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On the momentum of potential waves
Physica 83A (1976) 471-485 © North-Holland Publishing Co.
O N THE M O M E N T U M OF P O T E N T I A L WAVES L. J. F. BROER
Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands
Received 21 October 1975
Protracted discussions on concepts like the momentum of electromagnetic waves in matter and the pressure of sound indicate that the interpretation of the momentum balance in nonrelativistic physics can be less clear than that of the energy balance. It is shown that this can be understood from the theory of a special class of wave equations, called potential waves.
1. M o m e n t u m
The notions of momentum density and momentum flux have occassionally caused some difficulties in continuum physics. These difficulties have been discussed in the literature through many years. The best known case is that of electromagnetic waves in polarizable media. More or le~s similar discussions have envolved around mechanical waves. Usually the density and flux then are termed "phonon momentum" and "sound pressure". We will neither attempt to recapitulate these discussions nor to take parts. The only purpose of this paper is to point out the relation between these, real or apparent, ambiguities in the definitions of momentum and a certain property of a simple class of wave equations, which will be termed potential waves. Before considering these equations in section 2 a few preliminary remarks will be made. There is no difference of opinion upon the momentum of a system of particles interacting through mechanical forces, of electromagnetic fields in vacuum and of combinations of those systems. Therefore it can be, and has been, argued that the momentum problem really does not exist. One starts with a model of the nature indicated above and then performs some averaging process taken from statistical mechanics, in order to obtain a consistent continuum or macroscopic description of the system, see e.g. ref. 1. It then will be clear w h a t the averaged momentum of the ensemble really is. It seems obvious that nowadays enough is known about statistical mechanics to make this argument irrefutable. Nevertheless, in practice people still continue setting up continuum theories directly. When these theories 471
472
L. J, F. BROER
are used to interprete measurements of force or pressure the old phenomenological m o m e n t u m problem might come up again in some form. In purely mechanical theories as fluid mechanics and elasticity the use of continuum models is of long and respected standing. There is again no difference upon the interpretation of the energy- and m o m e n t u m balance equations given by these theories. Examples of these equations will be given in section 3. As acoustics is a special case of gas dynamics or elasticity it seems somewhat surprising that there can be a problem about the acoustic m o m e n t u m flux or sound pressure. The reason for this is that the acoustic equations are linearized or first-order approximations in gas dynamics whereas the relevant terms in the m o m e n t u m flux are of second order. The expressions found for this flux therefore could be dependent on neglected second order terms in the equations, that is on the way the linearization has been performed. This turns out to be the case indeed, we will consider this in some more detail in section 3. At first sight one might think that there is no direct relation between the problems of sound pressure and electromagnetic m o m e n t u m as the latter problem does not result from some approximation in the electromagnetic equations. However, there is an approximation involved. When a strong electromagnetic wave runs in, say, an polarizable elastic solid undoubtedly some motion of this medium will be caused. The approximation then consists in neglecting this motion when setting up Maxwell's equation. Looking closer into the interaction of field and motion we will meet the same "mixing of orders" effect which is at the bottom of the sound pressure problem. This effect is, as mentioned above, a property of some potential wave equations. These equations have other interesting properties. The stimulus to look first into the m o m e n t u m equation rather than these other relations came from some interesting discussions with H.B.G. Casimir on electromagnetic momentum. In the next section we will therefore consider only those few aspects of potential waves which will be needed for the m o m e n t u m problem, reserving a more general discussion for a later occasion.
2. Potential waves
When a set of wave equations can be derived from a lagrangian L depending only on derivatives of the independent variables we call these waves potential waves. In the most simple case we have only one variable q and one spatial coordinate x. Moreover we suppose that there is a lagrangian density:
L~' = S dx L (q,, qx), which means that non-local interactions are excluded. The wave equation then is: (G,), + ( G ) ~ = 0.
(2.1)
O N THE M O M E N T U M OF POTENTIAL WAVES P
473
A property of these waves is that after linearization there is no dispersion. For our purposes it is convenient to eliminate the potential q altogether and to express everything in terms of the fields: u = qx,
w = --qt.
In stead of the second order wave equation we then have the first order set: u, + wx = 0,
(2.2)
(Lw), - (L,,)~ = O.
(2.3)
Equation (2.2) is an identity. It depends only on the nature of the problem considered, not on the particular choice of L. Both equations have the form of conservation laws. For eq. (2.3) this is, according to Noether's theorem, a consequence of the fact that L is invariant for a constant variation 6q, it will therefore be called the q-rule. Equation (2.2) is not connected with any variation, it is an example of a so-called trivial conservation law. Two other conservation laws follow from the invariance of L for constant variations & and 6x. One easily finds for the t-rule: ( w L w - L ) t - - (wL,,)x = 0
(2.4)
and for the x-rule: (uL~), + ( L -
uGL
= O.
(2.5)
The, constant, quantity .~ff = ~ d x ( w L w
-
L)
is the hamiltonian of the system. It could be used to write the equations in canonical form. This is not necessary however for the present purpose and we will keep to the lagrangian description. The conservation laws (2.2) to (2.5) are of a rather special kind. Both densities and fluxes depend only on the fields, not on their derivatives or on the coordinates. It is an interesting property of potential waves that there is in general an infinity of these special conservation laws. We will not study this system of conservation laws here, we mention only that it can be constructed explicitely for the equations treated in section 3. It would be rash to conclude that the t- and x-rules correspond to the energy and m o m e n t u m equations respectively. In the cases mentioned in section l, viz. electromagnetic fields in vacuum and mechanical systems, it is not difficult,
474
L.J.F. BROER
although by no means necessary, to choose the variables in such a way that this is true. In other cases some uncertainties can arise as we will see. We now turn to the "mixing of orders" property referred to above. We first observe that constant values of the fields u and w of course provide trivial solutions of the equations (2.2) and (2.3). We now look for steady states u0, Wo which correspond to stationary values of the hamiltonian density. V=
wL~, - L .
As Vw --- wLw~, there can be two kinds of steady states. We will not consider here the possible solutions connected with L~w = 0. Taking Wo = 0 we find that Uo corresponds to the zero's of dLu (0, u)/du. In the sequel we restrict ourselves to systems having only one value for Uo. It then is possible to take H and L equal to zero in the state 0, Uo. When H is positive in all other states Uo corresponds to a stable equilibrium. Many of our formal results are independent of this stability assumption, in the interpretation of the formulae we will however, for physical reasons, assume its validity. Once the existence of a single stable equilibrium being granted it is natural to look for motions "in the neighbourhood" of this equilibrium. One then expands everything in terms of the small quantities w and u - Uo. It is obvious that linear terms in L drop out in the equation of motion (2.3) and therefore can be discarded throughout. Therefore one puts: L
= L (2) + L (3) +
....
Now it is seen that the operations leading to the q- and t-rules are homogeneous. E.g. L ~2~ yields the linear terms in (2.3) and the quadratic terms in (2.4) and so on.
