Copyright © 19 by Edwin A

Fig. 1

The wavelength is created in all directions spherically and propagated outwards from the source, (Fig. 1). If the source moves to the left at ½ C while it forms the wavelength, the wave front will be in the same place as it is in Fig.1 when the wave rear is completed ½² to the left (Fig. 2), that is 1² + ½² = 1 ½², so the wavelength is now 1 ½² long on the source 180° X axis.

Fig. 2

If the source moved to the right at ½ C while forming the wave, the wave front would be 1² to the right of the starting point, but the source would be ½² to the right while completing the wave rear, (Fig. 3), therefore, 1² – ½² = ½², so the wavelength would be ½² long.

Fig. 3

Since the wave propagation is always away from the source the wavelength is altered in a real way by source motion according to addition or subtraction of velocities. No intercept problem exists here. Propagation velocity is never altered really or apparently. Wavelength only is altered according to source motion.

OBSERVER MOTION

When a stationary observer receives a wavelength, the length of the wave is exactly the same as when it was formed and transmitted. Also, there is no apparent change in carrier propagation velocity.

Fig. 4

The 1² wave is carried to the observer at C (or any other carrier velocity). However, if the observer moves to the right at ½ C an intercept problem exists. The wave front is received and the observer starts timing and measuring. The wave rear would catch up with the observer in double the time that it would if the observer was stationary. (ie.- instead of 1 sec. It would take 2 sec.)

Fig. 5

Or, looking at this same condition on a 1 foot scale –

Fig. 6

If the observer is at the 1" mark and the source is at the 0" mark the same instant the observer receives the wave front and at the same instant the observer starts moving to the right at ½ C. The rear end of the wave would be at the 2" mark the same instant the observer arrived at the 2" mark, so the wavelength would appear to be twice as long as a standard wavelength. Again, if it took the wavelength 1 sec. to travel each 1" span, the observer begins timing when he receives the wave front, so, in 1 sec. the observer is at the 1 1/2" mark and the wave rear is at the 1" mark, and in 2 sec. the observer is at the 2" mark when the wave rear is at the 2" mark. Therefore, this becomes an intercept problem to determine wavelength (or apparent wavelength) alteration on the observer’s 180 deg. X axis. This can be solved with algebra. Note – this apparent wavelength alteration is due to observer motion causing an apparent alteration to carrier propagation velocity.

The formulae to solve for the four basic in-line source or observer conditions would be chosen from one of the four main transformation equations in this theory (further on), it in turn produces a transform factor which is used by multiplication or division to adjust any transformable quantity depending on whether the quantity varies in reverse, direct, or inverse, proportion to the addition or subtraction of velocities.

With the forgoing examples of source and observer influence in mind, consider the following motions.

Fig. 7

Fig. 8

For simplicity, let C = 2, and S_v and O_v = 1. The conditions of the wave leaving the front of the source are exactly the opposite as for the wave approaching the rear of the observer. Also the conditions of the wave leaving the rear of the source are exactly the opposite as for the wave approaching the front of the observer. Therefore, if this is truly the case, each of these opposite transform factors should cancel, and the equations to solve for the transform factors should be exactly the opposite or inverse one from the other. This is the case.

To demonstrate, to obtain the Stf for the 0 deg. X axis of the source –

By these examples it is demonstrated how for source action the result of the equations amount to addition or subtraction of velocities (reverse) (1 – ½ = .5 or 1 + ½ = 1.5), and for the observer it becomes an intercept of velocities (2 / 1 = 2 or 2 / 3 = `.6

The above equations would not break down at C (to agree with the Principle of Causality) so to get the same results properly and so the equations are sound mathematically, logically, and agree with the principles of physics that are established safeguards, they must be stabilized. The four main transformation equations (T.E. #1, T.E. #2, T.E. #3, and T.E. #4) are the correct ones to use for such as the previously described circumstances for in-line operations. They also agree with the four laws of relative motion, T.E. #4 is the inverse of T.E. #1, and T.E. #3 is the inverse of T.E. #2, and the laws and transformation equations agree with the geometric transformation methods for the X axis (where they begin and end). The initial relation, S_v / C or O_v / C must be retained as such, however the symbol C in the rest of each may be changed to the quantity of a l, a time, etc, and this will still produce the tf.

To use the equations (T.E. 1, 2, 3 ,4) four examples are given now to demonstrate the changes to a 24m l emitted from the front or rear of a S which is moving at ½ C, and to a 24m l which is received from the front or rear of an O which is moving at ½ C.

Use T.E. #1 –

Fig. 9 ( C – [ ( S_v / C ) x C ] ) / C.

Substituting quantities: ( 300 000 - [( 150 000 / 300 000 ) x 300 000] ) / 300 000 = .5

The tf obtained by subtraction of velocities is .5, so the l of 24m is 24 x .5 = 12m.

Use T.E. #2 –

Fig. 10 (C + [( S_v / C ) x C ] ) / C.

Substituting quantities: ( 300 000 + [( 150 000 / 300 000) x 300 000]) / 300 000 = 1.5

The tf thus obtained by addition of velocities is 1.5, so the l of 24m now becomes 24 x 1.5 = 36m.

Use T.E. #3 –

Fig. 11 C / ( C + [( O_v / C ) x C ]) .

Substituting quantities: 300 000 / ( 300 000 +[(150 000 / 300 000) x 300 000 ]) = .`6

The tf thus obtained is used now to alter the l, 24 x .6 = 16m.

Use T.E. #4 –

Fig. 12 C / ( C – [( O_v / C ) x C ] )

Substituting quantities – 300 000 / ( 300 000 – [( 150 000 / 300 000 ) x 300 000 ]) = 2.

By the same procedure, 24 x 2 = 48m.

To obtain Stf s for angles between 0 deg. and 180 deg. X axis the geometric addition or subtraction of velocities is used, and to obtain Otf s for angles off the 0 deg. and 180 deg. X axis the geometric intercept of velocities is used. These are both described in full a little further on. Mathematical equations are adjusted to replace the geometric methods when possible for more accuracy. The cosine is added to handle the angular operations to the X axes.

INTERFEROMETER RESULTS EXPLAINED

On the following illustrations, follow mathematically the laboratory 24mm wavelength as it is altered first by the S then by the O and it will become clear why the Michaelson-Morley interferometer failed to show anything but a null result due to wavelength change. However, the Sagnac and Michaelson-Gale interferometer experiments produced fringe displacement due to wave phase differences (see p19), because the experiments are designed in a way which brings one wave train to the target out of phase with the waves of the other beam. The laws should be noted here. Also, there is no change to a wavelength due to S or O motion when it is transmitted or received 90 deg. off the X axis.

