Jack Bazer.

Reflection and refraction of weak hydromagnetic discontinuities online

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Online LibraryJack BazerReflection and refraction of weak hydromagnetic discontinuities → online text (page 2 of 4)
Font size (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 Statement (5) is no longer valid in the exceptional cases,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 ÂŁ '

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 ÂŁ' from the N-axis differs
from that of ÂŁ 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 {\) ÂŁ' 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 ÂŁ with the A
and s^-curves lead to the following expressions:

sin 0.â€ž sin 9.

i2 1

}

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 ÂŁ

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 ÂŁ 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

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° ^

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

2 4

Online LibraryJack BazerReflection and refraction of weak hydromagnetic discontinuities → online text (page 2 of 4)