This is however not so for the x-rule as e.g. uLw = uoLw + (u - Uo) Lw. Therefore, the contribution of U 2~ to the x-rule divides between the first and second order terms. On the other hand, if one would require all second order terms in (2.5) one needs L ~2) as well as L t3) in order to write them down. The x-density, which might be the momentum, then reads in second order: ~,(3)
o'~
/(2)
+(U-Uo)~w
-
We notice that the "out of order" part of the x-rule corresponds to Uo times (2.3). Therefore we find by subtraction a homogeneous conservation law: {(u -
Uo)Lw}, + {c -
(u -
Uo)L.},
= 0.
(2.6)
This law corresponds to a constant variation of the quantity x - q/uo, we call it the strain rule. This name will be explained in the next section.
ON THE MOMENTUM OF POTENTIAL WAVES
475
3. Longitudinal elastic waves Mechanical waves can be described either in local or in material coordinates. Some properties of these descriptions and their mutual relation have been given in earlier papers 2' 3). We will state the relevant formulae here as results of potential wave theory without going into details of their derivation. As we will later on look into the coupling of mechanical to electromagnetic waves we settle here for local coordinates. The identity for these potential waves now is the mass equation. In local coordinates this reads:
Ot + (ov)x = O.
(3.1)
Therefore a potential m (x, t) is defined by: 0 = mx,
Qv = w =
-mr.
(3.2)
This potential can be visualized as the mass to the left of the point x. It is a convenient choice when material coordinates are preferred. In mechanical systems the lagrangian usually can be taken as the difference between kinetic energy and potential or deformation energy. We put therefore W2
Lm = - - oU(o), 2O
(3.3)
where U ( o ) is the energy per unit mass. Using the thermodynamic relations for isentropic changes we write for the pressure (opposite of the stress as used in solid mechanics) (3.4)
p = o2U o
and for the enthalpy i =
u + P
(3.5)
= (eU)o.
Q
We now apply the equations (2.3), (2.4) and (2.5) and obtain the equation of motion or m-rule:
w t +
(_;w2) 02 + I
x = 0
(3.6)
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L.J.F. BROER
the m o m e n t u m balance or x-rule
wt +
(w2 ) ,
(3.7)
=0
+p
9
:,
and the energy balance or t-rule
+ gU 9
+ ~
-- + wl 292
=0.
(3.8)
These equations become somewhat more familiar when v is used in stead of w. (3.6) then becomes v~ + vv~ + I x = O ,
which is the Euler equation from gas dynamics. This equation essentially states that, for an isentropic motion, - I is the acceleration potential. The m o m e n t u m equation is obvious when we remember that the Reynolds stress 9vv really stands for the convection of momentum. Equation (3.8) becomes more clear when we write for the total energy per unit volume. 1
W2
V=--~+oU 2 9
and use (3.5). The t-rule then transforms into: v, + ( v V + vp)~ = O,
which obviously is the energy equation as p v is the rate of work per unit cross section. When the equations are applied to solids or liquids there is clearly an equilibrium density 9o. Around 90 U is a second order quantity, its leading term is: U =--a° 2
9_.-- 90 90
(3.9)
where ao is the equilibrium sound speed, p and 1 are of the first order. When (3.9) would be the exact expression for the energy we would have 2
p = ao (9 - 90)
(3.10)
ON THE MOMENTUM OF POTENTIAL WAVES
477
whereas, up to the second order 2 ao
2
I = ~ (92 - 00). 29o For gases we need an additional condition in order to obtain a stable equilibrium. The simplest way is to prescribe a constant averaged density 9o. Using (3.10) we then take p - Po in stead o f p . The strain rule now reads:
[w( I- 0°~10/At-b[-~-(1-~00)q-P-9°l]~ =0.
(3.11)
This rule corresponds with a constant variation of the quantity x - m/Oo. Now rn/9o = Xo is the equilibrium location of the material element present at the point x. Therefore x - Xo is the overall deformation or strain, hence the name. It is to be noted that p
P - OoI = (9 - Oo) I - pU is of the second order. Therefore all leading terms in (3.11) are of second order. We will not enter upon the sound pressure problem again. We only mention that this pressure essentially is the time average of the m o m e n t u m flux at a fixed point. When one considers, e.g., standing waves the terms of odd order average out. Therefore the second order terms are the dominant ones and the result will depend on the assumptions made on the third order terms in the lagrangian (3.3). F r o m (3.9) and (3.10) we infer that, for a purely quadratic U , p (2) = 0. On the other hand, when we restrict ourselves to the quadratic terms in L,, this entails:
U=
a~2 90 (99
-- frO9o ) 2 ,
which yields p~2) =
(~ _ 90)2. 29o
In the case of an ideal gas one finds on expanding the isentropic law: P-Po
=a z(9-Oo)
+ -y - - 1 ao2 ( 9 _ ~ 0 ) 2 + . . ' , 2 90
where Y is the specific heat ratio. The differences in these results illustrate that some information on the nonlinear properties of the model is required to calculate the second order or acoustic approximation of the sound pressure.
478
L.J.F. BROER
4. Electromagnetic waves In this section we consider briefly electromagnetic waves polarised, say, in the y-direction. The potential q then is the y - c o m p o n e n t of the vector potential A, w is E~. and u is B~. We suppose that the waves run in an h o m o g e n e o u s isotropic m e d i u m with given constant polarizability properties. F o r our purposes it is convenient to consider the electric and magnetic polarization per unit mass. M o r e o v e r we will use units of field strength differing a factor e~ from the usual. We then take: L~ = L (E, B) = ½E z - ~'0"-" 1~2~2 + Qo W ( E , B )
(4.1)
where W is a function such that: P = WE,
M = WB.
(4.2)
We will see later that - W is a sort of electromagnetic enthalpy. The identity (2.2) now is Maxwell's first relation: B, + E~ = 0.