Fig. 13 (refraction is treated separately further on)

Fig. 14

Fig. 15

Fig. 16

In each of the four cases (90 deg. variations in direction) the various gaps remain constant. Michaelson and Morley assumed the beams would lose time traveling upstream and gain time traveling downstream, relative to the mirrors. I agree. The wavelengths are determined in a real way by S motion and by O motion, Apparent changes in propagation velocity mean nothing whatever to the final result BUT CAN CAUSE THE TWO BEAMS TO REACH THE TARGET OUT OF PHASE. It is a fact that wavelength changes are not always accompanied by apparent velocity changes due to basics. The interferometer screen shows real and apparent wavelength changes only and due to basics can display no apparent velocity changes except by phase shifts. However, it can display phase shifts between two beams of waves due to apparent differences in velocity caused by the beams moving in different directions through the wave carrier at V = C and relative to the moving interferometer.

The Michaelson-Morley experiment used 2nd order effects and are therefore scalar. 1st order effects are necessary as the Michaelson-Gale experiment. Scalar effects are not capable in theory of giving required data concerning the one way velocity of electromagnetic waves.

BASIC MOTIONS USING THE SLIDE RULE

Mathematical examples of four basic motions along the source and observer in-line X axis using the slide rule.

GEOMETRIC TRANSFORMATION

Fig. 20

Geometric method of solving for source angular transform factors.

1. Draw a circle of r = 1 l (enlarged or reduced for convenience). S is at center at A.

2. Draw X axis.

3. Locate F on AG line to represent S_v / C.

4. Draw line AC on tf angle desired.

5. Draw line CD perpendicular to AC. (This line always runs forward from C).

6. Draw line from point F parallel to line AC until it meets line CD. (This line never moves off F when AC and FE are drawn on a new angle).

7. The linear measurement AC represents 1 l at C.

8. The distance AF represents the distance the S moves during transmission of 1 l_o of distance AC.

9. FE represents the actual l transformation due to S motion, but when AC is referred to as 1, FE becomes a tf. (ie - .43, 1.26, etc.)

10. Note – there is no alteration to l perpendicular to the X axis.

11. Note – in-line transformations follow in-line transformation equations for S.

12. This geometry represents reverse addition or subtraction of velocities and agrees with the laws.

GEOMETRIC TRANSFORMATION

(ALTERNATE METHOD)

Fig. 21

Geometric method of solving for source angular transform factors.

1 Draw a circle of r = 1 l (enlarged or reduced to scale). S at center.

2 Draw X axis.

3 Locate B on AG line (or E on AH line) to represent S_v / C.

4 Draw small circles as shown of d = AB.

5 Draw line AD on tf angle desired.

6 AD represents 1 l, so subtract AC from AD to get l' or tf. (on the rear, add AJ to AF).

7 Note that no alteration to l occurs perpendicular to the X axis.

8 Note that in-line transformations follow in-line equations for source.

GEOMETRIC TRANSFORMATION

Fig.22

Geometric method of solving for observer angular transform factors.

1 Draw circle of r = 1 l (altered to scale), O at center at A.

2 Draw X axis.

3 Locate F between A and G to represent O_v / C.

4 Draw line AC at desired transform angle.

5 Draw line CD perpendicular to AC.

6 Locate BE parallel to AC so full scale measure on line BE is identical to fractional scale measure on line AB.

7 Linear measure of BE or AB will represent the l' for that particular reception angle at that particular O_v.

8 The rear angular transforms are found the same way, but instead of being a fraction of 1 (line A to F), it would be 1 (A to F) plus a fraction more, or 2 plus a fraction more, etc.

9 Note that perpendicular to the X axis there is no alteration to the l regardless of Ov up to lim : C, full scale measure and fractional scale measure are both 1. Also on 0 deg. and 180 deg. X axis the in-line transformation equations are used. This geometric method also (as well as source geometric transformation method) predicts the results obtained by the in-line transformation equations.

Fig. 23

When a source emits a wave, the wave whether very directional, circular, or spherical, will be carried away from the source at the standard velocity of the carrier, molecular (sound etc.), or electromagnetic (light, radio, etc.), which will be modified by density, temperature, structure of the medium. The length of the wave is mechanically determined by the oscillator or other cause in the source (frequency) and by any motion of the source. On the forward side of a moving source, the transmitter is chasing after the wave front as it forms the rest of the wave. On the rear side the wave front and transmitter are moving in opposite directions as the rest of the wave is formed. Source motion does not affect the carrier propagation in any real or apparent way whatsoever. It does change the wave length physically in a real way. After the wave length is determined and formed by the carrier propagation velocity, transmitter frequency, and source motion, that wave length will be propagated unchanged unless acted upon by another force, interference, or blocked completely or partially as by an observer, and will (I suggest) be deenergized with distance as per luminosity, absolute brightness, and spectrum red shifting.

Whatever the angular velocity transformation factor from an observer maintaining a steady gap from a source on a parallel track, the source angular velocity transform factor will be such that they reduce to standard wavelength regardless of velocity, lim 0 to C when source leads, and lim 0 to C when observer leads.

To match a moving and rotating source the observer must match the rotational velocity or balance it by constantly altering the gap to compensate for a different rotational velocity, or select the mathematical null to base measurements on.

From the previous illustration and solution (Fig. 23) it can be seen that when source and observer X axes (and space tracks) are parallel and the gap between them is constant, the observer will always detect a standard lab. condition wavelength from the source regardless of the source’s angular position at any point on the observer’s projected sphere. It is possible to balance source X axis angle and source velocity so the observer will detect the same wavelength for each, so the observer must balance the source angular velocity factors against the certain source motion (in radians for example) to obtain the true source X axis location, velocity, and distance. When using this method in astronomy, it must be checked against distance obtained from parallax measurements, absolute luminosity measurements, proximity to nearby cephids, and any other method, which will help to confirm the distance.

Of special interest

It is demonstrated that when the gap between S and O is constant regardless of velocity or 90 degree angular relationships on the three axes, the S and O wavelength transformation factors “together” return the wavelength to standard laboratory conditions. Therefore, rather than geometrically solve for both S and O tf’s for each change in angle and/or velocity, it is sufficient to establish the S transform factor, then obtain the Otf’s from the slide rule (scale C and C1) or by computer for more accuracy, and used with the D scale (or computer) to solve real or apparent wavelength changes.

