On the propagation of waves in an electromagnetic elastic solid

Int. J. &gng. Sci. Vol. 1, pp. 461-495. Pergamon Press 1963. Printed in Great Britain.

ON THE PROPAGATION AN ELECTROMAGNETIC

OF WAVES IN ELASTIC SOLID

J. W. DUNKIN Jersey Production

Research Co., Tulsa, Oklahoma, U.S.A.

and A. C. EIUNGEN Purdue University, Lafayette, Indiana, U.S.A.

Abstract-The coupling of electromagnetic and elastic waves elasticity and a linearized electromagnetic theory. The problem magnetostatic field is considered and couplings of the waves problem for a uniform electrostatic field shows that the usual

is considered from the standpoint of linear of plane waves traveling through a uniform are studied. An investigation of the same

plane waveS propagate without any change in their phase velocities but that the mechanical waves are accompanied by small fluctuating electromagnetic fields. The problem of the vibration of a free infinite elastic plate in a large magnetostatic field is examined under the assumption that the resulting electromagnetic fields are quasistationary. Frequency equations are obtained for both symmetric and antisymmetric vibrations and the damping caused by the field for both the first two symmetric and antisymmetric modes is obtained as a linear correction to the usual free plate frequencies. 1. INTRODUCTION

DURINGthe last decade the subject of magnetohydrodynamics has been rapidly developed and applied to numerous practical situations. Stimulated by the success of this merger of electromagnetic theory and fluid mechanics, more recently, interest has grown in the area of electromagnetic-elastic solid interactions. The purpose of this study is to investigate some of the dynamic interactions that can occur between electromagnetic and elastic fields in a homogeneous, isotropic solid. The

theory is essentially a combination of infinitesimal elasticity and a linearized electromagnetic theory. The interaction effects considered consist of the Lorentz body force, the modification of the electromagnetic constitutive equations and boundary conditions by the velocity of the material, and the surface tractions introduced by the fields. In Section 2 we present the basic equations which are derived from the conservation laws which apply to the theories of electromagnetism and linear elasticity in combination. The conservation laws in integral form are applied to appropriate boundary regions in order to deduce the boundary conditions which the electric, magnetic, and stress fields must satisfy. The electromagnetic theory used here follows that presented by Sommerfeld [I] and Msller [2] while the linear elasticity is typical of most books covering the subject, e.g. Sokolnikoff [3]. For a more sophisticated treatment of elasticity, electromagnetism, and other continuum theories the reader is referred to Eringen [4] and Truesdell and Toupin [5]. Static electroelastic theories considering finite deformations and constitutive interactions were presented by Toupin [6] and Eringen [7]. More recently Toupin [8] has published a dynamic theory with the same considerations. 461

462

J. W. DUNKIN and A. C. ERINGEN

The effect of a large, uniform, static magnetic field on the propagation of plane waves is considered in Section 3. Here the equations are linearized around the large magnetic field and frequency equations are obtained for harmonic plane waves. Special cases of infinite conductivity, the quasistatic electrical state, and low conductivity are considered. There is a considerable amount of literature available that treats the propagations of mechanical waves through a large magnetic field. In 1955 Knopoff [9] considered this problem from the standpoint of the effect of the earth’s magnetic field on elastic waves in the conducting core of the earth. He first treats the case of plane waves through a uniform, static, magnetic field and discusses the various couplings that can occur in imperfectly and perfectly conduction media. Subsequently the cases of a semi-infinite magnetic field and an infinite strip of uniform field in an infinite medium are considered. He concludes that the effects studied here are negligible in the earth’s core. Bafios [IO] has investigated the effect of the angle between the field and the direction of propagation on the coupled plane waves resulting because of a uniform, static, magnetic field. He also describes the dispersive effect of the field on shear waves propagated at an angle to the field. Both Knopoff and Baiios show how the waves in perfect conductors reduce to Alfvkn waves as the shear modulus tends to zero. A paper by Chadwick [I I] deals with this same problem but uses a slightly different mathematical technique. More recently, in a series of articles, Kaliski [12]--[I61has considered various mechanical problems coupled to electromagnetic effects through a large magnetic field. These include the motion of isotropic and anisotropic conductors in a magnetic field, Cauchy’s problem, and Rayleigh waves for certain limiting cases of the large magnetic field theory. In a brief note on the propagation of longitudinal and shear waves at an angle to a uniform, static, magnetic field, Rodriguez [17] mentions confirming experimental work by Galkin and Koroliuk [18] and by G. A. Alers. Buchwald and Davis [19] use a Fourier integral method due to Lighthill to estimate displacements at large distances from a harmonic point source in an isotropic elastic medium with infinite conductivity, subject to a uniform magnetic field. The effect of a large, uniform, static, electric field on the propagation of plane waves is treated in Section 4 in a manner similar to that of Section 3. In an effort to be more general a non-zero conductivity is allowed which implies a steady, uniform current. Such a current generates an axially-symmetric, magnetic field which increases with distance away from the axis. This effect requires that the plane waves be examined locally rather than on a gross scale in order to ignore the non-uniform magnetic field. Under these restrictions we find that, unlike the large magnetic field which both modifies the mechanical wave speeds and introduces a damping, the large electric field simply introduces extremely small, fluctuating, electrical fields which ride along with the mechanical waves. Plane electromagnetic waves are unaffected by the large electrical field. Finally in Section 5 we tackle the boundary value problem concerned with the damped motion of an infinite plate in a large magnetic field. Here we use the simplified quasistatic theory discussed in Section 3. All electric and magnetic fields can be properly extended into the surrounding space except the normal electric field which requires that one boundary condition be satisfied only in an averaged sense. The damping of the first two symmetric and antisymmetic mechanical modes is obtained. The damping of the first anti-symmetric mode is compared with that of a simple beam with eddy current damping.

463

On the propagation of wave in an ele&ornagneticelasticsolid 2. FIELD

EQUATIONS, CONSTITUTIVE RELATIONS, BOUNDARY CONDITIONS

AND

In the following analysis the electromagnetic elastic material will be treated from the standpoint of linear elasticity coupled to the electrodynamics of a moving medium by the motion of the material. The medium is assumed to be homogeneous and isotropic in the initial state of rest. Furthermore no interaction effects which might arise through couplings in the constitutive equations, except for velocity effects, are to be considered. Electromagnetic theory The electromagnetic

vectors E, H, D and B satisfy Maxwell’s equations (c.f. [ll, f2]) curl E+$=

div B = 0

0,

(2.1)

aD curlH-z=J,

div D = pe

where J is the current density, p, is the charge density, and all quantities are expressed in MKS units. When the material is at rest the electromagnetic constitutive equations of a homogeneous isotropic medium are D”=eEo,

B’=IcH’,

J”=aEo

(2.2)

where E and x are constants called the electric and ma~etic permeabilities and c is the constant electrical conductivity. These same equations are assumed to hold at each point in the reference frame moving with the velocity of the material point, i.e. the proper frame, but they are expressed in terms of the fields measured in the laboratory frame in which the motion is observed. For small velocities the proper quantities are related to the laboratory quantities by ([l], equations (34.3), (34.6) and (34.6a)) E” = E+vxB,

Do = D+e-?xH

H’=Ii-VXD,

B” = B-c-%xE

Jo = J-p,v,

PZ=Pe,

(2.3) c = I/&O%)

where KOand so are the free space permeabiities. If these relations are substituted into (2.2) and terms of the order ~559 and higher are dropped, there results D=eE+avxH,

CI= fc&-fc&

B=icH-ervxE,

J-p,v=

o(E+vxB).

(2.4)

For more of the details of electromagnetic theory, the reader is referred to the many textbooks that treat this seld, e.g. Sommerfeld [l], Msller [2], Stratton [ZO],etc. The ele~roma~etic boundary conditions are obtained in the following manner. First rewrite (2.1) in an equivalent form curl (E-t-vxB)

c= -

aB -;j;+vdivB-curlvxB >

curl(H-VXD)

=

g+vdivD-curIvxD

+J-p,v >

divB=0,

divD=p,.

(2.5)

J. W. DUNKINand A. C. ERINGEN

464

Then integral forms of these equations can be obtained by integrating the first two over a surface S’ composed of material particles and bounded by a curve C and the third and fourth over a volume V of material particles bounded by the surface S. Note that C, S’, S, and Vactually move with the material. After using Stokes’ theorem on the left hand sides of the first two of (2.5) and the Green-Gauss theorem on the third and fourth, we obtain

s C

(E+vxB).dC=

s

(H-vxD)*dC s C

B.dS’

-;

= ~~s~.dS’+~s(~-~=v).dS’

(2.6)

p.dV Y where we have also used the relation ([4] e.g. 20.6), +vdivA-curl(vx

]

A) ds’.

Select S’ to be a small rectangular area oriented perpendicular to the surface of the body such that one side lies inside the body and one outside (Fig. 1).

Fm. 1. The boundary and small area and volume elemats.

As the dimension of S’ perpendicular to the boundary tends to zero the first two of (2.6) become [E+vxBJ, = 0 [H-Y x II& = Js,-p:v,

(2.7)

where [A& means the difference in the values of the tangential component of A outside and inside the medium and .I; and p; represent the surface current and charge, respectively.

On the propagation

of waves in an electromagnetic

elastic solid

465

Here and afterwards subscripts t, m, and n will denote components of vectors in the directions t, m, and n which form a right handed orthogonal triad (Fig. 1). Now choose Y to be a small cylindrical volume whose axis is perpendicular to the surface of the body such that one of the circular ends lies inside the body and one outside (Fig. 1). As the axial dimension tends to zero the last two equations of (2.6) become IBan = 0,

PIln = P:

(2.8)

where [A], means the jump in the normal component of A. However, the conservation of charge applied to the region V gives the following normal boundary condition:

(2.9 Equations (2.7)-(2.9) constitute the complete electromagnetic boundary conditions. It is important to note that the portions of C and S which lie outside the body are assumed to have the velocity of the surface at that point .* Of course if the boundary joins two different materials, there is no problem since the velocity will be continuous across the boundary. Hastic equations The mechanical equations will be derived by applying the conservation of momentum to a volume of material, V, with bounding surface S, using the assumption that the only mechanical effect of the electromagnetic fields is the introduction of the Lorentz body force f’= p,E+JxB.

