Reflections from Linearly Vibrating Objects: Plane Mirror ... - IEEE Xplore

898

IEEE TRANSACTIONS

ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO.

5,

SEPTEMBER 1982

Reflections from Linearly Vibrating Objects: Plane Mirror at Oblique Incidence DANIEL DE ZUTTER

Abstracr-The reflections from a perfectly conducting plane in sinusoidalvibrationare evaluated. The incident wave is a timeharmonic plane wave at oblique incidence. The solution is essentially based upon expansions of thescatteredfields in infinite series of evanescentand propagating plane waves. The expressions obtained by this method are compared to the expressions obtained by the quasistationary method and to the expressions obtained by means of the special theory of relativity.

I. INTRODUCTION

I

F A MOVING BODY is illuminated by an incident wave, the scattered field is modulated by the motionof the target. Theradaroperatorshouldbeabletoextractthemotional Fig. 1. Vibrating perfectly conducting plane, immersed in plane wave properties of this target from the characteristics of the radar at oblique incidence. signal. However, before this synthesis problem can be solved, someconfigurationsmustbeanalyzedwhere the motion of The instantaneous velocityis therefore the scatterer has a known form. Much attention has been given u(t) = a d cos = PC cos to the case where the motion is a translation with uniform ve(2) locity. In this case the theory of special relativity can be apwhere plied to find the scattered fields and the associated Doppler spectra [ 11, [2 1. The scattering of a plane wave by a rotating circular cylinder [ 3 ] , [4]or by a rotating sphere [ 51 has been We start with the polarization shown in Fig. 1 , i.e., with an ininvestigated,using theinstantaneousrest-framehypothesis [4 ] . This hypothesis states that a scattering probleminvolving cident E-wave. The most general expressions for the incident fields in the laboratory are anacceleratedbodycanbesolvedusing the local boundary conditions and constitutive relations of special relativity. The constantvelocityappearingintheseequationsmust be replaced by the time and/or space-dependent velocity of the accelerated body. In the present paper this hypothesis will be applied to a sinusoidally moving and perfectly conducting plane. The scattering of a time-harmonic plane wave by this vibrating mirror has The fields are given in complex form. Their actual value is the [ 6 ] -[X]. In recently been investigated for normal incidence real part of the complex representation. The incident electric [ 8 ] it is shown how the results can be extended to a mirror is given by field has a unit amplitude. The phase angle moving arbitrarily along a direction perpendicular to its surface and to normal plane-wave incidence with arbitrary time~i=~t-~zcos~i-~sine,v+cu dependence. k =o/c Inwhatfollows, the mirror problem will be solved for a (5) plane wave at oblique incidence. Contrary to the case of normal incidence, finding the reflected fields in closed analytical form does not seem possible in this case. The solution is essenTime t = 0 corresponds to the central position of the mirror, tially based upon the representation of the reflected fields by plane-wave expansions. Analogous representations are found in hence an additional phase angle (Y must be introduced in This angle turns out to be trivial, however, and will be omitted the theory of reflection by gratings [91. The expressions are satisfy Maxwell's obtained for the scattered fields using plane-wave expansions in the following. The reflected fields equations in vacuo. These equations must be supplemented by andthencomparedwiththoseobtainedthrougheitherthe quasi-stationaryapproximationor an approximation based the radiation condition for Z +. -= and by the boundary contion at the air-metal interface. According to the instantaneous uponthe specialrelativity.Ourmethodscaneasilybeex[41, theboundaryconditionsata rest-frame hypothesis tended to mirrors endowed with arbitrary periodic motion. boundary point moving with instantaneous velocity c a r e t h e 11. SOLUTION IN THE LABORATORY FRAME same as if the body were in uniform translation with constant The vibrating plane is shown in Fig. 1. The displacement is velocity Fromthe special theory of relativity, it is found of the form that the boundary conditions at a perfectly conducting body, Z o = d sin (1) moving with constant velocity E, are [IO]

at

at

ai.

(z, z)

at.

Manuscript received April1, 1981;revised January 20, 1982. The author is with the Laboratory of Electromagnetism and Acoustics, University of Ghent, Ghent, B9000 Belgium. 0018-926X/82/0900-0898$00.75 0 1982 IEEE

R:

899

DE

In (6), iin represents the unit vector along the normal to the conductingbody.andarethetotalelectric field andthe total magnetic induction at the interface. This theory is generally accepted [4] as valid provided (v/c)' 1. At the moving mirror and in the case of an incident E-wave, (6) reduces to

s

(Exi - vByi) + (Exr - uBYr) = 0

az

+ B~~ = o u = pc cos at.

aHzr aHyr aExr ---- E o -. ay az at

at)

= exp [j(wt - k y sin B i

- kd

cos 8 i sin Rt)].

