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Solutions Manual for Continuum Electromechanics
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-M
-W
C')
ci~
v
ci,
w
-O
-)
M -e
11.1
Prob. 11.2.1
1
With the understanding that the time derivative on the left
is the rate of change of t' for a given particle (for an observer moving
with the particle velocity
=
+(1)-
(
=
Substitution of
' ) the equation of motion is
and dot multiplication of this expression with
v gives
(2)
yKcl
Because
is perpendicular to
,
1~~
•--IV
3t
)
By definition, the rate of change of
r
I
->
ý
I -)
+
(3)
with respect to time is
.
where here it is understood that
(4)
/
means the partial is taken
holding the Eulerian coordinates (x,y,z) fixed.
tive is zero.
I
Thus, this partial deriva­
It follows that because the del operator used in expressing
Eq. 3 is also written in Eulerian coordinates, that the right-hand side of
Eq. 4 can be taken as the rate of change of a spatially varying
respect to time as observed by a particle.
I
I
with
So, now with the understanding
that the partial is taken holding the identity of a particle fixed (for
example, using the initial coordinates of the particle as the independent
spatial variables) Eq. 3 becomes the desired energy conservation statement.
a)
+
0
(5)
1
I
I
U
11.2
Prob. 11.3.1
(a)
Using (x,y,z) to denote the
cartesian coordinates of a given electron between
t-
the electrodes shown to the right, the particle
equations of motion (Eq. 11.2.2) are simply
h!
e
m
=Co
(3)
There is no initial velocity in the z direction, so it follows from Eq. 3
that the motion in the z direction can be taken as zero.
(b)
To obtain the required expression for x(t), take the time derivative
of Eq. (1) and replace the second derivative of y using Eq. (2).
Thus,
When the electron is at x = 0,
I
0)
A
=0
(5)
So that Eq. 4 becomes
++
--
(6)
Iote that for operation with electrons,
I
(
(c)
V
<
0
This expression is most easily solved by adding to the particular
solution,
.V/
42?2,the combination of
S;ir
WAx
and
Cos w x(the
homogeneous solutions) required to satisfy the initial conditions.
However, to proceed in a manner analogous to that required in the text,
Eq. 6 is multiplied by
the form
I
IX[+Z
c 4 K/jt
and the resulting expression written in
?-
so that it is evident that the quantity in brackets is conserved.
satisfy the condition of Eq. 5, the constant of integration is zero
I
(7)
To
11.3
Prob. 11.3.1 (cont.)
(the initial total energy is zero) so it follows from Eq. 7 that
?_ %
,./
where
(8)
1
e V<O.
The potential well picture given by this
expression is shown at the right.
I
I
I
I
Rearrange-
ment of Eq. 8 puts it in a form that can be
integrated.
First, it is written as
+
Z
~c
eV9
X
t
(9)
Then, integration gives
.r _eV
Cos
'
-
-
M-•
•
= •t3
x-
oS Wt -,•
(10)
|
Of course, this is just the combination of particular and homogeneous
solutions to Eq. 6 required to satisfy the initial condition.
I
The associated motion in the y direction follows by using Eq. 10 to
evaluate the right-hand side of Eq. 2.
Then, integration gives the
(11)
C• V ()
e-
velocity
dt
( Co.
1
cj'
where the integration constant is evaluated to satisfy Eq. 5.
A second
integration, this time with the constant of integration evaluated to make
y=O when t=0,
gives
(note that
. =--
C/Am ).
Thus, with t as a parameter, Eqs. 10 and 12 give the trajectory of a
particle starting out from the origin when t=0.
Electrons coming from
the cathode at other times or other locations along the y axis have
similar trajectories.
1
11.4
Prob. 11.3.1 (cont.)
(d)
The construction shown in the figure is useful in picturing
particle motions that are the planar analogous of those found in cylind­
rical geometry in the text.
I
(e)
I
The trajectory just grazes the anode if the peak amplitude given
by Eq. 10 is just equal to the spacing, a.
The potential resulting from
this equality is then the critical one.
1
I
V~
(13)
-mL
2 (2
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123 Worksheet used tc- do calculations f'o:r 6.672 P10.8.2
Pote!ntial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
46
47
48
49
50
51
Integrand
midplane Phi of mid.
-3 1006766 0.250715
0 1.009043-2. 9676 9.748310 1.255904 0. 100820 7
4/2 ý
-2. 9514 9.592493 0.258556 0.0004167
-2. 9352 9.439193 0.261247 0.004210
- .919 9.288371 0.263979 0.004254
-2.9028 9.139986 0.266752 0.004298
-2. 8866 8.994000 0.269567 0. 004344
-2.8704 8.850375 0.272425 0. 004390
-2.8542 8.709072 0.275327 0. 004436
-2.838 8.570055 0.278276 0.1:004484
-2.8218 8.433287 0.281270 0.004532
-2.8056 8.298732 0.284313 0.004581
-
.38
-p;
0.008207
0.012374
0.016584
0.020839
0.025138
0.029482
0.033872
0.038309
0.042793
0.047325
0.051906
-2.7894 8.166356 0.287405 0.004630
-2.7732 8.036122 0.290548 0 .004681
-2.757 7.907998 0.293742 0 . 004732
-2.7408 7.781949 0.296990 0. 004784
-2.7246 71.657942 0.300293 0.004838
-2.7084 7.535945 0.303652 0.004891
-2.6922 7.415926 0.307069 0 . )004946
-2.676 7.297853 0.310546 0.005002
-2.6598 7.181696 0.314084 0.005059
-2.6436 7.067423 0.317686 0.005117
-2.6274 6.955005 0.321353 0.005176
-2.6112 6.844412 0.325087 0.005236
-2.595 6.735616 0.328891 0.005297
-2.5788 6.628588 0.332766 0.005359
-2.5626 6.523299 0.336715 0.005422
-2.5464 6.419722 0.340740 0.005487
-2.5302 6.317830 0.344844 0.005553
-2.514 6.217596 0.349029 0.005620
-2.4978 6.118993 0.353299 0.005688
-2.4816 6.021997 0.357656 0.005758
-2.4654 5.926581 0.362103 0. :005830
-2.4492 5.832721 0.366643 0.005902
-2.433 5.740391 0.371280 0 .005977
-2.4168 5.649568 0.376018 I. 0.06053
0.056537
5.560227 0.380859 0 .006130:
5.472346 0.385809 0. 006210
5.385901 0.3908"71 0.006291
5.300870 0.396050 0 .00637 4
5.217229 0.401351 0.006458
5.134958 0.406778 0. 006545
5.054035 0.412337 0.006634
4.974438 0.418033 0.006726
4.8 96146 0.A23872 0. 006819
4.819140( 0.429862 0.006915
4.743398 0.436007 0 .007013
-2.2224 4.668901 0.442316 0 .007114
4.595630 0.448797 0 . .107218
-2.19 4.523564 0.455456 S.(107324
0.135087
-2.4006
-2.3844
-2.3682
-2.352
-2.3358
-2.3196
-2.3034
-2.2872
-2.271'
-2.2548
-2.2386
-2.1q38 4.452686 0 .,-62304 0.007433
0.061219
0.065952
0.070737
0.:,75575
0.080466
0.085413
0.090416
0.095476
0.100593
0.105769
0.111005
0.116303
0.121662
0.127085
0.132572
0.138125
0.143746
0.149435
0.155193
0.161023
0.166926
0.172903
0.1?78957
0. 191297
0.197588
0.203962
0.210421
0.216967
0.223602
0.230328
0.237148
0.244063
0.251076
01.258191
0.2165A09
0.272733
0.280167
a. o1'
I ,I
. \40V
Courtesy of James Brennan. Used with permission.
-2.1576 4.382977 0.469350 0 . 007546
-2.1414 4.314417 0.476603 0.l007662
-2.1252 4.246990 0.484075 0.007781
-2.109 4. 180678 0.491777 0. 007904
-2.0928 4.115462 0.499722 0.008031
-2.0766 4.051327 0.507923 0.008161
-2.0604 3.988255 0.516395 0.008296
-2.0442 3.926230 0.525154 0.008436
-2.028 3.865235 0.534217 0.i08580
-2.0118 3.805255 0.543604 0.008730-1.9956 3.746273 0.553334 0.008885
-1.9794 3.688275 0.563429 0.009045
-1.9632 3.631244 0.573916 0.009212
-1.947 3.575167 0.584819 0.009385
-1.9308 3.520028 0.596170 0.009566
-1.9146 3.465812 0.608001 0.009753
-1.8984 3.412506 0.620n348 0. 009949
-1.8822 3.360096 0.633252 0.010154
-1.8866 3.308568 0.646757 0. :10368
-1.8498 3. 257908 0.660914 0.010592
-1.8336 3. 208103 0.675779 0.010827
-1.8174 3.159140 0.631416 0.011074
-1.8012 3.111006 0.707897 0.011334
-1.785 3.063688 0.725305 0.011608
-1.7688 3.017175 0.743731 0.011899
-1.7526 2.971453 0.763286 0.012206
-1.7364 2.926511 0.784092 0.012533
-1.7202 2.882337 0.806295'0.012882
-1.704 2.838920 0.830065 0.013254
-1.6878 2.796248 0.855602 0.013653
-1.6716 2.754310 0.883145 0.014083
-1.6554 2.713094 0.912981 0.014548
-1.6392 2.672591 0.945459 0.015053
-1.623 2.632789 0.981006 0.015604
-1.6068 2.593678 1.020155 0.016209
-1.5906 2.555247 1.063579 0.016789
1.112144 0. 17623
-1.5744 2.517487
-1.5582 2.480388 1 .166981 0.018460
-1.542 2.443940 1 .229610 0.".94_'
-1.52583 2.408134 1.302123
2."SC507
-1.5096 2.372959 1.387499 0.021785
-1.4934 2.338407 1.490157 0.023309
-1.4772 2.304469 1.616992 0.025167
-1.46! 2.271135 1.779507 0.027511
-1.4443 2.238398 1.998733 0.030603
-]1.4286 2.206248 2.318587 0.034970
-1.4124 2.174677 2.852775 0.041888
-',3962
143677 4.053033 0.055937
-1..2s 2.113240
ERR 0.034481
2.113240
Divide by Z-er-c. error
'•
eraclw tc
iv
0 .287714
0.295376
0.303157
0 .311 062
0.319093
0.327255
0. 335552
0.343988
0.352569
0.361300
0.370185
0.379231
0.388443
0.397829
0.407395
0.417149
0.427098
0.437252
0.447621
0.458213
0.469040
(0.480114
0.491449
0.503058
(1.514957
0.527164
0.539697
0.552580
0.565834
0.579488
0.593572
0.608120
0.623174
0.638778
0.654988
0.671g66
0.689489
. 707950
U.727363
".747870
0.769656
u.792965
0 .818132
0 . 845644
0. 876248
0.911218
0.953106
1. 009043
1.0143524
r(•)
Ný
P".v
-
(. I
-
-
man
Courtesy of James Brennan. Used with permission.
a
0
C
*1I
0.2
8,6
Position (delta/2) 1,8 :aidplane
8,4
Potential vs. Position
P10,8,1 inContinu Electromechanics
-m
-m
-
£
11.5
Prob. 11.4.1
The point in this problem is to appreciate the quasi-one­
dimensional model represented by the paraxial ray equation.
