE
q
.(
8
‑
1
2
2
]
,t
h
e
n
ψ 二回（x
I）叫（ギx ~t +
O
u
)
]
x0‑V
t
,
i
sa
l
s
oas
o
l
u
t
i
o
n
We next wish t
o show t
h
a
tt
h
e nonlinear Schrodinger equation
d
e
s
c
r
i
b
e
slarge‑amplitudee
l
e
c
l
lonplasmawaves.Theprocedurei
st
o
s
o
l
v
es
e
l
f
‑
c
o
n
s
i
s
t
e
n
t
l
yf
o
rt
h
ed
e
n
s
i
t
yc
a
v
i
t
yt
h
a
tt
h
ewavesdigbymeans
o
ft
h
e
i
rponderomotivef
o
r
c
eandf
o
rt
h
ebehavioroft
h
ewavesi
nsuch
ac
a
v
i
t
y
. The high‑frequency motion o
ft
h
ee
l
e
c
t
r
o
n
si
s governed by
PROBLEMS
340
equations[
4
‑
1
8
]
.[
4
‑
1
9
]
,and[
4
‑
2
8
]
,whichwer
e
w
r
i
t
ea
s
Ch αヤleγ
a
u
e~ 3KT,aη
a
t
m
mno a
x
一一一一一－
E
i
g
h
t
[
8
‑
1
2
6
]
ハリ
一一
o
u 一 x
qO 一円U
n
[
8
‑
1
2
7
]
η
ρ
ε
一
一
ハリ
司
一ε
JF
-2
町一間
n
十
k
0 一
m7
n 一2
[
8
‑
1
2
9
]
[8 田 135]
ハリ
叫
一一
n
。。
一2
、、、‘，，，，
l
－、
ESE
+
A
凶
+
[
8
‑
1
3
6
]
Herei
ti
sunderstoodt
h
a
ta
/
a
ti
st
h
etimed
e
r
i
v
a
t
i
v
eont
h
eslowtime
s
c
a
l
e
,althoughu c
o
n
t
a
i
n
sbotht
h
eexp(‑iw0t) f
a
c
t
o
rand t
h
es
l
o
w
l
y
varying c
o
e
f
f
i
c
i
e
n
tu
1
. We have e
s
s
e
n
t
i
a
l
l
y derived t
h
e nonlinear
Schrodingerequation [
8
‑
1
2
2
]
,buti
tremainst
oevaluateδηin terms
o
fI
u
d2
Thelow‑frequencyequationo
fmotionf
o
rt
h
ee
l
e
c
t
r
o
n
si
sobtained
byn
e
g
l
e
c
t
i
n
gt
h
ei
n
e
r
t
i
atermi
nE
q
.[
4
‑
2
8
]andaddingaponderomotive
f
o
r
c
etermfromE
q
.[
8
‑
4
4
]
a
n w~ a(
<
o
E
2
)
ーす－
'
i
!
x w~ a
x 2
0= ‑enE KT,
[
8
‑
1
3
7
]
[8・ 131]
2 、 ei
w
.
,
I
.
.