(4.3)
The q-rule yields the second Maxwell equation: (E + OoP)t + (cgB -- "~oM)x = O,
(4.4)
whereas the x- and t-rules now read: [c~B(E + .ooP)l, + [z~E2 + ½Co~O2 + q o W - e o S M l x = 0
(4.5)
(1E2 + ,:.22~o_ - e o W +
(4.6)
and OoE?), + [ E ( c o ~ - ,~,oM)lx = 0,
respectively. There is no doubt a b o u t the interpretation of (4.5) and (4.6) as m o m e n t u m and energy balances as far as the v a c u u m terms go. It will turn out later on that, this interpretation of (4.6) meets no particular p r o b l e m s even in presence of the W-dependent terms. The situation as regards in (4.5) is more difficult. Writing (4.4) in the form Dr + H x = O we see that (4.5) is a conservation equation for the quantity BD. It is tempting to consider it as a m o m e n t u m equation. However, the result of the discussions referred to in section 1 seems to favor the expression EH/c~ (in the present units) for the
ON THE MOMENTUM OF POTENTIAL WAVES
479
m o m e n t u m density rather than BD. As there are infinitely many conservation laws there might be one as well for EH. Unfortunately this turns out not to be the case except when E and D, B and H are proportional. At first sight this might be not so serious. We observed already that a strong electromagnetic wave passing through the medium will set it into motion to some extend. Therefore the electromagnetic momentum, whatever that may be, will not necessarily be conserved on its own. But, on the other hand, as L in (4.1) does not contain any dynamical variables connected with this motion of the medium, the foregoing argument tends to cast doubt upon eq. (4.4) as a good approximation. It seems therefore that we are faced with a sort of dilemma. Either the material motion is negligible, (4.4) is a good equation and one has to live with the untraditional expression BD for the electromagnetic m o m e n t u m or the motion is not quite to be neglected, EH/c~ could be the e.m. m o m e n t u m but eq. (4.4) is somewhat suspect. In order to investigate this point we will have to incorporate the motion of the medium into our equations. This will be done by coupling the wave equations considered in this and the preceding section. In this way we will find that the situation actually is a bit more involved than indicated above. The reason for this is that it is not possible to split the total m o m e n t u m density or flux into electrical and mechanical parts. The role of mixed interaction terms will turn out to be essential. Before tackling this problem we will make a few remarks upon the equations above. Simple choices for the function W would be a E or ½0~E2. The first corresponds to a piezoelectric effect, the second to a constant electric polarizability. The first type of interaction exist only in media which have no inversion symmetry. In order to simplify matters as much as possible we will exclude this case by assuming that Wis a function o f E 2 and B 2. Further more we will take W t o vanish for zero field. This then is the equilibrium state, therefore there is no difference as yet between x-rule and strain rule. Equation (4.4) then is of a kind which has been considered in connection with nonlinear optics, e.g. in4). One then could write e.g. W = ½o~E2 (I + EZ/EZ~).
(4.7)
In " h a r d " dielectrics E, is of the order of the atomic fields, say 10 l° volt/meter. In certain "soft" dielectrics it is considerably smaller. The question then is whether it makes sense to include terms of the order E, -2 into (4.4) and (4.5) and at the same time delete all reference to the dynamical variables involved. Finally we write out for later use the difference between both contesting expressions for the electromagnetic m o m e n t u m : ----7
-=
=50
We.B+
Co where K is some function defined by this equation.
W. .
= OoK (E, B ) ,
(4.8)
480
L.J.F. BROER
5. Coupled waves In order to obtain a formalism for coupled waves we could simply write L = Le ( E , B ) + Lm (w, 9) + Li,t (E, B, w, 9)
(5.1)
and apply the rules given in section 2. The resultant general formulae would be complicated and not very instructive. We will therefore use a special simple expression for L i n t . For coupling the mechanical waves to the vacuum field a plausible assumption would be that the specific electromagnetic entalpy W is independent of the density. Therefore :
Li,, = 9 W (E, B).
(5.2)
However, two adjustments must be made. In the first place the average value 9oW already occurs in Le, therefore it has to be taken out from (5.2). This is no problem as it does not contain the dynamical variables, only 90 as a material parameter. The second point is more incisive. Strictly speaking (5.2) applies in the rest system of the material, not in the x - t system. We will not attempt here a completely covariant formulation of the theory. We will use only the first order pseudocovariant theory as will be familiar from chapters on slowly moving matter in textbooks on classical electrodynamics. This means that, in the present contexl, the interaction lagrangian would be: Lint
=
o W (E
-
vB, B -- vE/c~)
with the restriction that only first order terms in v are to be kept. After Taylor expansion, taking out the average interaction and using (4.8) we find that in (5.1) we must write Lint = (9 -
90) W -
(5.3)
wK.
Of course the second term is very small. We know however from section 2 that small terms in Lint can be essential in considering the interpretation of the x-rule, therefore we keep it on. The resulting equations are still fairly complicated. We retain the mass equation and Maxwell's first equation as identities. Then there are two equations of motion, containing interaction terms and either mechanical or electromagnetic variable.% For the Euler equation or m-rule we obtain: w
+ t
w +I 292
)
-Kt-
Wx=0.
(5.4)
The last term explains why-W was termed an electromagnetic specific enthalpy.
ON THE MOMENTUM OF POTENTIAL WAVES
481
The second Maxwell equation or A-rule now reads Dt + Hx +
[(~
-
~o) P -
w K E ] t --
[(~
-- ~o) M
-
= 0.
wKB]x
(5.5)
In the conservation laws we now have three groups of terms corresponding to the three terms in (5.1). Applying the t-rule we find the sum of (4.6), (3.8) and the interaction part which is, using (2.4) and (5.3): [(o - Oo) ( E P
-
W) -
[(~ - ~Oo)E M
wEKe]t -
+ w (W
-
EKn)]x.
(5.6)
Proceeding in the same way the x-rule is found to be the sum of (4.5), (3.7) and: [(e -
~o) ( B P -
g)
-
+ [(~ - ~Oo)BM -
wBK~],
wBKs]x -
OoKt -
Oo W x .
(5.7)
Looking first at (5.4) and (5.5) we observe that these equations do have solutions for which the fields are zero. However, solutions for which the dynamical variables vanish throughout do not exist. The reason for this lack of symmetry lies of course in the form chosen for Lin t in (5.3). This quantity is linear in the dynamic variables and, at least, quadratic in the fields. Therefore the interaction terms in (5.5) vanish for zero field. When we would have admitted piezo-electric or odd terms in Wpure acoustic modes would not occur. Although the second Maxwell equation (4.4) thus is not an exact equation an order of magnitude estimation, given in section 6, will yield the unsurprising result that it is a good approximation in the present case. The same estimate would show that the density and flux contributions to (5.6) are small. Therefore no difficulties are met when the t-rule is interpreted as the exact energy balance of our coupled systems. Density and flux are the sum of electromagnetic parts as given by (4.6), mechanical parts as given by (3.8) and small interaction' terms. This division corresponds with the splitting of the lagrangian according to (5.1) and (5.3). More complicated is the situation for the x-rule. The total density is : BD
+ W -
~ o K + (~ -
~o) ( B P -
K) -
wBKe.
(5.8)
Using (4.8) this also can be written as:
EH/c~
+ w + (~ -
Qo) ( B P -
K) -
wBKe.
