Covariant transformations of wave equations for initialboundaryvalue problems withmoving boundary conditionsH. E. Wilhelm Citation: Journal of Applied Physics 64, 1652 (1988); doi: 10.1063/1.341785 View online: http://dx.doi.org/10.1063/1.341785 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/64/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the initial boundary value problem for a shallow water equation J. Math. Phys. 45, 3479 (2004); 10.1063/1.1765216 Initialboundaryvalue problem for diffusion of magnetic fields into conductors with external electromagnetictransients J. Math. Phys. 23, 1765 (1982); 10.1063/1.525227 InitialBoundaryValue Problem of the Formation of an Electrical Discharge in a Flow Phys. Fluids 15, 1328 (1972); 10.1063/1.1694085 Nonlinear InitialBoundaryValue Problem for Convection, Diffusion, Ionization, and Recombination Processes J. Math. Phys. 13, 252 (1972); 10.1063/1.1665965 Analysis of Nonlinear InitialBoundaryValue Problems in Recombination and Diffusion Kinetics J. Chem. Phys. 53, 1677 (1970); 10.1063/1.1674243

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Covariant transformations of wave equations for inilial .. boundary-value problems with moving boundary conditions

H. E. Wiihelm Department of Electrical Engineering. King Fahd University 0/ Petroleum and Minerals, Dhahran 31261, Saudi Arabia

(Received 14 December 1987; accepted for publication 12 April 1988)

It is shown that the wave equation 1/Jxx - t/lyy = 0 for the field ",,(x,y) in the domain R(xy) can be transformed into a wave equation \Ii S5 - \{I1ft1 = 0 for the field lI'(t,'I]) in the domain S(S'l]). The transformation is accomplished through a complex function F(x,y) = t(x,y) + i1J(x,y), which is not analytic. For the transformation to exist, the real transformation functions t = t(x,y) and rJ = 'I](x,y) have to satisfy wave equations in the domain R(xy) and the first-order partial equations $ x = ± lly and $y = ± 1]x [" ± " distinguishes transformations of the first ( + ) and second ( - ) kinds]. Thus, the hyperbolic transformation theory is different from the conformal mapping theory, where the real transformation functions satisfy the Laplace equation and the Cauchy-Riemann conditions. As applications, the linear Lorentz transformation and nonlinear mappings of time-varying regions into fixed domains are discussed as solutions of the indicated partial differentia! equations. Furthermore, an initial-boundary~value problem for the electromagnetic wave equation with moving boundary condition is solved analytically (compression of microwaves in an imploding resonator cavity).

I. INTRODUCTION

In many fields of science and engineering, initial~bound­ary~value problems for the wave equation with moving boundary conditions are encountered, e.g., the analysis of the compression of electromagnetic fields (magnetic flux, electric flux, and microwaves) between moving conductors (liners) requires the solution of hyperbolic initial-boundary­value problems with boundary conditions at least at one moving boundary.I-6 Simple problems, such as the micro­wave or optical frequency tuning by a conductor moving with constant velocity, have been analyzed to some extent, since the functional equation resulting from the moving boundary condition has a simple analytical solution. I In gen~ era!, however, the solution of initial-boundary-value prob~ lems with moving boundary conditions amounts to employ~ ing more general mathematical methods, e.g., Fourier expansion in space- and time-dependent eigenfunctions2

,3

(resulting in an infinite set of coupled, second-order differ­ential equations for the t-dependent Fourier amplitudes) and Laplace integral transformations4 (resulting in a com­plex integral equation for the s-dependent Laplace ampli­tude function). Also conformal mapping theory and the WKB method were used for the solution of such moving boundary-value problems.5

,6

The methods mentioned are mathematically rather ill­volved. 1--<> For this rea.son, we investigate whether hyperbo~ lie problems wi.th moving boundary can be solved by trans­forming the wave equation for the domain with moving boundary to a wave equation in a domain where the bound~ ary is fixed. We show that the desired transformation func~ tions are determined by two coupled linear partial differen­tial equations. For illustration, linear and nonlinear transformations which leave the wave equation form invar­iant are discussed, and an initial-boundary-value problem

for an electromagnetic resonator with transient waH motion is analyzed.

The presented partial differential equations for the de­termination of the covariant transformations of the wave equation are as important as the Cauchy-Riemann equa­tions in the transformation theory of the Laplace equation. These fundamental partial equations permit an analytical solution of hyperbolic initial-boundary-value problems with moving boundary conditions. In addition, they determine aU possible transformations which leave the wave equation form invariant. The Lorentz transformation turns out to be a simple linear transformation.

