The Generalized Doppler Effect and Applications*censor/JFI-Generalized-Doppler-Effect.pdf · ... to include certain classes of problems involving non-uniformly moving ... in this

I

The Generalized Doppler Effect

and Applications*

by DAN CENSOR

Department of Environmental Sciences Tel Aviv University, Ramat-Aviv, Israel

ABSTRACT: Scattering Doppler effect is generulized to include certain classes of problems

involving non-uniformly moving boundaries. The one-dimensional problem is contidered

for waves on a string and plane electromagnetic waves perpendicular to plane boundaries.

The related quantum-mechanical problem is considered, simple case of constant

velocity, in to out the di@dties involved in this cluss of problems. The solutions

are derived without using space-time transformations. This facilitates the analysis of

arbitrary modes of motion, e.g. harmonically moving, and uniformly accelerated boundarks.

Two methods are given for solving such problems. One method relies on the D’Alembert

solution for the one-dimensional wave equation, the other starts with a general spectral

representation, and the boundary conditions determine the exact structure of tti spectrum.

I. Introduction

A wide range of phenomena and applications are grouped under the title “Doppler effect”, e.g. see Gill (1) for a general description. Generally one refers to changes in wave parameters, frequency, wavelength, amplitude, produced by transport processes in the media (including boundaries), supporting the waves in question. Doppler (2) considered the problem of relative motion of radiation sources and observers. Here we are concerned with the scattering Doppler effect, i.e. we wish to study the effects of the motion of obstacles on the reradiation of incident waves. There are numerous studies in this area for the class of problems involving uniform motion. Even this class of problems is too extensive to be surveyed here. Usually the scattering Doppler effect is studied by applying an adequate space-time transformation to the incident wave, thereby reducing the problem to the simpler case of a scatterer at rest. Then the inverse transformation is applied in order to obtain the result in the frame of reference of the observer. An example is provided by reflection of electromagnetic waves from a moving mirror, as given by Einstein (3). However, this method fails or becomes impractical when nonuniform motion is involved. Therefore presently the boundary conditions are applied directly in the frame of reference of the observer. Consequently we are dealing with time-dependent boundary conditions arising from the motion of the obstacles.

* This work was supported by the Bat-Sheva de Rothschild Fund for the Advance- ment of Science and Technology, Jerusalem, Israel.

103

Dan Censor

The present study is confined to one-dimensional problems of waves on a string, electromagnetic plane waves perpendicular to plane interfaces and, to some extent, to the corresponding quantum-mechanical problem of scattering by a moving potential barrier.

We start with the relatively simple problem of waves on an ideal string. The boundary moves arbitrarily, provided that at all times, the velocity of the boundary with respect to the string does not exceed the Mach number value of one. Two methods of solution are considered in parallel. One method relies on the D’Alembert solution for the one-dimensional wave equation, the other starts with a general spectral representation and the boundary conditions at the moving boundaries determine the exact spectral structure of the scattered waves. Particular modes of motion are provided by constant velocity, constant acceleration and harmonic motion of the boundary. The case of constant velocity serves to introduce the new methods, and since the results are well known, it serves as a check on the methods, to some extent.

It is shown that for periodic motion of the boundary the reflected wave is “frequency modulated”, in the sense used by electronics engineers. This statement is true only as a first-order effect. If higher powers of the maximum Mach number are considered, the wave is much more complicated.

Energy considerations for the string problem show that if the boundary moves periodically, energy is pumped into the wave field. This effect is present in the electromagnetic case too.

The electromagnetic case is very similar to the case of waves on a string, except for the boundary conditions. In the electromagnetic case the boundary conditions are prescribed by the principle of relativity, applied to Maxwell’s equations. For this case some considerations are given for scattering by a moving refractive slab. It is argued that for nonuniform motion Doppler effects can exist in the transmitted wave as well as in the reflected one.

The quantum-mechanical problem of scattering of plane waves by a moving potential barrier is considered for constant velocity only. The case of arbitrary motion is not solved and remains, therefore, an open question.

