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(5) However, at negative angles of Incidence that become successively

â– \hether or not the associated modes of propagation are excited or not

depends upon the bo\:indary conditions and on the orientation of the

magnetic field.

A wave of type j due to an incident wave of type i is said to be

critically reflected when 0. . just becomes equal to +90 or -90 .

- 15 -

larger in absolute value ^ first the Alfven and then the slow wave is

critically reflected.

Figure 4(b) is a plot of , and against the fast angle

of incidence for r = 0.5^ 0â€ž = k^Â° and trr = 0. It was obtained by

n n

graphical means from an enlargement of Figure k{a) . Statements (l)-(5)

refer specifically to Figures i4-(a). However^ with obvious changes^

they apply for all values of v, ^^ and 0^^ the angles 0â€ž = +90Â°^ +l80Â°

n n n â€” â€”

being excepted. Statement (5) is no longer valid in the exceptional cases,

but (l)-(l4-) still apply with minor changes. We leave to the reader the

analysis of the special cases [see Figures 10(b) and llj .

To determine the angles of refraction we must find the Intersection

of the line Â£ '

1 11

pb ' sin0 = p . b ' sin0

1

i'Â°(|v|A.)Â°Â»b'Â°77\'

1/2 ^ / , / ^l/2 /o ^n^

T\ ' = p- = (p7pJ ' (2.30)

with the upper parts at the curves obtained by plotting

against 0. . on a polar diagram. Note that here (cf . equations (2.25) ^Jid

(2.25)) we have referred the disturbance speeds to the Alfven speed b' in

the region OJ and the quantity R ' =(p'/p) M^ is the Mach nianber appro-

priate to the region"^. Observe, in addition, that equation (2.50) is of

the same form as equation (2.27) except that in (2.3) the ratios c. ./^'

depend upon r' not r. But, If as we shall here and hereafter assume,

both 7[ and ^T^ consist of polytropic gases, then we can conclude that

15

The properties listed in the next subsection will be found useful for

this purpose.

- 15a -

FIGURE 4(b)

- 16 -

2

a

^2

7P

- 7P' a'2

2 2

(2.32)

since H and the pressiire p are continuous across a hydromagnetic

contact discontinuity. It follows that c , /b ' is the same function of

9., as c. ^ /b is of 9. , , that c,^/b' is the same function of 9.^ as

i4 il' il' ly 15

c /b is of 9^^ etc. Thus^ on plotting p .b ' versus 9 ^^ j = h ,'^ ,(i ,

for r = 1/2, 1); = say^ we obtain curves [see Figure 5] which are

identical with those in Figure i|(a) . However^ for a given angle of

incidence the distance (M, ') of the line Â£' from the N-axis differs

from that of Â£ by the factor r; ' = (p'/p) .

Referring now to Figure 5; we observe that for sufficiently

small angles of incidence or sufficiently large Mach numbers the line

f = i'(Nl ') will intersect the fast branch (labelled f , ) ^ the Alfven

branch (labelled A ) and the slow branch (labelled sA exactly once.

If the incident wave is fast, then whatever the angle of incidence,

there will always be a fast refracted wave unless p exceeds p ' in which

case there will exist a critical amgle of incidence at which the fast wave

is totally reflected. If the incident wave is an Alfven or a slow

wave and the angle of incidence is positive^ then fast, Alfven and

slow-waves will be totally reflected at successively larger angles

of incidence. For successively smaller negative angles of incidence

we find (l) the fast angle of refraction is -90 degrees, (2) i' intersects

the Alfven branch and slow branches each exactly once (3) i' intersects

the Alfven brajich twice and the slow branch once and {\) Â£' intersects

Since the angles of refraction are necessarily acute angles,, only those

intersections are counted that lie in the region N > 0.

17

At this angle the refracted wave front is normal to the surface.

"Â«=~rt

I

- 16a -

(Pi6.Â«i6)

(Pis.^is)

(mT' = v^'m:'

D D

FIGURE 5

17 -

both the slow and the AlfVen tranches twice. With obvious modifications^

these featvires of Figure 5 apply for all r, ^\f < 3t/2 and Â©â€ž 4 1 p â– > ^

and n. In the exceptional cases aU. of the above discussion again

applies with minor changes apart from statements (3) and (h) which are

no longer tinae.

2. Some General Properties of the Angles of Reflection and

Refraction

The angles of reflection and refraction enjoy several

general properties which are consequences of the particular way in

which the (c. ./^

ii L-Â©iiJ = - 1 , ^ \ ^ J = ^'2,5,

^ ^ ' -(:t + 9.), if 9^ < Oj

^6

Â®ji [^Â®ij> ^'^^ = Â®i ^ J = ^^5,6, P^

\\^,\ < \\^\ < \^,e\ ' ^9

In these relations^ parameters not shown are considered fixed and

unless specifically indicated otheivise the subscripts i and j assume

the values 1^2^3 and l;2j...^6, respectively. The angles 9. . appearing

in equations P^ and Pq are what we shall hereafter refer to as the

acute angles of reflection. They are defined as follows:

"ij

: - 9. ., if 9. . > i = 1;2,5,

(2.37)

â€¢(9^^ + rt), if 9.^ < j = 1,2,5.

19

Recall that the first index i refers to the incident wave and the

second index j to the reflected or refracted waves.

