From an analysis of the seasonal dependence of the absorption it follows that with respect to cosmic radio emission at a frequency of 5.8 Mc the D-region of the ionosphere makes a substantial contribution to the absorption. This contribution is of the order of or greater than the contribution of the F-region to the total absorption. This result, which apparently contradicts the conclusions drawn in [3], IS valid only for the period close to the m i n i m u m of solar activity, when the c r i t i c a l frequencies f0 F9 are much reduced and the contribution of the F-region of the ionosphere to the total absorption is correspondingly less. The results obtained in [3] relate to years close to the m a x i m u m of solar activity, when the F-region of the ionosphere is strongly developed (noon values of f o F z > 7 Mc) and therefore do not contradict the results of the present investigation. The author wishes to thank E. A, Benediktov for his constant interest and useful comments. REFERENCE 1. A. 2. E. 3. E. Radiofizika,

M. Svechnikov, Yu. K. Chasovitin, and N. A. Kochenova, Izv. VUZ. Radiofizika, 3, 375, 1960. Appleton and W. R. Piggott, L Arm. Terr. Phys., 5, 141, 1954. A. Benediktov, Yu. S. Korobkov, N. A. Mityakov, V. O. Rapoport, and L. N. Khodaleva, Izv. VUZ. 3, 957, 1960.

5 August 1964

Scientific-Research Radiophysical Institute, Gor'kii University

UDC 621. 371.22 R A D I A T I O N O F A S U R F A C E W A V E BY A C H A R G E THE INTERFACE BETWEEN TWO MEDIA

CROSSING

V. Ya. Eidman Izvestiya VUZ. Radiofizika, Vol. 8, No. 1, pp. 188-190, 1965 The radiation of e l e c t r o m a g n e t i c waves by a charge crossing the interface between two media, first investigated by Ginzburg and Frank [1], has since been examined by a number of authors, but so far only the radiation of volume waves has been taken into account. It is well known, however, that a surface wave can also exist at the interface between certain media. Thus, the Cherenkov radiation of a surface wave by a charged filament moving along an interface was investigated in [2]. In this note I e x a m i n e the radiation of a surface wave by a charge passing through the interface between a vacuum and a m e d i u m with a real dielectric constant g(w). In order for a surface wave to exist, it is necessary to assume that over a certain range of frequencies ~(~) < - 1 . Proceeding in the usual manner (see, for e x a m p l e , [3, 4], for the component of the e l e c t r i c field Ez parallel to the trajectory of the particle in the m e d i u m we have CO

E,.,z (~, z) = ei -- f

0

Jo - ('/'?) e i p,,,z--~o t] ~.~z d z d %

(I)

k2-r

where

J0 is a Bessel function, v is the velocity of the charge, k z = ~ + w~/c ~, the coordinates of the observation point in the plane yz are p and z (the Oz axis is perpendicular to the interface). Note that only fields whose existence is due to the presence of the interface are discussed below. The integration in (I) with respect to the variable ~ as p --> oo can be carried out using the saddle-point method. In this case, however, it is necessary to consider the pole of the integrand at =

k2 ..... Ek, =- 0,

(2)

and also the bifurcation point B = vo/c. The remaining singular points of the integrand in (1) correspond only to exponentially small terms as p --* co, It is usually assumed that ~ > 0 and therefore the singularity at ~ = 0 is not r e a l i z e d for real 138

values of %. At the same t i m e , it is obvious that r e l a t i o n (2) corresponds to the radiation of a surface wave, i . e . , a wave, in which the field decreases e x p o n e n t i a l l y in both directions from the boundary z = 0. Equation (2) for real z has a solution only when the above c o n d i t i o n s(c0) < - 1 is fulfilled. T h e n from (2) we have

"/'='/'~

71

1,'

eI

where ~t = ts I. Using the pole of the integrand d e t e r m i n e d from (2), we o b t a i n f~om (1) Ltle following expression for the field in the surface wave:

//:-~j'>(,.o)o,
ei

_ g--I

=,4

.......

"

. . . . . . . . . .

-.

ZC

---

1;%

(<)<--1

f

. . . . .

~' - -

tJ~}

do~,

,

I' n--t

(3)

(* ..... 1)

where A2o

(~1o~,'c)/~/~% - - I, Z ": O.

Following [5], it iS easy to c a l c u l a t e the contribution to the expressions for the field corresponding to the point z = = co/c (co > 0). This part of the field is l i n k e d with the quasi-stationary field of a charge l o c a t e d above the boundary plane between two m e d i a ; below, this part will not be t a k e n into account. In principle, this field c a n be o b t a i n e d with the ordinary formulas of electrostatics. In order to find the energy lost by the charge due to radiation of the surface wave, it is necessary to d e t e r m i n e the energy flux transported by the surface wave through the l a t e r a l surface of a circular cylinder with its axis directed along the trajectory of the particle. With the help of (8) and the corresponding expression for the m a g n e t i c field it is easy to c a l c u l a t e the a b o v e - m e n t i o n e d flux. Thus, for the energy of the r a d i a t e d surface wave in the m e d i u m 2 2 during the e n tire transit t i m e of the charge we get (to > 0): v

,=,---

zl

--Be

,

C

-

,

1+

~4el

2

-".,

% d.~,

(e~-1)3,'2

s {a,) < .- 1

~~

~

2-~

I k ~"

.

