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Reprinted from THE PHYSICAL REVIEW, Vol. 135, No. 6A, A1713-A1724,14 September 1964

Printed in U. S. A.

Macroscopic Theory of Helicons


Laboratory of Atomic and Solid State Physics, Department of Physics, Cornell University, Ithaca, New York

(Received 27 April 1964)

Methods for treating boundary-value problems involving helicon waves (whistlers in solids) are developed and used for infinite plates and cylinders. The magnetoplasma inside the solid is assumed to be "driven" by means of external coils, which set up an oscillatory field with sinusoidal variation along the two coordinates tangential to the surface of the sample. The results show that in surfaces parallel to the external magnetic field an unusual surface mode is present; in this mode (for small resistivities) the power absorption due to Joule heating fails to decrease as the resistivity is decreased, until the limit of anomalous skin effect is reached, in which limit the lossy mode disappears. Several remarks are made concerning the various geometrical and physical properties of helicons.


MAGNETOPLASMA oscillations obeying the same equations as atmospheric radio whistlers were first reported in solids (sodium) by Bowers, Legendy, and Rose; in the context of solid-state physics they are known as helicons. The name is due to Aigrain, who first proposed achievable experiments to detect them in solids.

Sets of resonant frequencies in various materials, in addition to Na, were observed by Cotti, Wyder, and Quattropani (In, Al, and Cu); Chambers and Jones

* This work was supported in part by the U. S. Atomic Energy Commission.

+ Present address: United Aircraft Corporation Research Laboratories, East Hartford, Connecticut.

1 L. R. O. Storey, Phil. Trans. Roy. Soc. (London) 246A, 113 (1953).

2 R. Bowers, C. R. Legéndy, and F. E. Rose, Phys. Rev. Letters

71,339 (1961).

P. Aigrain, Proceedings of the International Conference on Semiconductor Physics, Prague, 1960 (Publishing House of the Czechoslovak Academy of Sciences, Prague, 1961), p. 224.

However, see also O. V. Konstantinov and V. I. Perel', Zh. Eksperim. i Teor. Fiz. 38, 161 (1960) [English transl.: Soviet Phys.-JETP 11, 117 (1960)]. This article deals with electromagnetic waves in a metal, in a magnetic field. The authors apparently did not recognize the feasibility of experiments at frequencies below the collision frequency.

P. Cotti, P. Wyder, and A. Quattropani, Phys. Letters 1, 50 (1962).

(Li, Na, K, Al, In, and InSb) ; Taylor, Merrill, and Bowers (Cu, Ag, Au, Pb) ; Libchaber and Veilex(InSb, at microwave frequencies) ; Kanai (PbTe, at radio frequencies) ; and Khaikin, Edelman, and Mina (Bi, at microwave frequencies). Detailed experimental studies of the mode structure in rectangular parallelepipeds were made by Rose, Taylor, and Bowers (Na), and, with more refined detection techniques, by Merrill, Taylor, and Goodman (Na). Cotti, Wyder, and Quattropani attempted a theoretical justification for the semiempirical rule obeyed by the resonant frequencies, however, the present author disagrees with their formulation of the boundary-value problem. Chambers and Jones exploited the helicon resonance

6 P. Cotti, A. Quattropani, and P. Wyder, Phys. Kondens. Materie 1, 27 (1963).

7 R. G. Chambers and B. K. Jones, Proc. Roy. Soc. (London) A270, 417 (1962).

M. T. Taylor, J. R. Merrill, and R. Bowers, Phys. Rev. 129, 2525 (1963).

A. Libchaber and R. Veilex, Phys. Rev. 127, 774 (1962).

10 Yasuo Kanai, Japan. J. Appl. Phys. 1, 132 (1962).

11 M. S. Khaikin, V. S. Edel'man, and R. T. Mina, Zh. Eksperim. i Teor. Fiz. 44, 2190 (1963) [English transl.: Soviet Phys.-JETP 17,1470 (1963)].

12 F. E. Rose, M. T. Taylor, and R. Bowers, Phys. Rev. 127, 1122 (1962).

13 J. J. Merrill, M. T. Taylor, and J. M. Goodman, Phys. Rev. 131, 2499 (1963).

A1714   C. R. LEGENDY

phenomenon as a means of measuring Hall coefficients with high precision. An abstract proof for the existence of helicon modes (in samples with zero resistivity) was

given by Legéndy

The macroscopic treatment of helicons may be conveniently started from the equation

E+RjX B=pj ,   (1.1)

where E, B, and j are the electric field, magnetic field, and electric current; R and p are the Hall coefficient and resistivity, respectively (all in rationalized mks units).

This article is concerned with the consequences of (1.1). Equation (1.1) will be assumed to hold true inside the sample carrying helicons; the field outside the sample will be approximated by the product of a static field and the time-dependent factor exp (iw r). Equation (1.1) was derived from . the Boltzmann equation by Cotti, Quattropani, and Wyder

Besides the standard assumptions ensuring that (1.1) correctly relates steady currents and fields, we underline, for emphasis, the assumptions ensuring that it correctly relates currents and fields depending on position and time. These assumptions require the wavelength and time period of helicons to be much larger than the relevant parameters of the microscopic conduction mechanism. [The article of Chambers and Jones contains a thorough list of the assumptions involved in (1 1).]

When the wavelength becomes small, the helicon phenomenon becomes dependent on the microscopic properties of the medium. This case is beyond the scope of the present paper, it is treated in Refs. 15-26.

The dispersion law for short-wavelength helicons propagating along the magnetic field was derived by Sheard,15; starting from the results of Rodriguez'6 and Kjeldaas17 obtained for acoustic absorption. A thorough treatment of the short-wavelength and high-frequency limit was given by Kaner and Skobov.18 Taylor, Merrill, and Bowers19 observed an edge in the absorption of short-wavelength helicons in sodium and explained it in terms of the Doppler-shifted cyclotron resonance predicted by Stern20 (similar to the Dopplershifted cyclotron resonance connected with ultrasonic waves, discussed by Kjeldaas17). Kaner and Skobov,18 Miller,21 and Quinn22 predicted giant quantum oscillations in the absorption of helicons; Stern and Callen23 predicted interactions between helicons and magnons;

14 C.. R. Legéndy, J Math. Phys. (to be published).

16 F. W. Sheard, Phys. Rev. 129, 2563 (1963). 16 S. Rodriguez, Phys. Rev. 112, 80 (1958).

17 T. Kjeldaas, Jr., Phys. Rev. 113, 1.473 (1959).

18 E. A. Kaner and V. G. Skobov, Zh. Eksperim. i Teor. Fiz. 45,

610 (1963) [English transl.: Soviet Phys.-JETP 18, 419 (1964)]. 19 M. T. Taylor, J. R. Merrill, and R. Bowers, Phys. Letters 6,

159 (1963).