II. TRANSFORMATION OF WAVE EQUATION

We consider the wave equation for the scalar field ¢(x,t) = 1/J(x,y) , where x and t are the independent, real­space and time coordinates. If c designates the characteristic wave speed [e.g., c= (uo€O)-1/2 and c= (rpr/po) 1/2 for electromagnetic and acoustic waves, respectively J and y = ct, the scalar wave equation for the independent vari­ables x,y in the domain R is given by

a2t/l a2¢ --2 - -2- = 0, x,yeR. (1) ax ay

The initial and boundary conditions for Eq. (1) are not needed for the transformation theory (they vary with the respective applications and are specified there) . The possible transformations of the wave equation ( 1) shall be investigat­ed herein, which leaves this hyperbolic equation form invar­iant. In particular, we are interested in those covariant trans­formations which map a given time-varying domain into a fixed domain.

The real, nondegenerate transformations, which map

1652 J. Appl. Phys. 64 (4),15 August 1988 0021-8979/88/161652-05$02.40 © i 988 American Institute of Physics 1652

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every point (x,y) of the domain R into a point (s,1j) of the domain S, are, in general, of the form

g = g(x,y), 1] = 1j(x,y),

with

x=f(s,1]), y=g(S,1J),

(2)

(3)

as the corresponding real, inverse transformations. These map every point 5. 11ES into a point x,yeR. For the determi­nant of the transformation (2)

(4)

The two real transformations in Eq. (2) are mathematically equivalent to the single complex transformation

F(x,y) = g(x,y) + i1](x,y). (5)

Equation (5) maps the domain R of the plane x,y into the domain S of the plane 511. The conditions for a covariant transformation of the wave equation (1) will specify the pos­sible types of functions F(x,y).

Through the real transformations (2), or the equivalent complex transformation (5), the wave function 1f;(x,y) of the domain R becomes a wave function t/J(s,'1]) of the do­mainS,

(6)

Under the assumption of the existence of the first- and sec­ond-orderpar!ials ofl,b(x,y) and W(g,'~), Eq. (6) yields, by partial differentiations,

o 2r/J a 2\fJ ( o2\fJ 02'if1 ' ax? = a$ 2 5 ~ + 01J as + ag a11 )5 x 1j x

a 2\j1 2 a'IJ oW + ar/ 11x + 05 gxx + 01] 1/= , (7)

a2¢ _ alw 2 ( alii' a2\11 ) oy2 - dS 2 g y + as d1] + a11 as Sy1]y

aZ\jI? aw aw + arl 1/; + as Syy + a1] 1]yy' (8)

where sa=a~/oa,1ja=a1Jlaa, and S~=(Sa)2, 1]~ = (1Ja )2, for a = x,y. Substitution of Eqs. (7) and (8) into Eq. (1) shows that the arbitrary complex transforma­tion (5) transforms the wave equation ( 1 ) into the (compli­cated) partial differential equation of second order:

2 f: 2 a 2qJ 2 ( .. f:) a 2\11 (S" - ~ y) 052 + 5x1]x - ':Jy1]y ago'TI

2 2 a 2\11 f: f: all! + (11", - 'TIy) a1]2 + (~xx - ~yy) as

alii + (1]x;o; -1]yy) a11 = 0, (9)

where 0 2'111 as 07] = a 21{11 a7] Os has been assumed (imply­ing continuity of the second-order partial derivatives as a sufficient condition).

Weare interested in those complex transformations (5) which transform the wave equation ( 1 ) into a wave equation of the covariant form, namely,

a 2\11 a 2\11 - - -=0 S.1]eS. (10) ag z 01J2 '

1653 J. Appl. Phys., Vol. 64. No.4, i 5 August i 988

Comparison of Eqs. (9) and (10) indicates that Eq. (9)

reduces to Eq. (lO) if the real transformation functions in Eq. (5) satisfy the conditions

Sxx - 5yy = 0, 'TIxx - 'YJw = 0,

SxYJx - gy 1])' = 0,

S; - 5; = - (7]; - 1]~ ) # o.