(5.9)
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L.J.F, BROER
As the density contains the unsuspected mechanical and vacuum terms w + BE there is no doubt about the interpretation of the x-rule as the m o m e n t u m balance of the coupled system. The last two terms of (5.8) or (5.9) are small according to the estimate mentioned above. However, OoK and BD can be of the same order. In fact, both contain a term ooBP. Therefore, when we want to split, up the m o m e n t u m density and flux into electromagnetical, mechanical and interaction parts we are faced with a dilemma. Using (5.8), taking 0oK as an interaction term, we obtain a splitting which fits nicely to the splitting of the energy equation. The electromagnetic energy and m o m e n t u m density both derive from the term L in (5.1). However, the interaction is not a small contribution now. Essentially this is again a consequence of the "mixing of orders" effect as explained in section 2. In (5.3) wK is a small quantity, the corresponding therm OoK in (5.8) is significantly larger. When we split up the m o m e n t u m equation according to (5.9) we do obtain a situation in which the interaction m o m e n t u m is small compared to both the electromagnetic and mechanical momentum. This entails, however, that ofie has to use an expression for the electromagnetic m o m e n t u m which does not stem from the same part of the total lagrangian as the electromagnetic energy. We note that neglecting the second term in (5.3) or the use of (5.2) to couple vacuum fields to mechanical waves would lead to the conclusion that BD would be preferable as the electromagnetic m o m e n t u m density. Both arguments, small interaction and lagrangian connection with the energy equation, then would act in favor of (5.8). At the present stage we do not see any compelling reason to prefer one of these alternatives. It might be that (5.8) is somewhat more convenient for general reasoning, especially when we would set up the theory in lagrangian form in a completely covariant way. On the other hand (5.9) could be more convenient for the conceptual interpretation of experiments in thought with Simple models. In our opinion there is also no reason to feel very unhappy about this inconclusive situation. The complete balance equations of energy and m o m e n t u m are the hard facts of this segment of mathematical physics. When, for some purpose, a splitting is required there is no harm in the possibility to choose a splitting according to this purpose and the approximations involved by it. There is one point however which still needs some attention. Let us take for granted that often the Maxwell equation (4.4) is a good approximation to (5.5). According to the results in section 4 EH/c~ then is not a particular well conserved quantity. This is in agreement with the estimates in section 6 which will show that w in (5.9) also can be of the order OoBP and therefore of relatively more importance than the interaction terms in (5.5). However, ooK is at least of the same order, nevertheless BD is expected to be a better conserved quantity because it is exactly conserved when the motion of the medium is neglected from the outset as was done in section 4. When we look more closely at our formulae we see that the troublesome terms in (5.7) are proportional to the non-mechanical terms in (5.4) and hence can be
ON THE MOMENTUM OF POTENTIAL WAVES
483
eliminated. This amounts to applying the strain rule to (5.1) and (5.3). The result is:
[BD + (1+
[2
~)
w + (°~ - ~o) (BP - K) - wBKEIt
2
E 2 + c--L°B 2 + O ~ o ( W 2
BM) +
W2 ( ~1 -_ )
+ P - poI + (e - eo) B M - wBKE~ = 0. d
x
(5.10)
This equation is an exact consequence of (5.1) and (5.3). The seperation of terms goes in the same way as in the energy equation and corresponds to (5.1). The electromagnetic part is the same as (4.5). The mechanical part is (3.11). It is of the second order in the dynamical variables which means, as we will see, that it is really small. The same applies to the interaction part. Our conclusion is therefore that BD is a better conserved quantity than EH/c~. In the first case the leading terms of the interaction and mechanical contributions cancel out, moreover the dominant part of w, which can, as we shall see, be of the same order as ooBP, has been compensated in transferring from the x-rule to the strain rule.
6. Some estimates for travelling waves In order to estimate the order of magnitude of the various terms in the complicated relations derived above we ought to specify .the initial and boundary conditions which excite the field and the motion. Obviously the results will depend on the ratio of the mechanical and electromagnetical excitation. As our main aim is to find out what the quality of the no-motion approximation in section 4 is we assume from the start that there is no mechanical excitation at all. That is, there is only a forced motion due to the fields. A glance at eq: (5.4) learns us that the dynamical variables in a forced wave solution will be of the order of the fields squared. This looks promising for the Maxwell equation (5,5). In order to obtain some estimates we first restrict ourselves to the leading term of (4.7) and put:
W = ½~E 2,
K = ~EB.
Assume now that the dominant part of the field is a travelling wave
E = f ( x - ct).
(6.1)
Then, from Maxwell's equations,
B = E/e,
c 2 (1 + o~Q0) = Co 2.
(6.2)
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L.J.F. BROER
For the forced mechanical wave we put m -- 9o x = g ( x
-- ct)
and therefore : w = eg',
~ -
(6.3)
90 = g ' .
Keeping only linear dynamical terms in (5.4) we obtain the basic result: g, =
1 2
O%Oo f a _ c2 -
10~o fz
ao2
2
(6.4)
C2
as ao ~ c. F r o m (6.3) and (6.4) we deduce. 9 o r ~-- w =
-½~9oEB
=
-½9oPB,
which shows that the mechanical m o m e n t u m can indeed not be neglected compared with other second order field terms in the m o m e n t u m balance. In the second Maxwell equation the first interaction terms clearly are of the third order. It is of interest therefore to see how they compare with the third order dielectric terms derived from the second term in (4.7). The order of magnitude of e.g. the term w K ~ is: wKE ~ PB.
~xB ~" ~xz9 ( E 3 / c 2 ) ,
whereas the dielectric third order term is of the order O~9o ( E a / E 2 ) . Therefore the ratio between these two kinds of terms is of the order ~ E 2 / e 2. This is a rather small number, even for hard dielectrics. To understand this we observe that, in a very rough approximation, E, is of the order of the field where polarization energy and ionization energy become comparable. This is really another way to state what is meant by a hard dielectric. The ratio given above then becomes of the order of the square of the fine structure constant, say 10 -5. Therefore inclussion of these fourth order dielectric terms in the Maxwell equation (4.4) seems perfectly safe. One final remark concerns the fact that, in the approximation considered here: 9 -- 9o
v
~o
c
instead of v / a o as in acoustics. (Incidentaly, this entails that both terms in (5.3) are of the same order of magnitude here.) The reason for this is of course that for forced waves at a given frequency the wavelength is much larger than in a free
ON THE MOMENTUM OF POTENTIAL WAVES
485
sound wave. Once this being understood one could try to use this fact to simplify the equations for forced waves. This indeed possible but not by taking 9 = 90 everywhere. Essentially this would mean the use of a rigid body model, which is relativistically unsound. Moreover it is rather paradoxial to use a model where ao >> Co to obtain effects due to the fact that ao '~ Co. The correct way to proceed is to retain the mass identity and the density dependent terms in Lin t but to strike out U from Lm. In this way all terms involving U, I and p vanish and one obtains the complete equations corresponding to the limit ao ~ 0 in the acoustic approximation. The resulting equations will provide an adequate description of the fields together with the forced elastic waves although the free waves are degenerate in this approximation.