ZZ. Problem of the String

The relatively simple problem of Doppler effects on strings with uniformly moving boundaries has received much attention. (See Censor and Schoenberg (4) for a recent reference.)

incident reflected moving boundary

.-w-m a X

FIG. 1. Geometry for scattering by a moving boundary on & string.

Consider a semi-infinite ideal string extending from x = --co to the time- dependent position of the moving boundary z = x(t). (See Fig. 1.) The string is assumed to be lossless and of uniform density and tension.

104 Journal of Franklin Institute

The Generalized Doppler Effect and Applications

For the one-dimensional wave equation the displacement of the string can be represented by the D’Alembert solution

Y = f(t - x/c) + g(t + x/c), (1)

where f, g, are functions of the indicated arguments and c is the wave velocity. The incident wave is specified, the scattered wave we wish to derive, subject to the boundary condition that the total wave vanishes, y = 0, at the boundary specified at x = x(t), for all t. Thus at the boundary,

g[t + x(t)/c] = -f[t -x(t)/c]. (2)

In order to illustrate the method used subsequently for arbitrary motion, consider the (almost trivial) case of constant velocity x(t) = vt. Substitution in (2) yields

g[t(l+M)] = -f[t(l-M)], M = v/c. (3)

Now let .$ = t( 1 + M), hence,

g(6) = -f&z%1 - M)l(l + WII- (4)

Hence we have obtained g(t), whatever [ stands for. But we are interested in g(t + x/c), hence,

&+x/c) = -fU+x/c) W-M)lP+Wl). (5)

This is exactly the Doppler effect obtained by using the Galilean transformation, as described in the Introduction. However, by avoiding the method of transformation from one frame of reference into another, more complicated modes of motion of the boundary can be considered.

Thus far the problem is considered in the time domain. This method depends on the existence of the D’Alembert solution. But this is not available in a straightforward way for two- and three-dimensional problems. Hence, it is desirable to derive the same result, using a more general representation of the solution of the wave equation.

Let the incident wave be monochromatic, exp [ - iwi(t -x/c)]. As long as an arbitrary wave function can be represented as a superposition (sum or integral) of such plane waves, i.e. as long as the wave equation is linear, this is not a restriction on the generality of our method. Again let x(t) = vt at the boundary. Since the reflected wave is unknown, it is represented in a general way as

cl= s m exp [ - iw,(t +x/c)] F(o,.) dw,, (6) --oo

where wr, F(w,) remain to be determined by the boundary condition. For the wave to vanish at the moving boundary we substitute x = vt and obtain

exp[-iwit(l-M)] = - J

‘m exp[-iw,t(l+M)]F(w,)d+ (7) --a,

Vol. 295,No.2,February1973 105

Dan Censor

However, (7) is the Fourier transform, such that its inverse is

P(0,) = - (27r)-1 s

m exp{it(l+M)[o,-w&l-N)/(l+M)]}(l+M)dt --a,

= -S[w,- W&l-M)/(l+N)-j. (3)

Substituting (8) in (6), the S-function picks the frequency w,, = wi(l -M)/ (1 + M), which is the expected result, in accordance with (5).

We return now to the time-domain solution (l), and consider arbitrary motion of the boundary. Let us define average velocity between t’ = 0 to t’ = t, where t’ is the dummy variable

ii = Ii* dt’//;dt’, (9)

where the dot indicates the time derivative. Then instead of (3) we have for x(0) = 0,

g(V + 4) = -fV(l -&I, Ht = i&/c. I

(10)

The similarity to Eq. (3) is deceptive, since here i%?t is a function of time. According to (lo), one can make the statement that, in general, the Doppler effect depends on the average Mach number, which for uniform motion becomes a constant. From physical considerations it is deduced that only 1 M 1-c 1 is admissible for our model, since for M > 1 the wave never reaches the boundary, and for M < - 1 no reflected wave can leave the moving boundary. In order for the velocity to satisfy these bounds at any given instance, we have to have 1 &I < 1. Consequently, if we define < = t(1 +&), the inverse function t = t(f), is always a single valued function of [. It follows that for arbitrary motion, the analog of (4) is

s(5) = --f&5 - 5wam). (11)