Three O angular transforms to show the geometrics a little clearer.

Note that the transforms operate from minimum on the front to zero on the Y or Z axis to maximum on the rear as the pivot at the O is rotated through 180°.

Fig. 24

Source transform factors from 0° - 180° X axis at .5 C.

Observer transform factors from 0° - 180° X axis at .5 C.

Fig. 25

Comparison between S and O transformations for the same velocity

.Fig. 26

LAWS OF RELATIVE MOTION

The transform factors are derived by use of the transformation equations and are used by multiplication or division according to standard mathematical practice to alter laboratory quantities due to source and/or observer motion.

They are valid for use with any national or international system of measure.

Addition or subtraction of velocities is by basic arithmetic only.

First law of relative motion.

Due to source motion, the transform factor varies in reverse proportion to the addition or subtraction of velocities.

Second law of relative motion.

Due to observer motion, the transform factor varies in inverse proportion to the addition or subtraction of velocities.

Third law of relative motion.

Any given source transform factor will be exactly cancelled by the diametrically opposite observer transform factor for synchronized constant velocities, parallel X axes, and rectilinear paths.

Fourth law of relative motion.

Motion of a reflector, relay, or refractor, will alter the incident transform factor as an observer, and the reflected transform factor as a source.

TRANSFORMATION FORMULAE

T. E. #1 – S 0 degree X axis tf due to source motion –

T. E. #2 – S 180 degree X axis tf due to source motion –

T. E. #3 – O 0 degree X axis tf due to observer motion –

T. E. #4 – O 180 degree X axis tf due to observer motion –

T. E. #5 – Quanta energy transformation due to source or observer motion –

(a) Eq’ = (hf)/tf or (b) Eq’ = h(f/tf .

When T.E. #1, T.E. #2, T.E. #3, or T.E. #4, are used in wave mechanics, insert the cos q (as shown in these examples) into the T.E.s, and the formulae will now give the tf for any angle. For example: #1 will handle wavelength alteration due to S motion, from 0 deg. X axis to 90 deg. (measured from the forward X axis). T.E. #2 will handle alterations from 180 deg. S X axis to 90 deg. (measured from S rear X axis). The formulae for O motion is handled the same way.

Example - [C – ([( S_v / C ) cos q ] x C )] / C

Example - C / [ C + ([( O_v / C ) cos q ] x C )]

Note – If Eq is measured by a moving O, the hf takes care of the Eq' automatically. If a prediction is required for a certain velocity and its effect on a lab. measurement, the Eq' would be predicted by hf / tf.

Also – If a moving S radiates wavelengths, the frequency he measures is the lab. value, but the wavelength a stationary O will measure is altered. Therefore, the moving S must adjust the frequency and thereby the Eq, by the Stf to give the desired wavelength and Eq.

USE OF T.E.s #1 AND #3 FOR ALL QUADRANTS IN WAVE MECHANICS

Once the use of the four transformation equations to explain and predict Doppler phenomena in wave mechanics for in-line operations, and the use of the cos q for angular functions, is understood, it is possible to do this by using only T.E. #1 and/or T.E. #3 by using the value for cos q as a calculator presents it, and by measuring angles from 0 deg. to 360 deg. continuously in the same direction.

For example:

When using T.E. #1 for S, or T.E. #3 for O, conditions, from 0 deg. X axis to 90 deg., insert the positive (+) value for the cos q.

When using T.E. #1 for S or T.E. #3 for O conditions from90 deg. to 180 deg., insert the negative (-) value for the cos q.

This allows the operator to measure the q (continuously) from 0 deg. to 360 deg. and use only two transformation equations instead of four. A calculator would give the proper value for the cos q, but when tables and slide rule are used the operator must measure each quadrant in the correct direction (from forward or rear X axis) and insert the correct sign for the cos value.

Note – In T.E. #1, T.E.#2, T.E. #3, and T.E. #4, C represents both the velocity of light and the velocity of sound. Also, in place of C, any quantity may be used for special purposes, such as wavelength, time, period, distance, force, etc.

The tf is a factor used by multiplication to alter a wavelength, time, period, distance, or by division to alter a uniform constant speed, or force, or kinetic energy.

T.E. #6 – Stf from known wavelengths, for source 0 deg. or 180 deg. X axes –

T.E. #7 – S_v for 0 deg. X axis from Stf –

T.E. #8 – S_v for 0 deg. X axis from lo and ls –

T.E. #9 – Sv for 180 deg. X axis from Stf –

T.E. #10 – S_v for 180 deg. X axis from lo and ls –

BASIC SOURCE / OBSERVER MOTIONS

(a)

C – Apparent increase due observer motion.

l - Real increase due source, apparent decrease due observer. (Spectroscope would show no change).

f - Real decrease due source, apparent increase due observer, will appear standard.

Result- necessary to measure C' not relying on adaptations of f or l.

(b)

C – Unaffected.

l - Real lengthening due source.

f - Real decrease due source. The number of source vibrations/sec. remain the same, but the l is increased due source recession which lowers the f for the same propagation velocity.

Result – red shift.

(c)

C – Unaffected by source, apparent decrease due observer.

l - Real lengthening by source, apparent lengthening due observer.

f - Apparent decrease due source (as in b), apparent decrease due observer.

Result – large red shift.

(d)

C – Apparent increase due observer.

l - Real decrease due source, apparent decrease due observer.

f - Increase due source, apparent increase due observer.

Result – large blue shift.

(e)

C – Unaffected.

l - Real decrease due source.

f - Increase due source.

Result – blue shift.

(f)

C – Apparent decrease due observer.

l - Real decrease due source, apparent increase due observer.

f - Standard.

Result – measure C'.

(g)

C – Apparent increase due observer.

l - Apparent decrease due observer.

f - Apparent increase due observer.

Result – blue shift.

(h)

C – Apparent decrease due observer.

l - Apparent increase due observer.

f - Apparent decrease due observer.

Result – red shift. (Linear shift is larger for the same O_v than in g).

SPECTRUMS – IDENTICAL ASPECTS

Various directions and velocities cause apparent changes in the velocity, wavelength, and therefrom the frequency, of sound due to observer motion, and real changes in wavelength, and therefrom the frequency, due to source motion. The velocity of sound propagation remains constant under standard conditions and is only altered by molecular density, structure, temperature, etc.

