ABSTRACT: Scattering Doppler effect is generulized to include certain classes of
problems involving non-uniformly moving boundaries. The one-dimensional.
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
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
