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similar "spheres". It is easily seen that these points of tangency

do not lie in the plane of incidence and, in fact, are not even coplanar.

Nevertheless, Huyghens' construction may he made graphically. One

need only work with the projection on the plane of incidence of the

locus of points on Fresnel 's ray surface at which the normals^ are

parallel to the plane of incidence. Parametric equations of these curves

may be obtained from the ray-equations ( 3.^1-5) -(3- ^7) of Section III

by substituting the expressions for n and H^ given in equations (3-5)

and (3-6).

In graphical work, the method presented here is more accurate

than Huyghens ' method in that a point of intersection is more easily

located than a point of tangency. Moreover, since the angles of reflection

and the (reciprocal) disturbance speeds are determined simultaneously,

it yields more information. The avoidance of the cusped figures

should also be noted. Unquestionably, however, Huyghens ' construction

has more direct physical appeal.

- 52-

III. REFLECTION AT M INFINITELY CONDUCTING RIGID WALL

A. Summary of the Basic Equations

Let tS = J[ i'^) d.enote any propagating planar wave front and

n the unit normal at each point of ^ (t) pointing along the direction

of propagation. A region will be referred to as being in front of

or behind A (t) according as n or -n is directed into that region.

Let Q be the undisturbed value of some physical quantity and Q and

â‚¬L denote the values of this quantity in front of and behind ^(t) .

Here and hereafter, we shall take the state of the medium behind the

wave front as the state in front plus a small perturbation. Letting

&Q represent such a perturbation in the quantity Q , we therefore have

Q^ = SQ + Qq . (3.1)

In a given wave 0^^ may be regarded as the " umperturbed" value of the

physical quantity it represents. It should be stressed, however, that

owing to the presence of other perturbations, Qq might be the sum

of Q and a small perturbation. Whether or not Qq= Q, or Q plus

a small perturbation, it can be shown, on neglecting all but the

highest order terms that 5H, 5u, and 5p as Alfven, slow and fast modes

26

are given by the following formulas :

Alfven: 5H = -|r: Hx n ,

5u = + â€” Hxn , (3.2)

1 pc â€¢Â« K,

5p = 5S = ;

2S

K. 0. Friedrichs, "Nonlinear wave motion in magnetohydrodynamics",

Los Alamos Report No. 2105 (written 1954 - distributed 1957)- See also

later version of this work by K. 0. Friedrichs and H. Kranzer, Report

No. MH-8, AEC Computing and Applied Math. Center, Inst. Math. Sci. ,

New York University, New York (1958).

- 35

5H = - -^ nx (nxH) ,

fE^

6u = +-

T=Vt"') - #.^-'^'-^-'' "â– "

y^ \c^ / '^ ~ yp;

e , \' '^

6p == p \ -5 1

"j^

\c

In these formulas, H, H , i_i; p and S are, in the stated order, the

magnetic intensity, the component of magnetic intensity in the

direction of propagation, the specific magnetic inductive capacity,

the density and the specific entropy of the flow all in the absence

of any distuirbance whatsoever. The c's are the disturbance speeds

appropriate to the mode in question. They are the positive solutions

of the equation:

" '' ^ ^ = 0. (3.1^)

(pc^ - ^^ )

k ,2 â€ž2. 2 2 â€ž 2

pc - (pa + jiH jc + a M-H

Specifically, in equation (3-2), c is the positive root of the first

factor, while in equation (3-3) ^ c is the smallest or largest positive

root of the second factor according as the mode is slow or fast.

With n and H expressed in terms of the angles shown in Figures 1 and 2

I.e.

n = cos N + sin 9 x , (3'5)

-o

H = H cos iV^Xcos 0^^ N + sin 0,^ x )+ H sin \lfâ€žzâ€ž,(3.6)

vÂ«v n H '~' n *-0 ^il~-0'

we obtain for the slow, Alfven and fast disturbance speeds, the explicit

expressions simmiarized in equations (2.^4-) - (2.6). The disturbance

- ^k -

speeds along n are related to the speed of propagation U > of >^(t)

along n and to the trace velocity V as follows:

WW ^v^

Vn = U = u-n + c ; (j.?)

\AV VW

(see Section II-A) . The upper (lower) signs in equations (3.2) and

(5.5) correspond to the upper (lower) signs in equation (5>7)' Note

that there will be only one speed of propagation along n, specifically^

the one corresponding to the upper sign in the above equations, unless

the projection of the fluid velocity u on n exceeds the disturbance

speed G.

The absolute value of the quantity e appearing in (5-2) and

(5.3) measures the strength of the discontinuity at the wave front

of the mode in question. For |e| to be a true measure of this strength

it is necessary to nomialize the mode of propagation in a suitable

.anner. This is the function of the constant factor "j/e^ whose choice

for each mode is dictated by the following considerations: In

reference 1, a mode of propagation was defined to be a six- vector of

the fona

mi

R =

1

J (3.8)

('H. ' 2 5H [V , 2 gH , 5u , ou , 5u , ^

\P y \P z' n-' y' z' p

here, the x-axis is assumed to be directed along n. R represents

the total discontinuity in the medium at a given point of the wave

front. A total of six modes of propagation are obtained by substituting

the appropriate components from equation (3-2) or (3-3) into the

above expression. Now it is natural to define the strength of the

discontinuity to be proportional to (R â€¢ R) , the Euclidean length of R.

35 -

For reasons which will soon be apparent it is convenient to choose

Vp/s as the constant of proportionality. With (-^pR â€¢ R)^ as the

definition of mode- strength, it then follows that E^ must be chosen

to be

E

1 _

Â§ R^ Ri

Â£i^,aM, ,(,â€ž.,2^ (j.g,

II 1 ?

if |e| is to equal (^pR â€¢ R ) . In equation (3.9) R^, &1I, 5pi and

ffl* denote the values of R, 5u, Sp and 5 H obtained by setting â‚¬ = 1

in equations (5. 2) and (3.3). If one actually inserts the expressions

for 5i/. 5p^ and 8H^ in equations (3.2) or (3.3) into the last member

of equation (3-9)^ one finds that

Alf v^n :

1^ = y^ b I sin 6 I ,

(3.10)

Slow

or

Fast

T^ = Vp b Isin sl '

2w - (l+r)

(w^- r)

= yp^ i

w

1

2

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

(3.11)

In these equations 5 is the angle between n^ and H. Employing

equations (3.5) and (5.6) it follows that

WW vvv

|sinS| = Tl-cosiircos (9-9..)

