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Jack Bazer.

Reflection and refraction of weak hydromagnetic discontinuities online

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which may be expressed as



e - e ' = -e. .

1



ih.6)



eb + e'b' = e.b . (i^-.7)



The solution of this system is







1


e
^i


b-b'
b+b'


2
n + 1

1


t
e

^i


2b
b+b'


_ 2n^

1







ih.d)



n = p'/p . ik.9)



The positions of the incident^ reflected and transmitted wavefronts^
along the N = z-axis, are determined at each instant by

z^ = wbt , t < ,

z = -wbt , t > ^ (14-. 10)

Z ' = Wb 't ;, t > .



52



Plots of these ec[uations are shown in Figures 8(a) and 9(a). In
Figure 8(a), it has been assumed that p = i|p so that the transmitted
wave front travels at half the speed at the incoming and reflected
wave fronts. In Figure 9(9-) it has been ass\mied that p = -j- p so
that the transmitted wave front moves with twice the speed of the
incoming and reflected wave fronts. Figure 8(b) and 9{^) shows the
waveforms of the ratio 5Qrp/5Q. where 50^^ represents the total
disturbance of any quantity Q - e.g., the pressure or N-component
of velocity - and 50. refers to the disturbance of Q carried by the
incident wave. Evidently, 6Q_, in the neighborhood of the interface
c>Ois in absolute value greater or less than the 50. according as p
is greater than or less than p.

Corresponding to equation (5.55) we have the equation

which expresses the conservation of flux through J^. But in the
present case it follows readily from equations (5.^5) ^^^ (5-^7)



b >~< b .*v b



wtj



so that equation (4.11) reduces to

b'e'^ + b£^ = be^^ . (4.15)

It is easy to verify directly with the aid of equations (4.8) and
(4.9) that this relation is indeed satisfied.



- 52a -



z = w bt



/)'>4/)




*^SQ/SQj



FIGURE 8




(a)



P' 1!

If we remove the restriction that the deflection of the stream
be effected by shocks only and regard the formulas (5.1^) - (5 -16)
as giving the variation of the magnetic intensity, the velocity and
pressure across a weak simple wavej it then appears, since such waves
may be expansive, that the deflection can be effected by a slow expansive
simple wave when M, < 1 and a > and a fast expansive simple wave
when M, > 1 and a < 0. In this way it is possible to get rid of
shocks facing upstream if such shocks are found to violate other
requirements on the flow.

The expression for p in equation (5.26) and the corresponding
expressions for 5H and 5u apply with obvious changes to nose-on flow
past a symmetrical wedge with wedge angle 2a [See Figure 10(c)] . Indeed,
the expression for p in (5.26) already gives the pressure above and below



- 61 -

the wedge J no modifications are necessary. Moreover by applying
formula (5.26) at each comer of the symmetrical wing 'It/', made up
of linear segments [see Figure 10(d)] , we can obtain the pressure
distribution at each point of the wing. In fact, if we imagine the
number of segments to increase without number and to approach the
continuous profile shown in Figure 10(e) then the expression for
p in equation (5.26) gives the pressure on top and on bottom of
the wing at a distance x from the leading edge. Here, a must be
regarded as a function of x.

The above analysis is also applicable to flow past a thin
plate having an angle of attack a. In this case the pressure of
each point of the plate is



Pt = P +
1 o



o

2p u (a/c)a
00'



V^TT



^^J



(5.27)



The additional factor of two Is due to the fact that the

pressure forces below the wing add to that above. The lift coefficient

due to pressure force alone, namely,



2 o o



is therefore



C^(M) = ^(^/"^^ f 1 - -^ ^ (5.29)

fV? - 1




62



B. The Crossed-Fields Case

The crossed fields case can be treated by methods analogous

to those of Section A^ provided M. is sufficiently large. Consider^

for e.xample^ the case where the magnetic field is parallel to N and

hence perpendicular to the direction of flow - u x . In this case

o-o

the wave front angles are obtained from a figure like that of Figure
11. Obseirve that according as



\'



>



fr +1



or



\



-1 . 1



<



1/r +1



one or two disturbance waves are available for meeting boundary

conditions at the wall. The first relation defines the so-called

35
"hyper-liptlc" region, the second relation, the hyperbolic region.

