From: <Сохранено Windows Internet Explorer 8> Subject: Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies/Section III - Wikisource Date: Fri, 4 Nov 2011 15:57:52 +0600 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_001B_01CC9B0A.825E2D10" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.6109 This is a multi-part message in MIME format. ------=_NextPart_000_001B_01CC9B0A.825E2D10 Content-Type: text/html; charset="utf-8" Content-Transfer-Encoding: quoted-printable Content-Location: http://en.wikisource.org/wiki/Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies/Section_III =EF=BB=BF
=E2=86=90Section=20 II | Attempt=20
of a Theory of Electrical and Optical Phenomena in Moving=20
Bodies by Section III. Investigation of = oscillations excited=20 by oscillating ions. |
Section=20 IV=E2=86=92 |
Contents |
=C2=A7 30. Once the motion of the ions is given, known functions of = x, y, z=20 and t appear on the right-hand side of equations (A) and (B) = (=C2=A7 21); with=20 respect to the last variable, these are periodic functions if the ions = carry out=20 oscillations with constant amplitude and a common oscillation interval = T.=20 It is easy to see, that in this case the equations are satisfied by = values of=20 ,=20 ,=20 ,=20 ,=20 ,=20 ,=20 which also have the period T. Therefore, the important and almost = self-evident theorem is given:
If ion oscillations of period T take place in a light source, = then=20 =20 and =20 indicate the same periodicity at each point that shares the translation = of the=20 source.
The resolution of the equations leads to quite complicated = expressions. For=20 simplicity, it is advisable to calculate the components of the vector = =20 (=C2=A7 20) at first.
According to (VIb)
Accordingly, we want to multiply the second and third of equations = (A) by=20 =20 and =20 respectively, and then add them to the first of equations (B). We obtain = in this=20 way, under consideration of the importance of =20 (=C2=A7 19),
=C2=A7 31. In the following calculation, magnitudes of order =20 should be neglected. First, we neglect on the right-hand side the = terms=20 with two factors ,=20 =20 or ,=20 since we find a similar term in V2; and we therefore retain = only
Second, we write for the operation that has to be applied to = ,
The form of this expression suggests the introduction of a new = independent=20 variable instead of t
(34) |
and to consider ,=20 as well as =20 and ,=20 as functions of x, y, z and t'.=20 We denote the derivative that corresponds to this view by
and give to the sign =CE=94' = the meaning
It=20 is now
(35) |
and
so that we find for the determination of
A solution of these equations is easy to give. Namely, imagine three=20 functions =CF=88x, =CF=88y, =CF=88z that satisfy the = conditions
(36) |
and put
(37) |
Once =20 is found by that, equation (IIIb) provides us with the value = of =20 and thus also, as far as we don't use additive constants, the value of = .=20 From (VIb) it also follows ;=20 from (Vb) and (VIIb) it follows =20 and .=20 That in this way really all the equations are satisfied, can be = proven,=20 but should not be discussed here for brevity.
In contrast, in the next section the value of =CF=88x shall be given, and in = =C2=A7 33 the=20 solution for a special case shall be further developed.
It should also be remarked before, that the variable t' can be regarded as a time, counting = from an=20 instant that depends on the location of the point. We can therefore call = this=20 variable the local time of this point, in contrast to the = general time=20 t. The transition from one time to another is provided by equation = (34).
=C2=A7 32. The product =20 in the first of equations (36), as noted already, is a known function of = x,=20 y, z and t'. We = accordingly=20 set
and thus have
(38) |
a solution of (36)[1].=20 By that we have to imagine two points; first, the fixed = point=20 (x, y, z), for which we want to calculate =CF=88x and which we call = P;=20 second, a moving point Q, which has to traverse the = whole=20 space, where =20 is different from zero. r represents the distance QP, and = t' the local time of P at = the instant=20 for which we wish to calculate =CF=88x; furthermore we have to = understand by=20 =CE=BE, =CE=B7, =CE=B6, the coordinates of Q, and by = d=CF=84 an element of the just=20 mentioned space. The function =20 is the value of =20 in this element, namely, if the local time that is valid at this place, = is .