If a car is moving away from an observer with the horn blowing, the sound still travels from car to observer at the constant velocity of sound governed only by the conditions of the molecular wave carrier. Because the car’s horn is oscillating while moving away, the real (not apparent) wavelength is lengthened rearward resulting in a lower pitch. An observer ahead of the car would witness the reverse in wavelength and pitch, but the velocity of sound would be the same for both directions. If the car’s velocity was equal to the velocity of sound (for prevailing atmospheric conditions) the energy buildup traveling with the forward edge of the car would hit the observer in a larger compression wave, instead of the usual waves due to the horn, and the observer would hear a sonic boom. The waves due to the horn become absorbed in the compression wave buildup. This sonic boom is due to the car pushing the molecules at their maximum velocity for prevailing conditions and is a mechanical function of the car’s structure and the air’s molecular structure at that velocity and has nothing whatsoever to do with the operation or non-operation of the horn.

When a source moves through a carrier as it is emitting energy waves, the waves vary from the standard lengths radially on any single plane from shorter wavelengths ahead of the moving source to longer wavelengths in the rearward direction. Fig. 102 shows the wave pattern when the source is in motion. See also Fig. 34. Motion of the source alters the wavelengths front and rear from the Y axis (lateral) as addition or subtraction of velocities. Observer motion “apparently” alters wavelengths by intercept of velocities. (Fig. 97, `.6 & 2 sec.). The Doppler effect will always convert algebraically to real conditions by removing the effect of observer motion, and from there to standard laboratory or at rest conditions by removing the addition or subtraction of velocities and thereby removing the effect of source motion. The `.6 sec. intercept in the train and lightning hypothesis of Fig. 97 converts to standard velocity of light as does the 2 sec. hypothesis. In the first case, the light traveled a shorter distance, and when the time and distance are converted to speed, it becomes the standard velocity of light. The same applies to the longer distance in the 2 sec. case, but, the `.6 and 2 sec. times are due to “apparent” velocity or wavelength alteration due to observer motion. This apparent change due to observer motion comes from using an alteration in time without using the proper change in distance.

The motion of a source will cause a real alteration of wavelength because of the mechanical connection between source and carrier, but the carrier propagation velocity is constant and can not be altered in a real way by the motion of the source or observer. The motion of the observer will cause an apparent alteration of wavelength and propagation velocity due to real altered distance and time. The energy wavelengths are imposed on the carrier by the source and are not altered by motion of the observer, but the distance and time of intercept gives the illusion of changed wavelength and propagation velocity due to observer motion only.

Light from a star we approach at 20 km/sec. will travel from a planet used as a shutter to the eyepiece of a telescope at a greater apparent velocity than light from a flashlight 20 cm away (when basing judgment on wave length alteration or interference) because the flashlight and telescope gap is fixed. No relative motion, 3rd law applies, hence, no visible Doppler effect. Source and observer changes convert to standard because of a constant gap. See table of velocities and transform factors. In the sound or electromagnetic spectrums, with a constant gap between source and observer regardless of direction, or velocity, the wavelength is intercepted as standard in a static carrier, because the apparent wavelength alteration due to observer motion cancels real wavelength alteration due to source motion, and a standard wavelength is presented. Fig. 98 of source, carrier, and observer, (ignoring possible carrier motions to simplify conditions), and the relationships of all including various motions and propagation velocities become very clear.

The longitudinal axis of wavelengths, distance, and time, undergo in some cases real alterations, and in others apparent alterations, due to real motions of source and observer relative to the carrier. “If” a carrier motion exists it would add to real and apparent alterations. For the longitudinal axis, if the distance alteration is ignored (because it would have to be computed from observer velocity also) the time alteration would represent the apparent propagation velocity alteration due to observer motion.

Energy can be transmitted one wave alone as from an atom, or one vibration of a car’s horn, or with a continuous wave after wave quantity, with wavelengths according to source oscillating frequency and carrier propagation velocity under standard conditions, or with altered energy presentation according to departure from standard. Also, it would be possible to, in each type of carrier, form a compression wave or single energy wave ahead of a source which is moving at the carrier propagation velocity.

As no real or apparent change in wavelength or velocity of sound will be detected for various directions inside a moving aircraft which encloses the source and detector or observer at a constant gap, but the complete Doppler phenomena will be experienced for sound entering or leaving the moving aircraft, so the Michaelson-Morley experiment is invalid for the performance claimed, because being set up on the earth with fixed gaps, we would be as enclosed in a moving aircraft with no relative motion between source and observer, except that in the aircraft there is no flow of wave carrier over the instruments. (The result and analogy is the same even if there were). It will be essential to use solar system worlds or satellites and motions such as planetary-stellar occultation as gigantic instruments and shutters to measure apparent changes in C. (See Figs. 95, and section ²C' Apparent). But for wavelength, the apparent changes are read in a spectroscope. Remove the apparent wavelength alteration due to observer motion and observer X axis position, and real wavelength changes due to source motion and source X axis angle remain. Determine the wavelength alteration due to source, and time for the source to traverse a given arc, and source distance and velocity can be computed. This can be checked against the luminosity scale and nearby cephids to be sure of the distance.

When a wavelength is measured mathematically, the time or distance or velocity must be measured from the front of the wave to the rear of the wave as it travels past any given point, and compared to standard.

The sound and electromagnetic spectrums exhibit most of the same effects. They both show the Doppler effect in exactly the same manner, convection coefficients, standard propagation constants, they both obey the same fundamental time – distance – speed formula, both obey the same fundamental propagation velocity – frequency – wavelength formula, both show the same type of source motion influence, both show the same type of observer motion influence. Olaf Romer has already measured an apparent alteration to C for the electromagnetic spectrum (see section ²C' Apparent) so I would say that in this respect also both spectrums are similar in operation. It is very logical to believe that if a standard wave length which is being transported at the standard propagation velocity of the carrier can be apparently altered by observer motion, then it should also be possible to measure the same apparent change to the carrier propagation velocity which caused its transport from one point to another point.

SPACE TRACKING

It is essential to know how various motions track through the wave carrier to always know the exact direction the source X axis is pointing and moving, because what seems apparent may be totally different.