wheret
h
edots
t
a
n
d
sf
o
ratimed
e
r
i
v
a
t
i
v
eont
h
es
l
o
wt
i
m
es
c
a
l
e
.We
i
,
,whichi
smuchs
m
a
l
l
e
rthanw
i
i
u
1
:
mayt
h
e
r
e
f
o
r
en
e
g
l
e
c
ti
内2
[
8
‑
1
3
2
]
一
7一
Here we have setγ，＝ Is
i
n
c
et
h
e low-frequen 仁y motion should be
i
s
o
t
h
e
r
m
a
lr
a
t
h
e
rthana
d
i
a
b
a
t
i
c
.Wemays
e
t
ー w0Ut)
~＝ー （w~Ut 十 2iw0ui) e‑
i
w
o
'
a
t
"
I
anda
s
s
u
r
r
ou(
x
,t
)v
i
aE
q
.[
81
3
1]
,and
(
t
h
e
s
ebeingunderstood ），仁onvert backt
approximatew~ byli
nt
h
ef
i
r
s
ttermt
oo
b
t
a
i
n
r’
内・
・
<'.ZWoUt
ム ＝（wo 一 Wp)/wρ ＝ w~
E
(
U
t
、
Definingthefrequencys
h
i
f
t~
2
U 一
一X
at~
[
8
‑
1
3
4
]
[ ,
a
u
: 3 山 l ば－
" I‑o
→一一寸＋
n
'
)
u
;
]ei
w
;
,
i
ｷ= 0
a
t
' 2ax ・L 2
、。一 nd
[
8
‑
1
3
0
]
t
w
i
c
ei
nt
i
m
e
,weo
b
t
a
i
n
a
2
u ’”
E
f
f
e
c
t
s
[8 ・ 133]
8η ’＝ δη／η。
u ’= u(KT,/m) 1
1
2
qJ一oh
u
(
x
,t
)= U
1
(
x
,t
)e•wo'
で一τ ＝
x ’= x／λD
w ’= w/wp
2
一一
ハリ
、、‘E1 ノ
+
品一h
ω
q’hムy
，，SE
，‘、、
一X
u 一2
、川
U － qパU
2
叫
7
u 一2
+
o
n
s
i
s
t
so
fahigh‑frequencyp司 rt o
s
c
i
l
l
a
t
i
n
ga
tw0(
e
s
s
e
n
ｭ
Thev
e
l
o
c
i
t
yuc
t
i
a
l
l
yt
h
eplasmafrequency)andalow‑frequencyp
a
r
tu
1describingthe
q
u
a
s
i
n
e
u
t
r
a
lmotiono
fe
l
e
c
t
r
o
n
sf
o
l
l
o
w
i
n
gt
h
ei
o
n
sa
st
h
e
ymovet
oform
t
h
ed
e
n
s
i
t
yc
a
v
i
t
y
.Bothf
a
s
tandslows
p
a
t
i
a
lv
a
r
i
a
t
i
o
n
sa
r
ei
n
c
l
u
d
e
di
n111.
Let
Di 圧erentiating
二 O
Wenowtransformt
ot
h
en
a
t
u
r
a
lu
n
i
t
s
如一日目
内
Aυ－RAυ
九d 一a
t
J
3KT,a2u1 I 2
2
2
8
n
一一言＋ lw 。一 ω ー ωρ
u
i
Je
m a
x ＼
η。
J
Nonlineaγ
o
b
t
a
i
n
i
n
g
e
s
c
r
i
b
et
h
ed
e
n
s
i
t
yc
a
v
i
t
y
;t
h
i
si
st
h
e
Wenowr
e
p
l
a
c
en0byn0+8ηto d
onlynonlineare
f
f
e
c
tc
o
n
s
i
d
e
r
e
d
.Equation[
8
‑
1
2
9
]i
so
fc
o
u
r
s
ef
o
l
l
o
w
e
d
byanyo
ft
h
el
i
n
e
a
rv
a
r
i
a
b
l
e
s
.I
tw
i
l
lbeconvenientt
ow
r
i
t
ei
ti
nterms
fWp;thus
o
fuandusethede白 nition o
-
2iw0向＋
[
8
‑
1
2
8
]
,n
,and u a
r
e
,
where n0i
st
h
e uniform unperturbed d
e
n
s
i
t
y
; andE
r
e
s
p
e
c
t
i
v
e
l
y
,t
h
ep
e
r
t
u
r
b
a
t
i
o
n
si
ne
l
e
c
t
r
i
cf
i
e
l
d
,e
l
e
c
t
r
o
nd
e
n
s
i
t
y
,andf
l
u
i
d
v
e
l
o
c
i
t
y
.Theseequationsa
r
el
i
n
e
a
r
i
z
e
d
,s
ot
h
a
tn
o
n
l
i
n
e
a
r
i
t
i
e
sduet
ot
h
e
r
enotc
o
n
s
i
d
e
r
e
d
.Takingt
h
etimed
e
r
i
v
a
t
i
v
e
uｷVuandV ｷ（πu) termsa
o
fE
q
.[
81
2
7
]andt
h
ex d
e
r
i
v
a
t
i
v
eo
fE
q
.[
8
‑
1
2
6
]
,wecane
l
i
m
i
n
a
t
eu
andE w
i
t
ht
h
ehelpo
f[
8
‑
1
2
8
]t
oo
b
t
a
i
n
2
[
t ’＝ ω。t
一ハU
伽一ud庄一以
+
341
S
u
b
s
t
i
t
u
t
i
n
gi
n
t
oE
q
.[
8
‑
1
3
0
]g
i
v
e
s
[
8
‑
1
3
8
)
by s
o
l
v
i
n
gt
h
e high-frequen 仁Y e
q
u
a
t
i
o
n[
8
‑
1
2
6
] without t
h
e thermal
c
o
r
r
e
c
t
i
o
n
.WithE = -Vφand x= eφ／ KT" E
q
.[
8
‑
1
3
7
]becomes
i
J
ニ (x
a
x
1 m a "
‑l
nn ） 一一...：：：＿＿ー（u") = 0
2K7二 ax
[
8
‑
1
3
9
)
342
C
h
a
p
t
e
r
E
i
g
h
t
Comparing w
i
t
hE
q
. [
8
‑
1
2
2
]
, we s
e
et
h
a
tt
h
i
si
st
h
e nonlinear
icanben
e
g
l
e
c
t
e
dand
Schrodingerequationi
fL
I
n
t
e
g
r
a
t
i
n
g
,s
e
t
t
i
n
gn ＝ η 。＋ δπ ， and usingt
h
en
a
t
u
r
a
lu
n
i
t
s[
8
‑
1
3
4
]
,we
have
~（u2)=tlul2=x
ln(l+8n)=x 8n
p= 2 q＝一言伊て-;;JM)
Wemustnowe
l
i
m
i
n
a
t
exbys
o
l
v
i
n
gt
h
ec
o
l
d
‑
i
o
ne
q
u
a
t
i
o
n
s[
8
‑
1
0
3
]
and[
8
‑
1
0
4
