IMPACT OF A WAVE-PACKET

Another important source of error arises from the factthat the flux passing through the density sample does soat various angles and Beers' law of absorption may be

used to evaluate the correction since the optical paththrough the sample is greater than for normal incidence.*

The expression for calculating the total error becomes

C(antilog DL)

,pP A[r2-(x-p) 2 ] 1 COS2¢p D_ dydx

P-r 0 b ~~~~ antilog n-s n p)

,(15)

where DL is the nominal density level of the sample, n isthe index of refraction of the emulsion and support, andthe remaining symbols are the same as used above. Thisequation has not yet been shown capable of solution.

The difficulties in treating these errors do not seriously

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

detract from the simplicity of the method since theirintegrated effect may be evaluated directly by simplychanging the base distance and comparing results.

* Correction for this phenomenon is further complicatedby the scattering power of the silver grains.

VOLUME 33, NUMBER 3 MARCH, 1943

Impact of a Wave-Packet and a Reflecting Particle

HERBERT E. IvEsBell Telephone Laboratories, New York, New York

(Received January 4, 1943)

INTRODUCTION

THE impact of a wave-packet and a particlehas been studied by A. H. Compton, for

the special case of a light quantum striking anelectron. The frequency of the reflected or"scattered" radiation was deduced by Comptonalong relativistic lines by invoking the laws ofconservation of momentum and energy, andintroducing the known values of h and m. Ageneral study of the impact of wave trains ofdefinite length, or "packets," on reflectingparticles, using classical wave theory, does notappear to have been made. In the investigationwhich follows this is done, with the interestingresult that a derivation of the variation of masswith velocity is obtained, using only the principlesof conservation of mass, momentum, and energy.

PROBLEM AND SOLUTION

We start with the definition of a wave-packetas a train of waves of energy density E, emittedin time At; its total energy content is E(cAt).Our problem is to study the impact of such apacket on an initially stationary reflecting

particle. The result of such an impact will bethe reflection of the packet, with a change ofenergy content, and the recoil of the particle ata velocity dependent on its mass.

We attack the solution of the problem byfinding the energy content of the reflectedwave-packet, and the pressure which causes therecoil of the particle.1

Consider first the energy content of a wave-packet reflected from a uniformly moving mirrorwhose velocity is v in the direction of wavepropagation. The frequency of the reflectedwave system is

(1)

where c is the velocity of propagation of thewaves; its energy density

EC-V 2E=EoX\c+v

(2)

I For the following energy, pressure, and mass relationssee H. E. Ives, J. Opt. Soc. Am. 32, 32 (1942).

log

27rBI

163

cvV :--- VO 2

C+v

HERBERT E. IVES

the length of the packet after reflection

C+V IC+VI=lo( )=cW t ),_(3)

where the subscript 0 indicates in each case thevalue for a stationary mirror; the energy contentis therefore

c-vEl = E, (cAt) _ .(4)

C+V

To transform this result over to the case of amirror initially stationary, but which under theimpact of the wave-packet acquires the velocityv, we require the average value of the velocityof the mirror. This will be somewhere between0 and v; call it for the present f(v). The energycontent of the reflected packet is then

E'At) c-f (v) 1CLC +f (v)J

Noting that the duration of the impact ofwave-packet and particle is cAt/[c-f(v)] we setup the impulse equation, which is stated in (8),as follows, designating the mass equivalentcoefficient by m:

-c-f (v) cAt2E f _ =mE(cAt)c

c+f(v) -f(V)

+mE(ct)[f( cc+f(v)

from which1

M = -

C2

(9)

(10)

The mass equivalent of the incident wave-packetis thus

(cAt)E-

C2

(5)

(1 1)

Proceeding now to the pressure, we equate thenet flow of energy into the mirror per unit timewith the work done by moving the mirroragainst the pressure. Calling P the averagepressure we have

E[c -f (v)] - E () [c+f(v) ] = Pf (v) (6)

from which

P=2E 4 lC+ AV)]

(7)

The existence of a pressure due to the incof a train of waves furnishes the wave-]with an attribute characteristic of a mass-the reflecting particle in other worebe affected as though struck by another nparticle. We can utilize this fact for the scof our problem by determining the mass elent of the wave-packet. We do this by udefinition of pressure made in terms ofnamely,

Pressure =change of momentum

time

change of (mass Xvelocity)

time

of the reflected wave-packet

C2 c+f(v)(12)

Since the velocity of the wave-packet is c, wecan, from the definition of momentum as massXvelocity, ascribe to the wave-packet beforeand after reflection the momenta

(cAt) (cAt)E c=E

C2

Cand

idence)acketloving

____) __ __ - F (cAt) c -f (v) 1E-L a LE-[ JC2 C+A v) c c +f(v)

(13)

(14)

Is will Having thus assigned a mass equivalent and aloving momentum to the wave-packet we make use of,lution these through the assumption that the principles!quiva- of conservation of mass and of momentum aresing a applicable to all the masses and mass equiva-mass, lents, and to all the momenta involved in the

impact of packet and particle; that is, that"mass equivalent" and particle mass are con-stant in sum, and that the momentum of theparticle and the momentum of the packet dueto its mass equivalent are also constant in sum.