(11 )

(12)

(13)

By Eq. (11), the transformation functions s(x,y) and 1](x,y} must be solutions of wave equations in R(xy)' too. By Eq. (12), their first-order derivatives have the similarity

Sxl1/y = 5yl11x = h(x,y). (14)

For the determination of h(x,y), Eq. (14) is inserted into Eq. (13). Accordingly,

5; - 5; = h (X,y) 2 (1/; - 7];) = - (11; - 1J; ) # O. (15)

Hence,

h(X,y)2 = 1, h(x,y) = ± 1. (16)

It is now recognized that two different types of complex transformations exist, depending on whether h (x,y) = ± 1. By Eqs. (14) and (16), we define the two covariant transfor­mations (5) of the wave equation ( 1) by the foHowing fun­damental conditions:

Transformations o/thefirst kind, h(x,y) = + 1:

Sx = + 1]y' 5:; = + 1Jx; (17)

transformations o/the second kind, h(x,y) = - 1:

sx = - 7/y, 5y = - 1Jx . (18)

In addition to either Eq. (17) or (18), the transformation functions g(x,y) and 11 (x,y) have to be solutions of the wave equations in Eq. (11). Equations (17) and ( 18) are consis­tent with Eq. (11).

In view of the constraints ofEqs. ( 11 ) and ( 17) or ( 18), the transformations 5 = S(x,y) and 'TJ = 1] (x,y) in Eq. (5) are not the "conformal transformations of the first and sec­ond kinds" known from the complex mapping theory of the Laplace equation. 7 The latter transformation functions have to satisfy Cauchy-Riemann conditions and be harmonic (solutions of Laplace equations). 7

Thus, we have derived the important result that the wave equation ( 1) in R (xy) can be transformed into a wave equation (10) of the same form (covariance) in S(S'TJ) by means of the complex transformation F(x,y) =s(x,y) + i71(x,y) in Eq. (5), if the real transformation

functions s(x,y) and 7/(x,y) satisfy the fundamental partial differentia! equations (17) or (18) (transformations of the first and second kinds). The transformations of the first and second kinds permit the mapping of regions of the x-space (1) expanding and (2) shrinking with time (y = ct) into fixed regions of the 1/ space.

In the proposed complex transformation theory of the wave equation, Eqs. (17) and (18) playa similar role as the Cauchy-Riemann conditions in the mapping theory of the Laplace equation. It should be noted that F(x,y) in the transformation (5) is a complex function of x and y, but not a complex function of z = x + iy, since S(x,y) and 7/(x,Y) do not satisfy the Cauchy-Riemann conditions

H. E. Wilhelm 1653

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(Sx = ± 1]y,Sy = +7fx),1 but the new conditions in Eq. (17) or (18). Since F(x,y) #F(z) is not analytic. the exist­ing covariant transformations of the wave equation (1) are not conformal with respect to infinitesimal areas, as it is the case for analytic transformations.7

It is remarkable that an analogous transformation theo­ry can be establishedtl for the solution of parabolic initial­boundary-value problems with moving boundary condi­tions,9 e.g., for the time-dependent Schroedinger equation. 10

m. APPLICATIONS

The mathematical significance of the partial differential equations ( 17) or ( 18), which determine all covariant trans­formations of the wave equation, is obvious. As an illustra­tion of the practical importance of these fundamental equa­tions, some applications are given (with time variable 1" = ct==y).

A. Linear Lorentz transformation

The transformations 5 = s(x,r) and 11 = 1](x,r) are so­lutions of Eqs. (11) and (17) or (18), and are, in general, nonlinear in x and r. An example for a linear transformation, which leaves the wave equation form invariant. is the Lor­entz transformation between i.nertial frames (x,r) and (S,1]) with constant relative velocity v (Ref. ! 1 ) :

S=y(x-/3r),1]=y(r-/3x), (19)

where

(20)

The transformations (19) are solutions of the partial equa­tions (11) and (17) since

Sx = 11T = y, 57 = 11x = - /3r. (21)

and

Sxx = STr = 0, 1]xx = 1JTr = O. (22)

Thus, the linear Lorentz transformation (19) satisfies the partial equations (17) of the first kind and the wave equa­tions (11) in a trivial fashion. Similarly, the nonlinear Lor­entz transformation, discovered by introducing a minimum length (nonpoint particles) into relativity theory,12 can be shown to satisfy Eqs. (1 I) and (17).