First, observe
5
that it is not simply a one-dimensional version of the general equations of
motion.
Er,
The exact equations are satisfied identically in a region where
1E
.r are zero by the solution r = constant,
and
and a uniform motion in the z
=
constant
That the magnetic field,
direction, z=Ut.
Bz, has a z variation (and hence that there are radial components of B)
is implied by the use of Busch's Theorem (Eq. 11.4.2).
I
The angular vel­
5
ocity implicit in writing the radial force equation reflects the arrival
of the electron at the point in question from a region where there is no
magnetic flux density.
It is the centrifugal force caused by the angular
j
velocity created in the transition from the field free region to the one
where B
is uniform that appears in Eq. 11.4.9, for example.
Prob. 11.4.2
The theorem is a consequence of the property of solutions to
Eq. 11.4.9.
(1)
In this expression, "y
'•(t), reflecting the possibility that the Bz
varies in an arbitrary way in the z-direction.
t
d
Integration of Eq. 1 gives
i,
I
(2)
Because the quantity on the right is positive definite, it follows that the
derivative at some downstream location is less than that at the entrance.
Il
a
Prob. 11.4.3
5
(3)
•
For the magnetic lens, Eq. 11.4.8 reduces to
(1)
-e
Integration through the length of the lens gives
ItI
I,+d8[
·
5 Jd
a
11.6
Prob. 11.4.3 (cont.)
and this expression becomes
Ar
r
- - ee r -
(3)
On the right it has been assumed that the variation through the "weak" lens
of the radial position is negligible.
SI
The definition of f that follows from
Fig. 11.4.2 is
_ -•(4)
6
•
I
so that for electrons entering the lens as parallel rays, it follows from
Eq. 3 that
d -a
I
which can be solved for f to obtain the expression given.
I
observe for the example given in the text where B = B over the length
z o
As a check,
of the lens,
R
+•
(6)
and it follows from Eq. 5 that
I
1__M_
(7)
This same expression is found from Eq. 11.4.12 in the limit j•
1
Prob. 11.4.4
(<( 1
For the given potential distribution
•(1)
the coefficients in Eq. 11.4.8 are
3A
C
(2)
and the differential equation reduces to one having constant coefficients.
I
A +1_
=o
0Ir---
At z = z+, just to the downstream side of the plane z=0, boundary
conditions are
I
I
,(4)
(3)
1
11.7
Prob. 11.4.4 (cont.)
Solutions to Eq. 3 are of the form
+
Z>e
-
•
((5)
and evaluation of the coefficients by using the conditions of Eq. 4
results in the desired electron trajectory.
A
Prob. 11.5.1
A
In Cartesian coordinates, the transverse force equations
e
are
ea
(1)
eU
Le+
0
R
(2)
With the same substitution as used in the zero order equations, these
relations become
I
A
e.­
- dX
(3)
I
1
aim"
0
I1
I
_e" ~V
where the potential distributions on the right are predetermined from
the zero order fields.
For example, solution of Eqs. 3 gives
If the Doppler shifted frequency is much less than the electron cyclotron
frequency,
lc =
Typically,
A8/l
./
I3
l
.
and
'
so that Eqs. 4 and 11.5.5 show
11.8
Prob. 11.5.1 (cont.)
I
that
A
(J-(
-(-
so, if
I
J-kUj<L•3,
-
i)
(6)
then the transverse motions are negligible compared to the
longitudinal ones.
3
)
Most likely
W'-QU
"O•p so the requirement is essentially
that the plasma frequency be low compared to the electron cyclotron frequency.
Prob. 11.5.2
geometry.
(a)
Equations 11.5.5 and 11.5.6 remain valid in cylindrical
However, Eq. 11.5.7 is replaced by the circular version of Eq. 11.5.4
combined with Eq. 11.5.6
0
!15
(1)
Thus, it has the form of Bessel's equation, Eq. 2.16.19, with k-.-'.
1
The deriva­
tion of the transfer relations in Table 2.16.2 remains valid because the displace­
ment vector is found from the potential by taking the radial derivative and that
"
involves
1
and not k.
(If the derivation involved a derivative with respect to
z, there would be two ways in which k entered in the original derivation, and
I
could not be unambiguously identified with k everywhere.)
(b)
Using (c),
(d) and (e) to designate the radii r=a and r=+b
and -b respect­
ively, the solid circular beam is described by
DI
I
)
= Ef,(,
e
(2)
while the free space annulus has
S(3)
Thusin
Thus,
I
;]
:
61
(b,)
in view of the conditions that
: D=
and
[i
Eq
,Eqs.
2
1
2 and
3b show that
S__h,_(_6,_0___
0Cb,•,6
) ) (C_
(4)
•
11.9
Prob. 11.5.2 (cont.)
This expression is then substituted into Eq. 3a to show that
6
)(5)
which is the desired driven response.
(c)
The dispersion equation follows from Eq. 5, and takes the same form as
Eq. 11.5.12
For the temporal modes, what is on the right (a function of geometry and the
wavenumber) is real.
( ,Qb))
From the properties of the f
O for 406 and
Eq. 6 cannot be satisfied.
;,(}b,<)(O, so it
is
determined in Sec. 2.17,
clear that for Y
real,
However, for X=-d•where c1 is defined as real,
Eq. 6 becomes
-CL
___
_
(7)
This expression can be solved graphically to find an infinite number of solutions,
dm
.
Given these values, the eigenfrequencies follow from the definition of
given with Eq. 11.5.7.
S= 'P J -+
(8)
11.10
Prob. 11.6.1
The system of m first order differential equations takes
the form
!
•__,S-A
where i =1 ....
I
(a)
I
equation by
+
i,-3)
(1)
m generates the m equations.
Following the method of "undetermined multipliers, multiply the ith
"• and add all m equations
1
+
ýLx)=
a
= 0i
(2)
I
I
I
I
These expressions, j =di1 ....
.
m can be written as
equations in the
1
11.11
Prob. 11.6.1 (cont.)
M
The first characteristic equations are given by the condition that the
As '
determinant of the coefficients of the
i
vanish.
&-=­
(b) Now, to form the coefficient matrix, write Eq. 1 as the first m of
the 2 m expressions
N
F,,
F?.
K_,
G,,
,
F,.
G,,
Fm
-
G
G,,
F
F
.*
-
.*
V,.
,
XV
X~,0
a
6
drX,
*
U
o
G r-t
G..
0
a
0
I
I
I
ax,
The second m of these expressions are
I
3t
To show that determinant of these coefficients is the same as Eq. 6,
operate on Eq. 7 in ways motivated by the special case of obtaining
Eq. 11.6.19 from Eq. 11.6.17.
Multiply the(m+1) 'st
'
equation (the last m equations)
2mth
by dt-1.
equation through
Then, these last m
I
I
11.12
Prob. 11.6.1 (cont.)
rows (m+l....2m) are first respectively multiplied
by F1 1, F 1 2 ... lm
and subtracted from the first equation.
The process is
then repeated
using of F2 1 , F 22 "... F 2 m and the result subtracted from the second
equation, and so on to the mth equation.
Thus, Eq. 7 becomes
-F
o
C7
S
o
Gmt-, Ao
GI
*
.
& 0
0
r a
a
I
*
Now, this expression is expanded by "minors" about the
0
0o
i
is that appear
as the only entries in the odd columns to obtain
G,,-F, a
J-4
G," _,P
(10)
E4
Multiplied by (-1) this is the same as Eq. 6.
*t·'L
mm
1
11.13
Prob.
Eqs. 9.13.11 and 9.13.12, with
11.7.1
V=0
and
b=O
i
are
I
at
_ +_
at
aCI(01) )=O
(1)
(2)
In a uniform channel, the compressible equations of motion are Eqs. 11.6.3
and 11.6.4
I
These last expressions are identical to the first two if the identification
is male
1,--aw
,
6-X
0
and
a
-b-
4
(Eq. 11.6.2) the analogy is not complete unless
of
e
.
.
Because
O
/54
= tx
1,
is independent
This .requires that (from Eq. 11.6.2)
c19
atZ'
(3)
I
`I
I
11.14
Eqs.
Prob. 11.7.2
and
6 =
9.13.4 and 9.13.9 with A and f defined by
are
T•2
-s
=--((-E~
,)
+
1.
-p
Si
. .
s
r~~d
*La1
PC
40
These form the first two of the following 4 equations.
0
-%
_
____
11(3
rj3
(1+
At
The last two state that
I
d
n
I-
I
and
are computed along the
characteristic lines.
from requiring that the
The Ist characteristic equations f
determinant of the coefficients vanish.
To reduce this determinant divide the third and fourth columns by
and
At/?
dt
respectively, and subtract from the first and second respectively.
Then expand by minors to obtain the new determinant
Sdt
dt
(E- E.V
1 3
-Y
OIL
=O
11.15
Prob. 11.7.2 (cont.)
Thus, the Ist characteristic equations are
or
--t
_
on
_
C
(6)
The IInd characteristics are found from the determinant obtained by substituting
I
the column matrix on the right for the column on the left.
a.
o
0I
9
dt
do
o
(7)
l1
itdgo
dt
Solution, expanding in minors about
de
a19 and c
, gives
(8)
d)
tj
With the understanding the + signs mean that the relations pertain to C,
Eq. 6 reduces this expression to the IInd characteristic equations.
'Z D-
.
on
C
(9)
I
11.16
(a) The equations of motion are 9.13.11 and 9.13.12 with
Prob. 11.7.3
V= 0
and
C)
6=o
+
.
+
-
-o
These are the first two of the following relations
,It
0
'-'
o
The last two define
4,6
L
di
and
I~­
as the differentials computed in the
C
characteristic directions.