( 8)

This field has the following Fourier expansion:

Exi(Z = d sin at)

+= J-,(kd

To obtain this expansion we made from FM theory, viz.,

COS

Bi)eimnt.

(9)

use of a classical formula

,,,(K).

(10)

The Fourier expansion of the incident fields B y i , B z i follows directly,by simple proportionality,fromtheexpansion of E x i . The incident wave at the mirror clearly contains the frequencies wm = w f ma. The coupling between reflected and incident wave is expressed by boundary conditions (7). Due t o the fact that the velocity of the mirror is periodic with period 27r/a2, it follows that (7) can only be satisfied if the reflected wave contains the same frequencies wm as the incident wave. The above argument, and the fact that the physical situation only depends on the y coordinate by a factor eCikY si-n ' i , leads t o expressions of the following form for the reflected fields:

2

-j(k

inf IAm 1-1/'mI

> 1.

To obtain (15) we used the fact that elmxJ-

m=-m

E,' =

lim


k2

s n i'

Bi

(12)

(k + mK)'< k' sin2 Bi.

(13)

+ m K )2 ] 1/2 ,

Noticethat we havecancelled out afactor e - j k y s i n ' i on both sides of (29) and (30). These two boundary conditions are not independent: differentiation of the second condition with respect to the time t immediately yields the first one. In the following, the unknown coefficients A , will be calculated

900

IEEE TRANSACTIONS

ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 5 , SEPTEMBER 1982

starting from (20). To this end, the factor Jut will be suppressed, and the following Fourier expansions introduced in the equation:

+-

=

e-jkdcosOisinSZt

einatJ-n(kd cos S i )

where

- 200 COS 0 i - k &*)(t

@r = Yo2W(l

+ Y o 2 k (cos Si - 200 + po2

(21)

n=--cD

'

(2 -2,)

-to)

- ky

sin S i

cos Si)

+ W t o - k COS S i 2 0

(29)

+m

ejymdsinSZt

=

ejln

I=-

t~I(ymd>.

(22)

Yo = (1 -00 2 )- 1 / 2

= uo/c;

m

The final (infinite) setof equations for theA , takes the form

x

+m

J-,(kdcosSi)=

Po

, =-

jkA,Jn-,(y,d).

(23)

m

For evanescent waves ( 1 3), 7, is a negative imaginary number. In this case the Bessel functions in (22) and (23)can be written in terms of the first-order modified Bessel functions, using the general property [ 11 I :

J,(-~z) = ( j ) - "I&).

At time to the position of the mirror is 20 = d sin at0 and its velocity uo = PC cos ate. If we assume that the mirror has a uniform velocity equal t o UO, the reflected wave has the value (28) at all times t . As was the case for normal incidence [8], we can go one step further and setto equal to theobservation time t , i.e., we ignore the propagation effects. In the neighborvalues of 0 = v,,,/c the hood of the mirror and for small scatteredelectric field can be obtainedfrom(28)and(29), dropping terms of order p2 :

( 24)

E,' = (1

<
m,,,. The range of n values should also be limited t o In 15 mmax. It will be shown in Section V that the A , obtained by this procedure quickly converge to their fiial values for mmax = m if sufficiently large values of i n m a x are chosen to truncate (23). The plane-wave expansions (1 1)showthatthereflected plane wave has a spread both in frequency and spatial reflection angle. The numerical results of Section V clearly demonstrate these properties.

For small 0 (30) can finally be reduced t o

Ex

r --ejwt

-

e - j k s i n O i y e j k Z c o s O .le- 2 j k d s i n S 2 t c o s O i

[ 1 - 2 f l ~ 0 ~ S i ~ 0 ~ f i t - 2 j k f l ( Z - d S i n S l t ) ~ 0 ~ L(31) 2t].