]
.S
i
n
c
ewea
r
enowusingt
h
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)
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e FIGURE8
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(/
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APPENDICES
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5
RADIALPOSITION(cm)
10
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FIGURE 8‑30 Coupled e
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frequency.
PROBLEMS
8
‑
2
1
. Checkt
h
a
tt
h
er
e
l
a
t
i
o
nbetween t
h
efrequencys
h
i
f
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h
es
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l
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t
o
n
amplitude i
nE
q
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8
‑
1
5
4
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sr
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a
s
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n
a
b
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e by c
a
l
c
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t
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p
r
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s
i
o
ni
nt
h
es
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l
i
t
o
nandt
h
ecorrespondingaverage 仁hange i
UseE
q
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81
4
6
]andassumet
h
a
tt
h
es目： h2 f
a
c
t
o
rh
a
sanaveragev
a
l
u
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f＝を
overt
h
es
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l
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t
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nw
i
d
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.
)
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l
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r
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2
3
. Ad
e
n
s
i
t
yc
a
v
i
t
yi
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e
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s
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t
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t
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t
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h
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t
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n
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4ﾗ I
0
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h
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y
?
寸
il
－－
AppendixA
111
U
N
I
T
S
,
CONSTANTS
ANDFORMULAS
VECTOR
RELATIONS
UNITS A
.I
Theformulasi
nt
h
i
sbooka
r
ew
r
i
t
t
e
ni
nt
h
emksu
n
i
t
so
ft
h
eI
n
t
e
r
n
a
t
i
o
n
a
l
System (
S
I
)
.I
n much o
ft
h
er
e
s
e
a
r
c
hl
i
t
e
r
a
t
u
r
e
, however, t
h
ec
g
s
ｭ
Gaussiansystemi
ss
t
i
l
lu
s
e
d
.Thef
o
l
l
o
w
i
n
gt
a
b
l
ecomparest
h
evacuum
Maxwelle
q
u
a
t
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o
n
s
,t
h
ef
l
u
i
de
q
u
a
t
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o
no
fm
o
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