19

Equation P expresses the fact that the angles of reflection and

refraction are Independent of the direction of the component of H_^

which is perpendicular to the plane of Incidence. Equation P states

that these angles are also independent of the sense of H . Together,

P and P imply that the reflected and transmitted angles are

independent of the sense of II_. Equation P states that the slow and

fast angles of reflection and refraction are Invariant under the

transformation of r->-r whenever the incident wave is slow or fast.

From P, we conclude that if a wave incident at the angle 9. produces

scattered waves at the angles 9. . . then a wave incident at the angle

ij

-9. will produce scattered waves at the angles -9. . only if the direction

1 . ij

of H is reflected in the N-axls specifically, only if 9,, is replaced

"~p "^ H

^y -Â©XT' If Â©TT is not so replaced, then according to P^. the scattered

n n P

waves emerge at the new angles 9f . = - (S) . . 1-9. ;9â€ž1 , which differ, in

general, from -9. .. Only when 9 is , +90 and l80 do we find that

â€¢)(â–

9. . = -9. .. In other words, negative angles of incidence are not

physically equivalent to positive angles of incidence except for

special orientations of the magnetic field. The next two properties P^

and P verify the reversibility of the scattering process. Thus, for

example, P may be Interpreted as follows: If 9. . is an angle of

refraction associated with the angle of incidence 9. , then a wave in /(_,

incident at the angle 9. . will produce a transmitted wave in ('J^ of type

1 emerging with the angle 9.. In the two relations Po and P , it is

assumed that 9. . 's and 9, . 's are real: these relations affirm a fact

ij ij

to be expected on physical grounds, namely, that the fast reflected

and refracted waves emerge first, followed by the reflected and refracted

- 20 -

Alfv6n waves which are in turn followed by the reflected and refracted

slow waves (see Figure 2(a)).

3, Numerical Considerations - Anomalous Reflection and Refraction

For sufficiently large angles of incidence it will he found

that^ regardless of scale ^ the intersections of the lines i(R ) and

i'(M ') [see equations (2.25) and (2.29)] with the slow and Alfven

curves fa3JL outside the range of the graph. For such angles it is

necessary to supplement the graphical analysis with suitable approximate

formulas. Fortunately, the Alfven curves are linear and the slow

branches are almost linear in that they approach the straight lines

of equation (2.28) asymptotically. These facts may be employed to

obtain analytical expressions for the angles of reflection and refraction

associated with large angles of incidence. Consider, for example,

the problem of determining the angles at which the reflected slow and

Alfven waves emerge. In this case, the intersection of Â£ with the A

and s^-curves lead to the following expressions:

sin 0.â€ž sin 9.

i2 1

cos tjj |cos (Â©j_2-Qh^I (Cj_A)

}

sin 0.,

i5

^ \ 1/2 sin 9.

cos tjj Icos (9^^-9g)| V / (c^/b)

The first of these equations is simply equation (2.35) with c /b

replaced by the explicit expression given in the right member of equation

(2.11)' this equation is exact. The second equation is, on the other

hand, an approximate equation: it is derived by replacing, for each

- 21 -

0, ^, the quantity p in the equation c = p" by the approximating

value of p on the asymptote [see equation (2.28)] and then substi-

tuting the resulting expression for c . ., into equation (2.53)Â« Such

an approximating value always exists unless = 0^ in which case Â£

20

is parallel to the asymptote. From the above equations and the

trigonometric s-um formula for the cosine it follows that

sec tjj sec 0^^. (c^/b)

ctn 0^2 = + tan 0^ ,

sin 0.

1

â€” 1 1 /p

(l+r" ) ' sec f sec (c./b)

ctn 0^ ^ = + tan 0â€ž .

^ sm

In these equations^ the "+" sign is intended when the intersection of i

with the A- line or asymptotic line occurs in the half -plane into which

H is directed; otherwise the negative sign must be chosen.

It is worthwhile noting the possibility of a kind of reflection

which does not arise in conventional gas under similar circumstances.

For the sake of concreteness^ suppose that the incident wave is

slow and the relevant Figure for the reflection process is Figure i4-(a) .

When 0,, the angle of incidence^ is positive and sufficiently small

the disposition of the wave fronts is as shown in Figure 2(a). As

if.

is increased to "critical angle", 0^ say [defined as the angle at

which = 90Â°3 , the fast wave front approaches the normal direction

to the surface Jo . What we should like to stress here is the following:

20

It should be mentioned that it is not difficult to obtain explicit

formulas for 0. . when 0â€ž = 0.

ij H

â– ' - 22

By increasing the angle 9 slightly past the critical angle 9 , it

is possible to extend the manifold of fast wave solutions continuously

past 9 = 90 . Physically this means that the fast wave fronts

associated with such values of 9-, emerge in the same quadrant as the

5

slow incident wave front as^ for example^ the dashed line does in

Figure 2(a) . The reader can convince himself from an inspection of

Figure 4(a) that for negative 9^ one can pass over^ by continuously

decreasing 9 . to the case where both the Alfvln and slow reflected

waves emerge in the same quadrant as the incident wave. If one were

to construct these wave fronts by means of Huyghens ' method (see

Section B-5) , one would find that the rays which guide the "energy

in the mode" [see Section II-B-5 and III-A-lJ all have components

directed along -N,^ as do the rays associated with ordinary reflected

waves. Incidentally, in Figure \, the outward normals to the s, A

and f-curves, drawn at their points of intersection with i give the

directions of the rays [cf . reference l] . This "radiation condition"

together with the continiiity considerations support the inclusion

within the manifold of reflected waves those which emerge in the

same quadrant as the incident wave.