It is known among astronomers that our moon tracks wave fashion through space with the Z axis of the waves inclined about 5 degrees to the plane of the earth’s orbit. (See Fig. 90). It appears to people on this planet that the moon is in an elliptical orbit. By coincidental coordinates, it does both. However for use of the wave carrier we must know where the moon’s 0 deg. X axis is at all times, and by the wave track orbit it points roughly in the direction that the earth is moving in its orbit. (Actually, it oscillates slightly from side to side). If we thought the moon tracked an elliptical orbit, the X axis would be 180 deg. reversed half the time because when it was on the inside (between earth and sun) and moving opposite to the earth’s motion the moons 0 deg. X axis would point rearwards as seen from the earth.

Astronomers also understand various multi-bodied wave track orbits, multi-bodied systems tracking laterally or tracking C of G axis parallel to trajectory. The system tracking laterally would range from circular orbits for stationary C of G axis to elliptical orbits of various sizes as system velocity increased, to wave track orbits for system velocities which are above the critical track velocity for the affected components. Our planetary system tracks propeller fashion at 19.4 km/sec. toward Vega in the constellation Lyra but the moon tracks wave fashion. Sirius and the minor body track corkscrew fashion with sufficient orbital velocities to sustain component separation. These motions and more can be seen all around us and no doubt many more complications will be learned to add to what we know now. (Figs. 90 to 94 show some possible examples).

From the range of transformation factors for source and observer moving at 1/2 C it becomes clear why knowing X axis position is vital for both because the wavelength can be altered from .5 on the front side of the source to 0 on the Y and Z axis to 1.5 on the rear side, and for the observer`.6 front to 0 on Y and Z axis to 2 rearward.

TRANSFORMATION OF PERIODS

Period information such as binary star cycles, planet-moon occultation times, any orbital

Period information such as binary star cycles, planet-moon occultation times, any orbital periods, or cephid pulsing times, are subject to Doppler phenomena and are transformed due to source and observer motion.

Romer, in 1675, measured period changes in Io’s orbit around Jupiter due to earth (observer) motion. See – ²Exploration Of The Universe² by George O. Abell, p129. Romer was trying to measure the velocity of light, which he rightly believed was finite. The period is unchanged and gives the correct timing when the earth’s and Jupiter’s tracks are parallel and relative positions are at 90 deg. to the tracks regardless which side of the earth’s orbit the earth is on or how near or far to Jupiter it is. The only deviation from the correct timing comes when the earth’s motion causes it to approach or recede from Jupiter.

When the plane of the multiple body system’s orbit (in this case Jupiter and Io) is in line with the plane of the observer’s orbit (earth’s), an occultation occurs which is handy to time the period. Under these conditions as light travels toward the earth, it is broken into lengths by the absence of light from the hidden body. So in effect we have light continuously streaming toward us from Jupiter, but we can ignore this. On the other hand, we have Io’s light streaming toward the earth interrupted periodically by occultation according to the orbital period. If the earth and Jupiter were stationary and only Io moved around Jupiter, we would get a stream of light with a blank (occultation), another stream of light followed by another blank (another occultation), and so on. This situation is now precisely the same as one wavelength followed by another wavelength except for the periodic blank in the stream of light. From this point the motion of Jupiter and its small moon would transform the period or it would transform the occultation time exactly the same as a source would a wavelength, due to the Doppler effect. Also, observer motion would transform the above (period and occultation time) by observer motion due to the Doppler effect.

When any period, such as described in the first paragraph, is measured, where motion of the observer or source are to be accounted for, the laws, transforms, geometrics, in this theory would all apply just as if it were wavelengths being transmitted and received.

Keppler’s third law describes the planetary orbital mechanics relating orbital periods to semi-major axes of the orbits, but complete system motion (not stationary systems with planets in orbit) and observer motion will cause irregularities which can be removed or described by the transforms in this theory.

Example #1

Fig. 27

Result – source system stationary, observer stationary, source’s orbital periods and occultation times are normal length and follow Newton’s and Keppler’s laws.

Example #2

Fig. 28

Result – source system in motion. Period and occultation times are transformed according to the Doppler effect, and agree with Newton’s and Keppler’s laws when the effects of the doppler are removed or cancel.

Example #3

Fig. 29

Result – observer moves toward source system (following it) at 1/2 C, therefore Otf =`.6.

Occultation time transforms to 7.5 x `.6 = 5 minutes.

Period transforms to 1.5 x `.6 = 1 hour.

Thus, due to transforms, laws, geometrics, and Principle of Reciprocity, the period and occultation times are transformed back to stationary conditions, just as the famous interferometer showed.

If source and observer motion violated the third law, the Doppler effect would be measured in the period and occultation times.

BALLISTICS

During constant velocity the wavelength or projectile/particle length remains constant but is ²apparently² altered during reception if the observer is moving and the entire length of the projectile can be received by an instrument as a wavelength can. Both a wavelength and projectile have definite lengths as far as the observer is concerned. The intercept time from source to target (O) can be solved for by obtaining the Otf by comparing known projectile length with known projectile velocity to obtain tfs and using the tfs to convert known distance to intercept time.

Consider the projectile length as l_s, projectile velocity as C, target velocity as O_v, use the transformation equations to obtain the tf, use this tf to establish intercept time when the distance between S and O is known. When the projectile impacts, the E would follow the dictates of all the in-line and angular tfs.

During acceleration or deceleration, particle or projectile tfs would be figured by using the terminal velocity. During O acceleration or deceleration, Otfs, when receiving integrating lengths, would be based on the average terminal velocity during the reception of the length, such as a wavelength. The difference between particle or projectile and non-solid or wavelength would be in ballistics and impact problems as compared to communications and reception problems. This difference is because during reception of a wavelength by an accelerating O, the shortest time as a coordinate would be the complete length of time it takes to receive any given wave. For example – an O continuously accelerating at a uniform rate while receiving a stream of waves all the same length (S_v steady), would continuously add to the Otf. The Otf would be known during any O motion, so the balance is due to S motion. (ie. S steady, acceleration, etc.) . This situation is due to the fact that an O can be accelerating as it is receiving the wavelength, but the wavelength is always traveling at the constant velocity according to the medium it is moving through.

When using the Otfs and equations under these conditions, use the ²average² O_v during the reception of one wavelength at any specific intercept time for one wave ²length². The rest of the equation is standard procedure as for any constant O_v.