â– H. H

(3.12)

The quantity w is defined by the equation

w =

2 >

(3.13)

- 56 -

it is understood here that c represents slow or fast disturbance speed

according as the mode is slow or fast.

Note that E^ has the dimensions of energy density; E^ may be

thought of as the "energy density of the mode" R^. Aq Interesting

sidelight concerning such energy densities is the following. For modes

R of arbitrary strength [^see equation (5.8)] , one can show^ by means

of equations {3.k) and (5.2) or (5.5) [See Appendix IlJ that

1^2 1 a^Sp^ ^ 1 =,1x2 r-K DA

gPBu = 2 â€” ^â– ^2'^^^ ' (5.14)

so that the energy density of the mode is divided in equal parts between

the kinetic energy density of the mode on the one hand and the sum

of the internal and magnetic energy densities on the other. From

this equation it follows readily that

E = p5u^ , (all modes) (5-15)

|e = |p5u^ = ^m^ (Alfven mode) . (5-16)

Here, E denotes the energy in a mode of arbitrary strength.

B. Reflection at an Infinitely Conducting Rigid Wall

1. The Boundary Conditions

We now imagine that the upper half- space uC in Figure 1

is a rigid infinite conductor and that the unperturbed state of the

fluid in the lower half-space W is constant with the fluid velocity

- 37 -

u = 0. The (planar) incident wave Is assumed to propagate into

the unperturbed fluid and the perturhations in the regions between

the surfaces of discontinuity - i.e.^ the wave fronts and the interface

J^ - are assumed to be constant. At the interface (^, behind the

slowest reflected wave, we have the following boundary conditions

N â€¢ &u* = ,

BC: N â€¢ 5H* = , (3.1?)

N X 5E* = nN X (H X Bu*) = ,

â– X- *

where 5u and 5H represent the total pertiirbation at the interface.

The first condition expresses the rigidity of the wall; the second

the conse2rvation of magnetic flux and the third , the continuity of

the tangential electric field vector across ^. If , as will here and

hereafter be assumed, H is in the plane of incidence, then the boundary

conditions split up into two sets, namely,

N â€¢ 6u^ = ,

BC : N â€¢ 6H* = , (3-l8)

p -Â» ~p '

N X (5u XH) = ,

and

BC^: N^x(5u_L*X H) = . (3.I9)

The first set evidently involves only 6u and 6H , the components of

Su_ and 6H in the plane of incidence, while the second involves Su, ,

the component of u normal to the plane of incidence. Now, as an

- 38 -

inspection of equations (3.2) and (3-3) reveals, ou and 6H are

normal to the plane of incidence when the wave is an Alfven wave

and parallel to the plane of incidence when the wave is slow or

fast. Since the boundary conditions for the normal and planar

components are uncoupled it follows that Alfven incident waves can

produce only Alfven reflected waves and slow or fast incident waves

can produce only slow and fast reflected waves. Assuming for the

moment that the incident wave is slow or fast, we note that there

are two reflected waves and apparently four boundary conditions

in equation (3'l8). It will now be shown that the conditions BC

of equation (3.I8) reduce to

BC 5u = , if H â€¢ N ^ , (3-20)

and to

BC N â€¢ 5u = , if H â€¢ N = . (3-21)

P vw >~p ' ~Â» lÂ«.

First, we observe that equations (3. 20) and (5.21) follow

directly from the first and third boinidary condition of (3.17) â€¢

To complete the proof we must show that N â€¢ 5H vanishes. To this end,

we note the following relations which hold for any propagating wave

[see reference 26J

n^ â€¢ 5H = , (3-22)

c 5H + n X (6uxH) = .

Replacing c by V â€¢ n [see equation (3-7) J note u = o] , we find that

n x(5HxV + SuxH) =0. (3-23)

- 39 -

This relation, in turn, implies that

5HxV + 5uxH = , (5.2i+)

since each vector within the parenthesis is normal to the plane of

incidence when, as we have assumed, H is in the plane of incidence.

One such relation applies to the incident wave and to each of the

reflected waves. If we add these relations we find that

5H* X V X 6u* X H = . (5.25)

But 5u X H vanishes by hypothesis; hence sH x V vanishes, which

implies the desired result, namely N â€¢ 5H* = 0.

2. Strengths of the Reflected Waves

We shall employ the symbol 0. with the single subscript

i, to denote the value of a physical quantity Q in front of an

incident wave of type i and oQ. to denote a perturbation of Q. in

the sense of Section A. Similarly, Q. and gO. . will denote the

corresponding quantities in a reflected wave of type j due to an

incident wave of type i. We shall further allow i and j to take on

the "values" s(for slow), f(for fast) and A(for Alfven) in place of

the numerical values used earlier. Thus, for example, e denotes the

strength of a fast incident wave and e â€ž denotes the strength of a

sf

fast reflected wave produced by a slow incident wave.

Case 1: Alfven wave incident - Assiome first H â€¢ N 4 0- "^^ boundary

conditions for this case are J_see (3.19)_]

Sii^A + 5%^ = > (5.26)

UO

since ^ under the assiomed conditions, only an Alfven wave can be

reflected. Employing equations (5.2), (3.5)^ (3-6) and (3. 10), we

readily conclude that

"AAl

'A'

(3.24)

so that the strength of the reflected wave equals the strength of

the incident wave even when the acute angle of reflection [see equation

(2.3T)j does not equal to the angle of Incidence.

When H'N vanishes, the incident wave causes no additional

MM Â«W ^

distiirbance within the medium, since no boimdary conditions have to

he met [see equation (3.19)J' At the interface behind the wave,

however, we find a surface current density proportional in magnitude

to |&H.|.