The determinant of the coefficients gives the Ist characteristics.
Using the same reduction as in going from Eq. 11.6.18 to 11.6.19 gives
Sat
FA
t
cdt
A , I- -Lý t
I
d
jt
I T-CT)
n( (T)
= -a
I
11.17
Prob. 11.7.3
(cont.)
The second characteristics are this same determinant with the column matrix
on the right substituted for the first column on the left.
o
0
o
3­
I
E
a 13
(6)
13
o0
o
+ AdT~
I
-)
At
II
U
I
I
In view of Eq. 5, this expression becomes
13L
A =
+
o
Integration gives
*1
- tf
X (T)= C*
(b)
The initial and boundary conditions are as shown to the right.
C+
characteristics are straight lines.
On C from A-13
the invariant is
A
(9)
-At B, itfl-l
At B, it follows that
R
fro = CC-,
and hence from
j-
C+
(
(10)
I)s(
)
T2,( 5 S~,)X(P)
+
(F)
= Z. R (TS)'5~
(11)
-o
Also, from
c=
I
I
I
C
(12)
11.18
Prob. 11.7.3 (cont.)
Eq. 8 shows that at a point where C+ and C- characteristics cross
:-:
I
C++C.
R)
(13)
(14)
C+-C.
So, at any point on
1-'C
, these equations are evaluated using Eqs. 11
and 12 to give
I
-
SR(T)
'3
(15)
= R (T,)
(16)
Further, the slope of the line is the constant, from Eq. 5,
-aT
(17)
Thus, the response on all C+ characteristics originating on the t axis is
3
determined.
and
ft
(c)
For those originating on the z axis, the solution is
i'= 0
= 'Tr
Initial conditions set the invariants
3-+
ct
2-y
Ct
z(18)
The numerical values are shown on the respective characteristics in
IFig.
11.7.3a to the left of the z axis.
3
(d)
At the intersections of the characteristics,
from Eqs. 13 and 14
I
1Y
and
g
follow
11.19
Prob. 11.7.3 (cont.)
S-(C4 +C)
(19)
(20)
The numerical values are displayed above the intersections in the
figure as
( t1,
).
Note that the characteristic lines in this figure
3
are only schematic.
(e)
The slopes of the characteristics at each intersection now follow from
Eq. 5.
(d
(21)
_
The numerical values are displayed under the characteristic intersections
as
[A•)1
(
)
.
Based on these slopes,
I
the characteristics
are drawn in Fig. P11.7.3b.
(f)
Note (t.,
1) are constant along characteristics C* leaving the "cone".
All other points outside the "cone" have characteristics originating
where
~= I
the solution is
and
IL
function of z when
T=
=
f
I
I
(constant state) and hence at these points
and
= O
,
$ =I .
I
The velocity is shown as a
and 4 in Fig. P11.7.3c.
As can be seen
I
from either these plots or the characteristics, the wavefronts steepen
into shocks.
I
I
I
I
I
aI
-1
-
a
a
a
r-3 a
a
a
-I
a
C
H
H­
11.21
1
I
I
I
I
I
:I
I
I
11.22
t=o
m
i
m
D
1
5
g
g
·
A
i
I
I
I
I
I
1
1
I
2.
I~_-­
_L~L~
v
|--
B
i
L
1
a
I
I-
iI
4
Fig. P11.7.3c
I
11.23
Prob. 11.7.4
(a)
Faraday's and Ampere's laws for fields of the given
forms reduce to
7"
0
E
0
=
~-cE31
c)YEt
E+3
3 tEZ
o
0
0
The fields are transverse and hence solenoida 1, as required by the remaining
two equations with
(b)
= 0
.
The characteristic equations follow from
IVO
'o7
0
E 41E Z
at
AN
The I'st characteristic equations follow by setting the determinant
of the coefficients equal to zero.
Expanding by minors about the two terms
in the first row gives
...(
tt
j(
Io(-3~'1)
o(_
VA,(
+
E)2)
11.24
I
Prob. 11.7.4 (cont.)
SThe
IInd characteristic equations follow from the determinant formed
by substituting the column matrix on the right in Eq. 3 for the first column
on the left.
I
U
I
I
o
1
0o
o
(5)
=O
o
o
AN
cit
o
o
C(
Expansion about the two terms in the first column gives
-dA dAt-cdi (dcr)-r)=O4 Ae+
With
ct/d+
dc-t
(6)
given by Eq. 4, this becomes
UdE+
A'o
k
+
d o
(7)
This expression is integrated to obtain
w
N 6.(E) = C.±
(8)
where
3
I
(c)
At point A on the t=O axis t]
invariant follows from Eq. 8
as
I-
I
Sa(E
•{E4¾.
U
R(0)
1 3
11.25
Prob. 11.7.4 (cont.)
Evaluation of the same equation at B when
kZ- 0A
u)=-
=
6ý(o)4
E =o(t)
=- 9(0)
9
+
(R
Thus, it is clear that if H were also given (l4•t))at
then gives
.
(10)
z=0, the problem would
be overspecified.
On the C+ characteristic, Eqs. 8 and 11 and the fact that E=E
at B
serve to evaluate
1
c4 =
Because
t(
(o)0
-
.)=
is the same for all C
(E,) )(11)
+ 2-1
characteristics coming from the z axis,
it follows from Eqs. 8, 9 and 12 that
H +
6C(E)P =
-
-W
-
(0)
-
(
+- . J
(E~ )
(12)
)
(13)
So, on the C+ characteristics originating on the t axis,
- 6?(0)
E .(E)
(14)
(15)
X (E)
Because the slope of this line is given by Eq. 4
-t
(16)
evaluated using E inferred from Eq. 16, it follows that the slope is the
I
same at each point on the line.
For
6
= E =
, the C+ characteristics have the slopes
and hence values shown in the table.
These lines are drawn in the figure.
Remember that E is constant along these lines.
Thus, it is possible to
I
m
2-s;
mmm
mm
V
m
m
m
m
mm
t
0
a
I
V
m
m
11.27
Prob. 11.7.4 (cont.)
plot either the z or t dependence of E, as shown.
t
Eo
Note that the wave front tends to smooth out.
0
0
0.Z 8S
0.0o493
o,8s7
0.188
I-I
o.Bsl
1. 14
I
0.688
0.813
0.950o o.s9
1.71
2. o
Prob. 11.7.5
1.0
o.SO
(a) Conservation of total flux requires that
Thus, for long wave deformations, radial stress equilibrium at the interface
requires that
P
"P
=(-TVB
By replacing
IrT
icQ
- A(O)
o
-¶
(2)
, the function on the right in Eq. (2)
takes the form of Eq. 9.13.5.
Thus, the desired equations of motion are
Eq. 9.13.9
)A +
A A L
= o+
(3)
and Eq. 9.13.4
Ž2
÷,1
c
A
(4)
_
tI
where
1
w
C.
-t:7A)I
I
11.28
Prob. 11.7.5 (cont.)
Then, the characteristic equations are formed from
0
0
FAt
C
SA
­
0
0
dt
JI
9d
The determinant of the coefficients gives the I'st characteristics
dt
I
while the second follows from
0
o
A
o
dt
which is
dtRR LI(L
ý0)= 0
+d~i
with the use of Eq. 6, this becomes
dA
The integral of this expression is
-ot
(A)
(10)
where
atR
2Ao(
tot/
A
2I
-q
A
FN' - A
(11)
11.29
Prob. 11.7.5 (cont.)
Now, given initial conditions
=
A =A.9
()
where the maximum A (z) is Ama x
+=
(Aa)
(12)
, invariants follow from Eq. 10 as
- 6(Ac)
(13)
so solution at D is
2
Thus, the solution R
2
at D is the mean of that at B and C.
The largest
possible value for A at D is therefore obtained if either B or C is at the
maximum in A.
Because this implies that the other characteristic comes
from a lesser value of A, it follows that A at D is smaller than Aax
I
I
I
I
I
I
I
I
I
I
11.30
Prob. 11.8.1
A~-(At)
and H =
-
For "plane-wave" motions of arbitrary orientation, v =
(Xt)
, the general laws are:
Mass Conservation
+4(1)
Momentum Conservation (three components)
)(F 14 +'•
,)-
-
( lo°
(3)
Energy Conservation (which reduces to the insentropic equation of state)
(
=t
+O
(5)
The laws of Faraday, Ampere and Ohm (for perfect conductor),
Eq. 6.2.3
0
(6)
(8)
*
These eight equations represent the evolution of the dependent variables
( ', r
I
r,I,
J,-a+ HA, j-!,; H4)
From Eq. 6, (as well as the requirement that H is solenoidal) it follows
that H
x
is independent of both t and x.
Hence, H
x
can be eliminated from
Eq. 2 and considered a constant in Eqs. 3, 4, 7 and 8.
Equations 1-5, 7
and 8 are now written as the first 7 of the following 14 equations.
I
I
I
I
mI
Im
I
SP'
HP
a
p
ap
o
00
I
II
?'HM
YI
yeA
vd
m
0
o
0
o
m
0
0
o
0
m
0
o
o
o
0
o
I-
Co
a
rl-
0
o
0
0
>
C
o
0
QQ
0
0
-
0
I
0
0
0
0
0
0
Q
0
o
0
o
vt o,/
o'Hr/ o "H('0 o
0
0
0
0
0
0
0
0
0
0
m
0
0
o0
m
o
0
o
m
0
0
0
O
m
0
o
0
o a
1
o
a
o
0
­
a oj
0o
o
--
oo 0
0
O 0
o
o
o
o o
o
4
o
a?
0
o
o
o
o
0
o
o
0o
0
0
V
o
o
v
o
0
0
o0 0
0-
o
Ho
o
V
0 ) 0 00
0
o
o
v
0
-=
m
11.32
Prob. 11.8.1 (cont.)
Following steps illustrated by Eq. 11.15.19, the determinant of the
coefficients is reduced to
0
0
0
0
p
0
0
0
0
o
0
)
-At
0
4 x
o
o
cxt
- (6.P
o
-
a
(-)
-W1
o
The quantity
0
a
t
0
0
0,
o
o
-QL'.-•)
0
-0
(10)
o
o
-(
can be factored out of the fifth row.
)
1
That row is then
subtracted from the second so that there are all zeros in the second column
except for the A52 term.