During interaction between accelerating O (target) and an accelerating particle or projectile, the differences are that as soon as the front edge of the particle or projectile is received the event is concluded or timed. You can not work out the tf from average velocity during reception of the length of the projectile. You must work it out from terminal velocities of both O and projectile. When the intercept time is known, the Otf is derived by the proper equation using projectile terminal velocity in place of C, and O_v at the time of intercept.

Fig. 30

Use average O_v ²during² i/c t, and use C.

Fig. 31

Use O_v ²at² i/c t, and use projectile terminal V ²at² i/c t.

An O or target watching a particle or projectile approach on a collision course would witness the E transforms due to O motion in exactly the same manner as an O receiving wavelengths from any direction. The O motion would cause an apparent increase in E on the O 0 deg. X axis, unaltered by O motion on the Y and Z axis, and decreased on the O 180 deg. X axis. All the angular transforms in between these directions would also apply.

If a particle or wave enters a cloud chamber, bubble chamber, spark chamber, or any other detector which exhibits some measure of the track length, from the front end as the earth’s motion carries the chamber “towards” the specimen, the track length should show the Doppler shift and the track will be lengthened from standard. If it enters from the rear, the track will be shortened, if it enters from the side the length would be standard but curved rearwards. (Undetectable perhaps).

As two space ships cruising along together can act one as a S and the other as an O while communicating, or the basics could be reversed when the receiver starts transmitting a reply. Or one could bump the other from any angle, after both have attained the same E_k due to their like velocities. The transformation equations in this theory work well unless applied to impossible situations.

USE OF TRANSFORMATION EQUATIONS WITH IN-LINE BALLISTICS

The following four examples are to demonstrate the use of T.E. #1, #2, #3, and #4, with in-line examples of two moving objects which could be two automobiles, two spaceships, one ship and one torpedo, etc., etc.

The speed of the slowest object is always inserted in S_v or O_v. The speed of the fastest object is always inserted in C (under S_v or O_v only). The other Cs are replaced with d (distance) of separation at the start of timing, or d from start to finish (as in a race).

Variations would be possible, these four illustrations are merely to give the basics.

Use of T.E. #1 –

Fig. 32

Conditions of problem –

The 120 at the beginning is added to use with the .75 transform factor to give 90 which is the transformed distance which is the final distance of separation.

The fastest one traveled 120 km.

The distance of separation at termination is 90 km.

The slowest one traveled (120 – 90 = 30) 30 km.

The objects must each obey the same t = d/s formula, and each must take the same time.

The time for the fast one is 120/40 = 3 hours.

The time for the slow one is 30/10 = 3 hours.

Use of T.E. #2 –

Fig. 33

Conditions of the problem –

The 120 at the beginning is added to use with the 1.25 tf to give 150 which is the transformed distance which is the final distance of separation.

The fast object traveled 120 km.

The distance at separation at termination is 150 km.

The slow object traveled (150 – 120 = 30) 30 km.

So checking via the t = d/s formula –

The t for the fast one is 120/40 = 3 hours.

The t for the slow one is 30/10 = 3 hours.

Use of T.E. #3 –

Fig. 34

Conditions of problem –

The 120 at the beginning is added to use with the .888 tf to give 106.66 which is the transformed distance which is the distance moved by the fast object to reach the intercept point.

The fast object traveled 106.66 km.

Initial separation is 120 km.

The slow object traveled 120 – 106.66 = 13.33 km.

By the t = d/s formula –

The t for the fast one is 106.66/40 = 2.666 hours.

The t for the slow one is 13.33/5 = 2.666 hours.

Use of T.E. #4 –

Fig. 35

Conditions of problem –

BALLISTICS – GEOMETRIC INTERCEPT

Symbols used –

P Projectile or particle.

P_v Projectile/particle velocity.

P_ov Projectile/particle laboratory velocity.

i/c Intercept of projectile/particle and O or target.

O Observer or target.

S Source from which projectile/particle is fired.

d Distance.

E_k (Net) laboratory kinetic energy from the formula 1/2mv².

E_ks P kinetic energy due to S motion (E_k + source influence).

E_ko P kinetic energy witnessed by a moving O.

Acceleration and deceleration are omitted for simplicity.

The main differences between wave i/c and projectile i/c are, the waves are emitted spherically and will propagate outwards and i/c the O under normal Doppler mechanics even though the O location or motion is unknown, but, projectiles are not carrier functions, therefore, they must be timed and directed by the S when O motion is known, or the O (knowing S intentions and motion) must adjust his motion to i/c.

Of special interest –

The following points are inserted at the beginning instead of the end so they may be considered at will.

1 The ratio of P_v to O_v for an i/c is equal to the ratio of Pd to Od.

2 The t from P firing to i/c is the same for both P and O and the scales in both frames are equal.

3 When a P is fired from a moving S, the amount of S_v that is added to or subtracted from the P_ov is determined by the firing angle.

4 When a P is fired from a moving S, the P tracking vector which is equal to the P_ov vector (at a certain angle for each S_v), leans forward from the perpendicular as S_v is increased, even though the P is fired on a more rearward angle to maintain the P_ov result.

Two geometric methods can be used to determine the P_v when fired from a moving S, the velocity vector parallelogram and the velocity resultant circles.

Velocity vector parallelogram –

Fig. 36

1 The radius of the circle is any convenient size and represents the distance the P moves in one unit of time at laboratory velocity.

2 The F point is as the ratio in l geometry.

3 The parallelogram of vectors gives the firing angle, S to O distance, S and O motion, and P vector (this is used to measure d¢ the P will track, and P_v¢ due to S motion). The P resultant is used as a tf

Velocity vector circles-

Fig. 37

1 Draw the same circle, X axis, and mark the A (S) and F points on the X axis.

2 Draw another identical circle with the center at F.

3 Measuring the tf on any angle from the center of the first circle, add the forward d between the two circles or subtract the rear.

4 The points where the two circles intersect at the circumference is the P tracking angle which is caused by the P_v being equal to the laboratory quantity.

Refer to Fig. 38.

If an O is placed anywhere on the circumference of the S circle, and the O moves parallel to the S (line CD), at the same and constant speed, the i/c point is always at the end of the O track (line CE). Under these conditions the S and O will always be at the same starting d, and may be at any angular relationship, and no ballistic Doppler effect will be detected. The t for the P to move the d of the resultant vector is the laboratory value ensured by the d¢ and v¢. Also under these conditions, the firing angle is always directly at the O.