=C2=A7 33. To excite electric oscillations, a single molecule with = oscillating=20 ions shall serve; let Q0 be=20 an arbitrary fixed point in it =E2=80=94 for brevity, we say, "the = molecule is present=20 in Q0" = =E2=80=94, and for P=20 a place is chosen, whose distance from Q0 is much larger than the = dimensions of=20 the molecules. For distinction, Q0P =3D = r0.
We now want to replace the various distances r, that are = present in=20 formula (38), by r0 and=20 also neglect the differences of local times at the various points of the = molecule. In this way,
where=20 all occurring =20 are related to the same instant, namely to the instant when
is the local time of Q0.
Since =20 is equal for all points of an ion, then, if we write e for the = charge of=20 such a particle, the last integral transforms into
The sum is extending over all ions of the molecule.
Furthermore, if =20 is now the displacement of an ion from its equilibrium position, = then
and
This has a simple meaning. We can conveniently call the vector =20 the electric moment of the molecule and denote it by .=20 Then it is
after the things said here, we have to take the value of the = derivative for=20 the instant when the local time in Q0 is .=20 Obviously we can also write
where =20 means the first component of the electric moment in that very instant. = After (by=20 that and by two equations of the same from) we have found =CF=88x, =CF=88y, =CF=88z for the point (x, y, = z) and the=20 local time t' at this = place, the study=20 of the propagating oscillations is very simple. The equations (37) = give
(39) |
and=20 because we seek the value of =20 outside the molecule, (IIIb) is transformed into
or, due to (35), it is transformed into
If we bring the last two terms on the left side, then we just obtain = =20 or ,=20 as it can be seen by (Vb); since =20 and =20 only differ by magnitudes of order ,=20 we may replace the vector product (Vb) by .
From
we obtain =20 by integration; constants were omitted by us, since we are only dealing = with=20 vibrations.
We substitute the values (39) and put
It is then
(40) |
and namely, ,=20 ,=20 =20 are still related to the instant given above.
As to how the other magnitudes occurring in = (Ib)-(VIIb)=20 can be determined, can immediately be seen.
=C2=A7 34. Just some words on the error committed in the above = calculation. That=20 in (38) the factor =20 was replaced by ,=20 needs surely no justification. But we also haven't taken the values of = =20 for the function f at the the correct times. Once we have = replaced =20 by =20 in (38), then in the time when l is one of the dimensions=20 of the molecule, we have committed an error of order ,=20 secondly, the inequality of the local times at the various locations of = the=20 molecule were not considered, and in this lies an error of order =20 by (34). But even then, if we want to keep magnitudes of the order ,=20 we don't need to care about this second error, when already the first = may be=20 neglected. Now this is indeed the case when the dimensions of the = molecule are=20 much smaller than the wavelength of TV. Then also l/V is=20 considerably smaller than T, and the state in the molecule will = not=20 noticeably change in the time l/V.
=C2=A7 35. The formulas for the propagation of oscillations is = obtained, if=20 goniometric functions of time are substituted into the equations (39) = and (40)=20 for ,=20 =20 ,.=20 If, for example,
and, as a function of local time which is valid for the location of = the=20 molecule,
thus at an external point in the distance r and for the local = time=20 t' that belongs to it
If we eventually want to consider a stationary light source once, so = we=20 simply have to omit all accents. The formulas then are in accordance = with the=20 expressions, by which Hertz[2]=20 represented the oscillations in the vicinity of his vibrator. =
=C2=A7 36. Now we shall examine the oscillations in such distances = from the=20 luminous molecules, which are considerably larger than the wavelength. = It should=20 be noted that in (39) and (40), ,=20 ,=20 =20 are goniometric functions of
we namely want to write from now on r instead of r0. The assumption made about = the length=20 of this line justifies to consider only the variability of the argument = of any=20 goniometric function for all differentiations with respect to x, y, z, = but to=20 consider as constant all factors such as ,=20 or cos(r,x), by = which these=20 functions are multiplied.