Subsituting [ = t +x/c yields the scattered wave at any time along the string. Similarly, using the spectral representation (6) for arbitrary motion we

have at the moving boundary

exp [ - iq(t - x(t)/c)] = - J:m exp ( - iw, 5) F(q) dm,.,

i

(12)

5 = t + x(t)/c,

which becomes a Fourier transform when t = t[f] is substituted on the left- hand side of (12).

The analog of (8) is now

P(o,.) = -(2n)-l Lrn exp{iw,E-iwit[.$]+iwiz(t[[])/c}dE. (13) J-co

Proceeding formally, the spectrum of exp (- iwi t[f] +iwix(t[~J)/c} can be found as a Fourier series or integral. If it is permitted to change the order of

106 Journal of The Branklin Institute


integration, then (13) can be recast as

Jb,) = - (2+ [” Wp) [” exp Lib, - ,4 El dt dp J-w J-W

*co =- 3-4 S(P - 4 dp = - =%=(4- (14)

--co

The last result is consistent with (8) for the special case of constant velocity, leading to

- s

m 6[~--wi(l-M)l(l+M)16(1*--W,)d~ --a,

= -6(0J,-wi[(l-M)/(1+M)]}. (15)

If the integrals (13) and (14) can be evaluated, the B(w,) in (6) is available and the problem is solved. Let us consider the special case of uniform acceleration

x(t) = at2/2, t 2 0. (16)

where a is the acceleration of the boundary. The discussion is valid only for at/c < 1. The inverse function of 6 = t + z(t)/c is

t = (c/a) [- 1 1 J(I + 2di41, (17)

and we have to choose the positive sign for t > 0. A binomial expansion yields

t = (c/a) [at/c - (at/c)“/2 -I- (af/c)3/2 - 5(at/c)4/S + . . .I, (W

and for sufficiently small values of (a/c) (t +x/c) only a few leading terms must be retained in (18). From (17) it follows that t +x/c < 3c/2a, i.e. the reflected wave consists only of a finite wave packet, since as M = 1 reflection ceases to take place.

For many practical situations, especially for low velocities, it is very tempting to consider (5) even for varying velocities, assuming that M is a slowly varying function, and to replace M by M(t). Obviously this leads to a contradiction since g is no more a function of t + x/c, hence g is not a proper solution of the wave equation. A better way to do it is to take into account the retardation and replace M(t) by M(t +x/c). For the present case, to the first order in a.$/c (11) yields

g(t) = -f(4-a%2/c), (19)

and g(t +x/c) follows. In general, the first approximation is derived by assuming t = f in (11)) hence

9(& = -f(f-2x(0/c), (20)

where x(e) = x(t) describes the motion of the boundary, and g(t + z/c) follows as before. For the present case the exact solution of (11) is available. On the other hand, the spectral representation method of solving (13) is less trans- parent. To the first-order approximation, (13) in terms of (18) may be recast

Vol. 295, No. 2. February 1973 107

Dan Censor

as

s a Jyw,) = - (277-l exp [i(w, - wi) 6 + iwi at”/c] d[. -*

On the other hand, since (19) is available we know g(t)

s m g(t) = - exp [ - iwi(E - at2/c)] = - exp ( - iw, .$) F(w,) dw,. --a,

Hence we have at least shown that (21) and (22) are consistent.

Another interesting example is the vibrating boundary. Let

x(t) = (A/Q) sin fit,

which prescribes A/c < 1. Here

(21)

(22)

(23)

(24)

which for t = t(t) entails the solution of a transcendental equation. However, the problem may be solved by iteration, yielding a series which can be truncated according to the power of A/c needed for a given accuracy. Thus the zeroeth approximation is t = [. The first-order approximation is obtained by substituting t = l+fi(& A/c, which yields

t = t- (A/cCl) sin!L$. (25)

By adding f2(f) (A/c)~ to (25) and substituting in (24) we find

fs( t) = 4 sin 2!Cl[, etc. (26)

To the first order in A/c we have t = .$, hence from (20),

g(e) = -fK-- (2AP) sinW1, (27)

and g(t + x/c) follows. Similarly, higher-order approximations are readily available.