(2.10)

Thus the equation of global conservation of momentum in rectangular co-ordinates is (2.11) where oes is the stress tensor, fz is the mechanical body force, and e is the momentum per unit volume. Using the Green-Gauss theorem for the surface integral and the differentiation of the volume integral according to {[4], equation (20.9)}, (2.12) we get dV=O.

If mechanical momentum is conserved locally then (2.13) * This approach is contrary to Sommerfeld’s procedure ([l], p. 28), but it agrees with that of Mnlkr [2]. Note that if Sommerfeld’s idea is accepted results are obtained for the vacuum-material boundary which contradict (2.7).

466

On the propagation of waves in an electromagnetic elastic solid

In an elastic solid g (2.13) to

= pv, and the assumption of infinitesimal strains and rotations reduces

(2.14) The elastic boundary conditions on stress are obtained by applying (2.11) to appropriate differential elements. First, however, the body forces: will be replaced by (2.15) where T& are the Maxwell stresses and g: is the electromagnetic momentum according to Minkowski, T; = E,DBfH,BB-3(E,D~fH,B~)6,8 g’= DxB.

(2.16)

The volume integral containing fz can now be written as

s s f,dV Y

where

;

(&)

=

S

(T; +g:QnrdS

-

was added and subtracted and the Green-Gauss

convert thl volume integral to a surface integral. we obtain

theorem was used to

Using this expression and (2.12) in (2.11)

(2.17) which is a form that is appropriate for obtaining the boundary conditions on surface tractions. Now let S and V be the surface and volume of a small cylindrical element whose axis is perpendicular to the surface of the body such that one end of the cylinder lies inside and one outside the body (Fig. 1). If the boundary separates a material from vacuum then the velocity of the outer portion must be taken as the surface velocity just as before. Applying (2.17) to this cylindrical region and allowing the axial dimension to approach zero, since the fr is assumed to remain bounded, (2.17) becomes

II%8+ T’$+g3&~ = 0

(2.18)

where once again the double bracket represents the difference in the values of the quantity on = 0 outside the both sides of the surface. In the case of a body surrounded by a vacuum CT.@ body and (2.18) reduces to C@P = [Tip +g:v,]np , on S

(2.19)

but the square bracket must be interpreted as the value of the quantity outside minus that inside the body.

467

On the propagation of waves in an electromagnetic elastic solid

The mechanical constitutive equations are taken to be the usual Hook’s law for a homogeneous, isotropic elastic solid, i.e. (2.20) where A and p are the Lame constants and U, is the displacement vector. In using these relations it is assumed that the stresses and strains for this combined system in the proper and laboratory frames are the same. Due to the fact that the system has been split into two parts, a mechanical part and the electromagnetic part as expressed by the Minkowski energy-momentum tensor, this question needs further consideration. For present purposes we simply assume that constitutive equations (2.20) for the purely elastic body are unaffected by the presence of the electromagnetic fields. For very large fields or finite deformations interaction terms will enter in the constitutive relations thereby coupling together the elastic and electromagnetic constitutive equations. * Finally, it must be remarked that in the examples to be considered later, large fields enter but their effect on the constitutive relations is neglected, attention being confined to the body force interaction. Summary of equations and boundary conditions Here we summarize the basic field equations and boundary conditions that apply to an electromagnetic elastic body in a vacuum. Field equations curl E+E curl H--

aD

at

= 0,

div B=O

= J,

div D = pc

(2.21)

Constitutive equations D=eE+crvxH,

cf = KE-ic()&O

B= rcH-CWXE,

J-p,v=a(E+vxB)

(2.22)

Boundary conditions

[H-vxD], BD], = P: .

[E+vxB],=O,

EB]. = 0

= J$-~;u~,

o(E+vxB),

=@r = &DP+HJ$-~&D~

* c.f. Touph [a] and Ehgen

ad

(2.23)

= at

+WhV,+@

[‘I] for the elastic solid subject to electrostatic field.

x

W.U&Q.

J. W. D-

468

and A. C. ERIN~XN

3. PLANE WAVES IN AN INFINITE MEDIUM IN THE PRESENCE OF A LARGE MAGNETIC FIELD

The linearized equations

In this section we investigate the propagation of plane, steady state, electromagnetic elastic waves through a large magnetic field in an infinite, homogeneous, isotropic medium. The starting point is the set of equations (2.21)-(2.23) which are linearized assuming that the magnetic field is composed of a large static field plus a small fluctuating field, while the electrical field and displacements consist solely of small fluctuating quantities. Solutions will be sought which have the character of plane waves traveling in the xl-direction of a right-handed rectangular frame of reference x,. A large, static, uniform magnetic field is assumed to be present in the form Ha = (HI, Hz, 0). The choice of a zero x3-component of the magnetic field is no restriction on the problem since the selected Ho has components both parallel and perpendicular to the direction of wave motion, see Fig. 2.

FIG. 2. Initial magnetic field and wave velocity.

The magnetic and electric fields may be expressed in the form H=&+h,

E=O+e

(3.1)

where e and h are the fluctuating fields and are assumed to be of the same order of magnitude as the particle displacement II. Equations (2.21) and (2.22) are linearized by inserting (3.1) and dropping products of e,, h,, u,, and their derivatives. Eliminating a+ D, B, and J from (2.21) and remembering that Ho is a constant vector, the following linearized equations are obtained : curle+tcE=O, +~u,~~+ (~+&?,L?.

curl h-s

~-CY $

xH,=o(e+rc$xH,)

(32)

1

=p (I:

a%, aP

’

469

On the propagation of wavesin an electromagneticelasticsolid D=se+a$xH,.

J=u(e+r$xl-q

B = rcH9 +I&,,

div b = 0

p,=sdiv

(3.3)

e+a div

Since solutions are being sought in the form of steady state plane waves moving in the a-direction, e, h, and u must be functions of x1 and t only and can be written as e, = N e,e ~(k-or),

jr, = i;hei(kxI-Pt), u, = N u,e i(kx,

-cd)

(3.4)

where the complex amplitude constantsTa, &, and rZ are to be detert$ned from the following linear equations obtained by introducing (3.4) into (3.2). We obtain hr = 0 and the following equations :

-ik

iua,

0

0

i;

ik

Pd

-V’,-POWI

0

ii

0

--K6H1

P, + irc2aoH~

m3H,

;

0

0

P,,

;

(P, -

PcW2

= 0

_I

I

im

fk

-ik

Pd

I I

&

0

0 (P, -P,)H,

G

-We-P&32

I

I

0

I

0

P, + irc20mHf

UUH,

-

I

- ix200H,H2

- irc2aoH, H,

I

I-

I,

==O

u2

-

I

I

UUH,

(3.5)

_ -

I

P, + iK2awH~

Ui

-

_

where we have defined P, spcu’-@+2&k’, P, s um2 f imco - k2,

P, zpco’-pk’, P,=c-iiso P, E rcO~,,w2- k2.

(3.6)

The equations have been rearranged so that the uncoupled frequency factors, which are enclosed in boxes, appear on the diagonals. Reading from upper left to lower right we find the characteristic equations associated with the x2-polarized damped electromagnetic wave, the +-polarized shear wave, the decay of er, the +-polarized damped electromagnetic wave, the x2-polarized shear wave, and the dilatational wave. Note that if either B or both HI and Hz are zero all waves are uncoupled.

470

J. W. DUNKIN and A. C. ERINGEN

The coefficients outside the boxes in (3.5) can be considered to be coupling quantities. Thus we notice that hs, e2, 243,and er are in no way coupled to hi, es, ~2, and ul. The x,-polarized shear wave is seen to be coupled to the x2-polarized electromagnetic wave by the Ht field and to the decaying et field by the Hz field. Furthermore, the Ht field alone couples the x3-polarized electromagnetic wave to the x2-polarized shear wave and the Hz field alone couples the xs-polarized electromagnetic wave to the dilatational wave while the HI and H2 fields acting at the same time couple all these waves together. The necessary condition for a nontrivial solution of (3.5) is the vanishing of the determinants of the coefficient matrices. This gives the characteristic equation that must be satisfied by w, i.e. (P,P,P, +

iic2aoH:P,P, +lccrH$c,&oo2P,)

*

+ i~~cxoH:P,P,

(P,P,P,

= 0.

(3.7)

(H, = 0)

(3.8)

(H2 = 0).

(3.9)

+ irc200H;P2P,)

Two special cases HI = 0 and H2 = 0, reduce this to P,P,(P,P,

+mH$cos,02)(P,P,

P,P,(P,P,

+ iK2adlfP,)2

+ irc2aoH~P,) = 0,

= 0,

It is interesting to try to predict the couplings that should be present on the basis of voltages that would be induced by the particle motion and on the basis of the Lorentz force caused by currents. For example (3.8) and (3.5) indicate that the u2 shear wave is not coupled to any other wave when only H2 is present. This seems reasonable since the u2 particle motion does not cut any flux lines while at the same time the currents associated with the xz-polarized electromagnetic wave do not contribute to the Lorentz force since they are parallel to the flux lines. The second factor in (3.8) which gives the coupling of the us shear wave to the decaying er field, can be argued from the standpoint that the motion in the x3-direction cuts the x2 flux lines causing a sinusoidal variation in the x1 direction of the el field. On the other hand a fluctuating el field generates currents which together with the x2 flux causes forces to be produced in the x3 direction. Hence the possibility of interaction exists. All the remaining couplings or lack of coupling can be argued similarly. The general solution to (3.5) can be obtained in the following way. First observe that the first factor in (3.7) is of fifth degree in o and fourth degree in k. Hence we would expect to find five different values of o/k which satisfy (3.7) and are associated with hs, e2, us, and et. Designating the values of o and k which make the first factor of (3.7) zero by w, and ki we can write the solutions for h3, ez. 24, and el in the form u3 =

Mi,

e2 =

cd,

h, = Bdi 9

e, = &#I~

(3.10)

where

4, = ei(kmr -WI)

(3.11)

and i takes the values 1, 2, . . .. 5. The second factor in (3.7) can be treated in the same way to give

On the

propagation of wavesin an electromagnetic elastic solid ~1

=A;cb;,

h2

=

qP;,

471

B;& e3 = OS&.