20 E. A. Stem, Phys. Rev. Letters 10, 91 (1963).

S1 P. B. Miller, Phys. Rev. Letters 11, 537 (1963). 22 J. J. Quinn, Phys. Letters 7, 235 (1963).

Y2 E. A. Stern and E. R. Callen, Phys. Rev. 131, 512 (1963).

Kaner and Skobov,18 Langenberg and Bok,29 and Quinn and Rodriguez 25 predicted interactions between helicons and phonons. The latter interaction was observed by Grimes in potassium.

The present paper is organized as follows : Sec. 2 . deals with helicons in an infinite medium; in Secs. 3-6 boundaries are introduced. In Sec. 5 the boundary-value problem is solved for an infinite plate perpendicular to the external magnetic field, an infinite plate, and an infinite cylinder parallel to the external magnetic field. In each case an oscillatory "driving field," sinusoidally varying along the two coordinates tangential to the boundary, is assumed, and the response field is computed as a function of the frequency and the tangential wave vector. In Sec. 6 is it shown that, ignoring anomalous skin effect, the Ohmic loss in the boundary surfaces parallel to the external field does not tend to zero in the limit of zero resistivity. Under anomalous skin-effect conditions the surface mode responsible for the loss disappears.

For the sake of symmetry and simplicity in what follows, we shall call all fields inside the sample helicon fields, instead of restricting the term to the freely propagating component.


Write B=Bo+b(r,t), where Bo is the (uniform and constant) external magnetic field. Then, by assuming Bo>>b, linearize (1.1)

E+RjX.Bo=pj.   (2.1)

Take the curl of both sides and combine with Maxwell's equations, neglecting displacement current. Letting the z axis point along the field Bo the result is:

u-o ab+uVXab+VX(VXb)=o,   (2.2)

P at   az

u=- -BoR/p=w.,7.

Assuming plane-wave solutions of the form b=b(0) exp[i(wt-k.r)],

k= (cr,Q,-y) ,   (2.3)

r= (x,y,z),

Eq. (2.2) becomes

(µo/P)iwb- wykX b+k2b= 0,

k2=a2+NR2+y2.   (2.4)

Written out in detail, (2.4) is a set of three coupled homogeneous linear equations for the three components of the constant vector b(0). The secular equation has

24D. N. Langenberg and J. Bole, Phys. Rev. Letters 11, 549 (1963).

26 J. J.. Quinn and S. Rodriguez, Phys. Rev. Letters 11, 552 (1963).

26 C. C. Grimes, Bull. Am. Phys. Soc. 9, 58 (1964).


one root leading to a physically unacceptable solution; dividing out that root, the secular equation becomes:

[(po/p)w-ik2]2-u2k2y2=0,   (2.5)

From Eq. (2.4), aside from an arbitrary constant factor,

(bo) x= ay+i (3/u'Y) [ (po/p)w- ik2] ,

(bo)u=(3'Y-i(a/u,Y)[(po/p)co--ik2],   (2.6),

(bo)== -a2-(32.

Factorizing (2.5), the dispersion relation may be rewritten in the following simpler form',': t

w= (p/µo) (uky+ik2),   (2.7)

and the solution2:

b= (ay+iki3,(3y-ika,-a2-(32) exp[i(wt-k.r)]. (2.8)

In the limit p-; 0 the product pu=-BoR remains unaltered and the second term in (2.7) tends to zero,

thus (2.7) becomes2-4

w=-µo -µd-'BoRky.   (2.9)

If k1, k, kt are any orthogonal components of k, the component y can be expressed in terms of these, and so can k2. Thus (2.5) interrelates the four (complex) quantities w, kI, k, k~, and if any three of these are specified, it can be solved for the fourth. In the cases to be treated below, w, k1, k„ are given real numbers; Eq. (2.5) is a quartic equation in k~, and therefore in general it yields four different complex roots. For each of these, k2 is well defined, but k has two values. The one to be used in (2.8) is the one satisfying (2.7). [Because each of the four kt satisfies (2.5), and (2.5) is merely the square of (2.7), one and only one square root of k2 for each kt necessarily satisfies (2.7).]

The above discussion should replace the remarks connected with a "f" alternative in Eq. (3) of Bowers, Legendy, and Rose'; the discussion concerning this point in Ref. 2 is confusing.

In the remainder of this section we shall make several simple remarks pertaining to helicons.

Direct computation from (2.8) shows that for a plane wave of helicons

V Xb=kb,

j=µ0 1®Xb=µo 5kb.   (2.10)

Thus, when k is real, the current associated with a single plane wave is everywhere parallel to the magnetic field. Let us multiply Eq. (2.2) through by p/µo, let p--).0, replace the operator a/az by -iy and the operator V by the vector -ik. Then (2.2) becomes:

ab/at=wXb,   (2.11) w=-µo 1BoRyk=constant vector.

Equation (2.11) can be recognized as the precession equation. There are two ways in which the vector w can be real: If all components of k are purely real, or if all

* See note 1 at the end of the article See note 2 at the end of the article

components of k are purely imaginary. A glance at the expression for w shows that when R<0, the scalar product w• Bo is positive in the former case and negative in the latter. This means that if the vectors rotate around the field lines in a sense agreeing with the cyclotron rotation of the carriers, the waves propagate freely; if they rotate oppositely, the waves are exponentially damped.

When k is real, the instantaneous spacial pattern of fields might form a right-handed screw, or a left-handed screw; the screw sense is determined by the sign of k. To show this, let a= (0,0,k2), k=k/k, Gi=kXa

(k(3,-ka,0), and G2=kX (kXa)= (ay,j3y,-a2-i(32). Clearly, when k and the components of k are real, G, and G2 are two real vectors of equal length, perpendicular to each other and to k. Now, Eq. (2.8) can be written as follows:

b=(G2+iG1) exp[i(wt-k•r)].

This, is the standard form of a circularly polarized wave; the screw sense of the instantaneous pattern is determined by the right- or left-handedness of the Cartesian coordinate system G1G2k. Note that changing the sign of k, changes the sign of G1, but leaves G2 unchanged. One can see at once that when k is negative, the screw sense is right-handed; when k is positive, the screw sense- is left-handed. [Note that, by Eq. (2.10), j and b are antiparallel when the screw sense is right-handed and parallel when it is left-handed.]