References I) S.R. de G-root and L.G.Suttorp, Foundations of Electrodynamics (North-Holland Publ. Comp., Amsterdam, 1972). 2) L.J.F.Broer, J. Eng. Math. 4 (1970) 1. 3) L.J.F.Broer and J.A.Kobussen, Appl. Sci. Res. 29 (1974) 419. 4) L.J.F. Broer and P.N.A.Sarluy, Physica 30 (1964) 1421,
O N THE M O M E N T U M OF P O T E N T I A L WAVES L. J. F. BROER
Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands
Received 21 October 1975
Protracted discussions on concepts like the momentum of electromagnetic waves in matter and the pressure of sound indicate that the interpretation of the momentum balance in nonrelativistic physics can be less clear than that of the energy balance. It is shown that this can be understood from the theory of a special class of wave equations, called potential waves.
1. M o m e n t u m
The notions of momentum density and momentum flux have occassionally caused some difficulties in continuum physics. These difficulties have been discussed in the literature through many years. The best known case is that of electromagnetic waves in polarizable media. More or le~s similar discussions have envolved around mechanical waves. Usually the density and flux then are termed "phonon momentum" and "sound pressure". We will neither attempt to recapitulate these discussions nor to take parts. The only purpose of this paper is to point out the relation between these, real or apparent, ambiguities in the definitions of momentum and a certain property of a simple class of wave equations, which will be termed potential waves. Before considering these equations in section 2 a few preliminary remarks will be made. There is no difference of opinion upon the momentum of a system of particles interacting through mechanical forces, of electromagnetic fields in vacuum and of combinations of those systems. Therefore it can be, and has been, argued that the momentum problem really does not exist. One starts with a model of the nature indicated above and then performs some averaging process taken from statistical mechanics, in order to obtain a consistent continuum or macroscopic description of the system, see e.g. ref. 1. It then will be clear w h a t the averaged momentum of the ensemble really is. It seems obvious that nowadays enough is known about statistical mechanics to make this argument irrefutable. Nevertheless, in practice people still continue setting up continuum theories directly. When these theories 471
472
L. J, F. BROER
are used to interprete measurements of force or pressure the old phenomenological m o m e n t u m problem might come up again in some form. In purely mechanical theories as fluid mechanics and elasticity the use of continuum models is of long and respected standing. There is again no difference upon the interpretation of the energy- and m o m e n t u m balance equations given by these theories. Examples of these equations will be given in section 3. As acoustics is a special case of gas dynamics or elasticity it seems somewhat surprising that there can be a problem about the acoustic m o m e n t u m flux or sound pressure. The reason for this is that the acoustic equations are linearized or first-order approximations in gas dynamics whereas the relevant terms in the m o m e n t u m flux are of second order. The expressions found for this flux therefore could be dependent on neglected second order terms in the equations, that is on the way the linearization has been performed. This turns out to be the case indeed, we will consider this in some more detail in section 3. At first sight one might think that there is no direct relation between the problems of sound pressure and electromagnetic m o m e n t u m as the latter problem does not result from some approximation in the electromagnetic equations. However, there is an approximation involved. When a strong electromagnetic wave runs in, say, an polarizable elastic solid undoubtedly some motion of this medium will be caused. The approximation then consists in neglecting this motion when setting up Maxwell's equation. Looking closer into the interaction of field and motion we will meet the same "mixing of orders" effect which is at the bottom of the sound pressure problem. This effect is, as mentioned above, a property of some potential wave equations. These equations have other interesting properties. The stimulus to look first into the m o m e n t u m equation rather than these other relations came from some interesting discussions with H.B.G. Casimir on electromagnetic momentum. In the next section we will therefore consider only those few aspects of potential waves which will be needed for the m o m e n t u m problem, reserving a more general discussion for a later occasion.
2. Potential waves
When a set of wave equations can be derived from a lagrangian L depending only on derivatives of the independent variables we call these waves potential waves. In the most simple case we have only one variable q and one spatial coordinate x. Moreover we suppose that there is a lagrangian density:
L~' = S dx L (q,, qx), which means that non-local interactions are excluded. The wave equation then is: (G,), + ( G ) ~ = 0.
(2.1)
O N THE M O M E N T U M OF POTENTIAL WAVES P
473
A property of these waves is that after linearization there is no dispersion. For our purposes it is convenient to eliminate the potential q altogether and to express everything in terms of the fields: u = qx,
w = --qt.
In stead of the second order wave equation we then have the first order set: u, + wx = 0,
(2.2)
(Lw), - (L,,)~ = O.
(2.3)
Equation (2.2) is an identity. It depends only on the nature of the problem considered, not on the particular choice of L. Both equations have the form of conservation laws. For eq. (2.3) this is, according to Noether's theorem, a consequence of the fact that L is invariant for a constant variation 6q, it will therefore be called the q-rule. Equation (2.2) is not connected with any variation, it is an example of a so-called trivial conservation law. Two other conservation laws follow from the invariance of L for constant variations & and 6x. One easily finds for the t-rule: ( w L w - L ) t - - (wL,,)x = 0
(2.4)
and for the x-rule: (uL~), + ( L -
uGL
= O.
(2.5)
The, constant, quantity .~ff = ~ d x ( w L w
-
L)
is the hamiltonian of the system. It could be used to write the equations in canonical form. This is not necessary however for the present purpose and we will keep to the lagrangian description. The conservation laws (2.2) to (2.5) are of a rather special kind. Both densities and fluxes depend only on the fields, not on their derivatives or on the coordinates. It is an interesting property of potential waves that there is in general an infinity of these special conservation laws. We will not study this system of conservation laws here, we mention only that it can be constructed explicitely for the equations treated in section 3. It would be rash to conclude that the t- and x-rules correspond to the energy and m o m e n t u m equations respectively. In the cases mentioned in section l, viz. electromagnetic fields in vacuum and mechanical systems, it is not difficult,
474
L.J.F. BROER
although by no means necessary, to choose the variables in such a way that this is true. In other cases some uncertainties can arise as we will see. We now turn to the "mixing of orders" property referred to above. We first observe that constant values of the fields u and w of course provide trivial solutions of the equations (2.2) and (2.3). We now look for steady states u0, Wo which correspond to stationary values of the hamiltonian density. V=
wL~, - L .
As Vw --- wLw~, there can be two kinds of steady states. We will not consider here the possible solutions connected with L~w = 0. Taking Wo = 0 we find that Uo corresponds to the zero's of dLu (0, u)/du. In the sequel we restrict ourselves to systems having only one value for Uo. It then is possible to take H and L equal to zero in the state 0, Uo. When H is positive in all other states Uo corresponds to a stable equilibrium. Many of our formal results are independent of this stability assumption, in the interpretation of the formulae we will however, for physical reasons, assume its validity. Once the existence of a single stable equilibrium being granted it is natural to look for motions "in the neighbourhood" of this equilibrium. One then expands everything in terms of the small quantities w and u - Uo. It is obvious that linear terms in L drop out in the equation of motion (2.3) and therefore can be discarded throughout. Therefore one puts: L
= L (2) + L (3) +
....
Now it is seen that the operations leading to the q- and t-rules are homogeneous. E.g. L ~2~ yields the linear terms in (2.3) and the quadratic terms in (2.4) and so on.