The method of Section A applies in general only to the hyperbolic

region.

In the above, in keeping with the assumption that the problem
is a two-dimensional one, it was implicitly ass\imed that -u x , N
and the prevailing magnetic field H were co-planar. When H is
not in the plane of -u x and |[, then, as the reader may readily
verify for himself, there will be a hyperbolic region of flow in

which Aljfven as well as slow and fast waves are available and in fact

necessary for meeting the boundary conditions.
_

See reference 51



- 62a -



r =


•5. eH=o«


i


> N- axis








^H = 0°






1















1









s


^—


1










A


/^


^










II , , .... II

— Hyper-liptic -►


(


A


^-"Hyper-


. . II
iptic -























Vr+i


/












fV


^


/










A




1










s —







.














4—


-K












Hyp


erbc


)lic









FIGURE II



- 63 -

APPENDIX I



We prove that



V • n. = U. , • (1)



V ■ n. . = U. . , (2)



and that n. , n. . and W are coplanar. Our proof does not require
the wave fronts to "be planar.

We "begin by expressing the equations of all wave fronts in
the form



W(x) - (t-t^) = . ' (5)



Defining p by the equation

p = vw/|vw| , (h)

we see that

n = p/p . (5)

It is also easily proved that the speed U of the wave front along
n^ is related to p by

U = p-^ . (6)

At each instant the incident and scattered wave fronts
intersect the interface (lO in a common curve W(t) - the so-called
trace - which sweeps but a portion of i/J . Let



- Gk -

be the equation of a trace for each t. Without loss of generality,
it may be assumed that for fixed i, the curve y = y(|,t) is orthogonal
to the family of traces ol (t) , t > t , so that we may write



dt



V = ^ . . (7)



I'low; since ^(t) is on all wave fronts, it follows that

Wi[y(i.t)] = w..[^(^,t)] = t-t^ , (8)

for each i and j. Differentiating first with respect to | and then
with respect to t, we find that






£i' ^ = £ij* 1 = ^ ' (10)

which, using equations (5) and (6), become

n. • -rf = n. . x^ = , (9)
n. -V n. .-V

^ = T^ = 1 ■ (10)

Equations (l) and (2) now follow immediately from equation (lO).

Moreover, as IT is proportional to the cross-product of ^ and



Ti



V = -^ it follows from (9) and (lO) that



N X n. = N X n. . , (ll)

which implies that the n. . are in the plane of W and n..



65



APPENDIX II



We sketch here a derivation of the relations



1 _ 2 1 2 5p" 1 _„2
2^ = 2 p~ + 2^5H



1 _ 2 1 a „ 2 1 ^„2
5E s 2 p&u + 2 ~ ^P ^ 2*^



p5u



5F = a 5p5u + Li(Hx5u)x 5H + 5Eu = 5Es ,



(1)

(2)
(3)



which hold for all modes of propagation. In (5) ^ denotes the ray
vector at each point of the vave front of the mode of propagation
in question. We shall also show how to derive the formulas

2



}e^ = h Vp |sin 6|



2w - (1+r)



2
w - r



w



by^ r



w



(w^- cos^5)(2w^- (l+r))



T 1
2



2 2/2
w = c /a ,



for slow and fast modes and



y?=TP



b sin 5



(5)



for Alfven waves.

It will be enough to give the derivation for the fast and
slow waves since the procedure for Alfven waves is quite analogous
and in fact much simpler.