) (8) For the conservation of mass we then have,designating the mass of the particle by M,

164

IMPACT OF A WAVE-PACKET

before impact(cAt)

. E +M,C

2

after impact(cAt)-f(v)

E - I +M.C2 Lc+f (V)

(15)

approaches asymptotically the value

1

g

or

(16)

It is at once evident that the conservation ofmass+mass equivalent, which we are assuming,cannot hold unless the mass of the particle afterimpact is greater than before the impact.Calling the new mass of the particle after impactMg we then set up the equations for conservationof mass and momentum as follows:

(cAt) (At)l ~c f (v)E- + M=E c+f(v) + Mg, (17)

C2 C2 11+fv

(cAt) (cAt)[c-f (v)

c c Lc+f() +Mgv. (18)

In these equations are two unknowns, f(v) and g.Solving first for f(v) we find

c2 (g- 1)AV)= .g (19)

gv

Now we know that f(v) must be between 0, theinitial velocity of the particle, and v, its finalvelocity. For a very short wave-packet f(v) willapproximate to v/2. Putting this value in (19)we obtain

g= 1- (V2/2C2 ) (20)

By the use of successive approximations 2 it maybe shown that the complete expression for g

2 The formula for any interval between two velocities v1and v2, derived by the same reasoning as above is

f(v1, v2) = Ec2 (g 2 -g1 )]/(v 2 g2 -vg 19).

If we put f(v0v 2 ) =v1+[(V 2 -v1)/21, and put vl=v 2/2, wehave for gi, in term of V2,

gi= /[1 - (V22/8C2)].

Putting this in the above expression and solving for g2,

we getg2 /[-(V22/2)-3(2/4]

Repeating this process for successive equal increments weget values for the coefficient of v

4/c

4, of 1, 15/128, 3/25,

etc., which asymptotically approach the value 8; andvalues for the coefficient of V6/c6 which similarly ap-proach 16

(21)

(22)

I1-(V2/2C2)- (V4C4 -) . . .

1

=[1 - 2C)J

which is the Lorentz value for the coefficient ofvariation of mass with velocity of an electricallycharged particle.

DISCUSSION

The first point to demand discussion is theprincipal assumption made in this study, namely,that the "mass equivalent" of radiation-ascribed to radiation because radiation, likemoving matter, exerts a pressure-is to partici-pate like real mass in the conservation of massin impact of radiation and particle. The assump-tion finds its justification in a survey of the finalresult. Having arrived at

Mov

[1 - (V2/C2)]i

as the momentum of a reflecting particle of massMO traveling with the velocity v, we note thatthe kinetic energy of the particle, by a well-known process of differentiation and integration'is

Mc{ 1 1

L(' -9YiThis kinetic energy is to be equated to the netrate of flow of radiant energy toward the particle(no energy is dissipated by absorption or otherprocess) or to

Em c~t) I -(c-fv))](C+A v))

E(ct)1 -[ lgiving f~v

See F. K. Richtmyer, Introduction to Modern Physics(McGraw-Hill, New York, 1934), second edition, D. 723.

165

HERBERT E. IVES

This by rearrangement gives

(cAt) (cAt) c -f(v)1E +Mo=E I +Mog

C2 c2 c+f(v)

which is our assumption, which is thus shown toexpress the conservation of energy.4

A second point to be noted is that while wehave used the pressure of a wave system in ourdevelopment there is nothing in the final resultwhich restricts the pressure to being so produced.The equation for the conservation of momentum(18) can in fact be re-expressed in the simpleform:

Change of momentum of the particle=pres-

4If we assume, contrary to our result, the M is invariantwith velocity we would have for the conservation ofmomentum

E(6tA~c+E(cAt)In Ct[-Av) C=P1VC2 LcTf(vfJ

where m' is a new (variable) coefficient of mass equivalenceof radiation. The conservation of mass would then be ex-pressed by

giving

E (CAt) + MI/ = E (clot) It,/ [C-f(n) i + M,

, 1

c2~~~~~c

M' = Icf(v)]i

Mv = 2 c

The kinetic energy of the particle corresponding to mo-mentum MV is 1Mv

2, giving

1 Mve = E(cAt)-,

which obviously is not equal to E(cAt) [1-cf(v)]; so that

conservation of energy and momentum would not hold, inthe impact process here studied, for an invariant M.

so that

sureXtime. The pressure may equally well beexerted by another particle.

Our method of derivation of the factor1/El - (v

2/c2)]! through the use of a wave systembrings in the factor c2, c being the velocity ofwave propagation in the medium, which would,in an attempt to solve the problem from purelymechanical considerations, be unlikely to besuspected.

The derivation of the variation of mass withvelocity here given is to be contrasted withthat from the Theory of Relativity. In thatderivation, as given for instance by Eddington'the necessity of the variation only appears whenthe platform carrying the impacting system is setin motion, or is viewed by a moving observer,("let the observer witness this collision as heflies over in an airplane" 6 ), and the occurrenceof the factor c2 comes about through the Fitz-gerald contraction of the rods and the variationof rate of the clocks7 used in making the observa-tions of velocity under these artificial conditions.'In the present derivation the theory of relativitydoes not enter, the observer needs no airplane,the result follows from the classical propertiesof wave systems and the Newtonian conservationprinciples.

5 A. S. Eddington, Mathematical Theory of Relativity (Cam-bridge University Press, London, 1937), second edition,p.31.

P5 F. K. Richtmyer, reference 3, p. 719.7 Of interest in this general connection is an article by

P. S. Epstein [Am. J. Phys. 10, 1 (1942)] in which it ispointed out that, given the variation of mass with velocityhere deduced, the variation of clock rate may be derived.It thus appears probable that all the behaviors charac-teristic of the restricted theory of relativity may be ob-tained from earlier principles.

166