Online Library → Jack Bazer → Reflection and refraction of weak hydromagnetic discontinuities → online text (page 1 of 4)

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AFCRL-268

NEW YOi^K UNiVtRSlTY

INSTITUTE OF MATHE,\\ATICAL SCIENCES

LIBRARY

{i WashintUKi Place, h>kw YoA 3, N. Y.

JUN 2 8 1961

^E ET PR^^

iWl

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

'^Zirrryyi'^ Division of Electromagnetic Research

RESEARCH REPORT No. MH-11

Reflection and Refraction of

Weak Hydromagnetic Discontinuities

JACK BAZER

Contract No. AF 19(604)6144

FEBRUARY, 1961

NEW YOEK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. MH-11

Reflection and. Refraction of Weak Bydromagnetic Discontinuities

Jack Bazer

Qgc^ l^jy:^

Jack Bazer

Morris Kline

Project Director

I The research reported in this document has heen sponsored by the

(^ ^p ifhi^ be the speed at in the normal direction n.

of the incident wave and let U. . > 0, j = 1, 2, . . . , 6 denote the local

normal speeds in the directions n^^ . of the scattered waves. Then, as in

geometrical optics and acoustics, it is easily shown [See Appendix ij

that (l) the n. . lie in the plane of incidence and (2) for any i

%=U, ,

(2.1)

V . n, . = U, . ,

^ -ij ij

j=l,2,...,6 (2.2)

In cont.-ast to geometrical optics, however, the U . 's are direction-dependent.

Q

The U 's are, in fact, given by

U. = u â™¦ n^ + c.

1 "^p ~ii â€” 1

i = 1, 2, 5

(2.5)

where^

2 2 2

(l+r) - kr cos ^if^ cos (O, - 9â€ž)

Jl In

, (2.i^)

~ = cos tjj I COS (Qg -0jj)

(2.5)

cf. the paper in reference 1, equations (5Â«5) - (5*8)

9

The existence of three disturbance speeds was discovered by N. Herlofson,

Nature, I65, 1020 (195O) and independently by H. C. van de Hulst in a

paper which appeared in Problems of Cosmical Aerodynamics (Central Air

Documents Office, Dayton, Ohio I95I) , p. k6.

- 7

2 2 2

(l+r) - kr cos \(râ€ž cos (O-.- 9^)

n. ^ n

^ 1

2

1

2

. (2.6)

In these expressions, u is the component of the unperturbed fluid

***p

2â€”1 1 /p

velocity in the plane at incidence: b = {[lE p ) ' is the Alfven

disturbance speed in the direction of the magnetic field Hj p is the

density in ?? j r = a /b is the squared ratio of the sound speed a

and the Alfven speed bj and c , c and c denote, in the order stated,

the slow, Alfven and fast disturbance speeds. It follows directly from

equations (2.5) - (2.7) that (l) c /b is an increasing function of \(f ,

while c^ /b and c /b are decreasing functions of \|r , r, and 0.,

j = 1, 2, 5 being assumed fixed; (2) for fixed \(r and r and all 9

and e.

-^ < min (l,r) < max (l,r) < ^ ;

(2.7)

(5) for fixed \|/ and r and all = =

Ci < Cg < c^

(2.8)

The corresponding expressions for the U_, . and c. . are

ij ij

U^ . = u . - n. . + c. . ,

Ij "v-pj ~ij - ij

j = 1, . .. , 6,

(2.9)

and

^ij

(l + r.) - li-r .cos i,^.cos (0. . - 0tt-)

, J=i^6,

(2.10)

c. .

-^ = COS ^iTgj |cos(9^j - Q^.)\ , j = 2,5. (2.11)

ii = __Ii _ ^ [fl+r.)^ - hv. cosV.cos^fQ. .- 9â€ž.)r

J

^ (l+r.)^ - kv. GOs\â€ž.cos^(9, .- 9^.) r

j = 5,^. (2.12)

Here, u . is defined by

.wpj

u .

r^ > ^ = 1.2,3,

lu^ . J = ^.5.6,

(2.15)

and the quantities r., b . , \|r and 9 are defined in an analogous

manner. In equation (2.15) the primed quantities refer to the region

cK. 3XiA the unprimed quantities to the region ^T- From equation (2.9)

it follows that in a given direction n. ., two scattered waves of type

j are possible whenever lu . * n . . I exceeds c. ..