Of course, when the wave vector is made complex, the geometrical clarity of the situation fades.

So far, no mention has been made of the electric field patterns. If the current density is specified, (2.1) gives the electric field explicitly; when p-+ 0, this relation takes the form

E= -RjX Bo.   (2.12)

As seen from (2.10), the currents in any given planewave form a pattern identical to the magnetic field pattern, except for a multiplicative constant. Thus all remarks made for magnetic fields can be repeated for currents. However, they cannot be repeated for electric fields. Equation (2.12) shows that when p -* 0, E cannot have a component along Bo. Thus, unless a=(3=0, the electric field has a longitudinal part, i.e., a part with V • E540, as well as a transverse part. If the reader has not encountered a similar situation before, he may wonder if this is compatible with the assumption of neglecting displacement currents. The latter assumption brings one of Maxwell's equations to the form V X H= j ; therefore, the electric current is represented as the curl of a vector and it can have no divergence. This means that space charges cannot periodically build up and disappear, thus the electric field cannot have a longitudinal component. The paradox disappears in the light of the actual magnitudes of the quantities involved. The current does have a longitudinal component, but it is about 1011 times smaller than the

A1716   C. R. LEGENDY

transverse component (the ratio between conduction current and displacement current densities at 10 cps is about 1018). Thus, space charges do periodically appear and disappear, but they are very small. The reason why the longitudinal component of electric field is still of the same order of magnitude as the transverse component is that E itself is very small; it is about 100 times smaller than that which would correspond with the same magnetic field in a freely propagating vacuum wave. The number 100 is the ratio of the speed of light to the helicon phase velocity at 10 cps.

The dispersion relation (2.7) does not contain the assumption that the resistivity p is small. For one can rearrange Eq. (2.7) to read as follows:

-iwµo/P 1/2 k-C1+iu(y/k)J

In the limit u=w,T --). 0 this reduces to the standard skin-effect formula.

The following remark27 concerns helicons whose amplitude is not small. Suppose b is not negligible compared to Bo. Then a nonlinear term

-(koR/p)OX(jXb)   (2.13)

must be added to the left-hand side of (2.2). And yet, a single plane wave of form (2.8), obeying the dispersion relation (2.7), still satisfies the equation. For, by virtue of (2.10) in such a wave j=µo'kb, and in the nonlinear term (2.13)

jXb= (1/µo)kbXb-==0.

Thus, (2.13) identically vanishes. The sum of two solutions is, as usual in nonlinear equations, not necessarily a solution.


The problem of dealing with boundaries has been first considered by Cotti, Wyder, and Quattropani.5 They assumed that the boundary condition to be satisfied is that all three components of E must be continuous at the boundary, and no electric current should cross the boundary. In their second paper', the authors drop the former condition and retain only the latter. (Indeed, the normal component of E is, in general, discontinuous.) Chambers and Jones, in their treatment of driven oscillations (in an infinite slab) use the condition that the tangential components of magnetic field must be continuous across the boundary; in calculating frequencies of free oscillation they use the current condition (i.e., the requirement that currents do not cross the boundary).

We wish to make a few comments on these boundary conditions. Since all three articles deal with nonferromagnetic materials of finite conductivity, there can be no surface currents in either, and all three components

S7 E. F. Johnson (private communication).

of the magnetic fields must be continuous across the boundary. This boundary condition, together with the assumption that the vacuum fields are static, implies that the current condition is satisfied. (For, if a vector is continuous across a boundary, the normal component of its curl is also continuous, but inside the sample V X H is the current; outside the sample V X H is zero.) However, the assumption that a field satisfies the current condition clearly does not ensure the continuity of H. Furthermore, in Sec. 5, we shall be able to construct a solution satisfying the current condition in a finite cylinder for any given frequency. The latter construction dramatizes the criticism against identifying a frequency as a frequency of resonance merely on the ground that at that frequency there exists a helicon field satisfying the current condition.

In the present article we shall use the boundary condition that all components of the magnetic field are continuous. Since the problem is quasistatic, this implies that the boundary conditions on electric field are automatically satisfied. The latter statement may be verified as follows

Assume that the field b(r) exp(iwt) satisfies (2.2) inside the sample, satisfies V X b= 0 and V . b= 0 outside the sample, and is continuous at the boundary. Construct an electric field El defined outside the sample such that V X E1= -iwb(r). (This can always be done by use of the Green's function for the curl operator, i.e., in parallelism with the elementary calculation of a static magnetic field from the current distribution.) The electric field inside the sample, E;.,, is uniquely determined from b through (2.10) and (2.1) ; one can easily check that automatically, V X E1A,= iwb. Write the electric field outside the sample as E1+ E2. The boundary condition requires that the tangential components of electric field be continuous at the

boundary; thus, with El and E;,,, given, (E2)tang is specified at the boundary: (E2)tang=(Eins-Ei)tang.

From the continuity of the normal component of b it follows that the line integral of E2 over any closed curve lying on the surface vanishes. Thus a scalar potential ~o can be defined on the surface in such a way that

(E2)tang= (- V ')tang. (The definition can be made

unique by taking into account the net charge on the sample.) The problem then reduces to extending so into all space in such a way that V2,p=0 throughout and tp -> 0 at infinity. This problem is a well-known case of the Dirichlet problem,28 and can always be solved.

The problem of driven oscillations will be formulated in parallelism with the standard problem of "reflection and refraction" in optics.

Imagine the vacuum field decomposed into an "incident" component or "driving field" defined as the field set up by the driving currents alone (i.e., as if the sample were removed), and a "reflected" component

28 P. M. Morse and H. Feshbach, Methods of Theoretical Physics

(McGraw-Hill Book Company, Inc., New York, 1953), Chap. 5.


due to the currents and charges in the sample. (Note that the former may be singular at infinity, the latter may be singular inside the sample but not vice versa.) The "transmitted wave" is the helicon field in the sample so chosen as to satisfy the boundary conditions at the samples surface.

As indicated in the Introduction, in Sec. 5 we shall, consider three types of infinite samples : A plate perpendicular to the magnetic field, a plate parallel to the magnetic field, and a cylinder parallel to the magnetic field. For all three of these the driving field will be assumed to be a sinusoidal function of the two coordinates parallel to the surface; the corresponding components of the wave vector are the two given quantities referred to in the previous section as kt and k,,. With these specified, the field equations restrict the third component to a choice of two in the vacuum and a choice of four in the conductor. Of the former two, one leads to singular behavior at infinity-that must be chosen as the incident wave; of the latter four, some may lead to singular behavior inside the sample-those must be disregarded. The complex constants multiplying the allowed fields (counting the reflected field too) are the only unknowns of the problems, and for them the boundary conditions provide the necessary and sufficient number of equations.