Case 2; Slow or fast wave incident - The following formulas are the

starting point of o\ir discussion:

5u

-c

1^

5u = e Su

cos(9-9jt)cos9^

2

w

(3.28)

cos9

W +

cos(9-9â€ž)sinGâ€ž

Jl n

sin9

w

aso

E = 1^ b |sin(9-9^)

2w - (1+r)

2

w - r

= Vp b

(w^-cos^(9-9jj)(2w^- 1-r)

2

w - r

1

2

1

2

(3.29)

w

2/k2

c /b .

1+1 -

These formulas, which apply to both slow and fast waves, follow

directly from (3.5), (5-5)^ (3-6) and (3-11) on setting iir^ = 0.

Let us first consider the special case in which H â€¢ N =

W^ t^V

- i.e., in which 0â€ž = 90Â°- Here, only one boundary condition

n

need be met, namely N â€¢ 5u =0 [see equation (3.2l)]J. Using

equation (3.29), one can easily show that this boundary condition

can be satisfied for all angles of incidence, by a reflected wave

hairing the following properties:

(1) It is of the same type as the incident wave.

(2) It emerges at an (acute) angle of reflection which is

equal to the angle of incidence. Moreover, the relation

e,. = e^ , (3.30)

holds, where i is s or f according as the incident wave is slow or

fast.

Suppose that 4 j:90Â° - i.e., that E which is in the plane

of incidence is not in the plane of the interface. In this case,

we have two boundary conditions from the vector equation (3-20) and

(at least for angles of incidence which are sufficiently small in

absolute value) , two waves for meeting these conditions - one slow

and one fast. Employing equation (3.29), we may express the boundary

condition, behind the slow reflected wave, as

e. 6u. + e. â€ž 6ufâ€ž + â‚¬.8u. = , (5.51)

IS "^is if -â– if 1-1 '

where i is f or s according as the incident wave is fast or slow.

- 1+2

From this relation It follows directly that

is -"is "^if i ~a -if '

â‚¬.^ 6ufâ€ž X 5uf = -e. 5uf- X &uf- ,

if ~if - IS 1 -i - is '

which in turn implies that

(Suf X 5uf^ â€¢ M)

-1 -if -

(3.52)

^is =

s

^^^Is^'^'^tf

â€¢ M)

"if =

h

(SH|- X Su^f^

â€¢ M)

I'o.. â– Â» , , 0.. t_

.Â«N

(3.55)

(5uf- X 5u?-â€ž â€¢ M) '

^ -IS ~if *-'

Here, M is the unit vector normal to the plane of incidence. Since,

for fixed r and 9 ^ 90 ; numerical analysis shows that the numerators

and denominators in (3- 55) vanish at isolated values of 9., if at all,

one may conclude that multiple reflection is the rule and single

reflection is the exception. Observe that the vanishing of numerators

or denominators in (3' 35) means that the velocity vectors involved

in these expressions are parallel.

Combining equations (5-29) and (3- 3) one obtains, with the aid

of standard trigonometric identities, the following explicit formulas

for the e . . ' s :

is 1 f ' sf '

(3.5^)

^if = ^^i^s'/^if

it3

Here, D is given by

D

sf

V. w, _(Ef Ef_)2

IS if^ IS if

â€¢ v.Jv. -l)cos(9. -9â€ž)sin(0. â€ž-9â€ž)

if^ IS ^ ^ IS H' ^ if H'

+ w^3(l-wJf)cos(0.^-9jj)sin(O.^-9j^) \ j (5.35)

the corresponding expressions for D (d ) may be obtained from

S 1

this equation simply by eliminating the subscript f(s) in all terms

where it appears as a second subscript. The values of v. , w. . and

9. . appearing in the D's may be obtained by the graphical procedure

-'-J

described in the earlier sections, and E. , E^ etc. may be calculated

1 is

on the basis of these results. In this fashion one can obtain a

complete quantitative solution of the problem. Without resorting to

numerical or graphical methods, however, it is still possible to

arrive at some idea of the content of formulas (j.jJl) and (5.35) l^y

choosing special angles of incidence or special orientations of the

magnetic field, or both. We shall examine several such special cases

now.

Let us begin by assuming that Q. ^ and 9^ = - i.e. , that

the incidence is oblique and that H is normal to the plane of the

interface. In this case, one might suppose that the boimdary conditions

can be met by a single reflected wave emerging at an acute angle of

reflection equal to the angle of incidence. Assuming that a slow

wave is incident, this would imply [according to equation (3'53)j that

- kh

e â€ž vanishes . However^ on substituting Â© =n-0 or-rt+0

SI ss s s

[according as 9 is positive or negative] into the expression

for e , one finds that the numerator will in general not vanish}

the vectors &u' and 6u' in eqviation (5.52) are evidently not

parallel. Here too then, double refraction is the rule.

Assume now that 0. = and 0^^ 4 - i.e., that the incidence

is normal but the magnetic field is oblique. Setting 0. =

0.^ =0. = l80 in equation (5'53) and observing that v.. = w,

if IS ^ ^ ' '=' 11 1

and E. . = E. we find that

li 1

6. = e. 5^ 1 = s.f

IS 1 is

^If = h \f ^ = ^'^

(5.56)

where &. . is the Kronecker delta. Thus, when H is oblique, a normally

incident wave produces a single reflected wave of the same type and

strength as the incident wave.

Suppose now that 0^^ =)= 0. Let the angle of incidence approach

as a limit. Assume first that the incident wave is slow. It then

H

follows, on taking limits in equations (3.5)^ that the resulting slow

incident wave reduces to an ordinary sound wave when r = a /b is less

than unity. The same is true of a fast incident wave when r > 1.