Fig. 38

Fig. 39

Fig. 39 shows an i/c when both S and O X axes are parallel, and velocities are constant and synchronized. This demonstrates the fact that the O can be at any angular relationship to the S, and as long as the X axes are parallel and speeds equal and constant, no ballistic Doppler effect will be noticed.

Now, if the O track is at an angle (not parallel) to the S track, the i/c point will change. The velocity circles are still used according to the scales, but a system of velocity-time vectors are added and the O track and P track extended to i/c.

It is with gratitude that I acknowledge the assistance of Professor J. A. McCorquodale, Civil Engineering Department, University of Windsor, who provided the match line (line BL) for the following method of angular i/c solutions and passed it to me by private communications.

Fig. 40 This is an illustration only to label all the lines and points.

Fig. 40

The geometric intercept for various observer angles is constructed as the following. Refer to Fig. 40.

1 Draw the S velocity circles, X axis, mark direction of S motion, and locate O on the circle GH.

2 Draw O direction of motion, draw line JK parallel to S X axis. Circle J to K is r = O_v/ P_ov.

3 Extend O track outward toward M.

4 Draw line LB parallel to line S to O.

Note – Ot (O or target time component) must be equal to Pt (projectile time component).

5 Draw P track from S to I and extend to i/c with O track.

6 Line S to I is the P_vtf due to S motion and firing angle. Line S to i/c is Pdtf from firing point to i/c with target. Line O to i/c is O_d tf from starting point at O (the instant the P is fired) to i/c with P.

7 An extra advantage with this graphic method of i/c is, if the O fires a P of its own, the extra velocity circles can be drawn as used for the P which is fired from the S, and the S projectile to O projectile i/c can be solved quickly. However, when the O is tracking along a line O to M when it fires a P, to draw the velocity circles in the O frame, the compass would be moved along line OM instead of OJ.

Kinetic Energy.

When S and O X axes are parallel and velocities are constant and synchronized, ballistic Doppler

effects cancel. This includes E_k. The process would take the following form.

Let P_ov = 24 m/sec., Pm = 10 gm., S_v and O_v = 12 m/sec., P i/c tf = 1.12. (SeeFig.39).

Note – P i/c tf is also here the P_v tf.

E_k (laboratory value) = using formula E_k = 1/2mv², is 2880.

E_kS (due to S motion) = [(24 x 1.12)² x 10]/2 = 3612.872.

E_ko ( due to O motion) = 3612.872 / 1.12² = 2880.

The effects of S and O motion have cancelled and E_k is restored to laboratory quantity upon reception by the O.

Now if the O track is perpendicular to the S track when the S fires the P (in such a way that an in-line i/c occurs), the E_k transformation would take the following form.

P_vtf (due to S motion) = .87.

P_d tf (d moved by P from S to i/c) = 2.35.

E_k = 2880. (laboratory value).

Therefore –

E_kS = [(24 x .87)² x 10] / 2 = 2179.872.

Or alternately – 2880 x .87² = 2179.872.

E_kO = 2179.872 / 2.35² = 394.72557.

A ballistic Doppler effect is measured.

At this point many complications may be introduced and handled in the usual manner.

The remaining drawings are to make a couple of points clearer.

When the i/c is toward the rear of the S motion, the t and vectors are rotated through the Y axis and are drawn on the rear of the S and O X axis, as shown in Fig. 41.

Fig. 41

To clarify another point. When the P_d tfis used to obtain the E_k¢ due to O motion, the reason is, when the velocity circles are used at the S to obtain the P_v due to S motion, the E_k is altered by S motion and this quantity travels toward the O. Now to determine the E_k¢ due to O motion, the velocity circles are drawn in the O frame. The first one using the O starting point as center, and the second one along the O track using the i/c point as center. Now when these velocity circles (for the O frame) are used in the same manner as those for the S frame, the velocity tf = P_d tf Of course the add and subtract signs in the velocity circles are reversed for the O frame, and the tf derived is used by division instead of multiplication. This makes it possible to omit the velocity circles entirely and use the projectile distance transform factor to alter E_ks by using this tf as a velocity tf. This point is clearer when considering the 1.12 tf of Fig. 39. Therefore, because of this result, it merely becomes necessary to use the Pdtf as the velocity tf to determine v¢ when solving for the E_k the way the O will measure it at impact.

Fig. 42

Fig. 43

Fig. 43 is to visually show what half the intercept curve looks like for one set of basic parameters, S_v, P_ov, O speed, S to O distance at start, etc. The half of the i/c curve rear of the O Y axis would go off the paper. The variables are, O track direction (or X axis) compared to S X axis, P firing angle, P_v, and P resultant (track angle).

When the two in-line i/c points, one on each side of the target’s Y axis (circled and marked A and B), are solved for mathematically, they coincide with the geometric intercept curve exactly. The i/c point where no ballistic Doppler effect is noticed (as described previously) also falls exactly on the i/c curve.

Forces may be included in the geometry at this point but two different types of force must be considered. If the force is due to convection (wind etc.), the length of the vector describing the relocation of the i/c point would vary in proportion to the time because this type of force causes a new ²constant² speed. Or if the force is due to a field such as electric, magnetic, or gravitational, the length of the vector describing the relocation of the i/c point will vary in proportion to the square of the time because this force will cause a ²continuous change² in speed.

When a convective force is included in the motions of a geometric i/c problem, the force may be included in each frame (P and O) similar to the method used for the velocity circle. This allows an automatic geometric i/c to be performed. See Figs. 44 and 45.

Fig. 44

Fig. 45

In Fig. 45 the i/c t with nil force applied is the same as the i/c t with a constant convective force applied.

When a field force is included in the motions of a geometric i/c problem, the i/c point (locus) must be solved first as though no force were being applied, and the tail of the vector is applied on this point. The length of the vector is found using the time from the first i/c and the formula 1/2at². This vector drawn in the proper direction now gives the i/c point due to the influence of the field force. As in the other type of force, both times are the same. This is a terminal vector because the projection principle does not apply. Also the part of the hyperbola the P or O (O is target now) would follow may be omitted and instead a straight line used from start to terminal vector arrow head, and this vector would be used as a resultant displacement. See Figs. 46 and 47.

Fig. 46

Fig. 47

The linear measures which are used as tf s may be taken from the geometry of either type of i/c and force and used in the usual manner to check results of the X or Y components or resultants, remembering, of course, that the Vtf s in the geometry that includes a field force is an average (Vtf) for the Y component and resultant. The velocity on the X component is constant. The Vtf s from geometry including convective force are all actual because all velocities are constant.