(3.12)

-0Jt)

(3.13)

u2 =

where 4; = ,+x1

ki and are values of k and o which make the second factor of (3.7) zero, and j takes on values 1,2, . . .. 6. In (3.10) only five of the coefficients are arbitrary while in (3.12) only six are arbitrary. Let us choose A, and Aj as the arbitrary coefficients and express the remaining ones in terms of these by means of equations (3.5). Since there is an independent solution associated with each of the 4, and (Pj, we construct the characteristic solutions separately, i.e. for each i or j separately. Thus using the first, second, and fourth of (3.5) and the fifth, sixth, and seventh of (3.5) we obtain B,, Ci, and Di and Bj, Cj, and Dj by solving two sets of simultaneous equations. The solutions of these equations are aJ

B, = A,[ik(l - P,P,- r)HI] , Ci = Ai[i~w(l -P,Pl’)H,] Di = Ai[(P, - P,)P,- ‘H,]

3

(3.14)

evaluated at w = Wi, k = ki, and C$ = A;I:-ik(P,-P,)H2P2P-‘], B; =i A; [~u~~&~H~P,P-‘], DJ = AJ [im@,-P,)H,P,P-“1 ,

evaluated at w = o;,

(3.15)

k = kj , where P s P,P, + iu2aoHfP0.

The wave composition would be completely determined if values of wi/ki and wj,fkj (i and j not summed) were known from (3.7). These will not be obtained here for the general case but instead we shall proceed to study the special limiting case of a perfect conductor. Perjtctly conducting solids Let us examine the limit of the solution as CT+co. Characteristic equation (3.7) tends to (Pz +IcH&

+KH,ZIC,E,~~)(P,P~ +KH:P,P,

+icH;P,P,)

= 0.

(3.16)

The first factor once again is associated with (3.10) while the second gives values of o/k to be used in (3.12). Since rcH$,u and KH$~ will be small for most materials, it is apparent that waves which satisfy (3.16) will have speeds more Eke rn~hani~ waves than electromagnetic waves. Therefore we have PO = ~g@.+--k2 = -k2 and (3.16) becomes (P2 - rcHfk2)(P1P2 - IcH;P, k2 - KHfP, k2) = 0 or, in~~uc~g be written

(3.17)

the no~tion 5 = co/k, Vf a rcfiflp, Vz s ~~~~~~and V2 zs Vf+ V:, it can

Ce’44

-l-W][(C’ -

c:)g2- 4) - w2 -

c:,- vg2

- c$)] =

0*

(3.18)

J. W.

412

DUNKIN

and A. C.

EIUNCXN

The tirst factor has the solution (3.19) which applies to the x3-polarized shear wave. The second factor has the solution

fd v~+(c~“::)2 1*

t2 =3(c:+c:+v2>~,3(c:-c22) 1+2

(3.20)

For solids V/(c:---c$) is generally small so that we get approximate values t =

fJ(+tm,

5 = IkJ
(3.21)

These values of r are the speeds of modified dilatational and x2-polarized shear waves, respectively. All of these waves exhibit neither damping nor dispersion. Note that if ,u is set equal to zero, then the xs-polarized shear wave has a velocity equal to VI. This would be the Alfv& wave of fluids. Similarly, if Hz and ,u are set equal to zero then the x2-polarized shear wave also takes on the characteristics of the Alfven wave. On the other hand, if HI and L+2p are set equal to zero then a longitudinal AlfvCn wave exists whose speed is Vz. Thus we conclude that whereas in a perfectly conducting fluid the magnetic field makes it possible for a new wave to be exhibited, in a perfectly conducting solid the field simply modifies the two known elastic waves. However, there ale fluctuating electric and magnetic fields which accompany the mechanical waves in solids. They are obtained by taking the limit of (3.14) and (3.15) as o-+co, setting PO = -k2, and inserting the expressions so obtained into (3.10) and (3.12) respectively. The limiting forms of (3.14) and (3.15) are Bi = A,ik$l,

wherei=

,

Ci = A,ircw,H, ,

(3.22)

Dr = A,( - ilco,H,)

1,2and Bj = A; [ - rcHI H2k2(P2 - 7cHfk2)-1], D; = A; [iicoP,(P,

C; = A; [ - ikP,(P, -~i!Z;k~)-~]

-

fcHfk2)-l] ,

c3 23j

evaluated at o = CI$,k = k; (j = 1,2,3,4). Since there was a reduction in the order of the characteristic equation, the number of roots has been reduced also. The expression for B; given in (3.23) is valid only if both HI and H2 are not zero. If either is zero, the two mechanical waves having displacements ur and ~2 are uncoupled and the coefficients B; are independent of A’. It must be remarked that for perfectly conducting metals the modifications of the wave velocities due to the magnetic field are small. For example, a value of Vi = ,/(K/~)H~ or V2 = ,/(~/p)H2 for a large field in a paramagnetic material might by lo-50 m/s while ct and ca would be of the order 104 m/s. Thus rather careful experiments would be required to detect the difference in wave velocity and effects such as magnetostriction might be more important at the large fields necessary to demonstrate Vr and Vz. For solids with very small mechanical wave velocities, however, this effect may become appreciable.

On the propa~tioR of waves in an ei~t~~a~etic

473

elastic so&i

Quasistatic electricaljie&; dynamic mechanical behaviour The frequencies associated with mechanical vibrations or waves are much smaller than those associated with electromagnetic waves of the same wave length. Thus when dealing with dynamic mechanical problems it may be expedient to assume that a quasistatic electromagnetic state exists. Mathematically this is accomplished by dropping aD,Qt from Maxwell’s equations while retaining all other time derivatives. The second of equations (2.21) under this assumption is changed to (3.24)

curl H= J. The linearized equations equivalent to (3.2) are

and (3.3) is unchanged. The appropriate equations in this case can be obtained by setting a = E = se = 0 in the general linearized equations. This is equivalent to setting P,,=u,

P,=iouw-k2,

P,=

-k2.

(3.26)

Equations (3.5) reduces to iK0

0

0

- irruwHi,

0

-ik d

ik 0

-K(THI

P, + iu%wH;

0

0

&OH2

-0

lcaH, CT -

iuw

ik

-ik

d

iaKwHl

- iaxwH,

G

0

lcaH,

P, + irc2awHf

- iu20wH,H2

u;

0

KUH,

- ~K~~wH~H~

P, + iu2awHz

I_ -

1

0

0

(3.27)

-_

G

-I

I-

-0

c

and characteristic equation (3,7) now becomes [P2(k2- ia~~w)+ik~rc~awH:][P,P,(k~--ianw)+ik2ic20w(H:P,

+H;P,)]

= 0.

(3.28)

The first factor pertains to a wave which is essentially an xs-polarized shear wave accompanied by fluctuating er, e2, and h3 fields. The second factor pertains to slightly modified dilatational and +polarized shear waves which are coupled together and are accompanied by fluctuating es and h2 fields.

474

J. W. DUNKIN and A. C.

We can further distinguish two special cases: (a) infinite conductivity, lk12< > In case (a) we get the characteristic equation (P2 - k%H~)(~,P,

- k2rd@',

hlNC3EN

CKW.

- k%cH$P,) = 0,

(3.29)

which is the same as (3.17). The characteristic equation for case (b) is (Pz + iK’aoHf)(P,

f 2 + iu2awHfP, + irc2aoH$P2) = 0.

(3.30)

This approximation is equivalent to setting aB/& = 0 in Maxwell’s equations, and since it forms the basis of the later analysis of the damped plate, we give some of the details of the solution. The first factor in (3.30) is associated with a damped shear Kave accompanied by a fluctuating damped et field. Dropping the aB/& term results in e2 = 0 from the first of (3.27). The remainder of the first set of (3.27) then yields u3 = A, exp{ -dt+ik[x,

-J(ci--d2/k2)t]} eI = - ikoH,u,,

+A, exp(-dt+ik[x, h, = afcwk-‘u,

+,/cc:-d2/k2)t]} (3.31)

where 2d z rczaHf/p := tia Vf and A 1 and A2 are arbitrary constants. The second factor in (3.30) can be viewed as giving rise to a shear wave and a dilatational wave which are damped and coupled. If we define y E k/o,

y, 3 c;Z,

26, E Ic’aHf/po,

y2 = CT2

26, 3 K’aH:/(l+2~)0

then the second factor in (3.30) can be written as a biquadratic in y, Y4- [YI +y2 +W4

+62Ny2

+hy2

+2i(bh

+6272)

=

(3.32)

0.

We choose to regard w as real and k as complex, rather than the opposite, so that the characteristic equation can be solved more easily. Consequently the resulting waves will have a real frequency, but will decay in space. The roots of (3.32) are fy, and fyb where Y,’= 3[yl +y2+2i(b yb’= 3[rI

+y2

+2i(dl

+~2)1+3J[(y1-~2)2-~4(~1+~2)2+4~(~2-~1)(y1-y2)1 +a,)]

-&&I

-~~)~-4(b

+6212+4i(62

-My

I--y2)l.

(3.33)

For 61 and 62 very small compared to yr and y2 we have y,’ g CT’ +2iS,,

yf r c;’ +2iSl

and hence +y, Z c;’ +ic,6,,

+yb g ~;~+ic~iS~.