When r = 1, both the limiting slow and fast waves move with the speed

of sound, but the associated 6u Â»s and SH.'s have non-vanishing

components tangent to the wave front. However, a suitable linear

combination of these waves furnishes a so\ind wave in this case also.

i^5 -

Thus for all values of r^ one can assume that the incident wave along

H is a so\md vave. The point we wish to emphasize is the following:

A sound wave incident along H, can produce a pair of geniilne hydro-

magnetic waves. It is necessary of course to assume here that the

angle of incidence is small enough^ when the incident wave is slow^

in order that the fast angle of reflection he real. To obtain the

actual strengths of the reflected waves ^ one simply lets 9 approach

0^ in the numerators of the right member of equation (5* 5^)*

5. Energy Relations

Let u^ p and H be time-independent and constant over a

given region of space - q , say - and let u ' ^ p ' and H ' denote

continuous,, continuously differentiable perturbations of these

quantities in U â€¢ On multiplying the linearized hydromagnetic momentum

equation by u'- and combining the result with the remaining linearized

equations one obtains, as in gas dynamics, an "energy" equation,

namely,

||- + V â€¢ F' = , (5.37)

which expresses the relationship between the "perturbation energy"

per unit volimie

2

E' = I p (u'2 +^ p.2 + ^jj,2^ ^ (3_38)

and the "flux of energy"

- kS -

27

F' = a%'u' + [i(Hxu') XH' + E'u . (5.39)

Corresponding to equation (5' 37)^ we have the jump relation

-USE + SF-n = , ' (3A0)

associated with a discontinuity at a wavefront moving at each point

with the speed U > in the normal direction n. 5E and 5F here denote,

in general, the difference in the energies and fluxes, respectively,

in front of and behind the wavefront. Since we always regard a

disturbance behind a wavefront as a perturbation of the state ahead

28

we conclude, using equations (j.jS) and (5- 39)^ that

5E = "I p (6u^ + a^Sp^/p + laSH^) , (3-^1)

6F = a^SpSu + |i(HX5u) X 5H + SEu , (3-^2)

[see equation (3.l)J.

In Appendix II, It will be shown that

6F = 5E s j = s,A,f , (3A5)

27

_F' in the present theory plays a role analogous to that of the Poynting

vector of electromagnetic theory. The second term in equation (3- 39)^

namely |j.(Hxu') XH' = ExH, is in fact the Poynting vector.

The quantities a,p^H and u^ in these expressions will, in general,

differ from the values of those in the totally undisturbed flow by

first order perturbation terms. However, it is clear that we may

ignore the contribution of these first order perturbations to a2,p,

H, and u, since they contribute third order terms to 6E and 5F, which

are themselves of the second order.

- hi

Here^

J J

and s is the ray vector associated with the given mode of propagation.

Specifically^ in slow waves

b^r(H /H^)

s = JiiÂ±^.aÂ± ^ (H-Hn) , (5.^5)

â€”s Â»- â€” s-" â€” . -^ ' n'

c

s

in Alfven waves

Sa = JiÂ± (^^/P)^ sgnCH )H , (5.1^6)

and in fast waves

_ l3^r(H /h2)

Sf = iiÂ±c^ + i (H-H^n) . (3.1+7)

c^C

In these equations

,2

(1+r)^ - l+r(H â€¢ nf/E^

.1

2

(5.H5)

and sgn(x) is the sign of x. For all ray vectors s we find that

s-n = u+c = U. (5-^9)

v~ w. n â€”

As is the case with propagation of electromagnetic waves in crystalline

media^ hydromagnetic energy is guided along rays which are not in

general directed along the normal to the wave front [see reference 1,

Section B-5] .

Turning now to the problem of reflection at a rigid infinite

conductor^ we expect that^ at the interface "behind the last reflected

- 1^8 -

29

wave, the total normal flux will satisfy the equation

since neither electric nor mechanical energy may pass through the

surface. In this equation the summation is extended over all the

fluxes 5F . associated with the reflected modes of type j excited

by an incident mode of type i with flux bF^. Combining equations

(5.i|-5) and (3.50) ^^ fi^ obvious manner, we find that

^.SF. .s, .-N + SE.s.-N = 0, (3.51)

or, alternatively, that

S~.p6u^.s. .-N + p5u?s.-N = 0. (3.52)

But with our choice of the normalization factor, we have

2

5E = e

in any given mode. It follows directly from equation (5. 52) that

Y^ e% -1+ ef s -N = , (3-55)

the summation being extended over all reflected waves excited by an

incident wave of type i and of strength I e . I -

'^Formally, equation (5.50) is a special case of equation (3.^) with

both u and U set equal to zero. The analogous relation in electro-

magnetic theory expresses the vanishing of the normal component at

the Poynting vector at an infinitely conducting interface.

- k9 -

With, the aid of equations (j-'^?) - (3.^7) one can easily verify this

relation in the simpler cases where only one ways is reflected. For

example^ in the case of Alfven wave reflection^ equations {'^.hS) and

(3-55) yield

2 2^

since s.. = -s.. The truth of this equation now follows from the fact

^AA "A

that |e..| = le.l [see equation (3.27)]

'AAi ' A'

IV. REFLECTION AND REFRACTION OF N(M4ALLY nrCIDEMT SLOW ATO FAST WAVES

AT A CONTACT DISCONTINUITY .

Suppose that ^ In Figure 1 is a contact-discontinuity surface

separating two stationary polytropic ideal gases. As in Section II-B;

let it be assumed that the entropy in^ is the same as that in "^ so

that only the density is discontinuous acrossotT. When the excitation

is normally incident the reflected and transmitted waves propagate

along N and -E, respectively. Thls^ coupled with the equality of r

pop

and r' [see equation (2.32)] implies that the ratio w = c /h is the

same for all waves of the same type^ reflected or transmitted and has^

in fact, the value

1

w^ = |(l+r) -fi [(l+r)2 _ l,r cos^]^ . (l+.l)

In this equation, the upper or lower sign is intended according as the

wave is slow or fast.

It is assumed below that H is in the plane of incidence and that

L =|: 0, The boundary conditions at (^ are then:

50

fc B^V N = fy 5u'\ N ^ &u^, (1^.2)

(i^.i+)

21 a^Sp = ^ a'^5p . (1^.5)

The summations in the left members are taken over the incident and all

reflected waves while those on the right are, taken over all transmitted

waves. These conditions are simply statements of the continuity of

velocity, magnetic intensity and pressure disturbances across

do not lie in the plane of incidence and, in fact, are not even coplanar.