8. Nonlinear transformations of first and second kinds for linear boundary motion

For a spatial region between a fixed plane x = 0 and a moving plane x = s( r) with s( r) = So ± {Jor [linear bound· ary motion increasing ( + ) or decreasing ( - ) the x do­main], the integrals S = S(x,r) and TJ = 1J(x,r) ofEqs. (17) or (18) shall be found which reduce the variable region O<x<s( 1') to a fixed region 0<11<110. Using, as trial solu­tions, functionals of s( 1") ± POX. one finds [see Eq. (56)]

S = !In{[s( r) +,8oX J [s( 1') - PoX ]/sU , (23)

1J=i!n{[s(r) +,8oX]/[s(r) -PoX]}. (24)

where

s( 'I) = So ± /3or. ,80 = vole> O. (25)

Indeed, 11(X = 0, 1") = 0 and 1J[x = s( '1'),1"] = 1Jo (con-

1654 J. Appl. Phys., Vol. 64, No.4, 15 August 1988

stant). Thus, the transformed field space 0, 11< 1]0 does not change, i.e., is fixed, as required. In addition, Eqs. (23) and (24) satisfy Eqs. (11), (17), and (18) since

S" = ± 11T = -/3~ x/N, t,. = ± 1]x = ±PoS(r)/N, (26)

Sxx = Sr.- = - /3~ [s(r)2 + (PoX)2]!N 2,

TJxx = TJrr = + 2f3~ (/3OX)s( r)/Nz,

where the" ± ., signs refer to those in s( 1"), and

N= [s(r) +.BoxJ [s(r) -/3oX].

(27)

(28)

(29)

Equations (23) and (24) are examples of nonlinear trans­formations t = t(x,r) and 1] = 7j(x,r), of the first ( + ) and second ( - ) kinds [see Eq. (25) J, which render the wave equation form invariant. Comparison with the ± signs in the boundary motion (25) indicates that the trans­formations of the first and second kinds map regions O,x<s( 1") expanding ( + ) and shrinking ( - ) with time 1',

respectively, into fixed regions 0<11<110 of the (t, 1]) plane.

C. Electromagnetic initlal-boundarymvalue problem with moving boundary condition

Consider a one-dimensional (x) electromagnetic reso­nator bound by ideally conducting plates x = 0 and x = s( r), where s( r) = So - .Bor decreases linearly with time r[Eq. (25)]. The compression of the electromagnetic fields in the variable resonator cavity,

!. = _ e oA(x,1') B = _ e oA(x,r) (30) c Z or' Y ax'

is determined by the initial-boundary-value problem with moving boundary condition for the vector potential A =A(x,r)ez :

oZA 02A ai2 = ox2' O<x<s(r), (31)

A(x = 0,1") = 0, o < 1" < so//3o,

A[x=s(r),r] =0, O<'1'<sO/13o•

(32)

(33)

aA(x,r=O) _ B () 0 (34) ax - - 0 x, <X<So,

ClA(x,r=O} Eo(x} --.:--::-..-.:- = - --, 0 < x < so' (35)

ar c

Bo(x) and Eo(x) are the initial electromagnetic fields in the resonator O<x<so, Le., at the time T = 0 when the conductor surface x = s ( 1") is set in motion.

By means of the transformations S = s(x,r) and 'YJ = 7](x,'1') in Eqs. (23) and (24), Eqs. (31)-(35) with moving boundary condition (33) atx = s( 1') are reduced to an initial-boundary-value problem with fixed boundary con­ditions for the vector potential A (S,1]) == IlJ (S, 1]), since by Eq. (24)

'1}(x = 0,1") = 0, 7][x = s( 7),rJ = 710'

where

(36)

(37)

The solutions of the (transformed) wave equation A'I'1 =Ass are arbitrary functions A =/(S± '1}). Accord-

H. E. Wilhelm 1654

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ingly, the general solution for A (S,1/) in the fixed domain 0< 1/<110' Eq. (36), is sought in terms ofthe complex Fourier series

+00 A(t,1/) = L An (e i1Tl'(S+1I)/1/n _ei1rn (s-1/)/1/o) , (38)

which satisfies the boundary conditions A (t,l1 = 0) = 0 [Eq. (32)] and A (S,l1 = 710) = 0 [Eq. (33)]. By Eqs. (23) and (24),

S:t1J=<P(1'+x), <1>(;) =In[(so-po;)/soJ. (39)

Transforming back to the original variables x, r by means of Eq. (39), the vector potential is obtained in form of a func­tional Fourier series of <P ( l' + x).

- '" (40)

The expansion coefficients An foHow from the initial condi­tions (34) and (35), according to which

2 ' \-~ A i1Ttl<t>(X)/1/ d [4>(x)171o] Irr £.. n " e 0 -=--..:-...;--'~

-co dx

= + [Bo(x) + Eo(x)Ic].