The existence of the above type of anomalous reflection process

[and, in fact, of others not mentioned, for the reason that they are

less physicalljr admissible] makes it necessary to define the terms

reflected, refracted and even incident wave more precisely. Hereafter,

we shall admit to the class of reflected jjrefractedj waves only those

which (l) may be arrived at by continuously increasing the angle of

incidence from zero, and (2) are associated with ray vectors having

- 25 -

components that are directed along -N [along n] . Finally, a wave

which propagates toward Jj from below - i.e., a wave whose normal

n has a component directed along N - will be admitted as an incident

wave only if the associated ray vector has a component which is

directed along N. It should be mentioned that in Figure 5. only

those admissible angles of reflection which are associated with

normally [as opposed to anomolouslyj reflected waves are shown.

Another interesting kind of anomolous reflection (refraction)

process is the phenomenon of conical reflection (refraction) [See

Reference ij . Conical reflection (refraction) can occur only when

r = 1 and H^ is in the plane of incidence and. then only at suitable

angles of incidence. In Figure 3(a), for example, conical reflection

can occur only when the line I is in the position shown - that is,

only when Â£ passes through the triple point where the s, A and f

cuives intersect. Imagine that the disturbance on the incident wave

front is concentrated in a small disc of negligible diameter. Then,

employing considerations similar to those given in Reference 1 [p. 57^J

we can infer that this "point" disturbance will emerge as a ring

disturbance with an illximinated point in its center. The disc

determined by the illuminated ring and its center point is perpendicular

to H and moves parallel to H with the speed b = ViaH p ; its radius

t seconds after the incident wave has hit the interface can be shown to

be ^t.

k. Explicit Formulas for the Alfven Waves

We assume here that an AlTven wave is incident on the contact

2k -

discontinuity^ and we derive explicit formulas for the angles at

whlcli the reflected and refracted Alfven waves emerge. We begin by

combining equations (2.35) :> (2. 5^1-) and equations (2.5) j> (2.11) and

find, after cancelling the common factor cos i|/â€ž, that

sin 9^2 sin 9^

022=03(922-9^) a^c^os{Q^-Q^) '

sin 92i^ sin 92 . " 2

(2.58)

o^^zos{Q^^-Q^) o^zos[Q^-Q^)

(2.59)

In these equations the a's are +1 or -1 according as the neighboring

cosine factors are positive or negative. Employing the standard

trigonometric formulas for the cosine of a difference of angles, we

find easily that

ctn 9â€ž^ = -^^ (ctn 9â€ž + tan 9j - tan 9^ , {2.k0)

22. O d. Vl il

1

Ctn 9 , = n^ -^^ (ctn 9 - tan 9^^) - tan 9^^ . (2.1fl)

Now, according to property P of equation (2.56), no generality is

lost in restricting 9â€ž to the range -90Â° < 9â€ž < 90Â°. Assuming, for

n 11

the moment that this restriction has been made, we readily verify,

by referring to Figures (5) and (^) , that

0-22

^2

and

^24

^2

= 1

- 25 -

Introducing the acute angle of reflection^

Â« - 0^2 . if Â©22 ^ Â° ^

Â©22 = (2.1^0)

j^-(:t + 922) , if Â©22 < ,

into equation (2.58)^ we then obtain the formulas

Gtn 0^â€ž = ctn 9^ + 2 tan 9^ , (2.4l)

dd. d n

Ctn 92i^ = -^ ctn Q^ + {^ - l) tan 9^^ (2.1^2)

which are valid for all 9^, 4 Â±90.

It should be stressed that these formulas are also valid for

all values of \Jf + 90Â°j i.e., ^ need not lie in the plane of incidence.

It is therefore noteworthy that 9 ^ and 9p, depend only on Qâ€ž, the

angle that H , the projection of H on the plane of incidence, makes

with the normal. This Interesting fact seems to have been overlooked

2

by V.C.A. Ferraro who derived another somewhat more complicated pair

of equations.

To relate Ferraro 's equations to ours, we refer to Figure 1 and

note the following relations between 0,^ and the polar angles p and 7

n

employed by him:

H cos 9^ = H sin P

P H

H sin 9,, = H cos p cos 7 .

P ^

f

- 26

These, in turn, imply the relation

tan 9â€ž = ctn 3 cos 7

n

which, combined with equations (2.ifl) and (2. 14-2) yields Ferraro's

results - apart from obvious notational differences.

In Figure 6(a) we have plotted 9 against 0^ for 9^ = ,

9 = 1+5Â° and 9^ =~k^Â°- These are typical curves in the parameter

H H

classes 9â€ž = 0, 9_ > and 9^ < 0, respectively. In Figures 6(b) and

6(c) we show plots of 9 , versus 9 for the same set of 9 -values j

however, in 6(b) t) = i)- and in 6(c) , ti = l/k. The reader may wish

to check, properties P, - P against these curves.