Nevertheless, Huyghens' construction may he made graphically. One

need only work with the projection on the plane of incidence of the

locus of points on Fresnel 's ray surface at which the normals^ are

parallel to the plane of incidence. Parametric equations of these curves

may be obtained from the ray-equations ( 3.^1-5) -(3- ^7) of Section III

by substituting the expressions for n and H^ given in equations (3-5)

and (3-6).

In graphical work, the method presented here is more accurate

than Huyghens ' method in that a point of intersection is more easily

located than a point of tangency. Moreover, since the angles of reflection

and the (reciprocal) disturbance speeds are determined simultaneously,

it yields more information. The avoidance of the cusped figures

should also be noted. Unquestionably, however, Huyghens ' construction

has more direct physical appeal.

- 52-

III. REFLECTION AT M INFINITELY CONDUCTING RIGID WALL

A. Summary of the Basic Equations

Let tS = J[ i'^) d.enote any propagating planar wave front and

n the unit normal at each point of ^ (t) pointing along the direction

of propagation. A region will be referred to as being in front of

or behind A (t) according as n or -n is directed into that region.

Let Q be the undisturbed value of some physical quantity and Q and

â‚¬L denote the values of this quantity in front of and behind ^(t) .

Here and hereafter, we shall take the state of the medium behind the

wave front as the state in front plus a small perturbation. Letting

&Q represent such a perturbation in the quantity Q , we therefore have

Q^ = SQ + Qq . (3.1)

In a given wave 0^^ may be regarded as the " umperturbed" value of the

physical quantity it represents. It should be stressed, however, that

owing to the presence of other perturbations, Qq might be the sum

of Q and a small perturbation. Whether or not Qq= Q, or Q plus

a small perturbation, it can be shown, on neglecting all but the

highest order terms that 5H, 5u, and 5p as Alfven, slow and fast modes

26

are given by the following formulas :

Alfven: 5H = -|r: Hx n ,

5u = + â€” Hxn , (3.2)

1 pc â€¢Â« K,

5p = 5S = ;

2S

K. 0. Friedrichs, "Nonlinear wave motion in magnetohydrodynamics",

Los Alamos Report No. 2105 (written 1954 - distributed 1957)- See also

later version of this work by K. 0. Friedrichs and H. Kranzer, Report

No. MH-8, AEC Computing and Applied Math. Center, Inst. Math. Sci. ,

New York University, New York (1958).

- 35

5H = - -^ nx (nxH) ,

fE^

6u = +-

T=Vt"') - #.^-'^'-^-'' "â– "

y^ \c^ / '^ ~ yp;

e , \' '^

6p == p \ -5 1

"j^

\c

In these formulas, H, H , i_i; p and S are, in the stated order, the

magnetic intensity, the component of magnetic intensity in the

direction of propagation, the specific magnetic inductive capacity,

the density and the specific entropy of the flow all in the absence

of any distuirbance whatsoever. The c's are the disturbance speeds

appropriate to the mode in question. They are the positive solutions

of the equation:

" '' ^ ^ = 0. (3.1^)

(pc^ - ^^ )

k ,2 â€ž2. 2 2 â€ž 2

pc - (pa + jiH jc + a M-H

Specifically, in equation (3-2), c is the positive root of the first

factor, while in equation (3-3) ^ c is the smallest or largest positive

root of the second factor according as the mode is slow or fast.

With n and H expressed in terms of the angles shown in Figures 1 and 2

I.e.

n = cos N + sin 9 x , (3'5)

-o

H = H cos iV^Xcos 0^^ N + sin 0,^ x )+ H sin \lfâ€žzâ€ž,(3.6)

vÂ«v n H '~' n *-0 ^il~-0'

we obtain for the slow, Alfven and fast disturbance speeds, the explicit

expressions simmiarized in equations (2.^4-) - (2.6). The disturbance

- ^k -

speeds along n are related to the speed of propagation U > of >^(t)

along n and to the trace velocity V as follows:

WW ^v^

Vn = U = u-n + c ; (j.?)

\AV VW

(see Section II-A) . The upper (lower) signs in equations (3.2) and

(5.5) correspond to the upper (lower) signs in equation (5>7)' Note

that there will be only one speed of propagation along n, specifically^

the one corresponding to the upper sign in the above equations, unless

the projection of the fluid velocity u on n exceeds the disturbance

speed G.

The absolute value of the quantity e appearing in (5-2) and

(5.3) measures the strength of the discontinuity at the wave front

of the mode in question. For |e| to be a true measure of this strength

it is necessary to nomialize the mode of propagation in a suitable

.anner. This is the function of the constant factor "j/e^ whose choice

for each mode is dictated by the following considerations: In

reference 1, a mode of propagation was defined to be a six- vector of

the fona

mi

R =

1

J (3.8)

('H. ' 2 5H [V , 2 gH , 5u , ou , 5u , ^

\P y \P z' n-' y' z' p

here, the x-axis is assumed to be directed along n. R represents

the total discontinuity in the medium at a given point of the wave

front. A total of six modes of propagation are obtained by substituting

the appropriate components from equation (3-2) or (3-3) into the

above expression. Now it is natural to define the strength of the

discontinuity to be proportional to (R â€¢ R) , the Euclidean length of R.

35 -

For reasons which will soon be apparent it is convenient to choose

Vp/s as the constant of proportionality. With (-^pR â€¢ R)^ as the

definition of mode- strength, it then follows that E^ must be chosen

to be

E

1 _

Â§ R^ Ri

Â£i^,aM, ,(,â€ž.,2^ (j.g,

II 1 ?

if |e| is to equal (^pR â€¢ R ) . In equation (3.9) R^, &1I, 5pi and

ffl* denote the values of R, 5u, Sp and 5 H obtained by setting â‚¬ = 1

in equations (5. 2) and (3.3). If one actually inserts the expressions

for 5i/. 5p^ and 8H^ in equations (3.2) or (3.3) into the last member

of equation (3-9)^ one finds that

Alf v^n :

1^ = y^ b I sin 6 I ,

(3.10)

Slow

or

Fast

T^ = Vp b Isin sl '

2w - (l+r)

(w^- r)

= yp^ i

w

1

2

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

(3.11)

In these equations 5 is the angle between n^ and H. Employing

equations (3.5) and (5.6) it follows that

WW vvv

|sinS| = Tl-cosiircos (9-9..)