From the results it is then possible, if desired, to obtain "resonance curves" by fixing kE and k„ and varying the frequency.


The following two simple boundary-value problems shall serve to illustrate the method outlined in Sec. 3. The solutions in Sec. 5 consist of straightforward synthesis of the observations made in 4A and 4B below.

A. Conducting Front in the x,y Plane;
a, g, w Specified

Suppose the region z__<-0 is filled with conductor of resistivity p, satisfying (2.1), and the rest of space is vacuum. Assume the driving field has a frequency w and its variation along x and y is wavelike ; a, (0 , and w are specified by the problem. The component a is real; without loss of generality we can set 3= 0.

The dependence of the fields on x, y, and t is described by the factor expi(wt-ax). We remark that the tangential phase velocity w/a is not the speed of light in vacuo, but is many times smaller. (In a typical experiment in sodium' it is of order 109 times smaller.) The frequency w is dictated solely by the oscillator connected to the driving coils, and the wave number a by the geometrical configuration of these coils. Since the problem has translational symmetry along x, y, and t, and is governed by linear equations, standard symmetry argument shows that the reflected field and the helicon field must have the same sinusoidal variation as the incident field.

We shall denote the incident and reflected field by bo and b,. respectively.

Since bo cannot become infinite as z -> - oo and b, cannot become infinite as z- oo, bo and b, must have the form:

Ibo}- Ibo}X (. ia,0,±JaI)e±I0 ,   (4.1)



where the upper line corresponds with the upper sign, and the lower line with the lower sign. The scalar quantities bo and b, are constants (possibly complex); bo is given, b, is unknown.

The components of wave vector called kt and k,, in Secs. 2 and 3 are here a and ,13; and, as was said there, for the third component kr (in the present case y) the field equations allow four different values which can be found from (2.5). They are

y1= -y3

1-E-P-4- =   (p2-q)112]112,   (4.2a)

y2= -y4

1+iu(w/wl)+2   [u(w/wl)-i]2

P=   1+u2   , 4=   1+u2   ,

wi= -yo 1BoRa2

Imyl>>=0, Imy2?0.   (4.2b)

To write down the helicon solutions (2.8) it is necessary to evaluate the quantities k corresponding to each y. These are "defined" in (2.7), which can be rearranged as follows:

k„= (1/uyn)[(µo/p)w-ikn2], n=1,2,3,4. (4.3)

From (4.2) and (4.3) it is seen that the four values k„ are pairwise connected by the relations

k3= -ki, k4=-k2

With these and (2.8), the four helicon waves are }bbi   b,

3}- }b3 }X~(±yi, Fikl,-a)r'r1`,


}b4}- }b4}X (~y2, Fikz,-a)e1i ,

S =e i(wt-ax).

One can check, by going to the limit p--> 0, that b1 and b2 differ in their direction of circular polarization; so do b3 and b4. By definition [see (4.2)], yl and 72 have positive imaginary parts. (For the case p=0, when yl and y3 are purely real, yl is chosen by means of a limiting procedure p--> 0+. Below, we shall assume that p is never strictly zero.) Therefore b3 and b4

A 171R "   C. R. LEGENDY

diverge at z-*- oo and must be excluded. (In Sec. 5A, where the sample is finite in the z direction, we shall have need for all four solutions.) The constants bland b2 together with br in (4.1) are the unknowns of the problem. For the three unknowns the boundary condition that b is continuous at z=0 furnishes three equations:

yib,-f y2b2+iabr=-iabo, -ikibi-ik2b2=0,

abi+ab2+ aIbr=IaIbo.

The solution is

-2a bo   2a   bo   - I a I A+A'

b1 ~aI A+A' ki, b2=    ~a~A+A' k2' br- IaIA+A' ,

1   1   iy1 iy2

A=---, A'=--+-.

ki k2   k1   k2

B. Conducting Front in the y, z Plane;
p, y, w Specified

Let the region x55.0 be filled with conductor, and the rest of space be vacuum. Assume that, similarly to the case in 4A, the driving field imposes a sinusoidal dependence of the fields on the tangential coordinates and time; the variation is characterized by the three real quantities, (3, y, and w. Note that, because the z direction is singled out by the vector Bo, neither 8 nor y can be set equal to zero without loss of generality. By the same arguments that lead to (4.1), the incident and reflected wave are

bo   bo

~brl- IbilX~(zl= K,-2(3,-i'y)C'",

K= (/32+y2)1I2,

~_ ei(-t-9v-7z) .

Equation (2.5) is most conveniently solved for k2; the expression for k2 only involves y and not 0.

4 w 112 z

k%= 4u2y2[lf(1+i--) J ,   (4.5)

U w2

w2=-µo 'BoRy2,

from this,

~a2=-a4j = t-K2 4u2y2[1±(1+ZUw2`1/z]2,112

Imai_0, Ima2>=0.   (4.6)

The corresponding four values of k are computed by means of (4.3); they are pairwise related as follows:

ki=k3, k2=k4.

For the purposes of the present arrangement it is desirable to replace.(2.8) by

(-K2,a@+iky,ay-ik8) exp[i(wi-k.r)], (4.7)

which differs from (2.8) only in a constant factor -K2/(ay+ik8). The four helicon waves are:

bi   bi


X (-K2,~a1~3+ikly,~aly-ik1Q)e, °1x, b3   INI


1 b2l   b2l b4 ={b4 XE(-K2,±a2/3+ik2y,±a2y_ik2R)e i 2x

Of these, b3 and b4 diverge when x-* - - and must be dropped. (In Sec. 5B, where the sample has a finite extension in the x direction, there will be need for all four solutions.) The coefficients bi and b2 together with b, are the unknowns of the problem; for them the boundary conditions furnish the following three equations

(-K2)bi+(-K2)b2- (-K)br=Kbo,

(ai13+ikiy)bi+ (a2,3+ik2y)b2- (-i#)br= -i(3bo, (a,y-iki(3)bi+ (a2y-ik2i3)b2- (-iy)br= -iybo.