66



We begin by observing that



f 2 2w2,2v h 2, 2^2, 2^2

(c -aj(c -b ) = c -c (a+bj + a d
n n n



2^>.2 -2.
= c (b - b )
n



(6)



c lap" (H-H^n) ,



in virtue of the equation



h ,2. ^2s 2 2^2

i - ( a + b ) c + aHD =
^ n



(7)



|_cf. equation (3.i|)J- Employing equation (3-3) anti (6), we find



5u



2 2

€ C <



2 2

€ G



b2\2 b ^ (H-Hn)2



n*



/



H



h



2\ 2



■^



V



n



b



1. 2 2
b c
n



1-



- -2



(8)



2 / b

2 2

c \ c



c - a b



Performing similar calculations, we can readily relate all quantities

2
appearing in the definitions of 5E and SF to p5u . The final results

are:



p c (c - a) 2
c -a b



(9)



67



2^2 a^Cc^-b^)
a 5p n _ 2

-f- = 1, P 2 P^^ '
c - a D



n



(10)



c5ff,2 _ 2^
n(HX5u)X5H = + -ff — „ _ % 5u n ,



La^



n
2„ „ a^c L 2 ^ 2 \ „ 2



a 5p5u = — ^-2- Tn h" ' ^ S. P^^



^-a^



(11)



(12)



n



Adding the first twiD of these equations, we obtain equation
(l) from which equation (2) follows immediately.

To prove (3), we observe that the ray equations (3A6) and
(3.47) may be expressed as follows:



a^ ^ (H - H n)



u + cn +



n n**



H (+ c C )
n^



(13)



Here, (l) the upper and lower sign in the denominator of the last
term applies when the rays are "slow" rays and "fast" rays,



respectively, (2) C is defined by



C = I



2 2 2. 2
(a + b ) - ka-b
n



1
2



(Ih)



[cf. equation {^.kQ)'] and (3) c is related to C by



2 2 2—2
2c = a + b + C



(15)



as may be seen by solving equation (7) for c . With the aid of
equation ( 7) ^ we find that



68



o2 2
+ C c



a 2 h

a b - c



n



(16)



from which it follows that



s = u + en +



a^ ^g(H- H n)
n n"'

H (a^ 2_ ^1.^
u n



u +



^ ^ _2^2



c - a



A^^



, aT3 H
4 n '^



(17)



Now adding equations (ll) and (12) , we find that



a 5pSu +h(HX6u)x5H = + -^ r — ;r

c - a D



a^2 -

+ n „

en- — H

n



P6u . (18)



Comparing the right memher of this expression with the last term in
equation (17) ; we conclude that



a 5p5u + u(Hx6u)x6H =■ p5u (s,-u) ;,



(19)



which^ using equation (2)^ leads directly to equation (j).



To prove (h) we must first calculate Rn'R-,' The simplest
procedure is to imagine that at a given point of the wave front a
special coordinate system has been introduced with the x-axis
directed along n and the z-axis taken along the direction of SH^which
is tangent to the wave front. It is then easy to show that



K^.K^ = .^..„^5 [flp^] M^ . A'-^^^ . (20)



T



- 69 -



where 5 is the angle between H and n. Now from {&) , it follows that



.^2 . 2.



/ 2 , 2v CD sin 5

^^ -\ ^ = 2 2
c - a



(21)



2 2
Substituting for (c - b ) in (20)^ we then find^ after simplifying,

n



that



\-«l= 2^%^



r r h



( ^ 2, 2
(c - a D
n_

2 2
c - a



(22)



which, employing (7); becomes



R«R = 2b sin 5



2c^- (a^ + b^)

2 2
c - a



(25)



Alternatively, employing (6), R-. 'K-. '^^^7 ^s expressed as



\-\ - 2



[2c2-(a2 + b2)-| [c^-b/j



(2l|)



Equation {h) now follows from Equation (25) and (24) on multiplying
p/2, introducing w for c /a and finally extracting roots.



NYU
MH-11



c.l



Bazer



Reflection and refraction
of... discontinuities



IJYU
Reflection and



c.l



refraction



'of ^_,_^A^mM^^^^\




N. Y. U. Institute of
Mathematical Sciences



New York 3, N. Y.

4 Washington Place



M I 8 I^^Lte due












HOV'Z


3199?














^m 30


1988






lulAY ^*^


m.






Wr^T








































































































GAYLORD






PRINTED IN U.S.A.






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Online LibraryJack BazerReflection and refraction of weak hydromagnetic discontinuities → online text (page 4 of 4)