â– H. H

(3.12)

The quantity w is defined by the equation

w =

2 >

(3.13)

- 56 -

it is understood here that c represents slow or fast disturbance speed

according as the mode is slow or fast.

Note that E^ has the dimensions of energy density; E^ may be

thought of as the "energy density of the mode" R^. Aq Interesting

sidelight concerning such energy densities is the following. For modes

R of arbitrary strength [^see equation (5.8)] , one can show^ by means

of equations {3.k) and (5.2) or (5.5) [See Appendix IlJ that

1^2 1 a^Sp^ ^ 1 =,1x2 r-K DA

gPBu = 2 â€” ^â– ^2'^^^ ' (5.14)

so that the energy density of the mode is divided in equal parts between

the kinetic energy density of the mode on the one hand and the sum

of the internal and magnetic energy densities on the other. From

this equation it follows readily that

E = p5u^ , (all modes) (5-15)

|e = |p5u^ = ^m^ (Alfven mode) . (5-16)

Here, E denotes the energy in a mode of arbitrary strength.

B. Reflection at an Infinitely Conducting Rigid Wall

1. The Boundary Conditions

We now imagine that the upper half- space uC in Figure 1

is a rigid infinite conductor and that the unperturbed state of the

fluid in the lower half-space W is constant with the fluid velocity

- 37 -

u = 0. The (planar) incident wave Is assumed to propagate into

the unperturbed fluid and the perturhations in the regions between

the surfaces of discontinuity - i.e.^ the wave fronts and the interface

J^ - are assumed to be constant. At the interface (^, behind the

slowest reflected wave, we have the following boundary conditions

N â€¢ &u* = ,

BC: N â€¢ 5H* = , (3.1?)

N X 5E* = nN X (H X Bu*) = ,

â– X- *

where 5u and 5H represent the total pertiirbation at the interface.

The first condition expresses the rigidity of the wall; the second

the conse2rvation of magnetic flux and the third , the continuity of

the tangential electric field vector across ^. If , as will here and

hereafter be assumed, H is in the plane of incidence, then the boundary

conditions split up into two sets, namely,

N â€¢ 6u^ = ,

BC : N â€¢ 6H* = , (3-l8)

p -Â» ~p '

N X (5u XH) = ,

and

BC^: N^x(5u_L*X H) = . (3.I9)

The first set evidently involves only 6u and 6H , the components of

Su_ and 6H in the plane of incidence, while the second involves Su, ,

the component of u normal to the plane of incidence. Now, as an

- 38 -

inspection of equations (3.2) and (3-3) reveals, ou and 6H are

normal to the plane of incidence when the wave is an Alfven wave

and parallel to the plane of incidence when the wave is slow or

fast. Since the boundary conditions for the normal and planar

components are uncoupled it follows that Alfven incident waves can

produce only Alfven reflected waves and slow or fast incident waves

can produce only slow and fast reflected waves. Assuming for the

moment that the incident wave is slow or fast, we note that there

are two reflected waves and apparently four boundary conditions

in equation (3'l8). It will now be shown that the conditions BC

of equation (3.I8) reduce to

BC 5u = , if H â€¢ N ^ , (3-20)

and to

BC N â€¢ 5u = , if H â€¢ N = . (3-21)

P vw >~p ' ~Â» lÂ«.

First, we observe that equations (3. 20) and (5.21) follow

directly from the first and third boinidary condition of (3.17) â€¢

To complete the proof we must show that N â€¢ 5H vanishes. To this end,

we note the following relations which hold for any propagating wave

[see reference 26J

n^ â€¢ 5H = , (3-22)

c 5H + n X (6uxH) = .

Replacing c by V â€¢ n [see equation (3-7) J note u = o] , we find that

n x(5HxV + SuxH) =0. (3-23)

- 39 -

This relation, in turn, implies that

5HxV + 5uxH = , (5.2i+)

since each vector within the parenthesis is normal to the plane of

incidence when, as we have assumed, H is in the plane of incidence.

One such relation applies to the incident wave and to each of the

reflected waves. If we add these relations we find that

5H* X V X 6u* X H = . (5.25)

But 5u X H vanishes by hypothesis; hence sH x V vanishes, which

implies the desired result, namely N â€¢ 5H* = 0.

2. Strengths of the Reflected Waves

We shall employ the symbol 0. with the single subscript

i, to denote the value of a physical quantity Q in front of an

incident wave of type i and oQ. to denote a perturbation of Q. in

the sense of Section A. Similarly, Q. and gO. . will denote the

corresponding quantities in a reflected wave of type j due to an

incident wave of type i. We shall further allow i and j to take on

the "values" s(for slow), f(for fast) and A(for Alfven) in place of

the numerical values used earlier. Thus, for example, e denotes the

strength of a fast incident wave and e â€ž denotes the strength of a

sf

fast reflected wave produced by a slow incident wave.

Case 1: Alfven wave incident - Assiome first H â€¢ N 4 0- "^^ boundary

conditions for this case are J_see (3.19)_]

Sii^A + 5%^ = > (5.26)

UO

since ^ under the assiomed conditions, only an Alfven wave can be

reflected. Employing equations (5.2), (3.5)^ (3-6) and (3. 10), we

readily conclude that

"AAl

'A'

(3.24)

so that the strength of the reflected wave equals the strength of

the incident wave even when the acute angle of reflection [see equation

(2.3T)j does not equal to the angle of Incidence.

When H'N vanishes, the incident wave causes no additional

MM Â«W ^

distiirbance within the medium, since no boimdary conditions have to

he met [see equation (3.19)J' At the interface behind the wave,

however, we find a surface current density proportional in magnitude

to |&H.|.