In terms of the spectral representation (6), for a monochromatic incident wave, (13) becomes

F(w,) = -(2rr)-1 s

* exp[iw,t-iwi.$+(2ioiA/cQ)sinL2,$+O(A/c)2]d.$. (28) -cc

Exploiting

exp [(2iwi A/&) sinRf = g ( - l)“rJ,(2w, A/&) exp ( - inCL$), (29) A=--OO

where J, are the nonsingular Bessel functions and changing order of sum- mation and integration yields

P(w,) = - 2 (-1)“J,(2wiA/cQ)6(w,-wi-nil). (30) ?&=-CC

108 Journal of The Franklin Institute


This is inserted in (6), yielding exactly the spectrum of a frequency modulated signal (as understood by electrical engineers), with the index of modulation 2AlcLl depending on the peak Mach number A/c. Note however that the statement that a harmonically moving boundary will frequency modulate the reflected wave applies only to small A/c. In general, additional spectral lines are present. By analyzing the reflected wave, the motion of the boundary can be sounded. It will be shown subsequently that in the electromagnetic case there is an additional first-order effect which makes the reflected wave differ from an ideal frequency modulated signal.

Energy consideration for the case of uniformly moving supports have been discussed previously ; Censor and Schoenberg (4) consider the cases of a semi-infinite string and the finite string with two supports, one at rest and the other moving with respect to the string. The energy density E consists of kinetic and potential components,

E(G t) = (a/2) (Re VU+ s)])2 + (T/2) (Re l&V+ s)1129 (31)

respectively ; E, T are the density and tension, respectively ; 8, I a/at, 8, = a/ax ; Re denotes the real part; f, g, are the incident and reflected waves, respectively; c = (T/s)*. Th e energy of the wave field in a system involving moving boundaries is not constant. Energy can be lost or gained by virtue of the boundary performing work as it moves while radiation pressures act on it. For f = sin [w(t - x/c] and a uniformly moving boundary we have

g = -sin[w(t+x/c)(l-M)/(l+M)].

Thus, (31) becomes

E(x, t) = &W2 {COG [w(t - x/c)]

+ru--M)l(1+w12 cos2[o(t+x/c)(1--)/(l+M)]), (32)

which is recognized as incident and reflected energy densities, moving with velocity c in the + x, - x directions, respectively. As the incident wave moves towards the boundary, receding at a velocity MC, the amount of energy absorbed by the boundary per second is c( 1 -M) times the incident energy density. Similarly the energy rate emitted by the boundary is c( 1 + M) times the reflected energy density. The difference is the work performed by the wave field on the boundary, hence the force F(t) is given by

F(t) MC = ed 2Mc[( 1 - M)/(l + M)] co9 [wt(l -M)], at x = Met (33)

and the time averaged force is

F = aw2(1 - M)/(l +M). (34)

For M = 0 (34) reduces to the radiation pressure found for supports at rest [cf. Morse and Ingard (5), pp. 103-104). Although the present formalism is non-relativistic, (34) has the same velocity dependence as the electromagnetic relativistic case (6). The fact that (34) depends on the sign of M implies that energy can be pumped into the wave field by moving the boundary to and fro, since

(l+M)/(l-M)-(l-M)/(l+M) = 4M/(l-M2), (35)

Vol. 295, No. 2, February 1973 109

Dan Censor

which is of first order in M. Conversely, a vibrating boundary, in the presence of a wave, will be damped, hence experiencing what may loosely be termed as “radiation viscosity”.