Upon comparing the values of 72and ~2with the coefficients of G and & in (3.27) we conclude that for small a, Vf, and Vi the x2-polarized shear wave is not coupled to the dilatational wave but that both waves decay. The assumption 8B/at = 0 reduces the last set of equations in (3.27) in such a way that the following solution results:

On the propagation

of waves in an electromagnetic

415

elastic solid

u1 = B,exp[-oc,b,x,

+io(c;‘x,-t)]+B,exp[wc,G,x,-io(c;’x,

+t)]

u2 = B,exp[-oc,6,x,

+io(c;‘xl-t)]+B,exp[wc,G,x,-io(c;’x,

+t)]

hz = -aic(c;’

+ic,G,){B,

exp[-oc,6,x,

+io(c;‘x,

-t)]

-I?, exp [oc,b,x,

+aic(c;’ +i~,6~){B,exp[-wc,ii,?c,

+io(c;‘x,

(3.34)

- iw(c;‘x, +t)]}

-t)]

-B,exp[oc,6,x,

-io~(c;~x,

+t)]}.

For larger values of 61 and & relative to yi and 72 values of y. and yb would have to be obtained by evaluating the square root in (3.33). Magnitudes of the magnetic interactions In order to get a feeling for the size of some of the interaction effects caused by the initial magnetic field we shall compute some quantities pertinent to the preceding discussion. Let us consider a maximum value of the magnetic field to be 10 W/M2 or 100,000 G. The values of the permeability IC in MKS units might vary from approximately 4nx 10-T RS/M for the paramagnetic and diamagnetic substances to 87~x lo-4 nS/M for ferromagnetic materials. A very large value of conductivity of the order lOtO( applies to a material such as silver at very low temperatures. The lower end of the range of conductivity can be taken as zero. The mechanical moduli can range all the way from zero for the shear modulus 1~of a fluid to about 3 x lOi* N/M2 for (d+2~) for some of the metals. First we consider the modification of the wave speed in perfect conduc$s in (3.18). One root of (3.18) is < = J(c$+ I’:) = cz+/l -t Vf/c& but V?/cz = KH$,u = B~/KP if we ignore the motion. For a nonferromagnetic material such as aluminum Vf/cz r l/300 while for iron Vf/cz r 05 x 10-e. The effect of the magnetic field on the mechanical stiffness or rigidity increases with both a decreasing permeability and a decreasing stiffness or rigidity, both of which tend to make Vf/cz greater for aluminum than for iron. However in neither material is the effect of much importance nor would it be in any of the solid metals. In order for the stiffness or rigidity contributed by the magnetic field to be important one would have to have an elastic material which is a perfect conductor, nonferromagnetic, and mechanically soft, e.g. a perfectly conducting rubber or soft plastic. The behavior of materials which are not perfect conductors in the strong static magnetic field is characterized by the damped wave solution (3.31). In this solution there exists the exponential factor exp{ -dt+ik[xl -J(cz-dz/kz)t]}. In this expression dean be viewed as the reciprocal of a time constant, i.e., to z d-1 = 2p/&H: which is indicative of the quickness of the decay of the associated wave. The phase velocity { also is affected by the factor d according to 5 = J(ci-dz/k2). In Table 1 are shown the results of a few simple calculations which should illustrate the extent of the phenomena considered here. In the second, third, and fourth columns representative values of density p, electrical conductivity 0, and shear modulus p are given in MKS units for five materials; the values are taken from [21] and [22]. In column five we have listed the values of the time constant to calculated for a 0.10 W/M2 flux density while in column six are listed the values of the critical flux density (~cHt),,, the flux density Th ese results apply to waves having wave which makes 5 = 0, (&?I),, = (4,upk+2)“4. lengths of 1 cm.

J. W. DUNKIN and A. C. ERINCXN

476

TABLE 1. TM CONSTANTAND CRITICALFLUXDENSITYIN MKS UNITSFOR A SHEARWAVE PASSING THROUGH A LARGE MAGNETIC FIELD

Material

Aluminum Lead Iron (cast) Carbon (graphite) Rubber (natural)

Density (K/M3)

Electrical conductivity (Q-‘M-1)

2700 11,340 7030 2250 920

3.54 x 107 4.55 x 106 I.0 x107 I .25 x 105 1.0 x10+5

4. PLANE WAVES TRAVELING ELECTRIC FIELD IN AN

Shear modulus (N/M2)

Time constant at 0.10 Wb/Mz (S)

Critical flux density 1 cm waves (WblM2)

2.37 x 10”’ 5,4 x109 8.4 x 101” 1.96x 109 2.4 x106

0.0152 0.498 0.146 3.6 1.84x 1020

16.8 46.4 55.2 145.0 2.42 x 101’

THROUGH A LARGE INFINITE MEDIUM

The linearized equations

In this chapter we investigate the coupling between the electromagnetic and elastic effects that exist because ofa uniform electrostatic field from exactly the point of view taken for the magnetic field in the last section. Once again all couplings that might arise in the constitutive relations, e.g. electrostrictive effects, are ignored. We select a rectangular frame of reference X, such that the xl-axis coincides with the direction in which the plane wave is moving ,and the electric field vector E and xl-axis define the (xl, x&plane. The equations that must be satisfied are (2.21) and (2.22).

DIrection of wave travel

FIG.

The presence flow whose order the moment the infinite, isotropic,

3. Static electric field and direction of wave travel.

of the uniform electrostatic field Eo causes a constant steady current to of magnitude depends on the conductivity of the material. Consider for magnetic field that would accompany a constant current field 4 in an homogeneous medium. The magnetic field would satisfy curlH=I,,

divH=O.

(4.1)

A solution to these equations is H = +I, x r

(4.2)

where r is the position vector. From this result it appears that a uniform current flow throughout all space requires infinite magnetic fields. However, if the conductivity is reasonably small, then there wilI exist a large region in which the static magnetic field caused by the small current flow can be considered small.

On the propagation

of waves in an electromagnetic

elastic solid

477

The assumptions of order of magnitude used for linearizing the basic equations are based on the preceding remarks. The electrical field is assumed to be E=E,+e

(4.3)

where E,-,is the large static uniform electrical field and e a small fluctuating field. The magnetic field H is assumed to be of the form H= ho+h

(4.4)

where ho is a small static magnetic field, not uniform but of the same order of magnitude as h, e, and u and is generated by uniform current jo = oEo. It should be remarked that ho actually would become large far from the origin and would exert its influence on the equations there. Thus the linearized equations with he considered small would not be valid throughout the space. However, it is felt that the plane wave studied under the assumption that hs is small actually would represent a good approximation to a plane wave in a finite body subjected to a large field and would indicate some of the features of the interaction between the electromagnetic and elastic phenomena caused by a large electrical field. Since jo is the source of ho we write ho = &joxr = &aEoxr. Then recognizing that ho does not depend on time and that j. is a constant vector and using (4.3) and (4.4), equations (2.21) and (2.22) become

ae ah a2u -a xE,=O, curl h--E - = ae at at at2

curl e+rc -

(4.6) J = j,,+ue, Note that ue is quite small but is retained to give damping in the electromagnetic waves. For the solution of (4.5) to have the form of a plane wave traveling in the x1 direction the vectors e, h, and u must be functions of x1 and t only and can be written in the form of (3.4). Inserting (3.4) into (4.5), dividing by the exponential, and using Eo = (El, Ez, 0) reduces (4.5) to the following set of homogeneous algebraic equations in the complex amplitudes : im&

+orc~~E,u~ =0,

ik&+im~2-aco2E1u~=0,

ikT2 - ircc& +aw2E2u^; - ao2El~2 I: 0 PA

= 0,

ikh<+P,g

= 0,

ik&-P&

=0

(4.7) (43) (4.9)

where PI, P2, and Pd are defined in (3.6).

J. W. DUNKINand A. C. EIUNGEN

478

Plane wave solutions

From the first of (4.8) we find that TI = 0 if Pd # 0, and consequently all electrical effects disappear from equations (4.9). The characteristic equation, which is the vanishing of the determinant of the equations (4.7)-(4.9), takes a particularly simple form. Except for trivial factors, it is P*PIP;P;

= 0

(4.10)

where P, = 7cso2+icnco-k2.

When PI = 0 the wave is a dilatational wave with speed m/k = cl = &+2,u)/p]; when P2 = 0 there can exist either an x2 or x3-polarized shear wave with speed w/k = c2 = &/p) ; when P, = 0 we can have x2- and x3-polarized, damped electromagnetic waves; finally the factor Pd = 0 gives the decay of the et field. Thus none of the wave speeds are modified by the electrical field. Turning back to (4.7)-(4.9) we find that the electromagnetic waves propagate without any modification. However, along with the mechanical waves there exist small fluctuating electrical and magnetic fields that travel at the mechanical speeds. For an x3-polarized shear wave given by u3

=

U,

exp [ik(x, - c2 t)] , (U, is arbitrary)

there also exist the field quantities h 1 = iakcgcS1E,U3

exp [ik(x, - c,t)]

h, = (a- iskc&k%~E,P, e3 = ik3&E,P;‘(k,

‘(k, kc,)U, exp [ik(x, -c&l

kcl)U3 exp [ik(q

-

(4.11)

c,t)] .

All other electrical and magnetic quantities are zero. For an x2-polarized shear wave given by u2 = U,exp [ik(x,-c,t)],

(U, is arbitrary)

there also exist fluctuating fields e2 = ik3cfaE1P;‘(k, h 3 = -(a-

kcz)Uzexp [ik(x, -c,t)]

iekcz)ak2c~P; ‘(k, kc&J,

exp [iI&

- czt)]

(4.12)

and all other components are zero. Finally a dilatational wave given by ul

=

U, exp [ik(q

-cIt)] , (U, is arbitrary)

is accompanied by e, = - ik3cfaE,P;‘(k,

kc,)U, exp [ik(x, - clt)]

h3 = (a- iskcl)ak2cfE2P;‘(k,

with all other components of e and b zero.

kc,)U, exp[ik(x,

-c,t)]

(4.13)

On the propagation

of waves in an electromagnetic

479

elastic solid

The fluctuating electric and magnetic fields which are associated with the meczanical waves are quite small. If in (4.13) we assume u = 0, K = K~,E = 2.50and P,(k, kc& = -k2, then es and h3, in magnitude, are approximately e,/Ep = U1 kcf/c’ ,

h3 = U, k(c:/c2)(~E2c,).