Expansion by minors about this term then gives
0
X
o
o
o
(x )
0
0
-
/"l
o -uM
(11)
O3
=0
0.
o
-
0
-
o
pN
1k, q-J
H
o
Multiplication of the second row by
-)
I(_)
0
o
RT
H4x
o
p0/
-jj-9)
and subtraction from the
11.33
Prob. 11.8.1 (cont.)
first generates all zeros in the first row except for the A12 term.
Expansion
about that term then gives
o
"
I
o
o
___
I
0
(12)
K)
-4"/
=0
0o
H
Io
k
a
aa
-( 0
~
I
K
0
Multiplication of the second column by /y
-_dx )
and addition to the
fourth column generates all zeros in the second row except for the A22 term,
while multiplication of the third column by
JoH_(-_)and addition to
the last column gives all zeros in the third row except for the A33 term.
Thus, expansion by minors about the A22 and A33 terms gives
-#1Y(
6
7>4.)ý{&( .c
_AX
cJ;d~l
I
2
H,-~
I
AH
(13)
I
0I
-(f
d%
I
I
I
I
11.34
Prob. 11.8.1 (cont.)
This third order determinant is then expanded by minors to give
*,,N-(&-
+?
(14)
This expression has been factored to make evident the 7 characteristic
lines.
First, there is the particle line, evident from the outset (Eq. 5)
as the line along which the isentropic invariant propagates.
c\X
(15)
The second represents the two Alfven waves
3
+KQ-*
~
S4I(16)
and the last represents four magnetoacoustic waves
4A
L64(17)
where
I
I
I
I
Olb
'ýe
= Fn Ti
+
11.35
Prob. 11.9.1
Linearized, Eq. 11.9.17 becomes
de
I
(1)
--
Thus,
(2)
and integration gives
,
+
e
=
Co~o•
(3)
•
where the constant of integration is evaluated at the upstream grid where
oo
n=0 and e=e..
Prob. 11.9.2
Linearized, Eqs. 11.9.9 and 11.9.10 reduce to
(1)
S--e
(2)
Elimination of e between these gives
A'¾--
(3)
(3O
+
The solution to this equation giving n=O when t=t
3
is
U
U
and it follows from Eq. 1 that
To establish A(t ) it is necessary to use Eq. 11.9.15, which requires that
-A+A(fI
A
For the specific excitation
V
0
)
,A:a /
(6)
0
\?'e.
(z)
it is reasonable to search for a solution to Eq. 6 in which the phase and
amplitude of the response at z=0 are unknown, but the frequency is the same
as that of the driving voltage.
I
I
11.36
Prob. 11.9.2 (cont.)
A= Oe A%
Observe that
·
if
C
A(t -U)e
-
and
e
-cei~i~
~a
V~\
2
AItz.
I_.
t
,2
(10)
Thus,
U
(11)
i
+
.S(-4I-'I)
Substitution of Eqs. 7, 8 and 11 into Eq. 6 then gives an expression that can
be solved for
A.
U4
( -
)U
3­
(I
+'-')'
Z
~(I-
(12)
02)
Thus, the solution taking the form of Eq. 4 is
by Eq. 12.
givenAis
where
where A is given by Eq. 12.
(13)
11.37
Prob. 11.10.1
P
With
S+
, Eqs. 11.10.7 and 11.10.8 are
= O
d e (AAi)=
+
+
4.6 -
Mi+
;
C
In this limit, Eq. 1 can be integrated.
Initial conditions are
eoz)0
. =
These serve to evaluate
()
o)
9,(',
C+
in Eq. 3
At
where
a point
theC characteristics cross Eq. 3 can be solved simultaneously(6)
At a point C where the characteristics cross Eq.
3 can be solved simultaneously
to give
4
i
C-
PM4I
[1 M4-I)c- (MAi •
S[C- -
C4
Integration of Eqs. 2 to give the characteristic lines shown gives
= (/
)
t ÷ :A
8
c­
,,,,
I
1I1.
Prob. 11.10.1
3
J
(cont.)
For these lines, the invariants of Eqs. 6 are
With
A
and
B
evaluated using Eq. 8, these
invarients are written in terms of the (z,t) at
point C.
C'+
+ (,AA
(A(10)
and, finally, the solutions at C, Eq. 7, are written in terms of the (z,t) at C.
I
(11)
|-(
e
I
I
I
I
=
AM-I*)
o[ -
-
(M•AI),(Me,
9, 4 - (M-I)t]4 (Mivi li
I)[,••(M-o)
t
- (M-')
61
(12)
11.39
Prob. 11.10.2
(a) With
Y= O , Eqs. 11.10.1 and 11.10.2 combine to give
IL
+UC£
1g
.
-
Normalization of this expression is such that
I
l=
/-r
= T/ OIL
= -/'-r
gives
k
· b-P
___
(I­
(i
rr
where
(b) With the introduction of v as a variable, Eq. 3 becomes
(k+ a,~tl
__
3~­
R=Lr
+
where
3tP~
r
The characteristics could be found by one of the approaches outlined,
but here they are obvious.
On the I'st characteristics
Am..
'= 4
the II'nd characteristic equations both apply and are
11.40
(cont.)
Prob. 11.10.2
A0
z-
(7)
j-
(8)
at
I
Multiply the left-hand side of Eq. 7 by the right-hand side of
Eq. 8 and similarly, the right-hand side of Eq. 7 by the left-hand side of
Eq. 8.
I
(c) It follows from Eq. 9 that
I z.?L
(10)
or specifically
1T
z
i ++T
-o
Phase-plane plots are shown in the first quandrant.
unstable nature of the dynamics, the trajectories are open for
a deflection that has
---
other of the electrodes).
0oo
as
4
P>
, showing
.-(the sheet approaches one or the
The oscillatory nature of the response with
is apparent from the closed trajectories.
I
I
I
I
I
i
Reflecting the
- ­
11.41
I
A
04
.2.
.4
--- q
11.42
Prob. 11.10.3
The characteristic equations follow from Eqs. 11.10.19­
11.10.22 written as
I
1
M,
0
I
ft
dt
A, M,-I
0
-1
0
0
0
o
o
o
a
o
o
o
o
o
o
U)
I
I
I
I
0
o
o
dt dz
o
a
o
o
a
0
o
o a
o
I M
o
o
o
o
0
o
o
0
ji,
ell
M•M-I
-I
d d1
0o
o
0
de,
(1)
P.:
Z
I
o
o
o
o
0
o
o0
t
lC
e,
VIit
eza
de,
Also included are the 4 equations representing the differentials
I
0,.
*
*
Aez.
These expressions have been written in such an order
that the lack of coupling between streams is exploited.
I
Thus, the determi-
nant of the coefficients can be reduced by independently manipulating the
first 4 rows and first 4 columns or the second 4 rows and second four
I
columns.
I
Thus, the determinant is reduced by dividing the third rows
by dt and subtracting from the first and adding the third column to the
second.
0
A
I
I
o
0
-aI
di
At
o
di
di
00
o
0
o
-I
o
o
At
Ak
o
A
dn
o
A-
dt
At
(2)
11.43
Prob. 11.10.3 (cont.)
This expression reduces to
,.zfI
.l..
.4
,4
.
- ',)Y -)-\
Cat) I ( ,i -"M ) -I I Ik "E
_
=
o
and it follows that the Ist characteristic equations are
Eqs. 11.10.24
and 11.10.26.
The IInd characteristics follow from
0
0
0
o
0
0
0
I
rA/
M.
0
o
I
-I
0a­
dt
dc
o
d•
o
o
dit
PI,
0
A9,1
d e,
M-Z
O
I
0
dt
0
0
0
da
0
0
0
0
Expanded by minors about the left column, this determinant becomes
+
Thus, so long as
•
O
0\
012 M,=t
(M,-0jID,
(not on the second characteristic equatioin)
Eq. 5 reduces to
In view of Eq. 2, this becomes
11.44
Prob. 11.10.3 (cont.)
Now, using Eq.
Sa,
AA.(Mt
) +(M-1)(AM
1(Mt
t
Ae,=?T I(M,+
\·1
At
and finally, Eq. 11.10.23 is obtained
de,
d-01 4 (/v\A,
= PSdt
These equations apply on C1 respectively.
characteristics, which apply where
•t)= O
To recover the IInd
and hence Eq. 4 degenerates,
substitute the column on the right in Eq. 1 for the fifth column on the
left.
The situation is then analogous to the one just considered.
The characteristic equations are written with Ad,-originating at A, etc.
Aon
A
CI
The subscripts A, B, C and D designate the change
in the variable along the line originating at the subscript point.
The
superscripts designate the positive or negative characteristic lines.
Thus, Eqs. 11.10.23 and 11.10.25 become the first, second, fifth and
sixth of the following eight equations.
0O
I 0
I
0
M&I
Mý-I
0
o
0
-P~gIAIzABbt
IA
eIB
o
o
0
O
M/I
0
0
0
o
M+JI
0
0
0
0
o
0
0
-I
0
0
0
-( eIAsI,3
- ( e
-1,9z- C-
-4
30
etc_~q,
0
0
1 -I
11.45
Prob. 11.10.3 (cont.)
The third, fourth and last two equations require that
+-
+
46E==
C
e
e
A
(10)
Clearly, the first four equations are coupled to the second four only
through the inhomogeneous terms.
Thus, solution for
4
A and
a
A
involves the inversion of the first 4 expressions.
The determinant of the respective 4x4 coefficients are
-
SI
2 1
(11)I
and hence
AITZ
/I-*I
0
I
0
+
IA
Z
So
PL(IIAj
fts
-( IA(
-I
r
A)&t
3)t-(
,~s)
e- A-4,3)
0
0
AM
-I
O
A11-I
0
0
0
o
o
I
-I
(
1
I
I
11.46
Prob. 11.10.3 (cont.)
which is Eq. 11.10.27
Similarly,
o
+
IA
P
ýlAbt
o0
I
-I
-( *IA"2LY)
-L9
0
0,M
O
-o -(~IA- ()
--
tI+(,d_
A7,1
3)
o
F4IAt+()A"
,,
-
(13)
4t
1
- (M,+
I)
-I
-( e IA-
e-IM)
-(CIA
which is the same as Eq. 11.10.28.