By means of the orthogonality relations,

f_+\1 exp(hr(m - n)(<t>171o) ld(<t>171o) = 28mn ,

Eq. (41) gives for the Fourier coefficients

(41)

An = - (4~n) ia+ (Bo(X) + Eo~X) )e-- i1Tn

4>(X)/1/0 dX,

(42)

where

a± = - (soI/3o)(e±1/o-l)~O (43)

are the integration limits of x corresponding to those of q,(x)171o= ± 1 [Eq. (39)]. Since a+ <0, the (periodic) initial mode function Bo(x) + Eo(x)1c in Eq. (42) is ex­tended into the region x < O. Equations (40) and (42) give the analytical solution of the EM initial-boundary-value problem (31 )-( 35) with moving boundary condition for ar­bitrary initial conditions Bo(x) and Eo(x) in the resonator.

Anyone of the eigenfunctions Er;" (x,1' = - 0) and B':,. (x,'1 = - 0) of the static resonator (1' < 0) can be cho­sen as initial state. By Eq. (30), these are derivable from the vector potential modes of the static resonator (O<x<so,1' < 0),

A! (x,1') = Re[A ± sin(mrrxlso)i"m'], (44)

where

(J)m = (mrrlso), m = 1,2,3,... (45)

are its eigenfrequencies. For A ± chosen imaginary (fA) or real ( - 1..4), two fundamental types of initial conditions ( 7 = - 0) result, namely ( + ) of electric type and ( - ) of magnetic type:

E 0+ (x) = (J)mcA sin(mrrx/so), B l (x) = 0, (46)

B 0- (x) = wmA cos (mrrxlso) , E 0- (x) = 0, (47)

where A is a real amplitUde. Accordingly, the electric ( + )

1655 J. Appl. Phys., Vol. 64, No.4, 15 August 1988

or magnetic ( - ) type initial values in Eq. (42) are of the form

± + I A sin I 1 B 0 (x) + E 0- (x) C = wmA (mrrx so), m = ,2,3, .... cos

(48)

Since the waH of the resonator can be moved only with in~ frarelativistic velocities, the following approximations hold extremely well: 710 = 130; 4> ( ± x) = =t= PoXl so; a ± = + so; and exp( - irrn$( ± x)171o) = exp( ± i1i'nxlso) since <P( ±x) =In(l=F,8oX/so) = +/3oX/so and O<x<so. for Po = vol c < 1. Thus, Eq. (42) gives for the Fourier ampli­tudes

( - 1) A

A;;: = i (sowmA 14rrn)5nm

= ( ~ l)(S#mA 14rrm) , n = m

=0, n¥m, (49)

where the coefficients ( - I,i)+'?( ± ) apply to the electric ( + ) and magnetic ( - ) type initial conditions (48). re­spectively. Since A n+- = real and A n- = imaginary,

A:I:" = ±A tn' n= 1,2,3, .... (50)

The simple result (49) shows that, for infrarelativistic cavity wall motion vo<c, the magnetic vector potential (and the associated EM field) is a degenerate Fourier series with only one essential term, n = m. Thus, we find from Eq. (40) for the magnetic vector potential in the imploding resonator cavity the closed-form solution (0<1' <soI,8o):

A ±(x,r) =1A [:~:(m?r<l>(r+x)/,8o)

- :~:( mrl>( r - X)/,8o)] , (51)

after combining the exponentials of imaginary argument to cos and sin functions. The logarithmic functional q, (1' ± x)

defined in Eq. (39) cannot be simplified in the case,8o = vol c< l,since.8orlso = vot Iso is not smaU compared with 1 dur­ing the compression period, O<t < solvo. The solution (51) satisfies all Eqs, (31)-(35) of the "moving" initial-bound­ary-value problem. The initial values A + (x,t = 0) = 0 and A - (x,t = 0) = - A sin(mrrxlso) are in accord with Eq. (44), and the equivalent initial conditions (46) and (47).

Furthermore, Eq, (51) satisfies the boundary conditions (32) and (33), since {3oS( 1')/so </30 < 1.