Taking Ulk = 1, which is unrealistically large, we see that for a material having a dilatational wave velocity 104 m/s the e2/E2 ratio is of the order of magnitude IO-9 and that the h3 field is also extremely small. Hence we would conclude that the large static electrical field as studied here does not introduce a significant coupling between dynamic electromagnetic effects and elastic effects. 5. THE

VIBRATION

OF AN INFINITE MAGNETIC

PLATE FIELD

IN A LARGE

STATIC

Formulation

The results of the last two chapters show that the magnetic field is much more conducive to interaction effects associated with the body force coupling than is the electrical field. Consequently we shall now investigate the problem of the vibration of an infinite plate in a large magnetic field and ignore the similar electrical problem. The problem of the plate is of interest for another reason, namely, for the first time we encounter a boundary value problem which requires careful consideration of electromagnetic elastic boundary conditions. Since our interest lies primarily with the motion of the plate the electrical behavior will be assumed to be quasistationary while the mechanical behavior is dynamical. Consider a plate of thickness 2T which extends to infinity in all other directions. A set of rectangular Cartesian co-ordinates (~1, x2, x3) are oriented such that x2 is perpendicular to the plate and the origin lies in the middle surface (see Fig. 4). A sinusoidal wave train is assumed to move in the xi direction parallel to a large magnetic field. This choice of the magnetic field represents a restriction on the problem but should contain all the phenomena associated with the coupling in the plate. Thus a magnetomechanical coupling comes into play because of the transverse motion of the plate cutting the flux lines. Another important problem, not considered here however, is the one in which the magnetic field is taken normal to the middle plane of the plate.

x3

Fro. 4. The infhlite

plate.

480

J. W. DUNKIN and A. C. ERINOEN

The field equations that must be solved are equations (3.25) in which ah/at is set equal to zero. This is a small concluctivity, high frequency approximation used in deriving (3.30). The more general quasistatic problem using the complete version of (3.25) is an important example which will be examined at a later date. Quantities e,, h,, and U, are assumed to be of the form h, = r=exp [Ax, + i(kx, - it)]

e, = T=exp [Ax2 + i(kx, -cot)] ,

,

(5.1)

u, = u:exp [Az+i(kx, -cot)]. The substitution of (5.1) into (3.25) in which ah/at = 0, making use of Ho = W,, 030)

(5.2)

and assuming fr = 0 yields G = 0 and the following set of algebraic equations in the complex amplitudes :

hl

~*-(&2p)k2

+pd]

ikA(l +p)

ikA(3, +/A) [(A +2,u)h2 - ,uk2 + ix2aoHf +pw2]

0

0

Ul

0

0

u2 &

0

- iawH 1

-A

ik

0

0

ik

A

=o

62

. -

(5.3)

&A’- k2) +*2aoHf

-

-icoH,

0

0

iaucoH i

-CT

0

-ik

0

ik

-A

0

0

0

--d

A

UC

+po2]

(5.4) The last equation in (5.3) comes from div B = 0 and is necessary since aB/at was dropped from the first of (2.21) thereby causing the three component equations to degenerate to two, namely z = 0 and the third of (5.4). The boundary conditions are given by equations (2.23) with J” = 0 for the body in a vacuum with no surface currents and a&% = 0 which is consistant with dropping aD/at in the field equations. Introducing E = e and H = Ho+h into (2.23) and linearizing the results we obtain the following linearized electromagnetic boundary conditions : [e+tcvxH,],=O,

[H,,+hB,=O,

[u@$,+h)]l,=O (e+fcvxH&

pt = fee+avxH,]..

= 0

(5.5)

481

On the propagation of wavesin an electromagneticelasticsolid

The last equation serves only to define the surface charge. If conditions (5.5) are to hold when the small fluctuating fields are zero, then He must satisfy the boundary conditions (5.6)

[HoIt = l~Ho1n = 0 Using this in (5.5) reduces it to [e +KV x Hoat = [[II],= frcb],,= (e + ICV x Ho), = 0,

pz = [se]” - a(v x Ho),,

(5.7)

where H,Jwithout the double bracket means the value of Ho inside the body. The stress boundary conditions of (2.23) upon linearization reduce to

and if nS is carried inside the bracket and boundary conditions (5.6) and (5.7) are used, then they become aQYn/¶ = ~KJniWoolll+KKl”P%ll +EHodKhn-3EKHo1Hg,~n,

- lMcl~~~B~, *

(5.9)

A portion of the right hand side of this equation has the form of a static traction which is is caused solely by the presence of Ho. Therefore if we redefine crapto mean the difference between the actual stress and the static stress S,, then we have ~@fl= ~&Kh,n+

lI&ll~~* - W%JJ,ll%

(5.10)

and

For the infinite plate the static stresses are solutions of

asQ80

(5.12)

-e: %

which satisfy (5.11) on the two faces x2 = &T and certain conditions at infinity which are not specified here. Using Ho = (HI, 0,O) and (5.6) the static stress state is found to be one in which S22

=

4(x

-

(5.13)

h)fc

the other normal stresses depend on the conditions at infinity, and the shear stresses are zero. The normal vector at the top surface is n, = (0, 1,O) and at the bottom surface is n, = (0, - 1, 0). Using these values and Ho = (HI, 0, 0) in (5.7) and (5.10) yields the following scalar boundary conditions which must be satisfied at both surfaces: el--1

(e) =

e2 +rcH,03 = 0

=0 e3-e3 (=)-(K-lcKO)H1tJ2

(x2= *n

~,_~~)=~~2_-Ko~~)=~3-~~)=0

012 =

c32

=

~22-(lC-&-JH~h~

=

(5.14)

0

where the superscript (e) denotes the external surface value of the quantity so labeled.

J. W. DUNKXN and A. C.

482

BUNOEN

Plate wave soiutiom A non-trivial solution of (5.3) and (5.4) wih exist when A is related to k and cv such that the determinants of the coefficient matrices are zero, i.e. when A is a solution of one of the characteristic equations associated with (5.3) and (5.4) which are, respectively (A2 -k2)(QlQz

+ k2awH:[Q,

-(A +#c~]}

(5.15)

= 0

and (5.16)

(A2 - k2)Q2 - irc2ac@k2 = 0 where Qi and Qz are deGned to be Qi ~(~+Z~)(A2-~z)~~#2,

(5.17)

Q, =JJ(A’-~~)-I-/R&

Introducing the notation e=A/k, 8 = ok/c,,

d, = lc%El:c&(A+2/4) p2 = c;,c; = (~+2~)~~

(5.18)

(5.15) can be written as

(e2-i)[e4-t(B282+8

2-2fidi@e2+(82(32-1)(S2-1)+id18j12(~2-1)]

= 0.

(5.19)

The roots of this equation are

e= fi,

fAl,

54,

(5.20)

where ng = -~,-.J($-z2) n: = -u,.i-&+-a,), 2c1, =(~2+l)~2-2+~d~~, a, ~~2~2-l~~2-l)+jd~~2~(~2-l). The characteristic are

(5.21)

functions associated with (5.3), except for the factor exp [z’(kxi-CM)], gfk

fL,kx~

fltkxz ,

e

re

(5.22)

*

Equation (5.16) can be written using the defined quantities of (5.18) as P+(/9262-2)02+1-/32S2-ij?2dl~

= 0.

(5.23)

The roots of this equation are

e= 45,

fr2,+

(5.24)

where Ai = -a,-J(af-a4) P3-- -acr,+J(az-aa, 2a, = p26=-2, a4 = 1-@262-i/32dlb.

(5.25)

00 the

propagation of waves in an electromagnetic elastic solid

483

The characteristic functions belonging to (5.16) can now be expressed as +i(kxl -ot)]

exp[*lJcxz

, exp[*A,kxz

+i(kx, -of)].

(5.26)

The general solution to (5.3) and (5.4) can be formed in terms of forty coefficients by allowing each of the functions to be a linear combination of the characteristic functions. The equations (5.3) and (5.4) for each fl, impose restrictions on all but ten of the coefficients. If we proceed in this manner the solution can be compressed into the form: -

_

1

1

0

1

a12

a22

0

--al2

h,

al3

a23

1

al3

h2

a14

a24

Ul u2

-

=

-i

1

0

-a22

0

a23 -a24

-ai4

1

A3ekxr

i

Bie-*‘kx’

_

d(kxl-mt)

(5.27)

Bze- A,‘=, Bsewkx*

1

113 e2

=

el h3

1 a42

‘32 a33

a43

a34

a44

1

1

a32

a42

-a33

-

a34

$(kxr-rot)

(5.28)

,

a43

L

a44

-

-

where

aj2 =

a

=

I I

iQ(W,cS2

-

1)

irckc1H16A~/(A~

-

1)

~KC~H~C~Q(A,)/(/~~ - l)($ - 1) Ickc,H,bAJ(l --AT)

- iUj3/Aj aJ4 = Icoc,H,G/(l -A$

j=l, 2 j=3,4 j = 1, 2 j=3,4

(5.29)

j=l,2 j-3,4

in which Q(A,) z 12j2-/P+/_?W and the Ai and Bi are arbitrary. For convenience in (5.27) and (5.28) we set (5.30)

484

J. W. DUNKINand A. C. ERINGBN

and ignore writing the factor exp [i(kxr -CM)]. Hence u1 = a,coshl,kx,

+a,cos&kx2

U2 = a,a,,sinM,kx, h, = ala,,cosM,kx,

+b,sinhR,kx,

+a,a,,sinM,kx,

-t-b,alzcosM,kxz

+a,a,,cosh&kx2

+a,a,,sinh&kxz

+b,a,,sinM,kx,

fa,cosh&kx,

fbssinhkx,

-a,isinhkxs

fb,al,cosMIkx, u3 = a,coshl,kx,

+b,az,cosh&kx,

fa,coshkx,

-t-bla,,sinMIkx, h2 = a,a,,sinM,kx,

$b,sinhd,kx,

-t-bza,,cosh12,kx:,-

+b3sinM,kxz

b,icoshkx,

(5.31)

+b,sinM,kx,

~a~a~~cosh~~k~~ +b~a~~sinh~~~~ +b~a~~s~~~kx~

e2 = a,a,,cosh&kx, e1 = a,a,,sinh&kx,

+a,a,,sinM4kxa

-I-b,a3,coshl,kx2

h3 = a,a,,cosh&kx,

+a,a4,cosM,kx2

+ b,a,,sinM3kxz

+ b,a4,cosM4kx,

+ b,a44sinhA*kx,.