Case 2; Slow or fast wave incident - The following formulas are the

starting point of o\ir discussion:

5u

-c

1^

5u = e Su

cos(9-9jt)cos9^

2

w

(3.28)

cos9

W +

cos(9-9â€ž)sinGâ€ž

Jl n

sin9

w

aso

E = 1^ b |sin(9-9^)

2w - (1+r)

2

w - r

= Vp b

(w^-cos^(9-9jj)(2w^- 1-r)

2

w - r

1

2

1

2

(3.29)

w

2/k2

c /b .

1+1 -

These formulas, which apply to both slow and fast waves, follow

directly from (3.5), (5-5)^ (3-6) and (3-11) on setting iir^ = 0.

Let us first consider the special case in which H â€¢ N =

W^ t^V

- i.e., in which 0â€ž = 90Â°- Here, only one boundary condition

n

need be met, namely N â€¢ 5u =0 [see equation (3.2l)]J. Using

equation (3.29), one can easily show that this boundary condition

can be satisfied for all angles of incidence, by a reflected wave

hairing the following properties:

(1) It is of the same type as the incident wave.

(2) It emerges at an (acute) angle of reflection which is

equal to the angle of incidence. Moreover, the relation

e,. = e^ , (3.30)

holds, where i is s or f according as the incident wave is slow or

fast.

Suppose that 4 j:90Â° - i.e., that E which is in the plane

of incidence is not in the plane of the interface. In this case,

we have two boundary conditions from the vector equation (3-20) and

(at least for angles of incidence which are sufficiently small in

absolute value) , two waves for meeting these conditions - one slow

and one fast. Employing equation (3.29), we may express the boundary

condition, behind the slow reflected wave, as

e. 6u. + e. â€ž 6ufâ€ž + â‚¬.8u. = , (5.51)

IS "^is if -â– if 1-1 '

where i is f or s according as the incident wave is fast or slow.

- 1+2

From this relation It follows directly that

is -"is "^if i ~a -if '

â‚¬.^ 6ufâ€ž X 5uf = -e. 5uf- X &uf- ,

if ~if - IS 1 -i - is '

which in turn implies that

(Suf X 5uf^ â€¢ M)

-1 -if -

(3.52)

^is =

s

^^^Is^'^'^tf

â€¢ M)

"if =

h

(SH|- X Su^f^

â€¢ M)

I'o.. â– Â» , , 0.. t_

.Â«N

(3.55)

(5uf- X 5u?-â€ž â€¢ M) '

^ -IS ~if *-'

Here, M is the unit vector normal to the plane of incidence. Since,

for fixed r and 9 ^ 90 ; numerical analysis shows that the numerators

and denominators in (3- 55) vanish at isolated values of 9., if at all,

one may conclude that multiple reflection is the rule and single

reflection is the exception. Observe that the vanishing of numerators

or denominators in (3' 35) means that the velocity vectors involved

in these expressions are parallel.

Combining equations (5-29) and (3- 3) one obtains, with the aid

of standard trigonometric identities, the following explicit formulas

for the e . . ' s :

is 1 f ' sf '

(3.5^)

^if = ^^i^s'/^if

it3

Here, D is given by

D

sf

V. w, _(Ef Ef_)2

IS if^ IS if

â€¢ v.Jv. -l)cos(9. -9â€ž)sin(0. â€ž-9â€ž)

if^ IS ^ ^ IS H' ^ if H'

+ w^3(l-wJf)cos(0.^-9jj)sin(O.^-9j^) \ j (5.35)

the corresponding expressions for D (d ) may be obtained from

S 1

this equation simply by eliminating the subscript f(s) in all terms

where it appears as a second subscript. The values of v. , w. . and

9. . appearing in the D's may be obtained by the graphical procedure

-'-J

described in the earlier sections, and E. , E^ etc. may be calculated

1 is

on the basis of these results. In this fashion one can obtain a

complete quantitative solution of the problem. Without resorting to

numerical or graphical methods, however, it is still possible to

arrive at some idea of the content of formulas (j.jJl) and (5.35) l^y

choosing special angles of incidence or special orientations of the

magnetic field, or both. We shall examine several such special cases

now.

Let us begin by assuming that Q. ^ and 9^ = - i.e. , that

the incidence is oblique and that H is normal to the plane of the

interface. In this case, one might suppose that the boimdary conditions

can be met by a single reflected wave emerging at an acute angle of

reflection equal to the angle of incidence. Assuming that a slow

wave is incident, this would imply [according to equation (3'53)j that

- kh

e â€ž vanishes . However^ on substituting Â© =n-0 or-rt+0

SI ss s s

[according as 9 is positive or negative] into the expression

for e , one finds that the numerator will in general not vanish}

the vectors &u' and 6u' in eqviation (5.52) are evidently not

parallel. Here too then, double refraction is the rule.

Assume now that 0. = and 0^^ 4 - i.e., that the incidence

is normal but the magnetic field is oblique. Setting 0. =

0.^ =0. = l80 in equation (5'53) and observing that v.. = w,

if IS ^ ^ ' '=' 11 1

and E. . = E. we find that

li 1

6. = e. 5^ 1 = s.f

IS 1 is

^If = h \f ^ = ^'^

(5.56)

where &. . is the Kronecker delta. Thus, when H is oblique, a normally

incident wave produces a single reflected wave of the same type and

strength as the incident wave.

Suppose now that 0^^ =)= 0. Let the angle of incidence approach

as a limit. Assume first that the incident wave is slow. It then

H

follows, on taking limits in equations (3.5)^ that the resulting slow

incident wave reduces to an ordinary sound wave when r = a /b is less

than unity. The same is true of a fast incident wave when r > 1.