'-PJ -*ij ' ij

Combining equations (2.5) and (2.9) with (2.1) and (2.2), we

find that for each i, i=l,2,5

(V - u ,) â€¢ n, = +c. , (2.li4-)

^Â«~' â€”pi - i â€” 1 -^

(V - u .) â€¢ n, . = +c, . , j = 1,...6, (2.15)

or equivalently that

V |cos(9^-a) I = c^ , (2.16)

. Y |cos(9^^-a)| = c^^ , J = 1,2,5, (2.1?)

- 9

V |cos(Â©^^- a')| = c^j . J = 4;5.6 . (2.18).

Here, V= Iv-u I , V'= (V-u'l and a and a' are the angles that the

relative velocities V-u and V-u'. respectively, make with the

s\irface normal N at 0. From equation (2.17) > we readily conclude

that an angle of reflection 9 . is real only if the component of

V-u along the direction of the normal n. . exceeds, in absolute value,

the disturbance speed along_^ .. The same statement holds for the

refracted waves., u' and a' replacing u and a, respectively.

Equations (2.l6) - (2.l8) can be reexpressed as

|cos(0 -a) I |cos(0 - a) I

-T^ = ^ = -^ . J = 1.2,3, (2.19)

ij i V

|cos(9 - a') |cos(e - a') 1

c. . c.

IJ 1

. J = i+,5,6, (2.20)

which have essentially the same form as Snell's laws for the scattering

of acoustic waves ! Introducing

p =-^ >0 , p = ^ >0 (2.21)

1 G^ IJ C^^

into these equations., we find that

p |cos(0 -a)| = p |cos(9 -a)| = -3- , j = 1,2,3, (2.22)

p. .|cos(9 - a') I = Pj_|cos(9 - a') I = -^ , J = ^,5,6. (2.23)

Since the (p .,9. .) may be regarded as the polar coordinates of points

-lo-

in a (p_,0) -plane ^ It follows directly from these equations that

(p^ -^6^ â€¢); J = 1^2,3^ all lie on the line I through (p^^Â©^) whose

equation is

i: p I cos (0- a) I =^ = p. I cos (0.-a)| (2.21+)

V

while (p .,

NEW YOi^K UNiVtRSlTY

INSTITUTE OF MATHE,\\ATICAL SCIENCES

LIBRARY

{i WashintUKi Place, h>kw YoA 3, N. Y.

JUN 2 8 1961

^E ET PR^^

iWl

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

'^Zirrryyi'^ Division of Electromagnetic Research

RESEARCH REPORT No. MH-11

Reflection and Refraction of

Weak Hydromagnetic Discontinuities

JACK BAZER

Contract No. AF 19(604)6144

FEBRUARY, 1961

NEW YOEK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. MH-11

Reflection and. Refraction of Weak Bydromagnetic Discontinuities

Jack Bazer

Qgc^ l^jy:^

Jack Bazer

Morris Kline

Project Director

I The research reported in this document has heen sponsored by the

(^ ^p ifhi^ be the speed at in the normal direction n.

of the incident wave and let U. . > 0, j = 1, 2, . . . , 6 denote the local

normal speeds in the directions n^^ . of the scattered waves. Then, as in

geometrical optics and acoustics, it is easily shown [See Appendix ij

that (l) the n. . lie in the plane of incidence and (2) for any i

%=U, ,

(2.1)

V . n, . = U, . ,

^ -ij ij

j=l,2,...,6 (2.2)

In cont.-ast to geometrical optics, however, the U . 's are direction-dependent.

Q

The U 's are, in fact, given by

U. = u â™¦ n^ + c.

1 "^p ~ii â€” 1

i = 1, 2, 5

(2.5)

where^

2 2 2

(l+r) - kr cos ^if^ cos (O, - 9â€ž)

Jl In

, (2.i^)

~ = cos tjj I COS (Qg -0jj)

(2.5)

cf. the paper in reference 1, equations (5Â«5) - (5*8)

9

The existence of three disturbance speeds was discovered by N. Herlofson,

Nature, I65, 1020 (195O) and independently by H. C. van de Hulst in a

paper which appeared in Problems of Cosmical Aerodynamics (Central Air

Documents Office, Dayton, Ohio I95I) , p. k6.