Boundary conditions (5.14) involve five external values of the field quantities at both surfaces, i.e. values of ey’, e?‘, h@ , , h’“’ 2 , and h$@’ as well as the six stress conditions. Although at first glance it might appear that these quantities could be arbitrarily specified thereby yielding more conditions than could be satisfied by the ten constants of solution (.5.31), actually some of these conditions merely serve to extend the electromagnetic part of the solution into the regions surrounding the plate. For the fully dynamic problem of electromagnetic fields in a body it is known (Stratton [20], p. 487) that the solution in nonferromagnetic materials is uniquely determined by specifying the tangential components of either E or H over the surface of the body. Therefore in the problem considered here we shall extend the solution of the electromagnetic field quantities into the region surrounding the body and, remem~ring that all the sources for the fluctuat~g fields are contained in the plate, are periodic in x1, and hence add up to zero, we shall insist that the solution vanish at x2 = &co. This will serve to define the solution. Outside the plate the external fields are solutions of curl e@)= div e@)= 0 curl h@) = div htet = 0.

(5.32)

(IX*] ’ T>

Solutions of these equations which vanish at x2 = rfr co and have the wave factor exp[i(kxt of)] are e@)= F,(l, i, O)exp[- kx, +i(kxl h@) = G,( 1, i, O)exp[ - kxz + i(kxl ecet = F,(l, - i, O)exp[kx, + i(kx, hce) = Gz(l, - i, O)exp[kx, -t i(kx, where F, and Gi are undetermined constants.

-ot)] - wt)] -cd)] - ot)]

(~2 > T) (x2 < -T)

(5.33)

At the boundary of the plate these solutions

are such that b~)+ib’,‘)=O, bfe)_ibyf = 0,

e~)+ie~)=O; ef”)-i@f = 0;

x2=T x2 = -7’

(5.34)

and hence, using (5.14) hi+im~1h2=0, h,-iwc~‘h2 = 0,

h,=O; h3 =O;

x2= T x2 = -T.

(5.35)

On the propagation

of waves in an electromagnetic

485

elastic sdid

The condition that h3 = 0 on the boundary is equivalent to the zero normal current condition as can be seen by examining the second of (5.4). The relations involving ey) and et) in (5.34) do not add conditions on the solution but describe the surface charge through the first of (5.14) and the last of (5.7). Because e3 = 0 everywhere we find that it is impossible to satisfy the condition e3--e~)-(~+c0)H1v2 = 0 at each point of the surface. This indicates that the problem stated here is not a legitimate two dimensional reduction of a three dimensional situation. However, since this boundary condition is met in an averaged sense because of the periodicity in the x1 co-ordinate, we shall proceed with the solution under the expectation that it will adequately represent the physical situation except for some details. By use of the mechanical constitutive relations, the last of (2.22), and /I* = (A+2p)/p the last three of (5.14) can be written as au,+%

_.

ax, ax,-

8% ’

o

K2”

(5.36)

These relations, together with (5.35), provide the conditions which define a, and b, in (5.31) and Fj and Gj in (5.33). If solutions (5.31) are substituted into (5.35) and (5.36) and the two equations obtained by evaluating each condition at the two surfaces are added and subtracted, the following equations result: .-

a,cosh&kT

0

a,,tanhl,kT

cl,,tanM,kT

azl

a22

--r

a31

aJ2

ag3

a,cosh&kT a,coshkT .-

a1 ,cotM,kT

a,,cothl,kT

azl B31

0

b,sinhR,kT

az2

--r

b,sinM,kT

B 32

833

= 0

(5.37)

= 0

(5.38)

-

b,sinhkT _ -

I-

I,sinM,kT

&sinM,kT

a,,cosM,kT

a,,coshl,kT

14cosM,kT

a,,sinM,kT

a,,sinhA,kT

(5.359

a4 _ -

I,cosM,kT

=o

a3

-

b3 x0 b4 - -

(5.40)

486

J. W. DUNKIN and A. C. ERINOBN

where for j = 1, 2 cC,j

-1 -

Uzj = i(/3’-2)+/?~jajz-raj3,

j+iClj,,

fljj = aja(1 +KKglAj-‘COthljkT),

U3j = aj,(l

C(~J= 1 +KK;%anhkT,

+KK~l~,~ltanMjkT)

&a = 1 +KK;%OthkT

(5.41)

in which r = (K-K~)HI/~L~ and the expressions for aij are given in (5.29). Solutions of (5.37) and (5.38), respectively, are associated with symmetric and antisymmetric motions of the plate relative to the middle surface. The respective frequency equations are obtained by setting the determinants of the coefficient matrices of (5.37) and (5.38) equal to zero. Thus symmetric waves have frequencies which are solutions of a,,(a,,a,,tanhl,kT-a,,u,,tanhl,kT)+r(a,,cr,,tanhl,kT-cc,,cr,,tanh;l,kT)

= 0

(5.42)

= 0.

(5.43)

and the antisymmetric waves have frequencies which are solutions of ~33(~,,~~2coth~,kT-ccr2t(~,coth~ZkT)+~(~,,~~Zcothjl,kT-a~2~31coth~~kT) For H, = 0 r

=0,

1, = J(l-S2),

1, =J(l-j3’b2)

and the frequency equations reduce to the well known frequency equations for the infinite plate (Ewing, Jardetsky, and Press [23], p. 283) tanh,/( 1 - P)kT tanh,/(l -fi2a2)kT

= 4J(l -S’)J(l

tanhJ( 1 - P)kT tanhJ(1 -p2d2)k7’

=

(2-@s)2

(5.44)

-j?2S2)

4J(l -#)J(l -/?‘S’) (2-/W)2 -

(5.45)

Due to the fact that 11 and AZare the roots of a biquadratic with complex coefficients, equations (5.42) and (5.43) would be almost impossible to solve as they stand. Therefore we shall look for ways of simplifying or specializing them. Symmetric waves

The frequencies of the symmetric plate waves are obtained from (5.42) which, using (5.41) and (5.28) and rearranging some of the terms, can be written as

fi2d16 +/Y-1

~-Ic~K-~ (A:-/?‘+/126’)(l A$--1

+~~;~A;‘tanhA~kT)

(5.46)

(B2-2)1’+B2(1-‘! ~2U2

-

1)

f12d,6 1

B” +82-l

(~~-~2+~2~2) -KoK-l

+p-1

If-1

(n:-/I2+/3%2)(1

+KK$I;‘tanMlkT)

= 0

On the propagation of waves in an electromagnetic elastic solid

487

where a33 does not depend on dl or 6 and for j = 1, 2 ~~j4+(B262+S2-2+~dlB)~~+(B2S2-1)(82-1)+id~S~2(82-1)=O

(5.47)

gives k1 and 22 as functions of S and dl. Rather than trying to solve (5.46) and (5.47) simultaneously for 6 we shall assume that the term dl is small and obtain the linear correction to the 6 for dl = 0. If we label the quantities obtained for dl = 0 with an asterisk, then 6* represents a root of (5.44) and from (5.47) ;I: =J(1-6*2),

r2;=J(l-p2),

(5.48)

Equations (5.46) and (5.47) can be viewed in the following functional form : .!I&,

A,,

dl,

6)

=

0

(5.46)

;

g,f&, dl, a> = 0;

j=

92V2,

j=2,

dl,

4

=

0 ;

1,

(5.47) (5.47)

If we consider dl to be a small increment in the dl variable, then the increments in RI, ,I* and S are related to dl by

af

a4

AA,+ gf

2

AA,+;

A&+$

d, =O.

(5.49)

1

The increments AlI and AhA2 can be expressed in terms of increments AS and di by applying the same procedure to gI = 0 and g2 = 0. Substituting these expressions into (5.49) gives

(5.50)

derivatives are evaluated at dl = 0. Equation (5.50) gives a systematic way of computing the ratio A6fdl. Carrying out indicated computations in (5.50) we get

in which all partial

AS = idIR,/R2

(5.51)

where RI and R2 are real and are given by R 1 = U: tax&$ kT+4i:‘kTsech21:

kT

+rl~-3(2-~25*2)[2-~26*2-2~4(~2-1)-’6*2~~2]tanhll~kT kT+212:2a;31(m;1-

+rtf-2kT(2-/126*2)2sech2Jf - [ZAF-‘tanhlr

kT(L$-‘tanhA:

-~2(2-~2~*2)~~-‘tanM~

kT- tanhkT)

kT(II:-‘tanh$kT-tanhkT)J

R, = 26*2[4~;-‘tanM:kT+4kTsech2L:

kT

+/?21f-3(2-~2Ci+2)(3j?2S*2-2)tanhrZ; - f.lakT$-“(2

-B2S*z)2sech2$

I)

kTl.

kT

(5.52)

488

J. W. DUNIUNand A. C. ERINOEN

Each of the quantities ~1, ~2, ht, and hz contain, along with other factors, the exponential

exp[i(kx, - wt)J = exp[(K*oHllp)(R,/R,)t

(5.53)

+ ii+, - c,b*t)] .