When r = 1, both the limiting slow and fast waves move with the speed

of sound, but the associated 6u Â»s and SH.'s have non-vanishing

components tangent to the wave front. However, a suitable linear

combination of these waves furnishes a so\ind wave in this case also.

i^5 -

Thus for all values of r^ one can assume that the incident wave along

H is a so\md vave. The point we wish to emphasize is the following:

A sound wave incident along H, can produce a pair of geniilne hydro-

magnetic waves. It is necessary of course to assume here that the

angle of incidence is small enough^ when the incident wave is slow^

in order that the fast angle of reflection he real. To obtain the

actual strengths of the reflected waves ^ one simply lets 9 approach

0^ in the numerators of the right member of equation (5* 5^)*

5. Energy Relations

Let u^ p and H be time-independent and constant over a

given region of space - q , say - and let u ' ^ p ' and H ' denote

continuous,, continuously differentiable perturbations of these

quantities in U â€¢ On multiplying the linearized hydromagnetic momentum

equation by u'- and combining the result with the remaining linearized

equations one obtains, as in gas dynamics, an "energy" equation,

namely,

||- + V â€¢ F' = , (5.37)

which expresses the relationship between the "perturbation energy"

per unit volimie

2

E' = I p (u'2 +^ p.2 + ^jj,2^ ^ (3_38)

and the "flux of energy"

- kS -

27

F' = a%'u' + [i(Hxu') XH' + E'u . (5.39)

Corresponding to equation (5' 37)^ we have the jump relation

-USE + SF-n = , ' (3A0)

associated with a discontinuity at a wavefront moving at each point

with the speed U > in the normal direction n. 5E and 5F here denote,

in general, the difference in the energies and fluxes, respectively,

in front of and behind the wavefront. Since we always regard a

disturbance behind a wavefront as a perturbation of the state ahead

28

we conclude, using equations (j.jS) and (5- 39)^ that

5E = "I p (6u^ + a^Sp^/p + laSH^) , (3-^1)

6F = a^SpSu + |i(HX5u) X 5H + SEu , (3-^2)

[see equation (3.l)J.

In Appendix II, It will be shown that

6F = 5E s j = s,A,f , (3A5)

27

_F' in the present theory plays a role analogous to that of the Poynting

vector of electromagnetic theory. The second term in equation (3- 39)^

namely |j.(Hxu') XH' = ExH, is in fact the Poynting vector.

The quantities a,p^H and u^ in these expressions will, in general,

differ from the values of those in the totally undisturbed flow by

first order perturbation terms. However, it is clear that we may

ignore the contribution of these first order perturbations to a2,p,

H, and u, since they contribute third order terms to 6E and 5F, which

are themselves of the second order.

- hi

Here^

J J

and s is the ray vector associated with the given mode of propagation.

Specifically^ in slow waves

b^r(H /H^)

s = JiiÂ±^.aÂ± ^ (H-Hn) , (5.^5)

â€”s Â»- â€” s-" â€” . -^ ' n'

c

s

in Alfven waves

Sa = JiÂ± (^^/P)^ sgnCH )H , (5.1^6)

and in fast waves

_ l3^r(H /h2)

Sf = iiÂ±c^ + i (H-H^n) . (3.1+7)

c^C

In these equations

,2

(1+r)^ - l+r(H â€¢ nf/E^

.1

2

(5.H5)

and sgn(x) is the sign of x. For all ray vectors s we find that

s-n = u+c = U. (5-^9)

v~ w. n â€”

As is the case with propagation of electromagnetic waves in crystalline

media^ hydromagnetic energy is guided along rays which are not in

general directed along the normal to the wave front [see reference 1,

Section B-5] .

Turning now to the problem of reflection at a rigid infinite

conductor^ we expect that^ at the interface "behind the last reflected

- 1^8 -

29

wave, the total normal flux will satisfy the equation

since neither electric nor mechanical energy may pass through the

surface. In this equation the summation is extended over all the

fluxes 5F . associated with the reflected modes of type j excited

by an incident mode of type i with flux bF^. Combining equations

(5.i|-5) and (3.50) ^^ fi^ obvious manner, we find that

^.SF. .s, .-N + SE.s.-N = 0, (3.51)

or, alternatively, that

S~.p6u^.s. .-N + p5u?s.-N = 0. (3.52)

But with our choice of the normalization factor, we have

2

5E = e

in any given mode. It follows directly from equation (5. 52) that

Y^ e% -1+ ef s -N = , (3-55)

the summation being extended over all reflected waves excited by an

incident wave of type i and of strength I e . I -

'^Formally, equation (5.50) is a special case of equation (3.^) with

both u and U set equal to zero. The analogous relation in electro-

magnetic theory expresses the vanishing of the normal component at

the Poynting vector at an infinitely conducting interface.

- k9 -

With, the aid of equations (j-'^?) - (3.^7) one can easily verify this

relation in the simpler cases where only one ways is reflected. For

example^ in the case of Alfven wave reflection^ equations {'^.hS) and

(3-55) yield

2 2^

since s.. = -s.. The truth of this equation now follows from the fact

^AA "A

that |e..| = le.l [see equation (3.27)]

'AAi ' A'

IV. REFLECTION AND REFRACTION OF N(M4ALLY nrCIDEMT SLOW ATO FAST WAVES

AT A CONTACT DISCONTINUITY .

Suppose that ^ In Figure 1 is a contact-discontinuity surface

separating two stationary polytropic ideal gases. As in Section II-B;

let it be assumed that the entropy in^ is the same as that in "^ so

that only the density is discontinuous acrossotT. When the excitation

is normally incident the reflected and transmitted waves propagate

along N and -E, respectively. Thls^ coupled with the equality of r

pop

and r' [see equation (2.32)] implies that the ratio w = c /h is the

same for all waves of the same type^ reflected or transmitted and has^

in fact, the value

1

w^ = |(l+r) -fi [(l+r)2 _ l,r cos^]^ . (l+.l)

In this equation, the upper or lower sign is intended according as the

wave is slow or fast.

It is assumed below that H is in the plane of incidence and that

L =|: 0, The boundary conditions at (^ are then:

50

fc B^V N = fy 5u'\ N ^ &u^, (1^.2)

(i^.i+)

21 a^Sp = ^ a'^5p . (1^.5)

The summations in the left members are taken over the incident and all

reflected waves while those on the right are, taken over all transmitted

waves. These conditions are simply statements of the continuity of

velocity, magnetic intensity and pressure disturbances across

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