+
The expressions for
VZ
+
_
and
et, are found in the same way
from the second set of 4 equations rather than the first.
is the same except that
A -- C ,
--• D)
1-
2 and
The calculation
11.47
Prob. 11.11.1
In the long-wave limit, the magnetic field intensity above
and below the sheet is given by the statement of flux conservation
Thus,
x-directed force
the per unit area on the sheet is
Thus, the x-directed force per unit area on the sheet is
Ai)
, .[(-/ol,[
+ Ad)
T
This expression is linearized to obtain
To ,lD-.)44
i 2-(-A
-L~~~~~i·.~-~
3
-74;2~YH3-;B~
o.I
CL
Thus, the equation of motion for the sheet is
'Z
C)
·pi~+
UA_ T=/
Z~f
OL
+ Za4°A_
9
A
Normalization such that
ý
=
t -T ) ? = ýZ
VE
236'/~(
gives
(ztC
V 3t
2.
_Lr
(-rv
2.~­
X
2
T
which becomes the desired result, Eq. 11.11.3
-~+?
where
~17)
t
4loC
p,_zU/
+
CA
2
,)
= AAo/,
Z/IIUH>2AJ
t4
11
3
I
Prob. 11.11.2
1.0
.LI
'4
The transverse force equation for the "wire" is written
by considering the incremental length ~t shown in the figure
L
and in the
Divided by
I
Divided by
dt
imit
and in the limit
6a
I
-.
6
-- o
tabt
,
this expression becomes
nI2
=L
(2)
4T
The force per unit length is
I
(3)
+X,
Evaluated at the location of the wire, Y
-
=
, this expression is inserted
into Eq. 2 to give
This takes the form of Eq. 11.11.3 with
VV
I
I
I
I
I
I
I
_(5)
and
/VM=O and
ý=o with f=••'
,
11.49
A
A
Prob. 11.11.3
The solution is given by evaluating
Eq. 11.11.9.
With the deflection made zero at
following two equations is obtained (
- = Y
A
and
a =
=
=
B
in
, the first of the
where
X
./ 1"V )
-i 2
C
(1)
d
The second assures that
(1)
A
b
*clj
6e".
A
Solution for
and
gives
A-A
-
Aa
A
-;,a-Ca
and Eq. 11.11.9 becomes
ee-a,k
6e ýA
+e
e
.R,-dlh,
e
t
cW.
C.
With the definitions
Me -l
-A
Eq. 3 is written as Eq. 11.11.13
-
For
is real.
CO
>p
(z
I)
9'e 5{.j
(sub-magnetic,
The deflection is then as sketched
U
< 0 and
eI
M
'''
(5)
),
11.50
I
Prob. 11.11.3 (cont.)
I ON
-
--
-
-1-zt/
I
I
a
Note that for
M
< 1. ,
<
The picture is for the wavelength of the envelope greater
-z direction.
( zrr/I)2T/
than that of the propagating wave
I
and the phases propagate in the
The relationship of wavelengths depends on C.
I
< 171
, as shown in the figure,
and is as sketched in the frequency range
I
P-
,
I
.
e (J<
<
C
For frequencies
, the deflections are more
complex to picture because the wavelength of
the envelope is shorter than that of the
traveling wave.
U
I
With the frequency below cut- j*~
-I
off, - becomes imaginary.
Let
0 =a
.
and Eq. 5
becomes
I
I
Q.
(6)
Now, the picture is as shown below
i
I
I
Again, the phases propogate upstream.
The decay of the envelope is likely
to be so rapid that the traveling wave would be difficult to discern.
11.51
Solutions have the general form of Eq. 11.11.9 where
Prob. 11.11.4
A­
+ Z
Ae
g.= Je(
.e
°
.
)
e.
Thus
-dl
e
7-dL
/ý
Z
evaluated at
)
U°
and the boundary conditions that
ýI
z
e
0
) -­
be zero require that
Z =0
AT
-ý
I
A
A
so that
i.
-_
_
kZ - @I
and Eq. 1 becomes
Ae
4Z,
.
e
~i1
Z -
)
a
Q 1
With the definitions
a
Eq. 5 becomes
/
M_ -I
M -- I
I
(cont.)
Prob. 11.11.4
.
_
---
-(7)
I
z/
CO
For
I
imaginary,
+
(
'ji
.
f
(super
(<
electric below "cut-off")
)
- -
is
Then, Eq. 7 becomes
..
At
7
Ct
Note that the phases propagate downstream with an envelope that eventually
Iis
is
an in
an inc
I
I
L
I
U
This is illustrated by the experiment of Fig. 11.11.5.
is so high that
C0 4
- /()
If the frequency
, the envelope is a standing
)O
wave
Note that at cut-off, where 4),=
I
3
I
I
infinite wavelength.
(a
-I)
, the envelope has an
As the frequency is raised, this wavelength shortens.
This is illustrated with
--
O
by the experiment of Fig. 11.11.4.
11.53
The analysis is as described in Prob. 8.13.1 except
(a)
Prob. 11.11.5
that there is now a coaxial cylinder.
Thus, instead of Eq. 10 from the
solution to Prob. 8.13.1, the transfer relation is Eq. (a) of Table 2.16.2
I
because the outer electrode is an equipotential.
with ~=O
^a
'
(1)
I
Then it follows that (m=l)
Rz
PR
In the long-wave limit,
(b)
(2)
Kj=
01
(
=
0o,
(3)
= 0O
(<ci and
and in view.of Eqs. 28, for
(oj
(4)
-
To take the long-wave limit of
l(c•,3)
, use Eqs. 2.16.24
1
_S
1
2
j
to evaluate
c. _•)+
R (a
so that Eq.
(6)
2 becomes
Z
P
I
I
@,U)
R+
Z
+
(C
I
A(7)
The equivalent "string" equation is
L )
*
(8)
3
Normalization, as introduced with Eq. 11.11.3, shows that
-
-
' V1io
-
-(9)
.n)
I
i
11.54
5
Prob. 11.12.1
The equation of motion is
and
temporal
the and spatial
and the temporal and spatial transforms are respectively defined as
I
2T
2ff
-0t
i(*,,=
-
C
(1)
(2)
(3)
•¢•,j
8C•,,,,•
• c& ,•=
(3)
00COZ
i
The excitation force is an impulse of width At
and amplitude
O
in space
and a cosinusoid that is turned on when t=O.
I
(
1)•=w a U" (
) ý" Cs SC "0• ie
(4)
It follows from Eq. 2 that
(5)
&-a U
I
In turn, Eq. 3 transforms this expression to
I
With the understanding that this is the Fourier-Laplace transform of f(z,t),
it follows from Eq. 1 that the transform of the response is given by
where
I
Now, to invert this transform, Eq. 3b is used to write
44
_I
I
I
I
I
LI
(C~~-"') -
I
_
_(9)
-x
3
11.55
Prob. 11.12.1 (cont.)
1
This integration is carried out using the residue theorem
5
=
M• +
2N.
d = zw•
N(k&)
CU(k)
It follows from Eq. 7 that
-
h
(10)
--
=-
3
and therefore
-The open integral
-called
for
with Eq.
8 is
equivalent
to the closed contour
The open integral called for with Eq. 8 is equivalent to the closed contour
integral that can be evaluated using Eq. 9
in
Fig. 11.12.4.
Poles,D (Lj) *=0,
in
I
U
I
I
the
k plane have the locations shown to the
right for values of C) on the Laplace
contour, because they are given in terms
of CW
by Eq. 10.
3
The ranges of z assoc­
iated with the respective contours are
those reauired to make the additional
parts of the integral added to make the contours closed ones make zero
contribution, Thus, Eq. 8 becomes
e
r 7 e~
Here, and in the following discussion, the uppei
and lower signs respectively refer to
2 <0 ana
The Laplace inversion, Eq. 2b, is evaluatE
using Eq. 12
_I?
I%
I
I
I
I
I
11.56
Prob. 11.12.1 (cont.)
C
I
.
.
._•
I
(13)
Choice of the contour used to close the integral is aided by noting
Sthat
5
+
•
(ti(••
e
-
-wa(t+
(14)
e(
and recognizing that if the addition to the original open integral is to
be zero,
t
- ->0
- +t
on the upper contour and
V
<O0 on the lower one.
\Vt
The integral on the lower contour encloses no poles (by definition
so that causality is preserved) and so the response is zero for
3
(15)
V
Conversely, closure in the upper half plane is appropriate for
>
3
?-(16)
>
'V
By the residue theorem, Eq. 9, Eq. 13 becomes
W dt
g
C
+ WI
C.
/
-J)
'0()= W4-(OA- w)
e
O
This function simplifies to a sinusoidal traveling wave.
I
To encapsulate
Eqs. 15 and 16, Eq. 17 is multiplied by the step function
P- )
I
I
I
I
(17)
5itrw(t
I)
(t01;
(18)
11.57
Prob. 11.12.2
The dispersion equation, without the long-wave approximation,
is given by Eq. 8.
Solved for CO it gives one root
That is, there is only one temporal mode and it is stable.
This is suffi­
cient condition to identify all spatial modes as evanescent.
The long-wave limit, if represented by Eq. 11, is not self-consistent.
This is evident from the fact that the expression is quadratic in CJ and it
is clear that an extraneous root has been introduced by the polynomial
approximation to the transcendental functions.
terms must be omitted to make the
In fact, two higher order
-k relation self-consistent, and Eq. 5.7.11
becomes
+
Solved for 0 , this expression gives
w=IQ
which is directly evident from Eq. 1.
To plot the loci of k for fixed
values of
Wr as O-
I
I
I
I
I
I
t
U-u
I
1
I
0r
-o.5
I
I
I
goes from 60
to zero, Eq. 2 is written as
U = o.2
(4)
The loci of k
the figure with
are illustrated by
U= 0,. ­
CWr= -05
0.5
I
I
I
I
a
11.56
Prob.ll.13.1
With the understanding that the total solution is the super­
position of this solution and one gotten following the prescription of Eq.
11.12.5, the desired limit is
e(tt)=x.-
M"
tc
r
C
C
where Eqs. 11.13.8 and 11.13.9 supply
Aecsa
t
) =
(W
-
--
%)
-=
IE J)63-
v
The contour of integration is shown to
the right (Fig. 11.13.4).
It
CL
Calculated here is the
response outside the range
z<o,#)Iso that the
summation is either n=l or n=-l.
For the
particular case where P ) 0 and M < 1 (sub-elect
C'%
Eq. 11.13.16 is
rrr\
r
(4)
Note that at the branch point, roots k
coalesce at k
nEq.
11.13.15,
in the k plane.
From
Eq. 11.13.15,
t --
(5)
as shown graphically by the coalescence of roots in Fig. 11.13.3.
the contributions to the integration on the contour
go to zero.