This is however not so for the x-rule as e.g. uLw = uoLw + (u - Uo) Lw. Therefore, the contribution of U 2~ to the x-rule divides between the first and second order terms. On the other hand, if one would require all second order terms in (2.5) one needs L ~2) as well as L t3) in order to write them down. The x-density, which might be the momentum, then reads in second order: ~,(3)
o'~
/(2)
+(U-Uo)~w
-
We notice that the "out of order" part of the x-rule corresponds to Uo times (2.3). Therefore we find by subtraction a homogeneous conservation law: {(u -
Uo)Lw}, + {c -
(u -
Uo)L.},
= 0.
(2.6)
This law corresponds to a constant variation of the quantity x - q/uo, we call it the strain rule. This name will be explained in the next section.
ON THE MOMENTUM OF POTENTIAL WAVES
475
3. Longitudinal elastic waves Mechanical waves can be described either in local or in material coordinates. Some properties of these descriptions and their mutual relation have been given in earlier papers 2' 3). We will state the relevant formulae here as results of potential wave theory without going into details of their derivation. As we will later on look into the coupling of mechanical to electromagnetic waves we settle here for local coordinates. The identity for these potential waves now is the mass equation. In local coordinates this reads:
Ot + (ov)x = O.
(3.1)
Therefore a potential m (x, t) is defined by: 0 = mx,
Qv = w =
-mr.
(3.2)
This potential can be visualized as the mass to the left of the point x. It is a convenient choice when material coordinates are preferred. In mechanical systems the lagrangian usually can be taken as the difference between kinetic energy and potential or deformation energy. We put therefore W2
Lm = - - oU(o), 2O
(3.3)
where U ( o ) is the energy per unit mass. Using the thermodynamic relations for isentropic changes we write for the pressure (opposite of the stress as used in solid mechanics) (3.4)
p = o2U o
and for the enthalpy i =
u + P
(3.5)
= (eU)o.
Q
We now apply the equations (2.3), (2.4) and (2.5) and obtain the equation of motion or m-rule:
w t +
(_;w2) 02 + I
x = 0
(3.6)
476
L.J.F. BROER
the m o m e n t u m balance or x-rule
wt +
(w2 ) ,
(3.7)
=0
+p
9
:,
and the energy balance or t-rule
+ gU 9
+ ~
-- + wl 292
=0.
(3.8)
These equations become somewhat more familiar when v is used in stead of w. (3.6) then becomes v~ + vv~ + I x = O ,
which is the Euler equation from gas dynamics. This equation essentially states that, for an isentropic motion, - I is the acceleration potential. The m o m e n t u m equation is obvious when we remember that the Reynolds stress 9vv really stands for the convection of momentum. Equation (3.8) becomes more clear when we write for the total energy per unit volume. 1
W2
V=--~+oU 2 9
and use (3.5). The t-rule then transforms into: v, + ( v V + vp)~ = O,
which obviously is the energy equation as p v is the rate of work per unit cross section. When the equations are applied to solids or liquids there is clearly an equilibrium density 9o. Around 90 U is a second order quantity, its leading term is: U =--a° 2
9_.-- 90 90
(3.9)
where ao is the equilibrium sound speed, p and 1 are of the first order. When (3.9) would be the exact expression for the energy we would have 2
p = ao (9 - 90)
(3.10)
ON THE MOMENTUM OF POTENTIAL WAVES
477
whereas, up to the second order 2 ao
2
I = ~ (92 - 00). 29o For gases we need an additional condition in order to obtain a stable equilibrium. The simplest way is to prescribe a constant averaged density 9o. Using (3.10) we then take p - Po in stead o f p . The strain rule now reads:
[w( I- 0°~10/At-b[-~-(1-~00)q-P-9°l]~ =0.
(3.11)
This rule corresponds with a constant variation of the quantity x - m/Oo. Now rn/9o = Xo is the equilibrium location of the material element present at the point x. Therefore x - Xo is the overall deformation or strain, hence the name. It is to be noted that p
P - OoI = (9 - Oo) I - pU is of the second order. Therefore all leading terms in (3.11) are of second order. We will not enter upon the sound pressure problem again. We only mention that this pressure essentially is the time average of the m o m e n t u m flux at a fixed point. When one considers, e.g., standing waves the terms of odd order average out. Therefore the second order terms are the dominant ones and the result will depend on the assumptions made on the third order terms in the lagrangian (3.3). F r o m (3.9) and (3.10) we infer that, for a purely quadratic U , p (2) = 0. On the other hand, when we restrict ourselves to the quadratic terms in L,, this entails:
U=
a~2 90 (99
-- frO9o ) 2 ,
which yields p~2) =
(~ _ 90)2. 29o
In the case of an ideal gas one finds on expanding the isentropic law: P-Po
=a z(9-Oo)
+ -y - - 1 ao2 ( 9 _ ~ 0 ) 2 + . . ' , 2 90
where Y is the specific heat ratio. The differences in these results illustrate that some information on the nonlinear properties of the model is required to calculate the second order or acoustic approximation of the sound pressure.
478
L.J.F. BROER
4. Electromagnetic waves In this section we consider briefly electromagnetic waves polarised, say, in the y-direction. The potential q then is the y - c o m p o n e n t of the vector potential A, w is E~. and u is B~. We suppose that the waves run in an h o m o g e n e o u s isotropic m e d i u m with given constant polarizability properties. F o r our purposes it is convenient to consider the electric and magnetic polarization per unit mass. M o r e o v e r we will use units of field strength differing a factor e~ from the usual. We then take: L~ = L (E, B) = ½E z - ~'0"-" 1~2~2 + Qo W ( E , B )
(4.1)
where W is a function such that: P = WE,
M = WB.
(4.2)
We will see later that - W is a sort of electromagnetic enthalpy. The identity (2.2) now is Maxwell's first relation: B, + E~ = 0.