"

|ell a 2

d,,, (~2,% n~--()2

c4) "

Here 7 - s~ ..... l; 71 .= I -}- ~2/'~; % _ | ~ ~2s2/],, [~ ::vie. The energy flux s of the surface wave in a v a c u u m , i . e . , for z < 0, is c a l c u l a t e d analogously. Thus, for the total energy lost by the charge due to radiation of the surface wave we have ~,~ %' 2

e2 32

,::

[(1+

-'

C o (21 s ( m ) < -1

-1)],2L\

-,

a2 ~,a -i ..... ^l,..7.~Z~

;t" [

1+ ,

.........

cs)

(32~1-f-'/) 2"

'

T h e energy fluxes in the two m e d i a are oppositely directed: s 1 > 0, P,= < 0, and the resultant e n e r g y flux is always dir e c t e d away from the trajectory of the particle (n > 0). Note that the radiated energy does not depend on the direction of m o t i o n of the electron (see [5]). Of course, formula (5), o b t a i n e d without taking a t t e n u a t i o n of the wave into account, is valid at the threshold of the effect when 0, -~ ~ 0 / 1 / ~ only for frequencies satisfying the r e l a t i o n ~ , , o / l / - 2 - - - ~ o >

, , w h e r e u is the effective n u m b e r

of collisions for the plasma electrons. However, if this fact is not formally t a k e n into account, if follows from (8) that although the spectral radiation density has a singularity at 7-~ 0 (v,, ~ ;,-U2), the total r a d i a t e d energy E has no singularity. When .... , - , , , o / I / 2 - ,

t h e r a d i a t e d energy in any finite or i n f i n i t e s i m a l frequen.cy interval is zero.

In the n o n r e l a t i v i s t i c case formula (5) simplifies to Z(~2 <<1)

e2 e~ F =

C

J

q (1 + at) d,~ ~1/2(}2 ~1 } ~ ) 2 I

e2mo 41 v !

~ I/2

/ '"02\ / e = 1 - ,,,N / '

(6)

s ( m ) < --1

It follows from (6) that at s m a l l charge v e l o c i t i e s the r a d i a t e d e n e r g y ~ -- I v t- ' and, in particular, s ~ ~o as v --~ 0. As already noted, this is also c o n n e c t e d with the fact that in the i d e a l i z a t i o n considered a t t e n u a t i o n of the wave is not taken into account. Therefore, when v --> 0 the t i m e of i n t e r a c t i o n of the charge with the spatially inhomogeneous field of the surface wave b e c o m e s infinite, and i n f i n i t e energy c a n be transferred to the wave. In fact, e v e n disregarding coilisions, formulas (.5) and (6) are v a l i d thanks to spatial dispersion only when v >> v T, where v T is the m e a n t h e r m a l v e l o c ity of the plasma electrons. At a r e l a t i v i s t i c charge v e l o c i t y the radiated energy tends to a finite l i m i t as 8 2 + 1:

139

'2 (~'~ ~ 1) - , '

t;9 C s, (! + ei) dlo e2 mo .... ' .... A - e -~] "/I/2 (S2-t- el- II c (~)< - - I

(7)

where 1

1 1......

r

1

'-'./'~

)

---

:~-i- l -4. . . . .

,; =:- 1/ 2 + V D-;

']

;I arctg--if -"-" 0.7;

~, ..... 1 / V 5 - - ' 2

.

Here attention is drawn to the following. It was noted above that although the radiated energy of the wave tends to zero, in a certain interval of frequencies as . . . . . . . , , / V 2 , the spectral energy density Xw-~ ~. At the same t i m e , for instantaneous transit of the boundary the work done by the field on the charge must be zero, since the fields on both sides of the boundary are then equal in magnitude and opposite in direction (s ~ - 1 ) . In fact, the transit t i m e is always finite (v < c); but formally from (6) as v -~ ~ it actually follows that r w = 0. From (7), in particular, it follows that in the case of a relativistic particle v e l o c i t y the surface wave radiation energy constitutes only a small addition to the forward radiation energy [6]. On the other hand, for frequencies with s(co) < - 1 , the radiation of the surface wave is decisive for the energy losses of the moving charge. Formula (5), of course, also follows from the expression for the work done by the field Ez,su r on the charge: 0

co

A = AI-'~ A.,= evRe { ~ Eoz~surdt + ,t' E~ --00

= --x,

0

where E0zisur = Ezisu r ~0 = 0, z = vt), i = 1, 2. Then, for e x a m p l e , the following relation holds for the work done by the field in the medium: A2 .

c

.

. . 7W2(7 + ~ 2 e l ) ( 7 t - ~2ee)

1+ =1%

72=1=2([

1 ,%2~,) d ....

(~)< - 1

Note that whereas A = - Z , A i ~ - Z i. In the relativistic case, when S --~ 1, the quantity A2--~ 0, while - A 1 tends to an expression coinciding with (7), that is, the work, roughly speaking, is done only during motion of the charge in the v a c uum. In conclusion, I should point out that the above formulas for the energy lost by a moving charge can also be obtained employing an expansion with respect to normal surface waves (see [7]), The author is obliged to V. L. Ginzburg for suggesting the subject and also to N. G. Denisov and M. A. Miller for making valuable comments. REFERENCES

1. 2. 3. 4. 5. 6. 7.

V. L. G. V. L. G. L.

L. Ginzburg and I. M. Frank, ZhETF, 16, 15, 1946. S. Dolin, Thesis, Gor2di University, 1959. M. Garibyan, ZhETF, 33, 1403, 1957. Ya. Eidman, Izv. VUZ. Radiofizika, 5, 478, 1962. D. Landau and E. M. Lifschitz, The Mechanics of Continua [in Russian], Gostekhizdat, Moscow, 1954. M. Garibyan, ZhETF, 37, 526, 1959. A. Vainshtein, Electromagnetic Waves [in Russian], I z d . Soy. r a d , , Moscow, 1957.

11 December 1963; after revision, 12 September 1964

Scientific-Research Radiophysical Institute, Oor'kii University

140