The balance of energy is now computed for harmonic motion of the boundary, in the presence of the above incident sine wave. The first-order velocity dependent approximation (25) is used, this is substituted in (11). To the second-order in A/c,

g(f) = -.I%- PAN-4 sin !L$ + (A2/c2 Cl) sin 2Q25] E --f(q) ; 8 = t +x/c. (36)

For f = sin [w(t -x/c)], the energy density of the incident wave, at the instantaneous position of the boundary, as defined by (23), is

ef 2 = ew2 cos2 [wt - (wA/cQ) sin fit]. (37)

The energy absorbed by the boundary during the time dt is c( 1 -M,) 8f2 dt, where CM, = A cos Qt. Similarly for the reflected wave, the energy density at the support is given by

&42 zz &w2[1 - (2A/c) cos fig + (2A2/c2 cos 2sZ.9” cos2 (~7)) ;

( = t + (A/h) sin Qt. (33)

The energy emitted by the support is c( 1 + Mt) aj2dt. The net energy loss is equal to the work performed on the moving boundary, hence,

F(t) CM, dt = c( 1 - Mt) 8f2 dt - c( 1 + M,) qj2 dt

= 2~w~{cos~ [wt - (wA/cfi) sin fit]) Mt( 1 - 23~) c dt, (39)

to the second-order in A/c on the right-hand side. Suppose that Q<w, then the average value of the expression in braces in (39) can be approximated by 4. If we now integrate the right-hand side of (39) over a period of 27r/Q of the motion of the boundary, then the net energy gain to the wave field is

ew2 c(A/c)~ 2,/Q. (40)

This is equivalent to a constant force ~w~?rA/c moving along the actual path 2A/SJ covered by the boundary during one cycle.

III. Corresponding Electromagnetic Problem

In this section the corresponding problem of plane electromagnetic waves, perpendicular to plane obstacles will be studied. In the first section the motion of the boundary has not affected the constitutive properties of the string, i.e. the tension and the density. In the present case this corresponds to plane interfaces, moving through a medium. Such idealized situations have been considered before (7, 8); however, here the obstacles will be assumed to move in free space (vacuum), therefore this difficulty does not arise.

In the first section the Doppler effects have been obtained without taking resort to the space-time transformations. However, it must be stressed that there is involved a physical assumption relating to frames of reference in



relative motion. Namely, it has been assumed that at the boundary the displacement of the string vanishes, for all observers. In the electromagnetic problem for uniform motion, the boundary conditions in the primed frame of reference, attached to the obstacle, and in the unprimed one, in which the observer is at rest, are not identical. The principle of relativity, as applied by Einstein (3), prescribes that the electric and magnetic fields be related by

Ei; = E,,, EL = y(E,+~,,v x H), H;, = H,,, H; = y(H,-QVXE), (41) y = (l-/P)-+, B = V/C, 1

where ,_, ,,, designate components perpendicular and parallel, respectively, with respect to the velocity v; c = (~,,.z,,-* is the speed of light, in free space characterized by Q, p,,. The derivation of (41) involves the Lorentz transformation; therefore, in the following, it may be said that instead of applying the Lorentz transformation to the waves, it should be applied to the boundary conditions.

d ‘E % incident reflected

.- - 1

f” %

0 hE

transmitted h

.-

Lx q’

FIG. 2. Geometry for scattering of electromagnetic waves with respect plane boundary.

The problem scattering of electromagnetic waves uniformly moving scatterers, free space, been considered by (9, 10) and Twersky (11). These rely on application of the Lorentz transformation to the waves in question. Here we prefer to use the formalism given for the string, and investigate problems involving non-uniform motion. Strictly speaking, non-uniform motion involves acceleration and therefore should be discussed in terms of the theory of general relativity. [See, for example, MO (12).] H owever, for low accelerations, such that the effect on the wave field is negligible, it is heuristically assumed that special relativity is valid, locally and instantaneously. Analytically it is assumed that “at the boundary” (41) is valid with v = u(t) substituted instead of the constant velocity.

