I
The Generalized Doppler Effect
and Applications*
by DAN
CENSOR
Department of Environmental Sciences
Tel Aviv University, Ramat-Aviv, Israel
ABSTRACT: Scattering
involving
Doppler
for waves on a string and plane
The related quantum-mechanical
velocity,
in
to
are derived
without
arbitrary
for
are given for
space-time
solving
and the boundary
certain
classes of problems
problem is contidered
waves perpendicular
is considered,
to plane
simple
moving, and uniformly
such problems.
wave equation,
conditions
This facilitates
One method
The solutions
the analysis
accelerated
relies on
boundaries.
case of constant
involved in this cluss of problems.
transformations.
e.g. harmonically
the one-dimensional
representation,
problem
to include
The one-dimensional
electromagnetic
out the di@dties
using
modes of motion,
Two methods
solution
effect is generulized
non-uniformly moving boundaries.
the D’Alembert
the other starts with a general
determine
the exact structure
of
boundarks.
spectral
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
ment of Science and Technology,
Jerusalem, Israel.
103
Fund for the Advance-
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
in the sense used by electronics engineers. This
“frequency
modulated”,
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
.-w-m
FIG. 1.
reflected
moving
boundary
a
X
Geometry for scattering by a moving boundary on & string.
Consider a semi-infinite ideal string extending from x = --co to the timedependent 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
be represented by the D’Alembert solution
Y = f(t - x/c) + g(t + x/c),
of the string can
(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].
In order to illustrate the method used subsequently
consider the (almost trivial) case of constant velocity
in (2) yields
g[t(l+M)]
(2)
for arbitrary motion,
x(t) = vt. Substitution
M = v/c.
= -f[t(l-M)],
(3)
Now let .$ = t( 1 + M), hence,
g(6) = -f&z%1 Hence we have obtained
in g(t + x/c), hence,
&+x/c)
g(t), whatever
= -fU+x/c)
M)l(l + WII-
(4)
[ stands for. But we are interested
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=
sm
exp [ - iw,(t +x/c)]
--oo
F(o,.) dw,,
(6)
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)]
Vol.295,No.2,February1973
= -
‘m exp[-iw,t(l+M)]F(w,)d+
J --a,
(7)
105
Dan Censor
However,
P(0,)
(7) is the Fourier transform,
such that its inverse is
= - (27r)-1 m exp{it(l+M)[o,-w&l-N)/(l+M)]}(l+M)dt
s --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’,
where the dot indicates the time derivative.
x(0) = 0,
g(V + 4)
= -fV(l
(9)
Then instead of (3) we have for
-&I,
Ht = i&/c.
(10)
I
The similarity to Eq. (3) is deceptive, since here i%?tis 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
1M 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 -
(11)
5wam).
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,.,
5 = t + x(t)/c,
which becomes a Fourier transform
hand side of (12).
The analog of (8) is now
P(o,.) = -(2n)-l
when t = t[f]
(12)
i
is substituted
on the left-
Lrn exp{iw,E-iwit[.$]+iwiz(t[[])/c}dE.
J-co
(13)
can be
Proceeding formally, the spectrum of exp (- iwi t[f] +iwix(t[~J)/c}
found as a Fourier series or integral. If it is permitted to change the order of
106
Journal of The Branklin
Institute
The Generalized Doppler Effect and Applications
integration,
then (13) can be recast as
Jb,)
[”Wp) [”exp Lib, - ,4 Eldt dp
= - (2+
J-w
J-W
*co
=-
3-4 S(P- 4 dp = - =%=(4-
(14)
--co
The last result is consistent with (8) for the special case of constant velocity,
leading to
-
m 6[~--wi(l-M)l(l+M)16(1*--W,)d~
s --a,
=
-6(0J,-wi[(l-M)/(1+M)]}.
(15)
If the integrals (13) and (14) can be evaluated, the B(w,) in (6) is available
and the problem is solved. Let us consider the special case of uniform
acceleration
x(t) = at2/2, t 2 0.
(16)
where a is the acceleration of the boundary. The discussion is valid only for
at/c < 1. The inverse function of 6 = t + z(t)/c is
t = (c/a) [-
1 1 J(I +
2di41,
(17)
and we have to choose the positive sign for t > 0. A binomial expansion yields
t = (c/a) [at/c - (at/c)“/2 -I-(af/c)3/2 - 5(at/c)4/S + . . .I,
(W
and for sufficiently small values of (a/c) (t +x/c) only a few leading terms
must be retained in (18). From (17) it follows that t +x/c < 3c/2a, i.e. the
reflected wave consists only of a finite wave packet, since as M = 1 reflection
ceases to take place.