The solutionsare

-2 1   2 1   -KA+A'

bi-   -bo, b2=   -bo, br=   ,

KA+A' ki   KA+A' k2   KA+A'

1   1   -iai ia2 A----, A'- +-.

ki k2   ki   k2

Finally, consider Eq. (4.6) in the limit u>>4w/w2. At the frequency w= (K/y)w2 the quantities a2 and a4 vanish, which means that the helicon wave vector is tangential to the boundary. Above this frequency a2 and a4 are real, but below it they are imaginary. The phenomenon is recognized as a phenomenon familiar from geometrical optics: total reflection; below the frequency (K/y)w2 the tangential phase velocity of the artificial vacuum wave becomes lower than can be matched by helicons.

The same does not occur when the conducting surface is parallel to the x,y plane. In the limit u --* co, p -- 0 the phase velocity is given by Eq. (2.9) : w/k =-µo 1BoRy. When y is fixed by the driving field, the phase velocity is fixed, but if only a and i are fixed by the driving field, the phase velocity can be made smaller than any arbitrary quantity by choosing y small enough.


Sections 5A, 5B, and 5C contain the solutions of three resonance problems that can be solved exactly. The term "resonance problem" is intended to underline the fundamental difference between these three problems


and the two described in Sec. 4. It can be verified at once that, in each of the problems below, for a fixed wave vector of the driving field there exist nonzero (complex) frequencies for which the secular determinant vanishes; the same is not true in 4A and 4B.

A. Infinite Plate Perpendicular to Bo

Let the region -a<=z<=a be filled with conductor and let the rest of space be vacuum. As in Sec. 4A, the two specified components of the wave vector are a and (3; a is real; without loss of generality we set 0=0. The allowed values of y are those given in (4.2).

Suppose the incident field is of the form

bo=boE(-iacoshaz,O,asinhaz), jzI_a,

~= ei(.i-ax) .   (5.1)

Arguments similar to those used in the previous section show that the reflected field in the two vacuum regions z > a and z<-a is, respectively:

br-br15(-ia,0,-IaI )e lal(z-a), in region z>a,

=br2 (-ia,O,jaielal(=+a),   in region z<-a, (5.2)

where brl and br2 are constants, as yet undetermined. Of the four helicon fields (4.4) all four will be needed; their amplitudes bl, b2, b3, b4 are further unknowns of the problem. For the six unknown constants the boundary conditions furnish six linear equations; three express the condition that at the surface z=a, all three components of the magnetic field are continuous; the other three express the same for the surface z=-a. For the sake of illustration, the six equations are written out below:

yle iYlabl+y2e iy2ab2-yieiYlab3-y2ei72ab4 -ia(-b,,)= -ia (coshaa)bo, -ikie iylabl-ik2e iY2ab2+ikleiriab3

+ik2eiY2ab4= 0 ,

-ae iylabi-ae iy2ab2-aetiylab3-aeiY2ab4

- Jai (-bri)=a (sinhaa)bo,   (5.3)

yleaYlabl+T2ei72ab2-yle i71ab3-y2ei12ab4

-ia(-b,2)= -ia (coshaa)bo,

-iklei7labl-ik2eiY2ab2+ikie i'ylab3

+ ik 2e__ i721= 0 ,

-aeiYlabl-aeiY2ab2-ae zYlab3-ae iy2ab4

+ I a I (-b,2) _ - (sinhaa) bo.

It can be seen at once that the trial relationships

b1=-b3, b2=-b4, brl=br2 split the set into two

identical sets of three equations. The physical explanation of this is given in the following symmetry argument.

A boundary value problem involving helicons can be said to possess reflection symmetry about a plane if reflection of the sample and B0 about the plane,

followed by reversal of the vector Bo, turns both the sample and Bo into itself. The reason for including B0 as a part of the system rather than the field is that in the equations [namely in (2.1)] Bo appears merely as a geometrical property of the system, rather than a part of the magnetic field; however, the pseudovector nature of B0 shows up in the vectorial product-hence the reversal of sign upon reflection. When a problem has reflection symmetry about some plane, it possesses solutions symmetric and solutions antisymmetric under reflection about that plane. Because all our solutions relate to magnetic fields, we shall arbitrarily use the terms "even" and "odd" to denote solutions in which reflection leaves the magnetic fields b unchanged and changes the signs of the magnetic fields b, respectively. (The symmetry of the currents and electric fields is opposite to the symmetry of the magnetic fields b.) The problem of this section is clearly symmetric about the plane z=0. (Note that it is not symmetric about the plane x=0, because the aforementioned transformation reverses Bo instead of leaving it unchanged.) Because the driving field (5.1) is even, so must be all the other fields, which explains the simplification of the set (5.3).

Solving the three equations is quite straightforward; the solution is conveniently written in the following form

bH=bH:(A'(z),-iAo(z),iaA(z)),   (5.4)

where ~=ei(1t-ax) ,

sinylz   siny2z

A (z) _   -   

kicosyla k2cosy2a


A' (z) -A (z) ,

cosy 1z cosy2z

A o(z) _   - cosyla cosy2a

The constants bH in (5.4) and br1, br2 in (5.2) are a coshaa+ I a I sinhaa

bH--Z    Ia I A (a)+A'(a) b°'

- I a I A (a) coshaa+ (a/ I a I )A' (a) sinhaa

b,,= b,2=   bo.

I aI A(a)+A'(a)

In the special case a=0, (4.2) and (4.3) yield

'Y1=_'Y2*=k1=ik2*=-(co o/pu)112(1+i/u)-1(2'

and if

b0= (1,0,0)e" ,

the helicon field is

bH= (zA'(z). (i/2)Ao(z),0)eiw`

A1720   C. R. LEGÉNDY

10    f,   I   1'   1   I



Z -   -

a -   -   Fic. 1. Resonance curves for an

F 5 _   i =10   infinite plate-parallel to the x,y plane,

-   with a=0. The frequency scales have

5 -   been adjusted to bring the first peaks

into coincidence (courtesy of M. T.

X      Taylor). -




The average of the x component taken over the whole slab is

1   A (a) tanyla tanhyla

~_- ~ A'(z)dz=    =    +

4a J-a   2a   2y1a   2y1a

in agreement with the corresponding result of Chamber; and Jones.' (We remark that these authors ignored the existence of a "reflected field, but in the special case a=0 this leaves the shape of the frequency response curves unaffected, and the theoretical curves of Chambers and Jones are in good agreement with experiment.) The imaginary part of cp, plotted against frequency, goes through maxima, corresponding tc resonances (Fig. 1). One can see from the above expression that rp is infinite at those (complex) frequencies where y1a=±ir/2, ±37r/2, - - -, or yea=±-ir/2; ±37r/2, - - - ; these are the roots of the determinant of (5.3). When a; 0, the roots of this determinant are the roots of I a I A (a)+A' (a), the common denominator in the expressions for bH, br1, and br2. We calculate the correction to the root y1a=7r/2 to first order in aa. When aa<<l, A' (a) ti 2, and A (a)- (tanyla)/y, +a(tanh2)/(27r)= (tanyla)/y1+0.58a, the desired root y1a is given as the first root of the transcendental equation

tanyla/yxa= - 2/aa- 0.58.