The incident wave is chosen as a plane, linearly polarized electromagnetic wave, specified by its electric and magnetic fields fx(t -z/c), fH(t-X/c), respectively. [See Fig. 2.1 Similarly, the reflected wave is denoted by g,(t +x/c), gH(t +x/c). For the time being, the boundary is assumed to be a perfect reflector, i.e. a perfectly conducting plane. The electric and magnetic components are related by Maxwell’s equations in free space, prescribing that fE, fH and the direction of propagation x define a right-handed Cartesian triad of vectors, and that ) fx j / [fH/ = (&E,,)*; similarly, for the reflected wave. The boundary condition for a perfect reflector is that E’ = 0, in the

Vol. 295, No. 2. Pebruarr 1973 111

Dan Censor

frame attached to the scatterer. The spatial relation of the vectors is as in Fig. 2. By substitution in (41) of the incident and reflected fields, and following the argument leading to (a), we derive

R&+X/C) = -[(1-B)/(l+B)lf~{(t+x/c)[(1-B)/(1+8)1}. (42)

This agrees with Einstein’s (3) result. For arbitrary motion, the analog of (11) is

(43)

and the reflected wave is obtained by substituting E = t +x/c. In terms of the spectral representation, instead of (13) we now have

(44)

where iw, + i$( .$) stands for the exponent in (13). Thus (14) will be valid S(p) is the transform of

if

-P -BWN 0 +B(t[O)IY exp P&31.

As an example, consider the case of a harmonically vibrating mirror. It is clear from the previous section how to derive the reflected wave to a desired accuracy in powers of A/c, which is the maximum value attained by /3(t).

It is realized, however, that in practical situations we seldom encounter velocities comparable to c, and therefore only the first-order approximation is of importance. It must be kept in mind that in scattering (e.g. of laser light) by fast particles, we are dealing with relativistic velocities and higher- order approximation might be necessary.

In many cases, naive derivations of the electromagnetic Doppler effect disregard the amplitude factor [l - fl(t + x/c)]/[l - #l(t +x/c)] [see Censor (13, 14)], arguing that these effects are small, since only frequency changes are measurable to a high degree of accuracy. We shall show now that although the effect is justly neglected, the reason for neglecting it is different. Thus to the first order in A/c only n = 0, k 1 in (29) are retained. According to (30), this yields a carrier frequency wi and two sidebands of frequency wi +_ Sz, whose amplitude is given by 1 J,,(~w~ A/&) 1 N wi A/&. For the present case we have a factor 1 - Z(A/c) cos Clt (for x = 0, say). Accordingly, instead of (29) we have

[ 1 - 2(A/c) cos fit] 2 ( - 1)” J,(~w, A/&) exp ( - inQ& 7&=--a)

= ,g, ( - 1Y exp ( - inQ0 LA + (A/c) VA +&+A1

= ,=$, ( - 1)“exp ( -inLL$) J,(2wi A/&Z) [l + (nQ/wJ]. (45)



Consequently, for “narrow band frequency modulation”, neglecting the “amplitude modulation” effect amounts to assuming Q2/wi < 1, which is justified at high frequencies, e.g. in the optical range, but would not be justified when Q, wi are comparable.

We now turn our attention to the case of a penetrable non-dispersive reflector. Again it is assumed that the acceleration of the medium has negligible effect on the waves, otherwise a general relativistic formalism must be used, see MO (15). Let the scatterer be a half space characterized by the constitutive constants E,P. The geometry of the problem is given in Fig. 2. On the boundary, in the proper frame of the boundary at rest the boundary conditions are that the A x E’, fi x I-I’, the tangential components with respect to the boundary, be continuous across the boundary. The transmitted wave in the frame of reference comoving with the boundary is denoted, at the boundary, by hE(t’), h=(t’), where t’ is the time in this frame of reference. The boundary is referred to as x’ = 0, the origin in the comoving system. According to the Lorentz transformation we have t = yt’, which is the relativistic time dilatation. Inasmuch as most applications involve small /?, the dilatation effect is neglected subsequently. From (41), the usual reflection and transmission coefficients are derived, to first-order in /3,

%&+4t)l = fEP -awl r1 - ?wl(1 -%/-wl +&I/a hi&‘) =f&--W/cl P -/WI 2/P -t-&/z),

1 (46)

20 = (P&0)“> 2 = (P/&P.