For any of the magnitudes ,=20 ,=20 ,=20 ,=20 ,=20 =20 - we will call them =CF=86 =E2=80=94 it can therefore be found an = expression of the form
(41) |
where A and B are indeed dependent on the length and = the=20 direction of line Q0P =E2=80=94 Q0 is the location of the = molecule, and=20 P is the considered external point =E2=80=94, but, if r = were just big=20 enough, it may be regarded as constant in a space that comprises many=20 wavelengths. While x, y, z are the coordinates of P, we = denote by=20 =CE=BE, =CE=B7, =CE=B6 the coordinates of Q0, and by bx, by, bz the direction = constants of the=20 connection-line Q0P.=20 If we now replace in the formula (41) r by
and t' by the value = (34), we=20 obtain
=20 | (42) |
In=20 an area that isn't too extended, we may also regard bx, by, bz as constant, and thus = regard=20 the motion as a system of plane waves. The direction constants b'x, b'y, b'z of the wave normal = are=20 obviously to be determined from the condition
(43) |
For ,=20 b'x, = b'y, b'z fall into bx, by, bz, and the waves are=20 perpendicular to Q0P. This is not the = case if the=20 light source is moving. From (43) follows, that the waves are = perpendicular to=20 the line that connects P with that point at which the light = source was at=20 the moment, when the light was sent that reaches P at time = t.
=C2=A7 37. In a point that moves together with the luminous molecule = =E2=80=94 and thus=20 also for an observer who shares the translation =E2=80=94 the values of = =20 are changing, as we have seen (=C2=A7 30), as often in unit time as it = corresponds to=20 the actual period of oscillation T of the ions.
We can also examine, with which frequency these values in a = stationary=20 point are changing their sign. This frequency causes the oscillation = period=20 for a stationary observer. The question can be solved immediately, = if=20 instead of x, y, z we introduce new coordinates ,=20 ,=20 ,=20 which refer to a stationary system of axes. If the two systems = have the=20 same directions of axes and the same origin at time t =3D 0, = then
(44) |
and by (42) for =20 we obtain expressions of the form
where
is=20 the component of =20 with respect to the connection line Q0P.
The "observed" period of oscillation is thus
what is in agreement with the known law of Doppler[3].=20 If=20 the law, as it is usually applied, should be given, it must of course = still be=20 assumed, that the translation does not change the actual period of=20 oscillation of the luminous particles. I must abstain from giving an = account=20 of this hypothesis, since we know nothing about nature of the molecular = forces=20 that determine the oscillation period.
=C2=A7 38. The case that the light source is at rest and the observer = progresses,=20 allows of a similar treatment. If namely, as above, ,=20 ,=20 =20 are the coordinates based on stationary axes, then in a distant point = P,=20 any of the magnitudes =20 shall now be represented by
(46) |
We most conveniently describe the perception of motion by means of a=20 co-ordinate system, which shares the translation =20 of the observer. Here, again the relations (44) are applicable, and (46) = transforms into
from which it is given for the "observed" period of oscillation
(45) |
Here, ax, ay, az are the direction = constants=20 of the wave normal, while V is the velocity of propagation.
If we now want to know, by which frequency =CF=86 (in a = stationary point)=20 its sign is varying, then we have to introduce coordinates ,=20 ,=20 =20 with respect to stationary axes. By using the relations (44), (45) = transforms=20 into
where
are the components of =20 with respect to the wave normal
For the observed oscillation period we now obtain
What we have already stated without proof, namely that the period = T=20 exists throughout in the medium, is nothing else than what Petzval, in his attacks = against Doppler's theory, called the = law of=20 the immutability of the oscillation period (Wiener Sitz.-Ber., vol 8, = p. 134,=20 1852). He only forgot to notice, that this law only would apply, if we = consider the phenomena as a function of t and the = relative=20 coordinates.
The proof of the theorem is, by the way, easy to give, when the=20 oscillations are infinitely small, and when we have to do with = homogeneous=20 linear differential equations.
As regards the acoustic phenomena, = the=20 problem was discussed in detail by Was (Het beginsel van Doppler in de geluidsleer, = Leiden,=20 Engels, 1881).