For a square plate with a ratio 15:1 between edge length and thickness, aa^-ir/1542 and y1a''1.025(br). This corresponds to a 5% correction in frequency which may explain the discrepancy between theory and experiment reported by Chambers and Jones.7

Unfortunately, because of the necessity to deal with complex numbers, plotting graphs such as Fig. 1, or computing roots to the secular determinant is usually extremely lengthy; when as and p are not small, such calculations call for numerical work. When the driving field is odd in z, i.e.,

bo=boE(-a sinhaz,0,a coshaz), I zI >=a.

The equations (5.2) and (5.4) still correctly describe

the fields br and bH but bH, b,1, br2, A (z), Ao(z), must be


cosylz   cosy2z

A (z) _   -   


k1 siny1a k2 siny2a d

A' (z) =-A (z) ,

sinylz siny2z Ao(z)=- .   +   , siny1a siny2a

a sinhaa+ I a I coshaa

bH=-i I a I A(a)+A'(a) bo,

- I a I A (a) sinhaa+ (a/ I a I )A' (a) coshaa

br1= -br2   bo. I a I A (a)+A' (a)

B. Infinite Plate Parallel to Bo

Let the region -a<x<--_a be filled with conductor, and let the rest of space be vacuum. As in Sec. 4B, the two specified components of the wave vector are 9 and y, both real; neither of them can be set equal to zero without restriction of generality. The allowed values of a are those given in (4.6).

One can check by direct computation that to a driving field

bo=bo~(K sinhKx,-i,8 coshxx,-iy coshKx), I xI >=a, The response is

bH= bHt(ii2A (x),13A' (x)+'yAo (x),'yA' (x)-13Ao (x)) ,

in -a<x<a,

br=brl (-K,-i,Q,-iy)2 K(~a), for x>=a, =br2t(K,-0,--iy)e'(-'+a), for x<==-a,




bH=-i icA(a)+A'(a)    bo,

-KA (a) coshKa+A' (a) sinhKa

brl=brz=   bo,

KA (a)+A' (a)

sinalx   sinalx

A(x)=   -

k1 sinala k2 sinala


A' (x) _-A (x) ,

sinalx sinalx Ao(x)=   - sinala sinala

It can be seen at a glance that the helicon field bH is a linear combination of the four helicon fields (4.8). When the driving field is antisymmetric,

bo=boE(K coshKx,-,Q sinhKx,-y sinhKx), +xj? a.

A (z), A' (z), Ao (z), bH, br1, b,2, must be redefined as follows


bH -i KA (a) +A'(a) bo,

-KA (a) sinhKa+A' (a) coshKa br=   bo, KA (a)+A' (a)

cosaix   cosa2x

A(x)=   -   

k1 cosa1a k2 cosa2a


A' (x) -A (x) ,

sinalx sinalx Ao(x)=   -

sinala sinala

Note that the reflected field possesses a symmetry about the plane x=0 which corresponds with the symmetry of the incident field, but the helicon field has no such symmetry.

C. Infinite Cylinder Parallel to Bo
[Given y, n (= k9)]

Adaptation of the formula (2.8) useful to problems of cylindrical symmetry, can be obtained by formally summing solutions (2.8) over all k= tan-' (13/a) with a weighting factor exp (in ip). Through the formula


J e2c" e -. as„e>dB=21r(-1)nJ„(z),


this introduces Bessel functions. Expressed in terms

of k, n, y, and the cylindrical components, the solutions have the form

~R (krr)= ~(Rr,Rw,R$) ,




R,= (k-y)J„-1(krr)+(k+y)J„+1(krr), R ,= i (k- y)Jn-1(krr) - i (k+y)J„+1(krr) , Ra= - 2ik,J„ (krr),

k2= k .2+,y2.

Independent solutions are obtained by replacing the Bessel functions by Neumann functions; but these are singular at r= 0 and are of no interest to us.

Let the cylindrical region r < a be filled with conductor and let the rest of space be vacuum. The driving field fixes the quantities y and n; ,y is real, n=O,f1,f2,.... The allowed values of k2 and kr are identical with the allowed values of k2 and a given in (4.5) and (4.6) with the substitution (3= 0 used in the latter equation corresponding to the fact that k2=kr2+y2. Because of the symmetry properties of Bessel functions, reversing the sign of kr leaves the whole solution unchanged except at most for sign, thus, rather than . four, there are only two independent acceptable helicon solutions. There is only one possible incident field and one reflected field satisfying the usual requirements:

bo= bo~('YI,+' ('Yr), (n/r)I n (-yr),-iyI„ (7r)) , br= br~('YKn' (yr), (n/r)K„ (-tr),-iyK„ ('Yr)), where

a   d

I, (z)=~ n+1[Jn.(iz)+iN„(2z)], I,+'(z)=-I, (z),


K,, (z) = i nJ,, (Zz) ,   K,,' (z) =-K,,(z) .


Thus the problem is reduced to three equations expressing that at r= a the magnetic field is continuous. Letting b1 and b2 denote the amplitudes of the two allowed

helicons we have at r= aR,ib1+Rr2b2-yKn'br= yI n'bo ,

n   it Rwlbi+R„2b2--K„ br=-I„ bo,

a   a

Rz1b1+R,2b2+iyKnb,= -iyI„ bo.

Again the roots of the determinant of the set give the conditions for oscillation or free propagation.

The problem of 5C has reflection symmetry (in the sense of 5A) about all planes perpendicular to the z axis; therefore, if, under reflection about any such plane the incident magnetic field is antisymmetric, all of the magnetic field will be antisymmetric, and all the


currents symmetric. Reflection symmetry of currents about a plane means that the currents never cross that plane. Choosing the incident field to be the sum of two fields of form (5.5) differing in the sign of y, it is possible to set up a standing wave pattern in the z direction; such a pattern possesses a set of fixed planes about which the said reflection symmetry exists at all times, i.e., planes which are never crossed by currents. Thus, as promised in Sec. 3, for any given frequency we can construct a helicon field satisfying the "current condition" (i.e., the requirement that currents do not cross the boundary) for a finite cylinder.