The angles

9*2 = ctn"^ [2 tan 9 J (2.1^3)

and

1

9*^ = ctn"^ [(ti^ - 1) tan 9 J (2.1^)

introduced in Figures 6(a) - (c) are the maximum positive angles of

reflection and refraction on the cu37ves- of the class 9^-^ > 0, t] > 1 and

the mlniminn negative angles of reflection and refraction on curves of the

class 9^ < 0, Ti > 1.

n

The angles

9^= ctn"^[-2 tan 9j , (2.1l5)

and ..

9^"= ctn"^[(Tl ^ - 1) tan 9 ] , (2.46)

26a

^2=ctn-'[-2tan^J

^^=ctn-'[-2tan^H]

FIGURE 6(a)

- 26b

02 = ctn"'[(77^i)taneâ€ž]

FIGURE 6 X

(b)

0. The distance

of 0' from is therefore V. On the assumption that r. *â€ž and 0^

11 n

are. 5; and k^ , respectively^ the "sphere" through can he shown

22

to he the surface of revolution obtained by revolving the f , A and

s -curves of Figure 7 about the H-axis. The "spherical" wave fronts

thus traced out are known collectively as Fresnel's ray surface through

0. The W. 's in Figure 7 represent the three possible types of incident

wave fronts which can have the trace velocity V, and the W. . 's represent

â€¢^ ij

the associated family of reflected wave fronts. The W. . 's and the

dashed extensions of the W. 's are tangent to the f , A and s -curves

since by constmction these wave fronts are the envelopes of the

"spherical" wave fronts emitted earlier and up to one second later than

the "sphere" shown. It shoiild be stressed that the "hemisphere"

through which lies in the region N > of Figure 7 - i-e.^ in ^ -

indicates merely where the wave fronts emitted from would have reached

were there no discontinuity across *C7 . The "hemisphere" appropriate

to IK does, in fact, differ from that shown when Jj is a surface of

discontinuity and, as in geometrical optics, can be employed to construct

21p,

arametric equations of this surface are given in both the paper and

report of Reference 1.

T?he Alfven wave fronts evolving from consist of the two points

labelled A in the figure. For the purposes of Hioyghens ' constr

these two points should be regarded as small spheres.

- 28a -

X

29

the refracted wave fronts by Huyghens ' procedure.

We shall now explain the relationship of Huyghens ' construction

to the graphical procedure of the preceding sections. To convey the

essential elements of this relationship^ it will suffice to show how

to obtain Figure ka from Figure 7- We assert that Figure ii-amay be

23

obtained from Figure 7 first by constructing the pedal curves to the

curves of Figure 7 and then by inverting these pedal ciirves with

respect to the unit circle. To prove this we first observe that the

f , A and s of Figure 7 are the curves of intersection with the

2k

plane of incidence of the "sphere" through 0. On the other hand^

the f^ , A and s_, curves of Figure k are the curves of intersection of

12 3

the plane of incidence with the surface of wave normals (b serving as

the unit of speed) . Since the surface of wave normals can be obtained

from the pedal surface of the corresponding Fresnel ray surface by

25

inverting with respect to the unit sphere^ it follows that the curves

of Figure i)-(a) may be obtained from the pedal curves of those of Figure 7

by an inversion with respect to the unit circle.

To complete the picture we consider the pencil of lines through

The pedal curve of a curve "G, whose equation is x = x(^ ) , is constructed

as follows: Let T(|) be the tangent to~C, â€¢ From the origin 0^ draw

the perpendicular to T(^ ) and let y = y(l) denote the vector from to

the foot of this perpendicular. The curve traced out by y as | varies

over its domain of definition is the pedal wave of i^ . '~*'

2.k

This statement is true only when H is in the plane of incidence,, as is

the case here.

25

For a proof; see e.g., the lecture notes by E. K. Luneberg "Propagation

of Electromagnetic Waves"; New York University^ New York (19'^9) â€¢

- 50

0'. This family of lines includes among others the lines representing

the six wave fronts. Let be the angle that the nonaal to a line

of the family makes with the vertical N-axls through 0'. In Figure 7;

for example, 9., is the angle that the slow wave front makes with the

K-axis. Then, it is easy to verify that (l) the pedal cxirve of each

line of the family consists of a single point with the polar

coordinates (s,0) = (V sin Q, 9) and (2) the locus of these points

V

is a circle of diameter â€” whose equation is

s = V sin 9 .

If we invert this circle in the unit circle - i.e., if we replace s

by â€” , we find that

P

pb sin 9 = ^

which is precisely the equation of the line i of equation (2.19)

The present discussion applies with obvious changes when

the parameters 9â€ž, V/b and r are varied. Thus varying 9 has the

effect of rotating the f , A , s -curves rigidly about 0. Varying

â€” has the effect of moving 0' closer or farther from and at the

b

same time of governing the question of critical reflection. For example, if

0' falls within the f -oval, then a fast wave is not reflected as

there are no tangents to the f -curve. Moreover, if 0' coincides with 0''

we obtain a family of wave fronts disposed in a way alluded to in subsection

- 31 -

B-5; specifically^ an incident slow wave front at ' ' induces slow

and Alfven reflected wave fronts wMch lie in the same quadrant as

the incident wave front itself. Finally^ varying r has the effect

of changing only the detailed features of the cirrves of Figure 7 so

that the above discussion applies with little or no change for all

r > 0.

Note that increasing it^ from zero has the effect of turning

n

the H-axis, the axis of revolution for the Fresnel ray surface^

out of the plane of the page. When \lfâ€ž > 0^ Huyghens ' construction

therefore involves finding the points of tangency to a family of

â– \hether or not the associated modes of propagation are excited or not

depends upon the bo\:indary conditions and on the orientation of the

magnetic field.

