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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'^ + bÂ£^ = 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 =

â€¢5. eH=oÂ«

i

> N- axis

^H = 0Â°

1

1

â€”

s

^â€”

1

A

/^

^

II , , .... II

â€” Hyper-liptic -â–º

(

A

^-"Hyper-

. . II

iptic -

â–º

Vr+i

/

fV

^

/

A

1

s â€”

.

4â€”

-K

Hyp

erbc

)lic

FIGURE II

- 63 -

APPENDIX I

We prove that

V â€¢ n. = U. , â€¢ (1)

V â– 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

Â£i' ^ = Â£ij* 1 = ^ ' (10)

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

n. â€¢ -rf = n. . x^ = , (9)

n. -V n. .-V

^ = T^ = 1 â– (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 _â€ž2

2^ = 2 p~ + 2^5H

1 _ 2 1 a â€ž 2 1 ^â€ž2

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

â‚¬ C <

2 2

â‚¬ G

b2\2 b ^ (H-Hn)2

n*

/

H

h

2\ 2

â– ^

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 â€” â€ž _ % 5u n ,

La^

n

2â€ž â€ž a^c L 2 ^ 2 \ â€ž 2

a 5p5u = â€” ^-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 2â€”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 â€” ;r

c - a D

a^2 -

+ n â€ž

en- â€” 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 =â– 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^ = .^..â€ž^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

\-Â«l= 2^%^

r r h

( ^ 2, 2

(c - a D

n_

2 2

c - a

(22)

which, employing (7); becomes

RÂ«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.

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'^ + bÂ£^ = 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 =

â€¢5. eH=oÂ«

i

> N- axis

^H = 0Â°

1

1

â€”

s

^â€”

1

A

/^

^

II , , .... II

â€” Hyper-liptic -â–º

(

A

^-"Hyper-

. . II

iptic -

â–º

Vr+i

/

fV

^

/

A

1

s â€”

.

4â€”

-K

Hyp

erbc

)lic

FIGURE II

- 63 -

APPENDIX I

We prove that

V â€¢ n. = U. , â€¢ (1)

V â– 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

Â£i' ^ = Â£ij* 1 = ^ ' (10)

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

n. â€¢ -rf = n. . x^ = , (9)

n. -V n. .-V

^ = T^ = 1 â– (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 _â€ž2

2^ = 2 p~ + 2^5H

1 _ 2 1 a â€ž 2 1 ^â€ž2

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

â‚¬ C <

2 2

â‚¬ G

b2\2 b ^ (H-Hn)2

n*

/

H

h

2\ 2

â– ^

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 â€” â€ž _ % 5u n ,

La^

n

2â€ž â€ž a^c L 2 ^ 2 \ â€ž 2

a 5p5u = â€” ^-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 2â€”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 â€” ;r

c - a D

a^2 -

+ n â€ž

en- â€” 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 =â– 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^ = .^..â€ž^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

\-Â«l= 2^%^

r r h

( ^ 2, 2

(c - a D

n_

2 2

c - a

(22)

which, employing (7); becomes

RÂ«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

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N. Y. U. Institute of

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Online Library → Jack Bazer → Reflection and refraction of weak hydromagnetic discontinuities → online text (page 4 of 4)