- 7

2 2 2

(l+r) - kr cos \(râ€ž cos (O-.- 9^)

n. ^ n

^ 1

2

1

2

. (2.6)

In these expressions, u is the component of the unperturbed fluid

***p

2â€”1 1 /p

velocity in the plane at incidence: b = {[lE p ) ' is the Alfven

disturbance speed in the direction of the magnetic field Hj p is the

density in ?? j r = a /b is the squared ratio of the sound speed a

and the Alfven speed bj and c , c and c denote, in the order stated,

the slow, Alfven and fast disturbance speeds. It follows directly from

equations (2.5) - (2.7) that (l) c /b is an increasing function of \(f ,

while c^ /b and c /b are decreasing functions of \|r , r, and 0.,

j = 1, 2, 5 being assumed fixed; (2) for fixed \(r and r and all 9

and e.

-^ < min (l,r) < max (l,r) < ^ ;

(2.7)

(5) for fixed \|/ and r and all = =

Ci < Cg < c^

(2.8)

The corresponding expressions for the U_, . and c. . are

ij ij

U^ . = u . - n. . + c. . ,

Ij "v-pj ~ij - ij

j = 1, . .. , 6,

(2.9)

and

^ij

(l + r.) - li-r .cos i,^.cos (0. . - 0tt-)

, J=i^6,

(2.10)

c. .

-^ = COS ^iTgj |cos(9^j - Q^.)\ , j = 2,5. (2.11)

ii = __Ii _ ^ [fl+r.)^ - hv. cosV.cos^fQ. .- 9â€ž.)r

J

^ (l+r.)^ - kv. GOs\â€ž.cos^(9, .- 9^.) r

j = 5,^. (2.12)

Here, u . is defined by

.wpj

u .

r^ > ^ = 1.2,3,

lu^ . J = ^.5.6,

(2.15)

and the quantities r., b . , \|r and 9 are defined in an analogous

manner. In equation (2.15) the primed quantities refer to the region

cK. 3XiA the unprimed quantities to the region ^T- From equation (2.9)

it follows that in a given direction n. ., two scattered waves of type

j are possible whenever lu . * n . . I exceeds c. ..

'-PJ -*ij ' ij

Combining equations (2.5) and (2.9) with (2.1) and (2.2), we

find that for each i, i=l,2,5

(V - u ,) â€¢ n, = +c. , (2.li4-)

^Â«~' â€”pi - i â€” 1 -^

(V - u .) â€¢ n, . = +c, . , j = 1,...6, (2.15)

or equivalently that

V |cos(9^-a) I = c^ , (2.16)

. Y |cos(9^^-a)| = c^^ , J = 1,2,5, (2.1?)

- 9

V |cos(Â©^^- a')| = c^j . J = 4;5.6 . (2.18).

Here, V= Iv-u I , V'= (V-u'l and a and a' are the angles that the

relative velocities V-u and V-u'. respectively, make with the

s\irface normal N at 0. From equation (2.17) > we readily conclude

that an angle of reflection 9 . is real only if the component of

V-u along the direction of the normal n. . exceeds, in absolute value,

the disturbance speed along_^ .. The same statement holds for the

refracted waves., u' and a' replacing u and a, respectively.

Equations (2.l6) - (2.l8) can be reexpressed as

|cos(0 -a) I |cos(0 - a) I

-T^ = ^ = -^ . J = 1.2,3, (2.19)

ij i V

|cos(9 - a') |cos(e - a') 1

c. . c.

IJ 1

. J = i+,5,6, (2.20)

which have essentially the same form as Snell's laws for the scattering

of acoustic waves ! Introducing

p =-^ >0 , p = ^ >0 (2.21)

1 G^ IJ C^^

into these equations., we find that

p |cos(0 -a)| = p |cos(9 -a)| = -3- , j = 1,2,3, (2.22)

p. .|cos(9 - a') I = Pj_|cos(9 - a') I = -^ , J = ^,5,6. (2.23)

Since the (p .,9. .) may be regarded as the polar coordinates of points

-lo-

in a (p_,0) -plane ^ It follows directly from these equations that

(p^ -^6^ â€¢); J = 1^2,3^ all lie on the line I through (p^^Â©^) whose

equation is

i: p I cos (0- a) I =^ = p. I cos (0.-a)| (2.21+)

V

while (p .,

Online Library → Jack Bazer → Reflection and refraction of weak hydromagnetic discontinuities → online text (page 1 of 4)