A wave of type j due to an incident wave of type i is said to be

critically reflected when 0. . just becomes equal to +90 or -90 .

- 15 -

larger in absolute value ^ first the Alfven and then the slow wave is

critically reflected.

Figure 4(b) is a plot of , and against the fast angle

of incidence for r = 0.5^ 0â€ž = k^Â° and trr = 0. It was obtained by

n n

graphical means from an enlargement of Figure k{a) . Statements (l)-(5)

refer specifically to Figures i4-(a). However^ with obvious changes^

they apply for all values of v, ^^ and 0^^ the angles 0â€ž = +90Â°^ +l80Â°

n n n â€” â€”

being excepted. Statement (5) is no longer valid in the exceptional cases,

but (l)-(l4-) still apply with minor changes. We leave to the reader the

analysis of the special cases [see Figures 10(b) and llj .

To determine the angles of refraction we must find the Intersection

of the line Â£ '

1 11

pb ' sin0 = p . b ' sin0

1

i'Â°(|v|A.)Â°Â»b'Â°77\'

1/2 ^ / , / ^l/2 /o ^n^

T\ ' = p- = (p7pJ ' (2.30)

with the upper parts at the curves obtained by plotting

against 0. . on a polar diagram. Note that here (cf . equations (2.25) ^Jid

(2.25)) we have referred the disturbance speeds to the Alfven speed b' in

the region OJ and the quantity R ' =(p'/p) M^ is the Mach nianber appro-

priate to the region"^. Observe, in addition, that equation (2.50) is of

the same form as equation (2.27) except that in (2.3) the ratios c. ./^'

depend upon r' not r. But, If as we shall here and hereafter assume,

both 7[ and ^T^ consist of polytropic gases, then we can conclude that

15

The properties listed in the next subsection will be found useful for

this purpose.

- 15a -

FIGURE 4(b)

- 16 -

2

a

^2

7P

- 7P' a'2

2 2

(2.32)

since H and the pressiire p are continuous across a hydromagnetic

contact discontinuity. It follows that c , /b ' is the same function of

9., as c. ^ /b is of 9. , , that c,^/b' is the same function of 9.^ as

i4 il' il' ly 15

c /b is of 9^^ etc. Thus^ on plotting p .b ' versus 9 ^^ j = h ,'^ ,(i ,

for r = 1/2, 1); = say^ we obtain curves [see Figure 5] which are

identical with those in Figure i|(a) . However^ for a given angle of

incidence the distance (M, ') of the line Â£' from the N-axis differs

from that of Â£ by the factor r; ' = (p'/p) .

Referring now to Figure 5; we observe that for sufficiently

small angles of incidence or sufficiently large Mach numbers the line

f = i'(Nl ') will intersect the fast branch (labelled f , ) ^ the Alfven

branch (labelled A ) and the slow branch (labelled sA exactly once.

If the incident wave is fast, then whatever the angle of incidence,

there will always be a fast refracted wave unless p exceeds p ' in which

case there will exist a critical amgle of incidence at which the fast wave

is totally reflected. If the incident wave is an Alfven or a slow

wave and the angle of incidence is positive^ then fast, Alfven and

slow-waves will be totally reflected at successively larger angles

of incidence. For successively smaller negative angles of incidence

we find (l) the fast angle of refraction is -90 degrees, (2) i' intersects

the Alfven branch and slow branches each exactly once (3) i' intersects

the Alfven brajich twice and the slow branch once and {\) Â£' intersects

Since the angles of refraction are necessarily acute angles,, only those

intersections are counted that lie in the region N > 0.

17

At this angle the refracted wave front is normal to the surface.

"Â«=~rt

I

- 16a -

(Pi6.Â«i6)

(Pis.^is)

(mT' = v^'m:'

D D

FIGURE 5

17 -

both the slow and the AlfVen tranches twice. With obvious modifications^

these featvires of Figure 5 apply for all r, ^\f < 3t/2 and Â©â€ž 4 1 p â– > ^

and n. In the exceptional cases aU. of the above discussion again

applies with minor changes apart from statements (3) and (h) which are

no longer tinae.

2. Some General Properties of the Angles of Reflection and

Refraction

The angles of reflection and refraction enjoy several

general properties which are consequences of the particular way in

which the (c. ./^

ii L-Â©iiJ = - 1 , ^ \ ^ J = ^'2,5,

^ ^ ' -(:t + 9.), if 9^ < Oj

^6

Â®ji [^Â®ij> ^'^^ = Â®i ^ J = ^^5,6, P^

\\^,\ < \\^\ < \^,e\ ' ^9

In these relations^ parameters not shown are considered fixed and

unless specifically indicated otheivise the subscripts i and j assume

the values 1^2^3 and l;2j...^6, respectively. The angles 9. . appearing

in equations P^ and Pq are what we shall hereafter refer to as the

acute angles of reflection. They are defined as follows:

"ij

: - 9. ., if 9. . > i = 1;2,5,

(2.37)

â€¢(9^^ + rt), if 9.^ < j = 1,2,5.

19

Recall that the first index i refers to the incident wave and the

second index j to the reflected or refracted waves.