In a similar way, the infinite plate of 5B can be driven so as to possess a similar set of planes perpendicular to z. However, because no problem involving helicons has symmetry about a plane parallel to the external magnetic field, analogously chosen driving field cannot achieve a similar set of planes parallel to z in the problem of 5A or of 5B. Thus, the current condition for a finite rectangular box cannot be satisfied by applying an appropriately chosen driving field to the plates of 5A or 5B.

Cotti, Wyder, and Quattropani5 obtained solutions satisfying- the current condition for an infinite rectangular bar, finite along y and z, by adding four freely propagating plane waves of helicons (or, more precisely, four helicons of that mode which in the limit p-4 0 is undamped). Chambers and Jones' obtained solutions approximately satisfying the current condition for a finite rectangular box, thin in the z direction, by adding eight such helicon waves. By using plane waves of both polarizations (sixteen plane waves in all) one can satisfy the current condition for a rectangular box exactly. As we pointed out in Sec. 3, these solutions do not, as a rule, correspond to proper modes of oscillation.


A closer look at the fields associated with the problems of Sees. 4B, 5B, and 5C reveals some unusual results.

In the problem of 4B, make the simplifying assumption fl= 0 and consider the limit of p --> 0 (with u ---> -, pu=const). Since in (4.6) and (4.8), u>>4w/w2, one can write

ahiuy, ki-iuy,

a2^• p'Y, k2"' (w/w2)y,   (6.1a)


W2= -.uo1B0Ry2,

p=-[(w/w2)2-1]1'2 for w/W2 1

=i[1- (w/w2)2]112 for CO/W2=< 1. (6.1b)



ib   b,

lb,,}= b,J(±1,0,-i('Y/IyI))ei(-t--)efl7lx,


bi/bo= (1/D) (- 2ew/w2) ,

b2/bo= (1/D) (2c),

b,./bo= (1/D)[p-W/wo-ie],

E=y/lyl, D=p-'(w/w2)+i€.   (6.2)   ..

Recalling Eq. (2.10), we have for a single plane wave of helicons

j=µ6- lkb.

The current corresponding to b1 is

ii= (1/so)(iuy)bl(0,1,i)ei(wt-ti.)euTx   (6.3)

and the joule loss (per unit area) in a surface layer of unit thickness due to ji is

0   0

W=1 1 2p(j1*.j1)dx-   2p(j1*.jl)dx=Cpu,


2y   (w/w2)2 z   

C= bo

Yo 2   1+[p- (w/w2)]2 }

where star denotes complex conjugate. This loss is confined to a narrow region near the surface x= 0 [see (6.3) and note u-> c]; it may therefore be termed "surface loss." The surprising feature of (6.4) is that if bo, w, and y are fixed, so is C; and, if then, the limit p ---). 0 is approached (in such a way that pu remains constant), W does not tend to zero. (Note that surface loss does not occur if the "tangential wavelength" 27r/,y is infinite.) The other two terms in the expression for the joule loss (per unit area) in a surface layer of unit thickness, f-10' p(j2*P

j2) and the "cross term"

f i zp(j1*..j2-f-ji•j2*), both tend to zero. In the

approximation (6.1) the integrand of the former term does not fall off toward the interior of the sample; in the same approximation the "cross term" is always negligible compared to the surface loss.

Suppose the direction of Bo is changed to make an angle 0 with the conducting surface. It can be shown that for small enough 8 the normal component of k1 becomes


al-      (6.5)

1-iu tanO

and in all other respects, including amplitudes, b1, b2, and br remain unchanged. (Note the sensitive dependence of ai on the angle between the conducting surface and the magnetic field, when u is large.) Denote the solution reducing to -al when 0=0 by a3 [see (4.6)]. One can show that for 0 0 the component a3 is not -al but becomes

as iuy/ (-1-iu tanG) ,

thus furnishing an example in which the pairwise "degeneracy except for sign" in the allowed values of the wave vector's normal component is split up.


energy in the electric fields at the surface tends to infinity as p --> 0. For, as was said above, the currents in the surface layer increase as u; thus, from (2.1) (in which p --> 0) so does the electric field. The square of the electric field multiplied by the thickness of the surface layer increases linearly with u. However, in a typical laboratory situation the magnetic field energy density in the surface layer dominates by about a factor of 1014 over the electric-field energy density, so that the anomaly of electric fields is merely of academic interest.

The narrowness of the surface layer confining the mode b, (of order 10,u in Ref. 30) and the fact that asymptotically (b1)v=i(b1),, can be used to advantage in reducing the number of equations necessary for solving boundary-value problems. Instead of using the ordinary boundary condition that all three components of b are continuous everywhere on the boundary one can assume that this is true everywhere except at surfaces parallel to Bo at which only bz and b,,-ib, are continuous (x being the direction of the outward normal), while a third quantity b„+ib, may suffer discontinuity. For the wave vector inside only a2 (and possibly a4) is allowed; a, and a3 are discarded. This viewpoint amounts to altogether disregarding the mode b, and retaining it on the record only to provide explanation for the surface singularity: "There are surface currents and therefore the tangential component of b may be discontinuous."

For the problem of Eq. (6.1) the remaining two boundary conditions provide only two equations, but, also, there are only two unknowns: b2 and b,. The method is applicable to the cylinder of 5C whenever b,,»1. The fact that the surface layer may be approximated by an infinite plane manifests itself in the asymptotic form of the Bessel function Jn(ip) for large,

real p: Jn(ip)_in(27rp)-1j2eP which is essentially an exponential function; J,(ip) and J,,,},(ip) differ only

in a factor i.


By combining the equation E+RjXB0=pj with Maxwell's equations, we arrived at the differential equation governing helicons, found its plane-wave solutions and dispersion relation. We pointed out that for plane waves the differential equation can be written in the form of a precession equation. When the medium is a perfect conductor, the mode rotating in the sense of the cyclotron rotation propagates freely (the other mode is exponentially damped), and the current and magnetic field are always either parallel or antiparallel (in the former case the spatial configuration forms a left-handed screw, in the latter case it forms a righthanded screw). Despite the incompressibility of the electron gas, the electric field has a longitudinal component of the same order of magnitude as the transverse component; the apparent contradiction was

In (6.4) the integral from (- oo) to 0 remains unchanged, as 0 is made to be different from zero, but its contributions come from a layer of thickness

(1+u2 tan2O)/guy rather than 1/guy and in the limit u--f oo this thickness tends to infinity rather than zero. The approximation equating the two integrals in (6.4) becomes invalid and in fact W --- 0 when p --~ 0.