If terms in 0 are dropped, (31) reproduces thevalue (27) given by the quasi-stationary method. The presence of a factor kZ implies that (31) has a limited region of validity. In Section V some numerical results will be given showing the differencebe111. APPROXIMATE SOLUTION IN CLOSED ANALYTICAL tween the series solution (1 l) and the approximations (27) FORM and (31). Contrary to the case of normal incidence [ 8 ] , finding the In the case of normalincidence [81 approximate expresreflected fields in closed analytical form does not seem possisions could be found, by solving the problem in the accelerble. Such a closed form is obtained if the problem is solved by ated frame of reference in which the scattereris at rest. A simthe quasi-stationarymethod. This approximation consists of ilar analysis could be undertaken in the present case. In [ 13 ] freezing the moving body in its track at time t and calculating this method is shown t o reproduce the value (31), if terms of the fields as if the scatterer were stationary. This yields a time order b2 are neglected. sequence of fields which isthe sought quasi-stationary approxmation. IV. SCATTERING O F AN H-WAVE The electric field reflected from a stationary plane at position 2, is Let us now consider an H-wave, i.e., a wave where is parallel with the x axis. The incident fields are E r =-&at - j k s i n O i y e j k Z c o s e . - 2 j k ~ ~ c o s 0 ~ e

X

le

.

(26)

According to the quasi-stationary method, the variationof 20 with time must be inserted in (26). This gives the following form for thereflected field: =--jar

e - j k s i n O i y e j k Z c o s O .Ie -2ZjkdsinCltcosei .

(27)

This result can be improved assuming that the object moves withauniform velocity j.i equal to the instantaneous vibrational value. Assume that the mirror is located at 2, at time to and that it moves with uniform velocity UO. The reflected wave is known from special relativity to be [ 121 :

Ex' =-yo2(1 - 200

COS

R,H~' = yo2 (COS di - 2p0

R,H~' = sin S i e j @ r

R cHx i -- - . i @ i ~~i

= -cos

E,' =

6jej@i

Ozei@i

( 3 2)

where @i, k, and R , are defined by (5). The analysis for the H-wave can be conducted along the same lines as that for the E-wave. The boundary conditions (6) obtained by the instantaneous rest-frame hypothesis now become

(Eyi f PC cos L2ttBxi) 4- (Ey' 4- PC COS QtB,') = 0.

(33)

Si + 002 )e i @ r

+ po2 cos Oj)ej*r

(28)

The series solutions (1 1) are also suitable for the solution of the H problem, provided that E,'is replaced by R a X r and that R,Hyr and R,Hz' are replaced by -Ey' and -Ez'. At

90 1

DE ZUTTER: REFLECTIONS FROM LINEARLY VIBRATING OBJECTS

the moving mirror, i.e., for 2 = d sin

[iy, +i(k ,=--co

at,(33) yields

responding angle of reflection8,"

+ mK)p cos

. A ejymdsinfLtejmfLt

sin Orrn =

m

- (p cos -

Notice that the common factor e-jky sin @iejwthas been cancelled on both sides of (34). Introducing the Fourier expan(22) into (34) leads t o t h e followingsetof sions(21)and equations for theA , :

To amve at (35), we used the following recursion relationship for theBessel functions [ 1 11

m +J-m-

(36)

I(u).

U

The series solution for the H-wave can be compared with an approximate solution obtained by means of the special relativity, as was done in Section 111 for the E-wave. We only quote the final result for Hxr,viz., c

r

- - j k s i n 9 i y e j w t ej k Z c o s O i , - Z j k d c o s B j s i n f L t

x --e

[ 1 - 2p cos eicos at

- ykp(2-d

sin

at)cos at]. (37)

Dropping terms of order p in (37) again gives the value predicted by the quasistationary method. V. NUMERICAL RESULTS

The unknown coefficients A , in the series solutions (1 1) can be found starting from (23) for the case of an incident Ewave and from (35) for the case of an incident H-wave. As explained in Section 11, the infinite set of equations must be truncated. We limit the number of equations to( 2 m m a x 4- 1). This is also the case for the numberof terms in the infiniteseries on the right-hand sides of (23) and (35). Calculations show that the A , quickly converge t o their final value of m m a x = m. The actual value of mmax for which sufficient accuracy is obtained depends both on kd and 0. Greater value of these parametersrequireagreatervalueof m m a x . If approaches one, a great many A , are different from zero. For values of p larger than one, corresponding to the nonphysical case of a maximumvelocityexceedingthevelocity of light, n o convergence can be obtained. For the following examples, however, a value of = 20 is always sufficient. As mentioned in Section 11, the series solutions (1 1) for the reflected wave consist of a discrete set of propagating or eva= w m a . For nescentplane waves withfrequencies a,,, every index m sdtisfying (12), Le., ( k m K ) k2 sin2 Bi, the correspondingplane wave will bepropagatinginadirection zirm , as indicated in Fig. 1 for the incident E-wave. The cor-

m,,,

+

>

+

~

.

sin

-

(1

ei

+ mQ/w)

From (38) the reflected wave can be seen to show not only a frequency spread but also a spread in spatial reflection angle. If, however, ( k f ~ z K