( W
= WC,4W&Jý
Eq. 1 (exp j CttXexp-&J
as t -•
at
J
.)
t)
As t--~c ,
just above the c),axis
makes the time dependence of the integrand in
and because W>)0, the integrand goes to zero
Contributions from the integration around the pole (due to f(w))
= •o are finite and hence dominated by the instability now represented
by the integration around the half of the branch-cut projecting into the lower
half plane.
The integration around the branch-cut is composed
of parts C 1 and C2
paralleling the cut along the imaginary axis and a apart C3 around the lower
branch point.
Because D' on C2 is the negative of that on C , and
C1 and C2
1
11.59
Prob. 11.13.1 (cont.)
are integrations in opposite directions, the contributions on C1+C 2 are
Thus, for C1 and C 2 , Eq. 1 is written in
twice that on C1 .
)(a
-2.o
terms of 4(*
=-8G)
(6)
2
In evaluating this expression approximately (for t-eo- ) let <, be the origin
by using G-4- as a new variable
o
~
-
Q
-
r--
~
(7)
O
(<
as the integration is carried out.
Thus, as
contributions to the integration are confined to regions where
3-
The remainder of the integrand, which varies slowly with
by its value at
j
Then, Eq. 6 becomes
Se
Note that
5
a
Eq. 7 becomes (kl-7
. Also,
=k i--i
Tat
0 is taken to
ks )
I
'C- -9•0.
,is approximated
so the integral of
co
I(ik't
iO N
(_ý
,
- . 0(8)
The definite integration called for here is given in standard tables as
(9)
The integration around the branch point is again in a region where all
but the
~
C' 4+
in the denominator is essentially constant.
iGJ
S'
, the integration on C3 of Eq. 1 becomes essentially
~
_-
Let
n
= R'Sp
Thus, with
g
3
4
nt
q"s
(10)
-,
and the integral from Eq. 9 becomes
In the limit R-- 0, this integration gives no contribution.
Thus, the
asymptotic response is given by the integrations on C + C2 alone.
I
I
5
11.60
Prob. 11.13.1
(cont.)
TOM
__
_
N-
_
__
The same solution applies for both z ( 0 and
renders the solution non-symmetric
in z.
(12)
_
<(z.
The z dependence in Eq. 11
This is the result of the convection,
as can be seen from the fact that as M-90, k--v 0.
s
Prob. 11.13.2
(a)
The dispersion equation is simply
Solved for CJ , this expression gives the frequency of the temporal modes.
I
I
72
+
Z
4
7
(2)
Alternatively, Eq. 1 can be normalized such that
=
WI
l
TTV1•
)-
=
(3)
and Eqs. 1 and 2 become
£
To see that
I
"small" 9 , Eq. 2 becomes
f >V (M
C' =
> i')
(OV)+
implies instability, observe that for
+()
M)
(5)
I
Thus, there is an
j
is based on an expansion of M about M=l, showing that instability depends
on having
I
I
I
I
/MI>1.
.4 O
A > I
.
Another examination of Eq.5
11.61
Prob. 11.13.2 (cont.)
.nmnl
,z
t.)
c
riin .t--nn
n
nf'
r•,l
I
'
4.,.
--
,
;
Fi
"11-e
V 4a
r.
11
,.
...
. L ..
M=2
C.),
k
Fig. 11.3.2a
(b)
To determine the nature of the instability, Eq. 4 is solved
for compllex k as a function of
C= C-)
M-I
or
c +
(C
M(4rar-'
,=
Mc.-·~\i~;~nnr;
~AK-I~~,·
W-
-?
a (AAf)]+
O--0 c
Note that as
-
and for
for
0
AK)
both roots go to
varying from
lower half plane.
Y[0,
-W
--
to zero at fixed
0-0.
I
Thus, the loci of complex k
CJr move upward through the
The two roots to Eq. 7 pass through the kr axis where c3
reaches the values shown in Fig. 11.3.2a.
Thus, one of the roots passes
into the upper half plane while the other remains in the lower half plane.
There is no possibility that they coalesce to form a saddle point, so the
instability is convective.
5
11.62
Prob. 11.14.1
(a)
P -
Stress equilibrium at the equilibrium interface
Eo
=;
(1)
=
In the stationary state,
(2)
P =Tb
I
and so, Eq. (1) requires that
-.'b
2-•
OL
-
(3)
FO
All other boundary conditions and bulk relations are automatically
satisfied by the stationary state where t-= rUC in the upper region,
1) = 0
in the lower region and
P=
I
(b)
9
2
(4)
The alteration to the derivation in Sec. 11.14 comes from
the additional electric stress at the perturbed interface.
I
The mechanical
bulk relations are again
ra
A L:
-P
[ co44i!ho*
C0+
-~~c
141
(5)
L
i
[:]
S
lSthh ,•J -I
5
fýb
The electric field takes the form
perturbations,
I
I
=-
e , are represented by
Lx
-=
I
Ib
III
I
eL -- -
(6)
and
11.63
I
Prob. 11.14.1 (cont.)
SSi
1
f(7)
in the upper region.
There is no E in the lower region.
Boundary conditions reflect mass conservation,
I
(8)
that the interface and the upper electrode are equipotentials,
-Lc
-
El
-~­
C
A
A
A
Z
0
_
1
(9)
and that stress equilibrium prevail in the x direction at the interface
The desired dispersion equation is obtained by substituting Eqs. 8 into
Eqs. 5b and 6a, and these expressions for
P
and p
into Eq. 10, and
j
Eq. 9 into Eq. 7b and the latter into Eq. 10.
-(
•,
C
o
•
,Y
11)
C)
A
, the term in brackets must be zero, so
To make
A/CB.+[ C"bc.",L
k
4
I'
(12)
This is simply Eq. 11.14.9 with an added term reflecting the self-fieldeffect of the electric stress.
In solving for C0,
-
~
a)C-OA i
•O).
'b~f+(,q p~g-e
-O
i;---3
I
1
I
group this additional
term with those due to surface tension and gravity B
(
+e
5
4-. ( pb-- .)%·
It then follows that instability
i
5
i
11.64
Prob. 11.14.1 (cont.)
results if (Eq. 11.14.11)
IeL'
For short waves
I
I>>i
k.(
) this condition becomes
The electric field contribution has no
g
dependence in this limit, thus
making it clear that the most critical wavelength for instability remains
the Taylor wavelength
I
F
I
j(15)
aI-1
Insertion of Eq. 15 for P
U
(
in Eq. 14 gives the critical velocity
+ - (-a
-QC-(
'E
(16)
By making
I
£
ý I Y
2-
(17)
the critical velocity becomes zero because the interface
is unstable in
the Rayleigh-Taylor sense of Secs. 8.9 and 8.10.
I
In the long-wave limit (I A
has the same effect as gravity.
Wea+
I
PE~
That is
and the i
(-
-g
the electric field
4
•
--­
dependence of the gravity and
Because the long-wave field effect can be lumped with that
due to gravity, the discussion of absolute vs. convective instability
given in Sec. 11.14 pertains directly.
I
)
electric field terms is the same.
(c)
I
V(E-ht&21
<< i
• ,
I I8I
11.65
Prob. 11.14.2
(a) This problem is similar to Prob. 11.14.1.
The
I
equilibrium pressure is now less above than below, because the surface
3
force density is now down rather than up.
=
•
o(1)
The analysis then follows the same format except that at the boundaries
of the upper region, the conditions are
D)
ik~,CL~c +5
I
A(2)
:o
and
I
=o
3)
Thus, the magnetic transfer relations for the upper region are
I
W+
4F1
s.nh
Fee,
1
1
S V1
The stress balance for the perturbed interface requires (VIA
+P -d ^&-+
+p H.; A
Substitution from the mechanical transfer relations
for
(Eqs. 5, 6 and 8 of Prob. 11.14.1) and for
=o0
P
(5)
1
£
and
^ d
Ad from Eq. 4 gives the
desired dispersion equation.
Z
(6)
Thus, the dispersion equation is Eq. 11.14.9 with
- ((h
-pc) 4/,74 '4 :
Co'
~Q/, . Because
+±÷~((
-(---.
the effect of stream-
ing is on perturbations propagating in the z direction, consider
=
I
I
a
Then, the problem is the anti-dual of Prob. 11.14.2 (as discussed in
B
Sec. 8.5) and results from Prob. 11.14.1 carry over directly with the
I
substitution
-
E
/4o
"a
5
11.66
Prob. 11.14.3
The analysis parallels that of Sec. 8.12.
There is now
an appreciable mass density to the initially static fluid surrounding the
I
now streaming plasma column.
Thus, the mechanical transfer relations are
(Table 7.9.1).
)
F0l
Gc
(1)
SG,,,
(2)
where substituted on the right are the relations t1
)
== •
CA- • U (-
same with d
I
5
and
The magnetic boundary conditions remain the
(no excitation at exterior boundary).
equilibrium equation (Eq. 8.12.10 with
p
is evaluated using Eqs. lb, and 2, for
p
8.127 and
IO
=-O
-
= 0.for
A
Thus, the stress
included)
and
p
and Eqs. 8.12.4b,
to give
V7/-Itc+
O
(*'
-/Ua1A~
(4)
+
I)Mc
L)
This expression is solved for w.
k •
to give an expression having the same form as Eq. 11.14.10
F
(5)
11.67
Prob. 11.14.3 (cont.)
(F., (o,j
I
Z(a0)
o, F
> o
,
The system is unstable for those wavenumbers making the radicand
negative, that is for
i
-z[10
Prob. 11.14.4
(a)
(OV=
/0 I(F.(I)01
(6)
1
The alteration to the analysis as presented in
1
Sec. 8.14 is in the transfer relations of Eq. 8.14.12, which become
[Inh
_
= Q3T)
UjO
(1)
I
where boundary conditions inserted on the right require that
AC
Ad
A
and
LK •
,
- =
(W - 9,U)T.
Then evaluation of the interfacial
stress equilibrium condition, using Eq. 1, requires that
kI
%( 6 -ea-
+
I
(2)
E.(
)+- E
66-9
(c
.1
(b) To obtain a temporal mode stability condition, Eq. 2 is solved
for
W.
I
Li
LT A
C
dBk k ý +
C
b
cokPeL
a
1+
i
I,
IZ
ýcr'i%
A(O-+CA
I
u-o'•
'-'-"
i•C. '
I
5I
I
11.68
Prob. 11.14.4 (cont.)