For many practical situations, especially for low velocities, it is very
tempting to consider (5) even for varying velocities, assuming that M is a
slowly varying function, and to replace M by M(t). Obviously this leads to
a contradiction since g is no more a function of t + x/c, hence g is not a proper
solution of the wave equation. A better way to do it is to take into account
the retardation and replace M(t) by M(t +x/c). For the present case, to the
first order in a.$/c (11) yields
g(t) = -f(4-a%2/c),
and g(t +x/c) follows. In general,
assuming t = f in (11)) hence
the first approximation
9(& = -f(f-2x(0/c),
(19)
is derived
by
(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 transparent. To the first-order approximation, (13) in terms of (18) may be recast
Vol.
295, No. 2. February
1973
107
Dan Censor
as
Jyw,) = - (277-l
a exp [i(w, - wi) 6 + iwi at”/c] d[.
s -*
(21)
On the other hand, since (19) is available we know g(t)
g(t) = - exp [ - iwi(E - at2/c)] = -
m exp ( - iw, .$)F(w,) dw,.
s --a,
(22)
Hence we have at least shown that (21) and (22) are consistent.
Another interesting example is the vibrating boundary. Let
x(t) = (A/Q) sin fit,
(23)
which prescribes A/c < 1. Here
(24)
which for t = t(t) entails the solution of a transcendental equation. However,
the problem may be solved by iteration, yielding a series which can be
truncated according to the power of A/c needed for a given accuracy. Thus
the zeroeth approximation is t = [. The first-order approximation is obtained
by substituting t = l+fi(&
A/c, which yields
t = t-
(A/cCl) sin!L$.
By adding f2(f) (A/c)~ to (25) and substituting
(25)
in (24) we find
etc.
fs( t) = 4 sin 2!Cl[,
(26)
To the first order in A/c we have t = .$, hence from (20),
g(e) = -fK--
(2AP)
sinW1,
(27)
and g(t + x/c) follows. Similarly, higher-order approximations
are readily
available.
In terms of the spectral representation (6), for a monochromatic incident
wave, (13) becomes
F(w,) = -(2rr)-1
* exp[iw,t-iwi.$+(2ioiA/cQ)sinL2,$+O(A/c)2]d.$.
s -cc
(28)
Exploiting
exp [(2iwi A/&)
sinRf
=
g ( - l)“rJ,(2w, A/&)
A=--OO
where J, are the nonsingular Bessel functions
mation and integration yields
P(w,) = -
108
exp ( - inCL$),
(29)
and changing order of sum-
2 (-1)“J,(2wiA/cQ)6(w,-wi-nil).
?&=-CC
Journal
(30)
of The
Franklin
Institute
The Generalized Doppler Effect and Applications
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)
Vol.
295, No.
2, February
1973
= 4M/(l-M2),
(35)
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/c2Cl) sin 2Q25]E --f(q) ;
8 = t +x/c.
For f = sin [w(t -x/c)],
the energy density of the incident wave,
instantaneous position of the boundary, as defined by (23), is
ef 2 = ew2 cos2 [wt - (wA/cQ) sin fit].
(36)
at the
(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
&42zz &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
110
Journal
of The Franklin
Institute
The Generalized Doppler Effect and Applications
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,,,
H;, = H,,,
y=
EL = y(E,+~,,v
x H),
H; = y(H,-QVXE),
(l-/P)-+,
(41)
1
B = V/C,
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.
‘E
d
.-
incident
%
0
hE
-
reflected
1
%
f”
FIG. 2. Geometry for
h
Lx
q’
transmitted
.-
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
S(p) is the transform of
-P -BWN
in (13). Thus (14) will be valid if
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 higherorder 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 1J,,(~w~ A/&) 1N 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]
112
2 ( - 1)” J,(~w, A/&)
7&=--a)
exp ( - inQ&
= ,g,
( - 1Y exp( - inQ0 LA+ (A/c)VA +&+A1
= ,=$,
( - 1)“exp ( -inLL$)
J,(2wi A/&Z)
[l + (nQ/wJ].
Journal
of The Franklin
(45)
Institute
The Generalized Doppler Effect and Applications
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
hi&‘) =f&--W/cl
20 = (P&0)“>
r1 - ?wl(1 -%/-wl +&I/a
P -/WI 2/P -t-&/z),
2 = (P/&P.
(46)
1
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 successively 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
+ Jymexp[-it(w,+k,
(+i + $,) lzCVl= exp [ - it(wi - bv)l
i.e. the total wave function vanishes at the boundary.
V)]F([)d<
= 0, (49)
Now we identify
.$ as
E = w,+k,v.
Hence (49) is a Fourier transform
F(t)
= -(277)-l
s
= +wi~kiv).
yields
O” exp [it( f - wi + ki v] dt
Therefore the reflected quantum-mechanical
in the opposite direction and wr, kr satisfy
w,+IC,v-coWi+kiv
114
(50)
and the inverse transformation
(51)
wave is a plane wave moving
= 0.
(52)
Journal of The Franklin
Institute
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
Abhandlungen der Kaniglich
BGhmischen Gesellschajt
Gestirne des Himmels”,
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
Doppler broadening diagnostics”,
IEEE
Trans. on
Censor, “Relativistic
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.
(13) D.
116
Journal of The Franklin
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