Once h&(t’) is known at x’ = 0, and it is known that h” satisfies the wave equation with the phase velocity C = (pa)*, the wave is known anywhere in the refractive medium. This is obtained by replacing t’ = t by the retarded value t’ - x’/C.

An interesting application is the case of a moving slab. Consider the relation between the reflected and transmitted waves, and the motion of the slab. In general, a slab, like other bounded geometries, constitutes a dispersive system, i.e. waves are multiply scattered within the system and then scattering products interfere with each other, and the overall result is frequency dependent. However (46), describing a single scattering process, is independent of frequencies. Therefore, rather than displaying the complicated overall result of scattering by a slab, we shall follow a few success- ively scattered modes.

For uniform motion /3 = constant it is well known that the Doppler effect cancels in the forward direction, in the sense that the only effect is introduced by the parameters E, p, d of the slab. If the incident wave is time harmonic with frequency w, then the only effect will be due to the fact that the moving slab is excited by w(l -/3). On the other hand, for arbitrary motion x(t), the waves launched into the slab depend on /3(t), and propagate in the slab according to /3(t’ - x’/C). For a certain mode of successive scattering the path covered by the wave is Z(Z = md, where m is a positive odd integer). Now we have @(t’ - Z/C). By this time the wave emerges at the x’ = d face of the slab, and is transformed into the frame of the observer by multiplying by a factor

Vol. 295, No. 2, February 1973 113

Dan Censor

1 +p(t+At) (where t is the time of entrance into the slab). The time-lapse is At = Z/C. Obviously the Doppler effect will not be cancelled. Moreover, in the optical approximation (i.e. to the first order of (1 - Z,/Z)/( 1+2,/Z) the reflected wave consists of the wave reflected from x’ = 0, according to (461, and the wave reflected by the other face x’ = d. Hence if 2dlC is a time comparable with the period T of the moving slab [say it is moving according to (23)], then the reflected wave will no longer be purely “frequency modulated”, and considerations similar to those leading to (45) should be incorporated.

IV. Related Quantum-mechanical Problem

The Doppler effect for moving particles is of importance in nuclear physics, especially in connection with neutron reactions. [See Bethe (16).] In the absence of a unique relativistic model for quantum theory, the present discussion is confined to a naive non-relativistic model.

A uniform stream of identical non-interacting particles will be described by the wave function

& = exp (ilci x - iwi t), (47)

where k, = P,/A, Pi is the momentum and A in the conventional notation is proportional to Planck’s constant h ; similarly wi = E,/E where Ei is the kinetic energy of a particle. Since for wave z,&, which the Schrijdinger a D’Alembert (1) does not exist, the spectral representation method must be used. Thus, the reflected wave is represented as

$7 =I* exp ( - krx - iw, t) F(c) dt, --co

(43)

where the parameter 5, related to the momentum and the energy of the reflected wave, will be defined subsequently.

Let the scattering obstacle be represented by an infinite potential barrier, moving according to x = vt, at a constant velocity v. The boundary condition is taken as

(+i + $,) lzCVl = exp [ - it(wi - bv)l + Jym exp[-it(w,+k, V)]F([)d< = 0, (49)

i.e. the total wave function vanishes at the boundary. Now we identify .$ as

E = w,+k,v. (50)

Hence (49) is a Fourier transform and the inverse transformation yields

s O” F(t) = -(277)-l exp [it( f - wi + ki v] dt

= +wi~kiv). (51)

Therefore the reflected quantum-mechanical wave is a plane wave moving in the opposite direction and wr, kr satisfy

w,+IC,v-coWi+kiv = 0. (52)


The Generulized Doppler Effect and Applications

If we assume ks := 2mw7, the classical relation between momentum and kinetic energy, then it is easily verified that (52) is consistent with

Ic, = k+( 1 - 2v/u), (53)

where u is the group velocity and therefore the particle velocity in the classical limit. This result is expected from a simple argument based on observations in the frame of reference of the incident wave and the frame comoving with the potential barrier.