Thus, instability results if
CA
5
I
ý
'h
Ratl,+ coý Ub)
g.
b cot+
4a
II
I
It
i
11.69
Prob. 11.15.1
Equations 11.15.1 and 11.15.2 become
Thus, these relations are written in terms of complex amplitudes as
=O
a
and it follows that the dispersion equation is
L(w -&4+
-
+
-&+P
M
P]?)j(
(3)
(4)
0
Multiplied out and arranged as a polynomial inQ , this expression is
Jr +
[z
-2
g(e•,'-
p-
T
1=0Z-,
I
3
(5)
Similarly, written as a polynomial in k, Eq. 4 is
• •,•-,1i÷+ f
P
- 2-C-•-,Z 6&I)+• 4,•f+2.• P+ X4--- o
These last two expressions are biquadratic in wJ and k respecively, and can
be conveniently solved for these variables by using the quadratic formula twice
't ',
-
O'4
)
-F
)
4
9 7)
A %
o IM C)-P~l~l
ý-=+ I
in
the limit
--+1*,
W--D t_
7.
,z
LMt
First, in plotting complex
Eq.
(
I
I
_')
(8)
for real k, it is helpful to observe that
7 takes the asymptotic form
(9)
(Mt 0)
These are shown in the four cases of Fig. 11.15.1a as the light straight lines.
Because the dispersion relation is biquadradic in both W
that for each root given, its negative is also a root.
and k, it is clear
Also, only the complex
i is given as a function of positive k, because the plots must be symmetric in k.
I
I
5
I
m -
m
1~
a
(c
~
CS
·
~-b%
$~
as a function of real k for subcritical electric-field (P=l) and magnetic-field (P=-l)
b,
-r3
coupled counterstreaming streams.
Complex
9
C
mI
-
I
-3
3
ao:
0
0
0
CV
o
s
n
-M
as a function of real k for supercritical magnetic-field (P=-l) and electric-field (P=1)
coupled counterstreaming streams.
Complex c
'd
1
o
5
11.72
j
Prob. 11.15.1 (cont.)
The subcritical magnetic case shows no "unstable" values of c. for
j
real k, so there is no question about whether the instability is absolute or
convective.
For the subcritical electric case, the figure
below shows the critical
5
00
b
plot of complex k as
is varied along the trajectory
U
U
I'
shown at the right.
The plot
makes it clear that the instabilities
are absolute, as would be expected' from the fact that the streams are subScritical.
Probably the most interesting
I
case is the supercritical magnetic one,
because the individual streams thel tend to be stable. In the map of complex
k shown on the next figure, there are also roots of k that are the negatives
of those shown.
I
Thus, there is a branching on the k axis at both k
r
r
and at kr 'F -.56.
Again, the instability is clearly absolute.
last figure shows the map for a super-electric case.
.56
Finally, the
As might be expected,
from the fact that the two stable (P=-l) streams become unstable when coupled,
I
I
this super-electric case is also absolutely unstable.
11.73
Prob. 11.15.1 (cont.)
3
'
'
P= 1
.4
.z
.3
i;
I0
o
oa•"
11.74
Prob. 11.16.1
With homogeneous boundary conditions, the amplitude of an
eigenmode is determined by the specific initial conditions.
JI
Each eigen-
mode can be thought of as the response to initial conditions having just
the distribution required to excite that mode.
To determine that distrib­
Sution,
For example,
one of the amplitudes in Eq. 11.16.6 is arbitrarily set.
suppose A 1 is given.
Then the first three of these equations require that
-A,
IAz
5
A,
3,
c?
AA
-4
JqL A4
- pqA
and the fourth is automatically satisfied because, for each mode,
I
CG
is
such that the determinant of the coefficients of Eq. 11.16.6 is zero.
With Al set, A 2 , A 3 and A 4 are determined by inverting Eqs. 1.
Thus,
within a multiplicative factor, namely AL, the coefficients needed to
I
I
I
I
evaluate Eq. 11.16.2 are determined.
11.75
I
Prob. 11.16.2
and IIA)
(
1
(a)
With M=-/Mi
3f
=AA
, the characteristic lines
are as shown in the figure.
if
Thus, by the
I
I
I
Is
arguments given in Sec. 11.10, Causality
and Boundary Condition , a point on either
boundary has two "incident" characteristics.
Thus, two conditions can be imposed at each
boundary with the result dynamics that do
not require initial conditions implied by
subsequent (later) boundary conditions.
The eigenfrequency equation follows from evaluation of the solutions
•,= "•-Q A.ete
h=1I
where (from Eq. 11.15.2)
_z2I.W-
•
~_A~p
P
I
I
'I
I;
I
a
Thus,
I
Ri
e
Q,
e"~
A.
q4
Qz
Q4,e
-ba
4P-
A3
A4
I
I
S
I
11.76
Prob. 11.16.2 (cont.)
Given the dispersion equation,
()8,
(4
D(c•.,~)
this is
an eigenfrequency equation.
(5)
Det (w) =
In the limit
~-.-o , Eqs. 11.15.1
and 11.15.2 require that
both of these equations are satisfied if
For
zJ,
and for
and it follows that
Q =I
)
Qq=
I
-
Thus, in this limit, Eq. 4 becomes
cI
e-
I­
e­
-
-I
- Z•,
I filla
(10)
5
11.77
Prob. 11.16.2 (cont.)
U
and reduces to
j
The roots follow from
I
'
Z
,3...
= I
--.
(12)
and hence from Eqs. 7 and 8
,
Instability is incipient in the odd m=l
P
For
(13)
mode when
r•
P
(b)
-
(14)
M)1,
the characteristics are as shown in the figure.
Each
I
I
boundary has two incident characteristics.
Thus, two conditions can be imposed at each
boundary.
In the limit where
--
O
I
,
the streams become uncoupled and it is most
likely that conditions would be imposed on
I
I
the streams where they (and hence their
associated characteristics) enter the region
of interest.
From Eqs. 11.15.2 and 11.15.5
4r~
- t
-tit
A,(15)
6?
4b
3tz
,B
AR A
t
(16)
I
I
5
11.78
I
Prob. 11.16.2 (cont.)
I
Evaluation of Eqs. 11.15.2, 15, 11.15.5 and 16 at the respective boundaries
where the conditions are specified then results in the desired eigen-
I
frequency equation.
1~3Q-3
I
Q4
"A,
f-4 P4,
A?
(17)
=O
€
e
ýze
ra
Given that
,
=
3
A,
AA
) , the determinant of the coefficients comprises a
b(
complex equation in the complex unknown, c
Prob. 11.17.1 The voltage and current circuit equations are
Ik%'t)
I
L Lc
c&~t)=
In the limit Ly-P
I
I
4
O, these become the first two of the given expressions.
(2))
In
addition, the surface current density is given by
K,
and in the limit
By Ampere's law,
1I
h
(1)
ti
N(
+ &V)-vi c~c~,
Ay--O, this becomes
" J=Kz and the third expression follows.
(3)
11.79
r%
prob.
,
.. ,r 1
-·
11­ ..L
,
.*.)---
-
L
A
-
witn amplituaes designated as
..
l
in the figure, the boundary conditions
representing the distributed coils and
I
transmission line (the equations summarized in
Prob. 11.17.1) are
L)
- Is = -.
•n
"*
(C
3)
The resistive sheet is represented by the boundary condition of
Table 6.3.1.
Eq. (a) from
^-B
^d
3)
The air-gap fields are represented by the transfer relations, Eqs. (a), from
1
I
I
Table 6.5.1 with '2- k.
5) I
These expressions are now combined to obtain the dispersion equation.
Equatiic
)ns
1
1 and 2 give the first of the following three equations
-CL-doM
o0C
A
f-i..
A
'C
AdC
-+t
T:ic,
.,,Cw
_e
o
-'C
o
,X~
a
.-9
__-
• ' •i , .4.f.•(,- •"('
,-·p
I
Ad
"÷A
~---7
The second of these equations is Eq. 5a with H4 given by Eq. 3.
The third is
Y
Eq. 5b with Eq. 4 substituted for
.
The dispersion equation follows from
the condition that the determinant of the coefficients vanish.
s)
I
I
I
I
I
I
I
5
11.80
Prob. 11.17.2
(cont.)
5(
(7)
·
As should be expected, as n-- 0 (so that coupling between the transmission line
and the resistive moving sheet is removed), the dispersion equations for the
transmission line waves and convective diffusion mode are obtained.
system is represented by the cubic obtained by expanding Eq. 7.
The coupled
In terms of
characteristic times respectively representing the transite of electromagnetic
waves on the line (without the effect of the coupling coils), material transport,
magnetic diffusion and coupling,
I
___
hZc&
~~CO-/4~Th-
and the normalized frequency and wavenumber
S'
= •/ /(9)
r'
"
the dispersion equation is
I
C'4
tt+
TVe
L i
(8)
£
11.81
I
Prob. 11.17.2 (cont.)
The long-wave limit of Eq. 10 is
N
'rfem
In the form of a polynomial in k, this is
:
-
L
--
t--iC
4
,
-,
_
- W
I
(12)
-C7
where it must be remembered that
<(
t
As would be expected for the coupling of two systems that individually
I
have two spatial modes, the coupled transmission line and convecting sheet
are represented by a quartic dispersion equation.
for real k are shown in Fig. 11.17. 2 a.
The complex values of
One of the three modes is indeed
g
unstable for the parameters used. Note that these are assigned to make the
material velocity exceed that of the uncoupled transmission-line wave.
It is unfortunate that the system exhibits instability even as k is
increased beyond the range of validity for the long-wave approximation
<
I
The mapping of complex shown in Fig. 11.17.2b is typical of a convective
instability.
Note that for 0,= 0.
the root crosses the k r axis.
Of course,
a rigerous proof that there are no absolute instabilities requires considering
all possible values of
r> O
I
11.82
Prob. 11.1 7 .2 a(cont.)
Fig. 11.17.2a
Complex tj
for real k.
11.83
Prob. 11.17.2 (cont.)
tI
I
£
I
~'F1
I
I
I
I
Prob. 11.17.3
The first relation requires that the drop in voltage across
the inductor be
Divided by
hI
and in the limit where
= L
-
dt-PO
this becomes
(2)
A•t
The second requires that the sum of currents into the mode at
+ •t
I
be zero.
where
0
I
is the net charge per unit area on the electrode
- ao•
Divided by
I
•
(4)
and in the limit
- - c L0 + W
T
3t
htI
O , Eq.
3 becomes
(5)I
at
3
11.84
Prob. 11.17.4
(a)
The beam and
air-gaps are represented by
Eq. 11.5.11, which is (%R=0
)
ACr
o.