However, for 0=0 Eq. (2.1) combined with Maxwell's equations yields a finite loss per unit volume even in the limit of perfect conduction. This unlikely result is no longer obtained when the microscopic processes of conduction are considered more carefully. When the resistivity becomes so low that the thickness of the surface layer becomes smaller- than the cyclotron radius, the finite surface loss disappears. It is easy to show that the details of the surface loss mechanism are the following. Right inside the surface there is an oscillating dipole layer consisting of surface charges on the boundary and space charge of opposite sign exponentially falling off (with skin depth 1/uy) toward the interior of the conductor. Between the two "charge layers" there is a strong electric field (perpendicular to the surface), which in turn gives rise (because of Bo) to a sheet of strong current parallel to the surface. The electric field as well as the current density are proportional to u, the thickness of the current sheet to u 1, the resistivity to u 1; and the current squared, times thickness, times resistivity remains constant, as u --* co. However, in

the extreme anomalous limit 1 a, j r,=yl»0 (where

r,=cyclotron radius, l=mean free path) the only electrons contributing to electrical conduction are those moving parallel to the conductor's surface 29 The effect of the electric field on these is expressed by a term proportional to E• v in the Boltzmann equation, which is zero when E is perpendicular and v is parallel to the conductor's surface. Then the strong electric field between the two charge layers mentioned above can have no first-order effect on the conduction thus it cannot bring about the strong currents responsible for surface loss, and the surface loss (or at least surface loss through the previously described mechanism) disappears. Deciding what actually does happen is beyond the scope of this paper.

This does not mean that the above remarks on surface loss can be disregarded. In the purest sodium samples available (of residual resistance ratio 8000), at liquidhelium temperatures, and magnetic fields around 50 kG, the value of u is 100 and the cyclotron radius is around 3 microns. For a "tangential wavelength" 27r/,y= I cm the thickness of the surface layer is about 15 microns, thus the extreme anomalous limit is not yet reached. It is in fact quite easy to make experimental samples in which most of the loss is surface loss. 30

One may mention the following anomaly, which directly follows from the macroscopic equations : the

29 M. Ia. Azbel' and E. A. Kaner, Zh. Eksperim. i Teor. Fiz. 32, 896 (1956) [English transl.: Soviet Phys.-JETP 5, 730 (1957)]. 11 J. M. Goodman and C. R. Legendy (unpublished work).

A1724   C. R. LEGENDY

resolved by simply noting that both field components were very small. The dispersion relation for helicons reduces to the ordinary skin-effect formula, if the magnetic field is made to be small, or the resistivity large. At the beginning of the calculation the equations have to be linearized by assuming that the helicon field is small, but, it turns out, single plane waves can propagate even if their amplitudes are large (though several plane waves would usually interact).

It was shown that if the field outside the sample is quasistatic, the boundary condition that all components of the magnetic field are continuous at the boundary implies that: (i) the electric current does not cross the boundary, and (ii) for the vacuum region it is possible to construct an electric field whose curl is -aB/at and whose tangential components join continuously with the field inside the sample.

We formulated the problem of driven oscillations in terms of a "driving field," a "reflected field" and a "transmitted field" (which is the helicon field); the first of these is the field that would be set up by driving coils if the sample were removed, the second and third can be determined by means of symmetry arguments, etc., except for multiplicative constants; these constants are determined from the boundary conditions. We solved the problem of driven oscillations for a few simple cases : Infinite plates, infinite cylinders, and semi-infinite regions. The response of plate samples was found to be quite sensitive to dependence of the driving field on the tangential coordinates; the approximate value of the fundamental resonance frequency for a square plate whose dimensions along x, y, z compare as 15:15:1, was found to be 5% higher than it would be if the first two dimensions were infinite. The effect is still more drastic in plates parallel to Bo. In this case

any variation along the z direction brings about a surface loss, which for small resistivities fails to decrease as the resistivity is decreased, until the limit of anomalous skin effect is reached, in which limit the loss disappears. However, for the purest of sodium samples presently available, at 4.2°K, and in a magnetic field of 50 kG, the anomalous skin effect only becomes marked for tangential wavelengths well below 1 cm. The surface mode causing the loss involves a thin oscillating dipole layer with a strong electric field and strong electric current between the charge layers; ignoring anomalous skin effect, the energy per cm2 in the electric field tends to infinity as p - 0. When the dipole layer is thin enough, the surface mode may be represented by an "equivalent boundary condition" on magnetic fields, and thereby the number of equations describing the boundary-value problem is reduced. The surface mode only appears in boundaries parallel to Bo.

The present work has three obvious limitations: (i) No attempt has been made to treat resonances in finite . samples exactly. (ii) We have not translated the theoretical results into immediately, useable graphs. (iii) The treatment of anomalous skin effect with a nonzero wave vector along the conducting surface has been bypassed and replaced by a simple reductio ad absurdum argument.


The author wishes to express his special thanks to Professor J. A. Krumhansl and Professor R. Bowers for their advice and guidance through the process of writing this article; thanks also go to J. M. Goodman for several critical comments, and to M. T. Taylor for helpful discussions.

Notes added in the Internet edition

Note I

The solution in question is the trivial solution of (2.4), for which k x b = 0 and (p/p)w+k2 = 0. For this solution, since k x b = 0, we have k•b = JkiIbi. Noting that b=b(0) exp[i((otk•r)], we also have div b = -ik-b = -rlkljbl = 0. This, in turn, means that either JkJ = 0 or IbI = 0, either of which makes the solution uninteresting.

Note 2

The reader used to the notation of Klozenberg, McNamara and Thonemann, J. Fluid Mech. 21, 545-563 (1965), may prefer to move from (2.5) to (4.5), which corresponds to Eq

(16) of Klozenberg, et al. My k and γ correspond to their β and k, respectively. The step from (2.5) to (2.7) superficially appears to replace an equation of the form "a²=b²" by "a=b", rather than by "a =±b", and to overlook a second solution. In fact, of course, the step amounts to noticing that the two branches of (2.5), which are "(μ0/ρ)ω-ik² = ukγ" and "(μ0/ρ)ωik² =-ukγ", both enumerate the same set of ω, k, γ values, in different order. Enumerating all solutions of both branches would be duplicative; it is enough to choose any one of the two branches, which is what was done in (2.7).