Notice that the linear correction to the symmetric plate frequency is purely imaginary and therefore represents a damping. Expression (5.51) furnishes values of the damping as a function of kT for any of the symmetric modes of the infinite plate once the dependence of S* on kT is known for that particular mode. Since the evaluation of (5.51) requires very accurate values of a*, it was necessary to use a digital computer to solve (5.44) before computing the ratio Rl/R2. In Fig. 5 a plot of the ratio of the phase velocity to the dilatational velocity is given for the first two symmetric modes versus kT and in Fig. 6 the coefficient of damping Rl/Rz versus k?‘. ~oth~figures apply to a Poisson material (82 = 3) which is also nonferromagnetic, i.e. #(x-l -_ 1 = 0. Since RI/R2 is always negative the magnetic field always damps out the waves. For the first mode the damping ratio is zero for kT I- 0, i.e. a very long wave length, and tends to the value -0.4665 for kT-+co. For the second mode R1/RZ is approximately -0.43 at kT = 0, has a minimum at approximately kT -:- 2.5, i.e. a wavelength equal to n/2.5 thicknesses, and as kT+cn tends to the value --4.

T

_.

i-

.- _.

/

-L 1

._i_

First mode

.l..,._._Ll i

F?

kl FIG. 5. Phase velocities of symmetric waves for a free infinite plate.

Since the magnetic field has only an x1 component the ~1 motion does not contribute to the damping and hence the transverse expansion and contraction in the lon~tudin~ magnetic field is the cause of the damping. For long waves of the first mode a unit width of the plate would behave like an infinite bar which is free on the top and bottom surfaces but clamped on the sides. The waves then would be such that transverse planes would move back and forth in the longitudinal direction, there would be simple expansions and contractions in the vertical direction causing a damping, and no motion would be observed in the lateral direction,

On the propagation

of waves in an electromagnetic

4

elastic solid

489

6

6

kT

FIG. 6. Damping of free infinite plate caused by magnetic field parallel to direction of propagation (symmetric vibrations).

The magnetic field can exert a very strong damping effect as can be shown by considering a decay time given by to = -(~2oH:/~)-l.(R2/R1). Thus a decay time of 1 s for aluminum, iron, and lead would require flux densities of 128, 388, and 730 G, respectively. These figures refer to the first symmetric mode for short waves and values of c and p were taken from Table 1.

Antisymmetric waves

The antisymmetric waves have frequencies that are examined more closely it becomes apparent that it could by coth ;IjkT(j = 1,2) and tanhkT by coth kT in (5.42). out the procedure for obtaining the linear correction to result can be written as A6 = idlRJR, where R, = 41:cothl:

kT-41:‘kTcsch2#’

roots of (5.43). If this equation is be obtained by replacing tanhljkT Making use of this fact in carrying the antisymmetric speeds, the final

kT

+1~-3(2-~26*2)[2-~28*2-2~4(~2-1)-1B*2~~2]coth~fkT -1~-2kT(2-j?2S*2)2csch2~; +[22:-‘coth$

kT+2J;2j?;,i(&1

kT(Af-‘cothl;

-/?2(2-/326*2)11:-1cothlZf

- 1)

kT-cothkT)

kT(.J:-‘cothl:

R 4 = 2b*2[41;-‘cothl:kT-4kTcsch21: +~2~~-3(2-/328*2)(3j?28*2-2)cotif +/!J2kT~,*-2(2-/926*2)2csch2d;kZJ.

kT kT

kT-cothkT)]

(5.54)

490

J. W.

DUNKIN

and A. C.

ERINC;EN

The imaginary character of the linear correction to the zero field phase velocity indicates a damping in time since ~1, u2, /I,, and h2 all contain the factor exp[i(kx, -wt)]

= exp[(K2aHI/p)(R,/R,)t+ik(x,

-c,6*t)].

(5.55)

Here also damping curves can be plotted for any mode once the dependence of 6* on kT is known for that particular mode. Fig. 7 is a plot of 6* versus kT for /P = 3 for the first two antisymmetric modes and resulted from digital computations of roots of (5.45). The corresponding damping ratios, RJR4 for both modes, are plotted in Fig. 8 for a nonferromagnetic material.

0

2

6

4

e

kT

FIG. 7. Phase velocities of antisymmetric

2

waves for a free infinite plate.

4

6

6

kT

PIG. 8. Damping of free infinite plate caused by magnetic field parallel to direction of propagation (antisymmetric vibrations).

On the propagation of waves in an electromagnetic elastic solid

491

For the first mode R3/R, as computed from (5.54) becomes large for small kT thereby violating an assumption of (5.50) concerning the smallness of A& Consequently it is necessary to obtain the behavior of the damping in this case directly from the antisymmetric version of (5.46), i.e. (5.46) with tanh A,kT replaced by coth l,kT. This is done by approxiusing (5.47), and finding the linear correction mating the coth lkjT with [3,ikT-$(ljkT)3]-l, to the velocity for no damping, i.e. dl = 0. The resulting velocity for small dl and small

WJ2 1

kT is

6 = & J(8/27)kT-

Therefore RJR,

id, $-- 6 [

.

(5.56)

for small kT behaves as RJR,

=

-

+(F2 [ 1

(5.57)

in this case. The damping factor Rs/R,, considering its correct behavior near kT = 0, is plotted in Fig. 8. Extensions of (5.57) and R3/R, computed according to (5.54) for small kTare shown in dotted lines. For the first mode R3/R4 = -3 at kT = 0 and tends to the same limit as Rl/RZ for the first symmetric mode, namely, -0.4665. The second antisymmetric mode has R3/R4 = 0 for kT = 0 and RsIR,+ -4 as kT+ co. The foregoing discussion is concerned with the waves which are solutions of equations (5.37) and (5.38). It remains to look for wave solutions of (5.39) and (5.40). We shall show that no such waves exist in the limiting cases of long and short waves and conjecture that, in all probability, none exist for all values of kT. This should not be surprising if one recognizes that there exists no such zero field, free plate wave and therefore the damping caused by the magnetic field is not apt to give rise to one. If any wave solutions belong to (5.39) and (5.40), then their frequencies will be obtained by solving the equations resulting by setting the determinants of the coefficient matrices equal to zero. These equations, respectively, reduce to A,( 1 - $) tanhL,kT L,(l-@=tanM,dT’

A,(1 -A:) tanhl,k? 1,(1-1~)=tanM,kT’

(5.58)

For waves that are long relative to the thickness 2T the hyperbolic tangents are replaced by their arguments giving for both equations, after some algebraic manipulation involving (5.23), n; = 1:

(5.59)

which reduces to 6’ = -4id,//3’.

(5.60)

One root is 6 = 3J(4dl/p) i which represents a damped solution but no wave. The other roots contain negative imaginary parts which are associated with an exponential increase in amplitude with time and hence must be discarded.

4Y2

J. W. DUNKINand A. C. ERINGEN

For the short wave approximation yielding in both cases

the hyperbolic tangents are set equal to unity (5.61)

A,( 1 - A$)= A.$(1-n;> which can be written as (a,-~,)(I:+a~+sa,-l)

= 0

(5.62)

the first factor set equal to zero is the same as (5.59) and hence contributes no waves. The second factor set equal to zero can be written as 63-j?-26fifi-2d,

= 0.

(5.63)

= 0

(5.64)

If we set 6 = ic, then (5.63) is reduced to c3+/T-2~-p-2dl

which obviously has a real positive root, say 1. Dividing by c--E then reduces (5.64) to which, together ~2+~~+12+~-2 = 0 which has the two roots i = -~~~~(3~2+4~-2) with 5 = I, correspond to 6= +tJ(312f4/?s2)-ii,

il.

(5.65)

Therefore we have the same type of roots as for long waves, i.e. ones which correspond to a damped, nonwave solution and two diverging solutions. ELEMENTARY

BEAM

THEORY

In this section we present an elementary theory that is an extension of the BernoulliEuler beam theory taking into account the damping caused by the magnetic field. The damping enters because the motion of the beam across the flux lines causes a transverse current which in turn gives rise to a retarding force. The energy is actually dissipated in the form of Joule heating.

Fro. 9. The

differentialelement

of the beam.

On the propagation

of waves in an electromagnetic

elastic solid

493

An element of a bent beam of length dxi, of unit width and of thickness 2T, is shown in Fig. 9. The vertical displacement 242is a function of x1 and t. The current density in the x3 direction resulting from a magnetic field Hi is J‘=mH1

au2 -.

(5.66)

at

The total current through the element introduces a retarding force per unit length,

au

fd = 2ThH:

2. at

(5.67)

The remaining forces acting on the beam are the distributed pressure and the inertia force per unit volume. Equating the total force in the x2 direction to zero yields 2 ‘3

aF+f,+p+2Tp 1

= 0.

(5.68)

The equation of moment equilibrium is

aM_y

(5.69)

aTI -

where V and M are the shear force and bending moment, respectively. If (5.69) and (5.67) are introduced into (5.68), and the expression of moment

M=E,~

(5.70)

1

is used, we obtain the differential equation for the beam, namely,

EI

‘21 +2Tlc2aH;

au,

~+2Tp

a%, = -p, at2

(5.71)

Since we are interested in steady state waves, uz is taken in the form u2(x1, t) = P,exp[i(kx, -cot)]

(5.72)

where ii2 is a constant. Substituting this expression into (5.71) withp = 0 gives the frequency equation co2 + i(rc2aH:/p)a, - (E1/2Tp)k4

= 0

(5.73)

- (ic2aH:/2p)‘].

(5.74)

which has the roots o = - r(lc2uH:/2p)fJ[(EZk4nTp)

494

J. W. DIJN~~N and A. C. EIUNGEN

If we expand the radical into a power series and retain only the terms that are linear in ~~cH:/2p, we may write the solution as u2 = C,exp{ - (rc2aH:/2p)t + ik[x, f,/(EZ/2Tp)kt]}

.