19

Equation P expresses the fact that the angles of reflection and

refraction are Independent of the direction of the component of H_^

which is perpendicular to the plane of Incidence. Equation P states

that these angles are also independent of the sense of H . Together,

P and P imply that the reflected and transmitted angles are

independent of the sense of II_. Equation P states that the slow and

fast angles of reflection and refraction are Invariant under the

transformation of r->-r whenever the incident wave is slow or fast.

From P, we conclude that if a wave incident at the angle 9. produces

scattered waves at the angles 9. . . then a wave incident at the angle

ij

-9. will produce scattered waves at the angles -9. . only if the direction

1 . ij

of H is reflected in the N-axls specifically, only if 9,, is replaced

"~p "^ H

^y -Â©XT' If Â©TT is not so replaced, then according to P^. the scattered

n n P

waves emerge at the new angles 9f . = - (S) . . 1-9. ;9â€ž1 , which differ, in

general, from -9. .. Only when 9 is , +90 and l80 do we find that

â€¢)(â–

9. . = -9. .. In other words, negative angles of incidence are not

physically equivalent to positive angles of incidence except for

special orientations of the magnetic field. The next two properties P^

and P verify the reversibility of the scattering process. Thus, for

example, P may be Interpreted as follows: If 9. . is an angle of

refraction associated with the angle of incidence 9. , then a wave in /(_,

incident at the angle 9. . will produce a transmitted wave in ('J^ of type

1 emerging with the angle 9.. In the two relations Po and P , it is

assumed that 9. . 's and 9, . 's are real: these relations affirm a fact

ij ij

to be expected on physical grounds, namely, that the fast reflected

and refracted waves emerge first, followed by the reflected and refracted

- 20 -

Alfv6n waves which are in turn followed by the reflected and refracted

slow waves (see Figure 2(a)).

3, Numerical Considerations - Anomalous Reflection and Refraction

For sufficiently large angles of incidence it will he found

that^ regardless of scale ^ the intersections of the lines i(R ) and

i'(M ') [see equations (2.25) and (2.29)] with the slow and Alfven

curves fa3JL outside the range of the graph. For such angles it is

necessary to supplement the graphical analysis with suitable approximate

formulas. Fortunately, the Alfven curves are linear and the slow

branches are almost linear in that they approach the straight lines

of equation (2.28) asymptotically. These facts may be employed to

obtain analytical expressions for the angles of reflection and refraction

associated with large angles of incidence. Consider, for example,

the problem of determining the angles at which the reflected slow and

Alfven waves emerge. In this case, the intersection of Â£ with the A

and s^-curves lead to the following expressions:

sin 0.â€ž sin 9.

i2 1

cos tjj |cos (Â©j_2-Qh^I (Cj_A)

}

sin 0.,

i5

^ \ 1/2 sin 9.

cos tjj Icos (9^^-9g)| V / (c^/b)

The first of these equations is simply equation (2.35) with c /b

replaced by the explicit expression given in the right member of equation

(2.11)' this equation is exact. The second equation is, on the other

hand, an approximate equation: it is derived by replacing, for each

- 21 -

0, ^, the quantity p in the equation c = p" by the approximating

value of p on the asymptote [see equation (2.28)] and then substi-

tuting the resulting expression for c . ., into equation (2.53)Â« Such

an approximating value always exists unless = 0^ in which case Â£

20

is parallel to the asymptote. From the above equations and the

trigonometric s-um formula for the cosine it follows that

sec tjj sec 0^^. (c^/b)

ctn 0^2 = + tan 0^ ,

sin 0.

1

â€” 1 1 /p

(l+r" ) ' sec f sec (c./b)

ctn 0^ ^ = + tan 0â€ž .

^ sm

In these equations^ the "+" sign is intended when the intersection of i

with the A- line or asymptotic line occurs in the half -plane into which

H is directed; otherwise the negative sign must be chosen.

It is worthwhile noting the possibility of a kind of reflection

which does not arise in conventional gas under similar circumstances.

For the sake of concreteness^ suppose that the incident wave is

slow and the relevant Figure for the reflection process is Figure i4-(a) .

When 0,, the angle of incidence^ is positive and sufficiently small

the disposition of the wave fronts is as shown in Figure 2(a). As

if.

is increased to "critical angle", 0^ say [defined as the angle at

which = 90Â°3 , the fast wave front approaches the normal direction

to the surface Jo . What we should like to stress here is the following:

20

It should be mentioned that it is not difficult to obtain explicit

formulas for 0. . when 0â€ž = 0.

ij H

â– ' - 22

By increasing the angle 9 slightly past the critical angle 9 , it

is possible to extend the manifold of fast wave solutions continuously

past 9 = 90 . Physically this means that the fast wave fronts

associated with such values of 9-, emerge in the same quadrant as the

5

slow incident wave front as^ for example^ the dashed line does in

Figure 2(a) . The reader can convince himself from an inspection of

Figure 4(a) that for negative 9^ one can pass over^ by continuously

decreasing 9 . to the case where both the Alfvln and slow reflected

waves emerge in the same quadrant as the incident wave. If one were

to construct these wave fronts by means of Huyghens ' method (see

Section B-5) , one would find that the rays which guide the "energy

in the mode" [see Section II-B-5 and III-A-lJ all have components

directed along -N,^ as do the rays associated with ordinary reflected

waves. Incidentally, in Figure \, the outward normals to the s, A

and f-curves, drawn at their points of intersection with i give the

directions of the rays [cf . reference l] . This "radiation condition"

together with the continiiity considerations support the inclusion

within the manifold of reflected waves those which emerge in the

same quadrant as the incident wave.