1
(~b)
(C)
a
(1)
(d'·
'
·.
:~
The transfer relations for the
region a-b, with
-O
T
require
(n)
that
D
(2)
A6
.A
With the recognition that
--i
AC
=
I
the traveling-wave structure
equations from Prob. 11.17.3 require that
A C
dRT
=;o
L
~
(3)
A C_
(
_A
A6
c-
The dispersion equation follows from substitution of Eqs. 1 and 2 (for
A6
A C
D
and
D,
) and Eq. 3 (for
4-g-
t
) into Eq. 4.
.. C +(5)
-.
As a check, in the limit where
L--+
D
C--pO
and
this expression
should be the dispersion relation for the electron beam (D=o in Eq. 11.5.11)
with a of that problem replaced by a+d.
In taking the long-wave limit of Eq.
(This follows by using the identity
) = tanhere
In taking the long-wave limit of Eq. 5, where
, )
e,< I
and
a<
(
and,
n
,
<I
11.85
Prob. 11.17.4 (cont.)
the object is to retain the dominant modes of the uncoupled systems.
These
I
Each of these is repre­
are the transmission line and the electron beam.
sented by a dispersion equation that is quadratic in tj and in
P
Thus, the appropriate limit of Eq. 5 should retain terms in
and
e3
I.
of sufficient order that the resulting dispersion equation for the coupled
system is quartic in
WJ and in
.
With
.C
4WG/
, Eq. 5
becomes
L-(6)
7
w
With normalization
- WL C
I
)
this expression becomes
(~C_C',.)
-
"•
S(7)
Written as a polynomial in CO
, this expression is
3
(8)­
(8)
_\
3 , [ja.
cZ_!
I.,Z
.+
YPP,,
iTJC
I
1
l
5
11.86
Prob. 11.17.4 (cont.)
This expression can be numerically solved for C
system is unstable, convective or absolute.
I
for real
unstable.
II.
I
I
I
I
I~
fi
I·
I
to determine if the
A typical plot of complex c
, shown in Fig. P11.17.4a, shows that the system is indeed
11.87
5
Prob. 11.17.4(cont.)
To determine whether the instability is convective or absolute, it
is necessary to map the loci of complex k as a function of complex
•'r
4d =
-
.
for the values of
trajectories
Typical
...
.
. .
. .
j
shown by
r
the
inset
are
shown
in
Fig.
11.17.4b.
I
I)
I
I
I
I
I
I
i
U
_I
I
Prob. 11.17.5
and
I
(a)
T
p
- C)
U
In a state of stationary equilibrium,
(
= constant, to satisfy mass and momentum conservation condi-
tions in the fluid bulk.
5
n_
Boundary conditions are automatically satisfied,
with normal stress equilibrium at the interfaces making
II
=
I
t
Z tA
H 0
I
.,
II
%_L
where the pressure in the low mass density media
I
surrounding the jet is taken as zero.
(b)
5
Bulk relations describe the magnetic
perturbations in the free-space region and the fluid
motion in the stream.
I
U
From Eqs. (a) of Table 2.16.1,
with
=
l
NC,
CSllh6
A
[A]~oko
I,
(2)
,+-= -V,+P
Sih1
iL -J
c
and from Table 7.9.1, Eq. (c),
iF, '1
6CL
Ce
(3)
JL.
[^:1
ii:
b^l "
iP
rS1
Because only the kinking motions are to be described, Eq. 4 has been
1
written with position (f) at the center of the stream.
Iof
that
_
=x' s=nk
3I
I
From the symmetry
the system, it can be argued that for the kinking motions the perturb­
ation pressure at the center-plane must valish.
I
(4);
so that Eq. 4a becomes
Thus, Eq. 4b requires
7)
e
b _b cSl fab
(5)
11.89
Prob. 11.17.5 (cont.)
Ae(6)
p
6f
s lnh%9b
(-O
or
)*Ib
.,
=
where the last equality introduces the fact that
=(I(c,-
|-revJ
,,i)
U)p.
Boundary conditions begin with the resistive sheet, described by Eq.
(a)
1
of Table 6.3.1.
(7)
W
which is written in terms of
AC
as (V).
_(8)
At the perfectly conducting interface,
^-
-
-
L
)
Stress equilibrium for the perturbed interface is written for the x component,
with the others identically satisfied to first order because the interface is
free of shear stress.
From Eq. 7.7.6 with
nph,
T j=
C
e-x
_'d(v.Th),
(10)
3
Linearization gives
Ae
Z
,A(11)
AA
where Eq. (d) of Table 7.6.2 has been used for the surface tension term.
With
-
, Eq. 5 becomes
e
0
(12)
Now, to combine the boundary conditions and bulk relations, Eq. 8 is
expressed using Eq. 3a as the first of the three relations
I
I
11
an
SProb.
11.17.5 (cont.)
5­
+
I
I
I
0
0TAa,
-C
Y
(13)
Ad
-0
The second is Eq. 9 with
with
4p
h,"
expressed using Eq. 3b.
The third is Eq. 12
given by Eq. 6.
Expansion by minors gives
I' '
o,,44U1
4
QJ
Co+~iP,.O..J
I
(14)
Some limits of interest are:
St--bo
so that mechanics and magnetic diffusion are uncoupled.
Then, Eq. 14 factors into dispersion equations for the capillary jet and
the magnetic diffusion
(co
(£'J
u)0
3
(15)
C,
(16)
The latter gives modes similar to those of Sec. 6.10 except that the wall
I
I
opposite the conducting sheet is now perfectly conducting rather than
11.91
Prob. 11.17.5
(cont.)
infinitely permeable.
q--wo , so that Eq. 14 can be factored into the dispersion
equations
(17)
3
This last expression agrees with the kink mode dispersion equation
(with Y--'PO) of Prob. 8.12.1.
In the long-wave limit, coth P-+I
/~
, tanh
-- 6L
1
and Eq. 14
becomes
)
In general, this expression is cubic in
W .
(19)
However, with interest
I
limited to frequencies such that
(20)
< i
and
where
I
, the expression reduces to
c.
r
and
-
OL'
,
//o)(
f
).
(21)
Thus, in this long-wave
low frequency approximation,
SI
(22)
I
11.92
Prob. 11.17.5 (cont.)
tu-
-
v'_v-)
St(UL 7 VL VL)
-
It follows from the diagram that if
/V+V
, the system is unstable.
To explore the nature of the instability, Eq. 21 is written as a polynomial
in
.
T- This quadratic in
q
-
-7..
.
(23)
is solved to give
(24)
With
=.-8d' , this becomes
Vu
(25)
11.93
I
Prob. 11.17.5 (cont.)
where
A -c)(V'+M
(c=
V)
CJr as
The loci of complex k at fixed
TZ>(VL
4
known that one of these passes through the k
temporal mode is unstable).
o
Or is varied from co to
However,
could be plotted in detail.
V.)
Vq.
)
4(V-VTVY
axis when
it
is
for
I
already
c6( O
(that one
To see that the instability is convective it
5
is only necessary to observe that both families of loci originate at
ý----C2
.
That is,
in
-
_
the limit
-_a-'
'_
V_ýv
t u'•-rV
and if
Cr-wco
"-Poo, Eq.
25 gives
'
(26)
VC
LI
V
.
Thus, the loci have the character of Fig. 11.12.8.
Vz
it follows that for both roots A- -6-t
"unstable" root crosses the kr axis
into the upper half-plane.
as
The
Because the
"stable" root never crosses the kr axis, these two loci cannot coalesce, as
f
required for an absolute instability.
Note that the same conclusion follows from reverting to a z-t model
for the dynamics.
The long-wave model represented by Eq. 21 is equivalent
to a "string" having the equation of motion
I& F _)L-2i
(27)
5
The characteristics for this expression are
\
_U
and it follows that if
convective.
tT
v,
z
t
(28)
, the instability must be
I
I
11.94
5
Prob. 11.17.6
i
electric field that is in common to both beams, the linearized longitudinal
(a) With the understanding that the potential represents an
force equations for the respective one-dimensional overlapping beams are
----
-
+
5
e
(1)
)ar e aH
V
(2)
To write particle conservation, first observe that the longitudinal current
I
density for the first beam is
Sand
e-k I.U,
S I =-
e- (
C,-
h
, + ho
)(3)
hence particle conservation for that beam is represented by
-_I+
U
oohj
+
(4)
Similarly, the conservation of particles on the second beam is represented
I
by
V17. + T
(5)
Finally, perturbations of charge density in each of the beams contribute
to the electric field, and the one-dimensional form of Gauss' Law is
1
(6)
The five dependent variables
Eqs. 1, 2, 4 and5 .
.
U
I
I
U
KC
I
19CI
,
1,
and t1
are described by
In terms of complex amplitudes, these expressions are
I
represented by the five algebraic statements summarized by
n
Se
-Ont-,
oo
0
o
o
0
0e, o
0
0e
0
n,,
A
o
a
A,
_
~1"2
0
0
(7)
11.95
Prob. 11.17.6 (cont.)
The determinant of the coefficients reduces to the desired dispersion
Z
equation.
)
(8)
where the respective beam plasma frequencies are defined as
(9)
f____oI
(b)
O
In the limit where the second "beam" is actually a plasma
(formally equivalent to making U 2 =0), the dispersion equation, Eq. 8,
becomes the polynomial,
Ri2.CJ k
where
(We,/W3
¶
mapping of complex
from
QT(-
c -- O
with
R
=
-I
/' p andW @
as a function of L
CO,
(10)
=
-
CJ,r
-
/Cj pz.
The
, d
varying
held fixed, shown in Fig. P11.17.6a, is that
characteristic of a convective instability (Fig. 11.12.8, for example).
(c)
In the limit of counter-streaming beams U 1 = U 2
E U, Eq. 8
becomes
•4
~
(V
z+ 4 (-Z)
where the normalization is as before.
L
%CL]
Y'+j+o=
(11)
This time, the mapping is as
illustrated by Fig. P11.17.6b, and it is clear that there is an absolute
instability.
(The loci are as typified by Fig. 11.13.3.)
W:
A --
_- /pa
1.01 I·o5
f?.
/1,If,
/:
I
I
-4
34
1 234
~iP~llli~
~-~a
I
--
-­
11.96
Prob. 11.17.6 (cont.)
Wtp
..
-,I
See, Briggs, R.J., Electron-Stream Interaction With Plasmas, M.I.T. Press (1964)
pp 32-34 and 42-44.