(4.3)
The q-rule yields the second Maxwell equation: (E + OoP)t + (cgB -- "~oM)x = O,
(4.4)
whereas the x- and t-rules now read: [c~B(E + .ooP)l, + [z~E2 + ½Co~O2 + q o W - e o S M l x = 0
(4.5)
(1E2 + ,:.22~o_ - e o W +
(4.6)
and OoE?), + [ E ( c o ~ - ,~,oM)lx = 0,
respectively. There is no doubt a b o u t the interpretation of (4.5) and (4.6) as m o m e n t u m and energy balances as far as the v a c u u m terms go. It will turn out later on that, this interpretation of (4.6) meets no particular p r o b l e m s even in presence of the W-dependent terms. The situation as regards in (4.5) is more difficult. Writing (4.4) in the form Dr + H x = O we see that (4.5) is a conservation equation for the quantity BD. It is tempting to consider it as a m o m e n t u m equation. However, the result of the discussions referred to in section 1 seems to favor the expression EH/c~ (in the present units) for the
ON THE MOMENTUM OF POTENTIAL WAVES
479
m o m e n t u m density rather than BD. As there are infinitely many conservation laws there might be one as well for EH. Unfortunately this turns out not to be the case except when E and D, B and H are proportional. At first sight this might be not so serious. We observed already that a strong electromagnetic wave passing through the medium will set it into motion to some extend. Therefore the electromagnetic momentum, whatever that may be, will not necessarily be conserved on its own. But, on the other hand, as L in (4.1) does not contain any dynamical variables connected with this motion of the medium, the foregoing argument tends to cast doubt upon eq. (4.4) as a good approximation. It seems therefore that we are faced with a sort of dilemma. Either the material motion is negligible, (4.4) is a good equation and one has to live with the untraditional expression BD for the electromagnetic m o m e n t u m or the motion is not quite to be neglected, EH/c~ could be the e.m. m o m e n t u m but eq. (4.4) is somewhat suspect. In order to investigate this point we will have to incorporate the motion of the medium into our equations. This will be done by coupling the wave equations considered in this and the preceding section. In this way we will find that the situation actually is a bit more involved than indicated above. The reason for this is that it is not possible to split the total m o m e n t u m density or flux into electrical and mechanical parts. The role of mixed interaction terms will turn out to be essential. Before tackling this problem we will make a few remarks upon the equations above. Simple choices for the function W would be a E or ½0~E2. The first corresponds to a piezoelectric effect, the second to a constant electric polarizability. The first type of interaction exist only in media which have no inversion symmetry. In order to simplify matters as much as possible we will exclude this case by assuming that Wis a function o f E 2 and B 2. Further more we will take W t o vanish for zero field. This then is the equilibrium state, therefore there is no difference as yet between x-rule and strain rule. Equation (4.4) then is of a kind which has been considered in connection with nonlinear optics, e.g. in4). One then could write e.g. W = ½o~E2 (I + EZ/EZ~).
(4.7)
In " h a r d " dielectrics E, is of the order of the atomic fields, say 10 l° volt/meter. In certain "soft" dielectrics it is considerably smaller. The question then is whether it makes sense to include terms of the order E, -2 into (4.4) and (4.5) and at the same time delete all reference to the dynamical variables involved. Finally we write out for later use the difference between both contesting expressions for the electromagnetic m o m e n t u m : ----7
-=
=50
We.B+
Co where K is some function defined by this equation.
W. .
= OoK (E, B ) ,
(4.8)
480
L.J.F. BROER
5. Coupled waves In order to obtain a formalism for coupled waves we could simply write L = Le ( E , B ) + Lm (w, 9) + Li,t (E, B, w, 9)
(5.1)
and apply the rules given in section 2. The resultant general formulae would be complicated and not very instructive. We will therefore use a special simple expression for L i n t . For coupling the mechanical waves to the vacuum field a plausible assumption would be that the specific electromagnetic entalpy W is independent of the density. Therefore :
Li,, = 9 W (E, B).
(5.2)
However, two adjustments must be made. In the first place the average value 9oW already occurs in Le, therefore it has to be taken out from (5.2). This is no problem as it does not contain the dynamical variables, only 90 as a material parameter. The second point is more incisive. Strictly speaking (5.2) applies in the rest system of the material, not in the x - t system. We will not attempt here a completely covariant formulation of the theory. We will use only the first order pseudocovariant theory as will be familiar from chapters on slowly moving matter in textbooks on classical electrodynamics. This means that, in the present contexl, the interaction lagrangian would be: Lint
=
o W (E
-
vB, B -- vE/c~)
with the restriction that only first order terms in v are to be kept. After Taylor expansion, taking out the average interaction and using (4.8) we find that in (5.1) we must write Lint = (9 -
90) W -
(5.3)
wK.
Of course the second term is very small. We know however from section 2 that small terms in Lint can be essential in considering the interpretation of the x-rule, therefore we keep it on. The resulting equations are still fairly complicated. We retain the mass equation and Maxwell's first equation as identities. Then there are two equations of motion, containing interaction terms and either mechanical or electromagnetic variable.% For the Euler equation or m-rule we obtain: w
+ t
w +I 292
)
-Kt-
Wx=0.
(5.4)
The last term explains why-W was termed an electromagnetic specific enthalpy.
ON THE MOMENTUM OF POTENTIAL WAVES
481
The second Maxwell equation or A-rule now reads Dt + Hx +
[(~
-
~o) P -
w K E ] t --
[(~
-- ~o) M
-
= 0.
wKB]x
(5.5)
In the conservation laws we now have three groups of terms corresponding to the three terms in (5.1). Applying the t-rule we find the sum of (4.6), (3.8) and the interaction part which is, using (2.4) and (5.3): [(o - Oo) ( E P
-
W) -
[(~ - ~Oo)E M
wEKe]t -
+ w (W
-
EKn)]x.
(5.6)
Proceeding in the same way the x-rule is found to be the sum of (4.5), (3.7) and: [(e -
~o) ( B P -
g)
-
+ [(~ - ~Oo)BM -
wBK~],
wBKs]x -
OoKt -
Oo W x .
(5.7)
Looking first at (5.4) and (5.5) we observe that these equations do have solutions for which the fields are zero. However, solutions for which the dynamical variables vanish throughout do not exist. The reason for this lack of symmetry lies of course in the form chosen for Lin t in (5.3). This quantity is linear in the dynamic variables and, at least, quadratic in the fields. Therefore the interaction terms in (5.5) vanish for zero field. When we would have admitted piezo-electric or odd terms in Wpure acoustic modes would not occur. Although the second Maxwell equation (4.4) thus is not an exact equation an order of magnitude estimation, given in section 6, will yield the unsurprising result that it is a good approximation in the present case. The same estimate would show that the density and flux contributions to (5.6) are small. Therefore no difficulties are met when the t-rule is interpreted as the exact energy balance of our coupled systems. Density and flux are the sum of electromagnetic parts as given by (4.6), mechanical parts as given by (3.8) and small interaction' terms. This division corresponds with the splitting of the lagrangian according to (5.1) and (5.3). More complicated is the situation for the x-rule. The total density is : BD
+ W -
~ o K + (~ -
~o) ( B P -
K) -
wBKe.
(5.8)
Using (4.8) this also can be written as:
EH/c~
+ w + (~ -
Qo) ( B P -
K) -
wBKe.