The existence of the above type of anomalous reflection process

[and, in fact, of others not mentioned, for the reason that they are

less physicalljr admissible] makes it necessary to define the terms

reflected, refracted and even incident wave more precisely. Hereafter,

we shall admit to the class of reflected jjrefractedj waves only those

which (l) may be arrived at by continuously increasing the angle of

incidence from zero, and (2) are associated with ray vectors having

- 25 -

components that are directed along -N [along n] . Finally, a wave

which propagates toward Jj from below - i.e., a wave whose normal

n has a component directed along N - will be admitted as an incident

wave only if the associated ray vector has a component which is

directed along N. It should be mentioned that in Figure 5. only

those admissible angles of reflection which are associated with

normally [as opposed to anomolouslyj reflected waves are shown.

Another interesting kind of anomolous reflection (refraction)

process is the phenomenon of conical reflection (refraction) [See

Reference ij . Conical reflection (refraction) can occur only when

r = 1 and H^ is in the plane of incidence and. then only at suitable

angles of incidence. In Figure 3(a), for example, conical reflection

can occur only when the line I is in the position shown - that is,

only when Â£ passes through the triple point where the s, A and f

cuives intersect. Imagine that the disturbance on the incident wave

front is concentrated in a small disc of negligible diameter. Then,

employing considerations similar to those given in Reference 1 [p. 57^J

we can infer that this "point" disturbance will emerge as a ring

disturbance with an illximinated point in its center. The disc

determined by the illuminated ring and its center point is perpendicular

to H and moves parallel to H with the speed b = ViaH p ; its radius

t seconds after the incident wave has hit the interface can be shown to

be ^t.

k. Explicit Formulas for the Alfven Waves

We assume here that an AlTven wave is incident on the contact

2k -

discontinuity^ and we derive explicit formulas for the angles at

whlcli the reflected and refracted Alfven waves emerge. We begin by

combining equations (2.35) :> (2. 5^1-) and equations (2.5) j> (2.11) and

find, after cancelling the common factor cos i|/â€ž, that

sin 9^2 sin 9^

022=03(922-9^) a^c^os{Q^-Q^) '

sin 92i^ sin 92 . " 2

(2.58)

o^^zos{Q^^-Q^) o^zos[Q^-Q^)

(2.59)

In these equations the a's are +1 or -1 according as the neighboring

cosine factors are positive or negative. Employing the standard

trigonometric formulas for the cosine of a difference of angles, we

find easily that

ctn 9â€ž^ = -^^ (ctn 9â€ž + tan 9j - tan 9^ , {2.k0)

22. O d. Vl il

1

Ctn 9 , = n^ -^^ (ctn 9 - tan 9^^) - tan 9^^ . (2.1fl)

Now, according to property P of equation (2.56), no generality is

lost in restricting 9â€ž to the range -90Â° < 9â€ž < 90Â°. Assuming, for

n 11

the moment that this restriction has been made, we readily verify,

by referring to Figures (5) and (^) , that

0-22

^2

and

^24

^2

= 1

- 25 -

Introducing the acute angle of reflection^

Â« - 0^2 . if Â©22 ^ Â° ^

Â©22 = (2.1^0)

j^-(:t + 922) , if Â©22 < ,

into equation (2.58)^ we then obtain the formulas

Gtn 0^â€ž = ctn 9^ + 2 tan 9^ , (2.4l)

dd. d n

Ctn 92i^ = -^ ctn Q^ + {^ - l) tan 9^^ (2.1^2)

which are valid for all 9^, 4 Â±90.

It should be stressed that these formulas are also valid for

all values of \Jf + 90Â°j i.e., ^ need not lie in the plane of incidence.

It is therefore noteworthy that 9 ^ and 9p, depend only on Qâ€ž, the

angle that H , the projection of H on the plane of incidence, makes

with the normal. This Interesting fact seems to have been overlooked

2

by V.C.A. Ferraro who derived another somewhat more complicated pair

of equations.

To relate Ferraro 's equations to ours, we refer to Figure 1 and

note the following relations between 0,^ and the polar angles p and 7

n

employed by him:

H cos 9^ = H sin P

P H

H sin 9,, = H cos p cos 7 .

P ^

f

- 26

These, in turn, imply the relation

tan 9â€ž = ctn 3 cos 7

n

which, combined with equations (2.ifl) and (2. 14-2) yields Ferraro's

results - apart from obvious notational differences.

In Figure 6(a) we have plotted 9 against 0^ for 9^ = ,

9 = 1+5Â° and 9^ =~k^Â°- These are typical curves in the parameter

H H

classes 9â€ž = 0, 9_ > and 9^ < 0, respectively. In Figures 6(b) and

6(c) we show plots of 9 , versus 9 for the same set of 9 -values j

however, in 6(b) t) = i)- and in 6(c) , ti = l/k. The reader may wish

to check, properties P, - P against these curves.

The angles

9*2 = ctn"^ [2 tan 9 J (2.1^3)

and

1

9*^ = ctn"^ [(ti^ - 1) tan 9 J (2.1^)

introduced in Figures 6(a) - (c) are the maximum positive angles of

reflection and refraction on the cu37ves- of the class 9^-^ > 0, t] > 1 and

the mlniminn negative angles of reflection and refraction on curves of the

class 9^ < 0, Ti > 1.

n

The angles

9^= ctn"^[-2 tan 9j , (2.1l5)

and ..