(5.75)

Thus we see that the elementary theory predicts a damping coefficient equal to --lc’%H2/2p or -_Bdl and a phase velocity equal to ,/(EZ/2Tp)k or ,/(5/18)kTq. For a comparison with the damping predicted by the exact plate theory for the first antisymmetric mode, the elementary beam damping is drawn in Fig. 8 as RJ/R, = -l/2. It is interesting to note that the two predictions of damping differ less than 10 per cent throughout all kT and they agree exactly in the limit as kT-+O. The wave velocity obtained from the beam theory is plotted in Fig. 7 as a straight line which is not quite tangent to the velocity curve for the first antisymmetric mode. However, if E were replaced by the corresponding plate modulus E/(1 -v2), then the two wave velocity curves would be tangent at kT = 0. Therefore the modified beam theory represents a good approximation for the exact plate theory in the range 0 S kTI l/2. 6. CONCLUSIONS

The present investigation is concerned with the propagation of waves in a homogeneous isotropic elastic solid subject to either a uniform magnetostatic or electrostatic field. It is found that the magnetic field is much more effective than the electric field in introducing couplings between the elastic and electromagnetic waves. The electric field does not change the wave velocities of the separate waves although it does cause a ripple in the electromagnetic fields which accompanies the mechanical waves. The magnetic field, on the other hand, introduces couplings which yield new wave velocities and which occur when the current associated with the electromagnetic wave is parallel to the current induced by the motion of the material. In the majority of the elastic solids which have large conductivities the elastic constants are so large that, except for the damping of the elastic waves by the magnetic fields, the coupling effects are very small. The magnetic damping of the mechanical waves in the free infinite plate is found to be quite important. For long wavelengths the plate theory for the first antisymmetric mode agrees with an elementary beam theory which includes eddy current damping. For smaller wavelengths the damping predicted by the elementary beam theory departs no more than 10 per cent from that predicted by the plate theory although the wave velocities in the two cases differ widely. ACKNOWLEDGMENTS This work was supported by the U.S. Office of Naval Research. The authors appreciate the assistance of Neal F. Jordan who checked the analysis and made several helpful suggestions. REFERENCES [l] A. SOMME~FXLD, Electrodynamics. Academic Press (1952). [2] C. MBLLER, The Theoryof Relativity. Clarendon Press, Oxford (1952). [3] I. S. SOKOLNXOFF,Mathematical Theory of Eiasticity McGraw-Hill,New York (1946). [4] A. C. ENNOEN, Nonlinear Theory of Continuous Media. McGraw-Hill,New York (1962). [5] C. TRU~SDELL and R. A. TOUPIN, Handbuch der Physik, Vol. 3/i. Springer-Verlag, Heidelberg (1960). [q R. A. TOUPIN,J. Rat. Mech. Anal. 5,6,849-915 (1956). [7] A. C. ERINGEN,Znr. J. Zhgn. Sci. 1, 127-153 (1963). [!I] R. A. Tours~, Znt. J. E.ngn. Sci. 1, 101-126 (1963). [9] L. KNOPOPP,J. Geophys. Res. 60,441-456 (1955). [lo] A. BANOS,JR., Phys. Rev. 104,2,300-305 (1956). [ll] P. CHADWICK, Ninth Znt. Congr. Appl. Me& 7,143-152 (1957) 1121 S. KALISKIand J. Parvxrawrcz, Prob. I%. Prob. 2 17-35 (1959).

On the propagation

of waves in an electromagnetic

elastic solid

495

S. KALI~KI,Proc. yib. Prob. 3, 53-67 (1958). S. KALJSKI,Proc. Vib. Prob. 4, 13-36 (1960). S. KALISKI,Proc. Fib. Prob. 2, 3(8), 237-249 (1961). S. KALI~KIand D. ROGULS,hoc. Fib. Prob. 5,63-80 (1960). S. RODRIGUEZ, Phys. Letters 2,6, 271-272 (1962). A. A. GALKINand A. P. KOROLIUK,SOV.Whys.JETP 7(34), 708-709 (1958). V. T. BUCHWALDand A. DAVIS, Muthematica 7, 161-171 (1960). J. A. STRATTON,Electromagnetic Theory. McGraw-Hill, New York (1941). Handbook of Chemistry and Physics, 34th Ed., Chemical Rubber (1952). W. E. FORS~THB,Smithsonian Physical Tables, 9th Ed., Smithsonian Institution, The Lord Baltimore press (1956). [23] W. M. EWING, W. S. JARDETSKY and F. PRESS,Elastic Waves in Layered Media, McGraw-Hill, New York (1957).

[13] [14] [15] [16] [17j [18] [19] [20] [21] [22]

R&me-L’association d’ondes electro-magnetiques et d’ondes dlastiques est exammee au point de vue de I’eIasticite lineaire et d’une th&orie Clectro-magnetique linearie; le probleme des ondes planes se deplacant dans un champs magnetostatique uniforme est Bgalement tram& ainsi que leur association. Considerant le meme probteme applique a un champ electrostatique uniformc, Ies auteurs montrent que lesondes planes ordmaires se propagent sans changer de vitesse de phase, tandis que les ondes m&aniques sont accompagn~s de Iegers champs electro-magnetiques fluctuants. Le probleme de la vibration dune plaque libre, inliniment

elastique, dans un vaste champ magnetostattque est t aite dans l’hypothese que les champs electromagnetiques r<ants sont quasi stattonnaires. On obtient les equations de la frequence aussi bien porn les vibrations symetrique que pour les vibrations anttsymetriques; l’amortissements cr& par le champ pour les deux premiers modes symetriques et antisymetrrques appatait comme une correction lineaire aux frequences usuelles dune plaque libre. Zusannnenfassung-Die Kopplung elektromagnetischer und elastischer Wellen wird behandelt, und zwar vom Gesichtspunkt der linearen Elastizitlt und einer linearisierten, elektromagnetischen Theorie aus. Es werden Bettachtungen angestellt iibet ebene Wellen, die sich iiber ein gleichformiges, magnetostatisches Feld verbreiten, und die Kopphmgen der Wellen werden untersucht. Eine Untersuchung des gleichen Problems mit einem aleichfBrmiaen. elektrostatischen Feld ergibt. dass die iiblichen ebenen Wellen sich ohne Verlnderung ih&r Phaseng&chwindigkeH verbreiten, dassaber die mechanischen Wellen von kleinen,

unregelmhsigen, elektromagnetischen Feldern begleitet sind. Das Problem einer freien, unendlichen elastischen Platte in einem grossen magnetostatischen Feld wird untersucht, wobei vorausgesetzt wird, dass die resultierenden elektromagnetischen Felder quasistationar sind. Frequenzgleichungen werden sowohl fiir symmetrische wie ftir asymmetrische Vibrationen aufgestellt und die von dem Feld verursachte Dampfung fiir die beiden ersten symmetrischen und asymmetrischen Typen wird als lineare Berichtigung der iiblichen freien Plattenfrequenz festgestellt. Sommario-L’accoppiamento di onde elettromagnetiche ed elasttche viene preso in considerazione da1 punto di vista deh’elasticita lineare e di una teoria elettromagnetica hnearizzata. Si prende in esame il problema delle onde piane the si spostano attravetso un campo magnetostattco uniforme, e si studiano gli accoppiamenti delle onde. Uno studio dello stesso problema per un campo elettrostatico uniforrne dimostra the le normali onde piane si propagano senza alcun cambiamento nelle loro velocita di fase, ma the le onde meccamche sono accompagnate da piccoli campi elettromagnetici fluttuanti. Si prende in considerazione il problema della vibrazione di una piastrina libera elastica infinita in un grande campo magnetostatico, presumendo the i campi elettromagnetici risultanti siano quasi-stazionari. Si rilevano le equazioni di frequenza per le vibrazioni simmetriche e per quelle antisimmetriche, e lo smorzamento provocato da1 campo per ambedue i primi modi, simmetrico e antisimmetrico, si ricava come correzione lineare alle normali frequenze della piastrina libera. A6CTpaHT - CBR3b 3BeKTpOMarHUTHMX H ynpyrHX BOJIH paCCMaTpUBaeTCR C TO=iKH 3p3HHR .TIHHetiHOHynpyrOCTH M nHHeapH3HpOBaHHOH 3neKTpOMarHHTHOti TCOpHH. PaccMaTpHBaeTcs npo6neMa nJIOCKHXBOJIH,npOXOBJHHHXH3p33 OjrHOpOAHOe MarHHTHO-CTaTHHCCHOC nOJI3H H3y%lrOTCR cBB3H B~JIH. klayqemre TOP-me npo6neMbt HnH o~riopo~rroro aneKTpocTaTHHecKor0 norm noKa3brBaeT 4TO o6bI’IHhIe nnOCKOCTHhI3BOnHhl paCnpOCTpaHHIOTCH6e3 H3MBHeHHHB HX @3HbIX CKOPOCTJIX,B To BpeMB KaK MexaHHHecKHe BOJIHMconpoBomBar0Tca He6OnbmHMH @ty~TyHpy~mm~H aneKTpoMarHHTHhtMH IIOJIHMH. %&=I%l BH6paqHB CBO~OHHOH, 633KOHOHHMX pa3MepOB, nJKICTMHKH, HaXOBHmeiCR B 6OJrbmOM MarHHTHO-CTaTH43CKOM nOJI3 paCCMaTpHBaeTCH npH ~OnymeHHH HTO nOny%%OTCR KBaaH-CTauHOHapHhte3Jt3KTpOM3I’HHTHbIO HOJtH. BMBoBHTcH ypaBHeHHR HaCTOTBnFI CHMMeTpHHHhtXH aHTMCHMMeTpH9HbrXBH6psLHHi H BeMn@HpOBaHHB npHHHHHeMO3 nOneM BJHi HBpBbIX HByX, KaK CUMMi?TpHHHMX TaK U aHTHCHMM3TpHHHbIX $OpM, nOnyHaeTCH KaK JHiH&iHiw HOUpaBKa K 06bIHHHM HaCTOTaMOBO6OAHOtinnaCTHHKU.