(5.9)
482
L.J.F, BROER
As the density contains the unsuspected mechanical and vacuum terms w + BE there is no doubt about the interpretation of the x-rule as the m o m e n t u m balance of the coupled system. The last two terms of (5.8) or (5.9) are small according to the estimate mentioned above. However, OoK and BD can be of the same order. In fact, both contain a term ooBP. Therefore, when we want to split, up the m o m e n t u m density and flux into electromagnetical, mechanical and interaction parts we are faced with a dilemma. Using (5.8), taking 0oK as an interaction term, we obtain a splitting which fits nicely to the splitting of the energy equation. The electromagnetic energy and m o m e n t u m density both derive from the term L in (5.1). However, the interaction is not a small contribution now. Essentially this is again a consequence of the "mixing of orders" effect as explained in section 2. In (5.3) wK is a small quantity, the corresponding therm OoK in (5.8) is significantly larger. When we split up the m o m e n t u m equation according to (5.9) we do obtain a situation in which the interaction m o m e n t u m is small compared to both the electromagnetic and mechanical momentum. This entails, however, that ofie has to use an expression for the electromagnetic m o m e n t u m which does not stem from the same part of the total lagrangian as the electromagnetic energy. We note that neglecting the second term in (5.3) or the use of (5.2) to couple vacuum fields to mechanical waves would lead to the conclusion that BD would be preferable as the electromagnetic m o m e n t u m density. Both arguments, small interaction and lagrangian connection with the energy equation, then would act in favor of (5.8). At the present stage we do not see any compelling reason to prefer one of these alternatives. It might be that (5.8) is somewhat more convenient for general reasoning, especially when we would set up the theory in lagrangian form in a completely covariant way. On the other hand (5.9) could be more convenient for the conceptual interpretation of experiments in thought with Simple models. In our opinion there is also no reason to feel very unhappy about this inconclusive situation. The complete balance equations of energy and m o m e n t u m are the hard facts of this segment of mathematical physics. When, for some purpose, a splitting is required there is no harm in the possibility to choose a splitting according to this purpose and the approximations involved by it. There is one point however which still needs some attention. Let us take for granted that often the Maxwell equation (4.4) is a good approximation to (5.5). According to the results in section 4 EH/c~ then is not a particular well conserved quantity. This is in agreement with the estimates in section 6 which will show that w in (5.9) also can be of the order OoBP and therefore of relatively more importance than the interaction terms in (5.5). However, ooK is at least of the same order, nevertheless BD is expected to be a better conserved quantity because it is exactly conserved when the motion of the medium is neglected from the outset as was done in section 4. When we look more closely at our formulae we see that the troublesome terms in (5.7) are proportional to the non-mechanical terms in (5.4) and hence can be
ON THE MOMENTUM OF POTENTIAL WAVES
483
eliminated. This amounts to applying the strain rule to (5.1) and (5.3). The result is:
[BD + (1+
[2
~)
w + (°~ - ~o) (BP - K) - wBKEIt
2
E 2 + c--L°B 2 + O ~ o ( W 2
BM) +
W2 ( ~1 -_ )
+ P - poI + (e - eo) B M - wBKE~ = 0. d
x
(5.10)
This equation is an exact consequence of (5.1) and (5.3). The seperation of terms goes in the same way as in the energy equation and corresponds to (5.1). The electromagnetic part is the same as (4.5). The mechanical part is (3.11). It is of the second order in the dynamical variables which means, as we will see, that it is really small. The same applies to the interaction part. Our conclusion is therefore that BD is a better conserved quantity than EH/c~. In the first case the leading terms of the interaction and mechanical contributions cancel out, moreover the dominant part of w, which can, as we shall see, be of the same order as ooBP, has been compensated in transferring from the x-rule to the strain rule.
6. Some estimates for travelling waves In order to estimate the order of magnitude of the various terms in the complicated relations derived above we ought to specify .the initial and boundary conditions which excite the field and the motion. Obviously the results will depend on the ratio of the mechanical and electromagnetical excitation. As our main aim is to find out what the quality of the no-motion approximation in section 4 is we assume from the start that there is no mechanical excitation at all. That is, there is only a forced motion due to the fields. A glance at eq: (5.4) learns us that the dynamical variables in a forced wave solution will be of the order of the fields squared. This looks promising for the Maxwell equation (5,5). In order to obtain some estimates we first restrict ourselves to the leading term of (4.7) and put:
W = ½~E 2,
K = ~EB.
Assume now that the dominant part of the field is a travelling wave
E = f ( x - ct).
(6.1)
Then, from Maxwell's equations,
B = E/e,
c 2 (1 + o~Q0) = Co 2.
(6.2)
484
L.J.F. BROER
For the forced mechanical wave we put m -- 9o x = g ( x
-- ct)
and therefore : w = eg',
~ -
(6.3)
90 = g ' .
Keeping only linear dynamical terms in (5.4) we obtain the basic result: g, =
1 2
O%Oo f a _ c2 -
10~o fz
ao2
2
(6.4)
C2
as ao ~ c. F r o m (6.3) and (6.4) we deduce. 9 o r ~-- w =
-½~9oEB
=
-½9oPB,
which shows that the mechanical m o m e n t u m can indeed not be neglected compared with other second order field terms in the m o m e n t u m balance. In the second Maxwell equation the first interaction terms clearly are of the third order. It is of interest therefore to see how they compare with the third order dielectric terms derived from the second term in (4.7). The order of magnitude of e.g. the term w K ~ is: wKE ~ PB.
~xB ~" ~xz9 ( E 3 / c 2 ) ,
whereas the dielectric third order term is of the order O~9o ( E a / E 2 ) . Therefore the ratio between these two kinds of terms is of the order ~ E 2 / e 2. This is a rather small number, even for hard dielectrics. To understand this we observe that, in a very rough approximation, E, is of the order of the field where polarization energy and ionization energy become comparable. This is really another way to state what is meant by a hard dielectric. The ratio given above then becomes of the order of the square of the fine structure constant, say 10 -5. Therefore inclussion of these fourth order dielectric terms in the Maxwell equation (4.4) seems perfectly safe. One final remark concerns the fact that, in the approximation considered here: 9 -- 9o
v
~o
c
instead of v / a o as in acoustics. (Incidentaly, this entails that both terms in (5.3) are of the same order of magnitude here.) The reason for this is of course that for forced waves at a given frequency the wavelength is much larger than in a free
ON THE MOMENTUM OF POTENTIAL WAVES
485
sound wave. Once this being understood one could try to use this fact to simplify the equations for forced waves. This indeed possible but not by taking 9 = 90 everywhere. Essentially this would mean the use of a rigid body model, which is relativistically unsound. Moreover it is rather paradoxial to use a model where ao >> Co to obtain effects due to the fact that ao '~ Co. The correct way to proceed is to retain the mass identity and the density dependent terms in Lin t but to strike out U from Lm. In this way all terms involving U, I and p vanish and one obtains the complete equations corresponding to the limit ao ~ 0 in the acoustic approximation. The resulting equations will provide an adequate description of the fields together with the forced elastic waves although the free waves are degenerate in this approximation.
References I) S.R. de G-root and L.G.Suttorp, Foundations of Electrodynamics (North-Holland Publ. Comp., Amsterdam, 1972). 2) L.J.F.Broer, J. Eng. Math. 4 (1970) 1. 3) L.J.F.Broer and J.A.Kobussen, Appl. Sci. Res. 29 (1974) 419. 4) L.J.F. Broer and P.N.A.Sarluy, Physica 30 (1964) 1421,