In a formal way the above problem can be extended to finite potential barriers, and square potential (analogous to the moving slab problem). However, the extension to arbitrary motion of the boundaries, e.g. harmonic motion according to (23) is not clear.

The main difficulty lies in the fact that for quantum-mechanical waves the wave velocity (phase or group velocity) depends on the behaviour of the barrier. In contradistinction, a proper wave depends on the properties of the medium. For example, consider the case of a potential barrier at rest, reflecting a steady wave & as in (47). At t = 0 it starts to move at a velocity v. Now a different wave will be reflected, and the front of this wave moves at a group velocity u - 2v. Hence the velocity of the boundary determines the wave velocity. Consequently this class of problems is not the analog of the problems in the previous sections.

References

(1) T. P. Gill, “The Doppler Effect”, London, Logos Press; New York, Academic

Press, 1965. (2) C. J. Doppler, “Uber das farbige Licht der Doppelsterne und einiger anderer

Gestirne des Himmels”, Abhandlungen der Kaniglich BGhmischen Gesellschajt

der Wissenschaften, Vol. 2, pp. 467-482, 1842. (3) A. Einstein, “Zur Elektrodynamik bewegter K&per”, Ann. d. Phys., Lpz., Vol.

17, pp. 891-921, 1905; English translation, “On the electrodynamics of moving bodies”, In The Principle of Relativity, New York, Dover, reprint.

(4) D. Censor and M. Schoenberg, “The problem of energy concentration on a rapidly wound cable”, Israel J. Tech., Vol. 9, pp. 531-534, 1971.

(5) P. M. Morse and K. U. Ingard, “Theoretical Acoustics”, New York, McGraw-Hill, 1968.

(6) D. Censor, “Energy balance and radiation forces for arbitrary moving objects”, Radio SC;., Vol. 6, pp. 903-910, 1971.

(7) D. Censor, “Scattering of a plane wave at a plane interface separating two moving media”, Radio Sci., Vol. 4, pp. 1079-1088, 1969.

(8) D. Censor, “Scattering of electromagnetic waves in uniformly moving media”, J. Math. Phys., Vol. 11, pp. 1968-1976, 1970.

(9) D. Censor, “Scattering in velocity dependent systems”, D.Sc. thesis (in Hebrew), Technion-Israel Inst. of Tech., Haifa, Israel, 1967.

(10) D. Censor, “Scattering in velocity dependent systems” (based on Ref. (9)), Radio Sci., Vol. 7, pp. 331-337, 1972.

(11) V. Twersky, “Relativistic scattering of electromagnetic waves by moving obstacles”, J. Math. Phy.s., Vol. 12, pp. 2328-2341, 1971.

(12) C. T. MO, “Theory of electrodynamics in media in noninertial frames and applications”, J. AT&h. Phys., Vol. 11, pp. 2589-2610, 1970.

Vol. 295, No. 2. February 1973 115

Dan Censor

(13) D. Censor, “Relativistic Doppler broadening diagnostics”, IEEE Trans. on Nuclear Sci., Vol. NS-15, pp. 27-30, 1968.

(14) D. Censor, “On Doppler broadening in velocity dependent random media”, Israel J. Tech., Vol. 8, pp. 395-406, 1970.

(15) T. C. MO, “Electromagnetic wave propagation in a uniformly accelerated simple medium”, Radio Sci., Vol. 6, pp. 673-679, 1971.

(16) H. A. Bethe, “Nuclear physics B. nuclear dynamics, theoretical”, Rev. Modern Phys., Vol. 9, p. 140, 1937.


Documents

The Generalized Doppler Effect and Applications*censor/JFI-Generalized-Doppler-Effect.pdf · ... to include certain classes of problems involving non-uniformly moving ... in this