Equation (51) indicates that an initial mode with wave number k m = mrr/so increases in intensity IA ± (x,or) I with time l' as the cavity space is reduced with speed Vo. Since the inward wall velocity Vo is very small in comparison with the velocity of light c, the compression of an EM mode field occurs quasistatically. In this respect, the compression of "microwaves" in an imploding cavity resonator is similar to the compression of static electric and magnetic fields by con­ducting liners.2

•3•13

The EM energy contained in the resonator cavity in­creases with time t, since the moving wall (conductor) per­forms work against the EM back pressure. As the cavity space is reduced, s(t) ."..0, EM wave fields with smaller and

H.E.Wilhelm 1655

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smaller wavelengths A = 211"/ k are generated, the maximum field intensities occurring at wave numbers k-rrls(t) , This follows mathematically from Eq. (51 ), by expanding A ± (x,t) in space- and time-dependent eigenfunctions2

,3

fn = sin [n11"x/s(t) ] of the instantaneous, variable resonator space O"x"s(t), where s(t) -.0 for t--solvo'

The above theory is of technical interest for the genera­tion of ultra strong microwave fields by imploding a conduct­ing microwave cavity by means of explosives. This process is very efficient, since the fiux of the EM wave fields can be compressed without significant losses (the implosion time t- 10-4 S is very short compared with the diffusion time 1" D

_10- 2 s for EM fields in good conductors such as cop­per).13

The solution ofthis EM initial-boundary-value problem with a moving boundary was accomplished by means of a form-invariant transformation of the second kind, Eq. (18), for the wave equation. In an analogous way, the transient microwave fields in an expanding resonator cavity can be analyzed by means of transformations of the first kind, Eq. (17).

IV. CONCLUDING REMARKS

Since the boundary values at the fixed boundary x = 0 and the moving boundary x = s( 1') are given, it is sometimes easier to deduce first the transformations x = x (;,rj) and r = r(;,rJ) from the "inverse" partial equations to Eqs. (17) and (18), namely

(52)

From the solutions x = x(;,rJ) and 1" = r(t,1J), the trans­formations 5 = s(x,r) and 1] = 1J(x,r) follow then by inver­sion.

The determination ofthe functions s(x,r) and rJ(x,l'), which accomplish the mapping of the variable domain O<x"s( 1") into a fixed domain O"1J<1Jo,

x = 0¢1] = 0, x =s(r)¢:>1] = 1]0. (53)

can also be reduced to the solution of a functional equation. Since rJ (x, 1") satisfies the wave equation ( I 1 ), this transfor-

1656 J. Appl. Phys., Vol. 64, No.4, 15 August 1988

mation is a function of the form

1](x,r) =!(l'-x) +f(r+x),

with

f( - ;) = - f( + s), S = 1" ± x,

JI l' - s( r)] + fI 1" + s( r)] = 17o,

(54)

(55)

(56)

according to the boundary conditions (53). By Eq. (55), the transformation function !(t;) is an odd function of t; = r ± x. Equation (56) gives the transformation function fct;) as a solution of a functional equation, in which the boundary motion s = s( 1') is known. The solution of func­tional equations is discussed elsewhere. 14 With 1] (x,r) thus determined, s(x,1') is obtained by integration ofEq. (17) or (18).

Further applications of the proposed transformation theory of the wave equation will be given in another paper. II

ACKNOWLEDGMENT

This investigation was supported by King Fahd Univer­sity of Petroleum and Minerals.

'R. I. Baranov and Y. M. Shirokov, SOy. Phys. JETP 26, 1199 (1968). 2H. E. Wilhelm, App!. Phys. 31 E, 173 (1983). 'H. E. Wilhelm, Phys. Rev, A 27,1515 (1983). "'H. E. Wilhelm, Radio Sci. 20,1006 (1985). 'K. A. Barsukov and G. A. Grigoryan, Radio Eng. Electron. Phys. 21, 46 (1976).

OK. A. Barsukov and G. A. Grigoryan, Radio Phys. 19, 194 ( 1976). 7R. Rothe, F. Ollendorff, and K. Pohlhausen, Theory of Complex Functions with Applications to Engineering Problems (Technology Press, Cam­bridge, 1933).

sH. E. Wilhelm (unpublished). 9H. E, Wilhelm, J. Phys. A 16, 2149 (1983). lOR E. Wilhelm and S. H. Hong, Acta Phys. Acad. Sci. Hung. 48, 425

( 1980). "S. P. Puri, Special Theory of Relativity (Asia Pub!. House, New York,

1972), "P. Winterberg, Atomkernenergie 44,238 (1984). I3H. E. Wilhelm, J. AppL Phys. 56,1285 (1984), 14J. Aczel, Functional Equations and their Applications (Academic, New

York,1966).

H. E. Wilhelm 1656

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