9^"= ctn"^[(Tl ^ - 1) tan 9 ] , (2.46)

26a

^2=ctn-'[-2tan^J

^^=ctn-'[-2tan^H]

FIGURE 6(a)

- 26b

02 = ctn"'[(77^i)taneâ€ž]

FIGURE 6 X

(b)

0. The distance

of 0' from is therefore V. On the assumption that r. *â€ž and 0^

11 n

are. 5; and k^ , respectively^ the "sphere" through can he shown

22

to he the surface of revolution obtained by revolving the f , A and

s -curves of Figure 7 about the H-axis. The "spherical" wave fronts

thus traced out are known collectively as Fresnel's ray surface through

0. The W. 's in Figure 7 represent the three possible types of incident

wave fronts which can have the trace velocity V, and the W. . 's represent

â€¢^ ij

the associated family of reflected wave fronts. The W. . 's and the

dashed extensions of the W. 's are tangent to the f , A and s -curves

since by constmction these wave fronts are the envelopes of the

"spherical" wave fronts emitted earlier and up to one second later than

the "sphere" shown. It shoiild be stressed that the "hemisphere"

through which lies in the region N > of Figure 7 - i-e.^ in ^ -

indicates merely where the wave fronts emitted from would have reached

were there no discontinuity across *C7 . The "hemisphere" appropriate

to IK does, in fact, differ from that shown when Jj is a surface of

discontinuity and, as in geometrical optics, can be employed to construct

21p,

arametric equations of this surface are given in both the paper and

report of Reference 1.

T?he Alfven wave fronts evolving from consist of the two points

labelled A in the figure. For the purposes of Hioyghens ' constr

these two points should be regarded as small spheres.

- 28a -

X

29

the refracted wave fronts by Huyghens ' procedure.

We shall now explain the relationship of Huyghens ' construction

to the graphical procedure of the preceding sections. To convey the

essential elements of this relationship^ it will suffice to show how

to obtain Figure ka from Figure 7- We assert that Figure ii-amay be

23

obtained from Figure 7 first by constructing the pedal curves to the

curves of Figure 7 and then by inverting these pedal ciirves with

respect to the unit circle. To prove this we first observe that the

f , A and s of Figure 7 are the curves of intersection with the

2k

plane of incidence of the "sphere" through 0. On the other hand^

the f^ , A and s_, curves of Figure k are the curves of intersection of

12 3

the plane of incidence with the surface of wave normals (b serving as

the unit of speed) . Since the surface of wave normals can be obtained

from the pedal surface of the corresponding Fresnel ray surface by

25

inverting with respect to the unit sphere^ it follows that the curves

of Figure i)-(a) may be obtained from the pedal curves of those of Figure 7

by an inversion with respect to the unit circle.

To complete the picture we consider the pencil of lines through

The pedal curve of a curve "G, whose equation is x = x(^ ) , is constructed

as follows: Let T(|) be the tangent to~C, â€¢ From the origin 0^ draw

the perpendicular to T(^ ) and let y = y(l) denote the vector from to

the foot of this perpendicular. The curve traced out by y as | varies

over its domain of definition is the pedal wave of i^ . '~*'

2.k

This statement is true only when H is in the plane of incidence,, as is

the case here.

25

For a proof; see e.g., the lecture notes by E. K. Luneberg "Propagation

of Electromagnetic Waves"; New York University^ New York (19'^9) â€¢

- 50

0'. This family of lines includes among others the lines representing

the six wave fronts. Let be the angle that the nonaal to a line

of the family makes with the vertical N-axls through 0'. In Figure 7;

for example, 9., is the angle that the slow wave front makes with the

K-axis. Then, it is easy to verify that (l) the pedal cxirve of each

line of the family consists of a single point with the polar

coordinates (s,0) = (V sin Q, 9) and (2) the locus of these points

V

is a circle of diameter â€” whose equation is

s = V sin 9 .

If we invert this circle in the unit circle - i.e., if we replace s

by â€” , we find that

P

pb sin 9 = ^

which is precisely the equation of the line i of equation (2.19)

The present discussion applies with obvious changes when

the parameters 9â€ž, V/b and r are varied. Thus varying 9 has the

effect of rotating the f , A , s -curves rigidly about 0. Varying

â€” has the effect of moving 0' closer or farther from and at the

b

same time of governing the question of critical reflection. For example, if

0' falls within the f -oval, then a fast wave is not reflected as

there are no tangents to the f -curve. Moreover, if 0' coincides with 0''

we obtain a family of wave fronts disposed in a way alluded to in subsection

- 31 -

B-5; specifically^ an incident slow wave front at ' ' induces slow

and Alfven reflected wave fronts wMch lie in the same quadrant as

the incident wave front itself. Finally^ varying r has the effect

of changing only the detailed features of the cirrves of Figure 7 so

that the above discussion applies with little or no change for all

r > 0.

Note that increasing it^ from zero has the effect of turning

n

the H-axis, the axis of revolution for the Fresnel ray surface^

out of the plane of the page. When \lfâ€ž > 0^ Huyghens ' construction

therefore involves finding the points of tangency to a family of

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