Fig. 48

X, Y, and resultant dtf s for P and O are shown in Fig. 48. The Vtf s in each case are measured to the alpha and beta points. In Fig. 48 All forces are omitted to avoid confusion due the extra lines.

Fig. 49

In Fig. 49, the dtf s for the X and Y components and for the resultant are the same as in Fig. 48 except the i/c is relocated to the force vector arrow head. The vtf s are relocated due to the force also, the O_vtf from a to a¢, the P_vtf from b to b¢.

Fig. 50

In Fig. 50 the i/c is relocated due to a field force. First, the t is found from the nil F i/c. The length of the F vector is then computed from 1/2at². Now the dtf s and the vtf s can be found when the i/c is completed. The F vector is from g to g¢.

Fig. 48 shows the P and O resultant displacements, P and O X components, P and O Y components. The method of obtaining the tf s for each component is shown also and when each set of distances and speeds are used all must give the same time. This checks the geometry three different ways and agrees with principles of physics.

The results of Fig. 48 may be checked using the language of mechanics to alter the velocity and distance in the following manner. If one wishes to check the t on the Y component this way the problem becomes as in Fig. 51.

Fig. 51

Where V_oy = V_o sin Ð, meaning, the initial velocity in the Y component is equal to the initial velocity times the sine of the angle. S (initial) is the d between P and O at the instant the P is fired. S_y means the d on the Y component. The formula 1/2at² may be changed to 1/2gt² when convenient.

Referring to Fig. 51 –

4 (a) P_sy = P_voyt – or + 1/2at².

4 (b) O_sy = (O_voy)^t – or + 1/2at².

4 (c) O_sy + S (initial) = d (initial) + (O_voy)^t – or + 1/2at².

Then at intercept –

4 (d) P_sy = O_sy + S (initial).

4 (e) Therefore- P_voyt = d + O_voyt.

4 (f) And- t = d / (P_voy – O_voy).

The t used in (a), (b), (c), and (e) is the t from the nil F i/c, but the formula in (f) gives the t from different relations. These two t’s are equal, and demonstrate the compatibility of the geometric and mathematical answers.

Refer to Fig. 52. For simplicity, the extra motions, and fields or forces are omitted.

A target can be detected, it’s distance, track, and speed, can be determined. The P speed when it is fired on an i/c course is also known. The P firing angle must be determined. From previous geometry we know the velocities on both X components must be equal when measured from the S to O reference line for an i/c to occur, so when we know the target velocity on the X component we also know what the P velocity must be on it’s X component in this case.

Fig. 52

Of course if timing is staggered this would not apply. Therefore, we can start here.

4 (g) O_vx = O_vo x sin a (50°) .7660 = 91.92.

4 (h) O_vy = O_vo x cos a (50°) .6428 = 77.36.

4 (i) Sin P firing angle = O_vx / P_ov = 91.92 / 240 = .383, from sine tables find .383 = 20° 50¢ = b. (P firing angle).

4 (j) P_vy = P_ov x cos b (20° 50¢) .9216 = 221.184.

Therefore using T.E. #3 –

4 (k) P i/c tf (res) =

This is the i/c tf on the resultant.

In the last equation, if we do not divide by .9216, we get the i/c tf on the Y component of .741432. This will agree with measurements in the geometry on the Y component, but the measurements in the resultant geometry is of different dimensions. To transform from the results on the Y component to results on the resultant we must divide by the cos b. Or –

4 (1) P i/c (res) = Py i/c / cos b.

Therefore – P i/c (res) = .741432 / .9216 = .8045052.

Pd = 240 x .8045052 = 193.08124.

P_v = 240.

T = d / s = 193.08124 / 240 = .8045051.

Note that the S that fired the P was stationary, therefore, P i/c t = P i/c tf (res).

The q a is always measured from a line drawn through the P and O starting points as shown in Fig. 53 (a) and (b).

When using the Y component geometry for the P, the i/c becomes –

P_vy = 240 x .925 = 222.

From geometric measureme made with a good engineers scale-

P_dy = 240 x .741 = 177.84.

.I/c t = 177.84 / 222. =.801081.

This shows agreement between i/c t’s on the P resultant and P Y component.

The i/c t on the X component agrees just as well when measurements are made from the geometry of the X component.

When we construct the geometry of the complete i/c problem, we can measure a and b directly. In this case, we choose the proper T.E. and find the cosines of a and b from tables and insert them. For example –

4 (m) i/c tf (res) =

[240 / [240 + ((120 x cos a / 240 x cos b) 240)]] / cos b.

In the i/c of Fig. 52, there was no S motion, and no field or convective forces acting. Now we introduce the motion of a S that fires the P at the target O. Refer to Fig. 54. Velocity circles omitted.

Fig. 52

Previously known quantities –

O_vo –120 m.

P_ov – 240 m.

S_v – 120 m.

d – 240 m.

a - 55 deg.

We can describe the i/c tf now from eight steps, this is used to find the P i/c d and with the P_v on the resultant will give us the i/c t. (These equations are modified further on).

The equations for the eight steps are –

4 (n) O_vx = O_vo x sin a.

4 (o) O_vy = O_vo x cos a.

4 (p) P_fq (P firing angle) = (S_v – O_vx) / P_ov = cos e.

When the Ðg is known we also know b.

4 (q) P_v res = (P_ov x [(cos + bq) - (cos - bq)]) + (S_v x cos g) – (S_v x cos - e).

4 (r) P (res) q = O_vx / P_v (res) = sin b.

4 (s) P_vy = P_v (res) x cos b. Or, P_vy = P_ov x sin e.

4 (t) tan g = P_vy / P_vx. Or, cot b = P_vy / P_vx. The O_vx is known and the P_vx must be the same.

4 (u) Pi/c tf (res) = (d / [d + [(O_vy / P_vy) x d ]]) / cos b. (Using T.E. #3).

The above steps are not in order.

Results from Fig. 52 are –

From geometry From equations

Pd – 200.64 201.04068

P_v – 256.8 256.56

I/c tf - .836 .8376695

I/c t - .7813084 .783601

Symbols used in Fig. 52 and later drawings and equations –

a - q (angle) between O track and S to O reference line.

b - q between S to O reference line and P resultant.

g - q between P resultant and S forward X axis.

d - q between P firing q and S to O reference line. (Or, when the P firing q and P resultant q are on the same side of S to O reference line, d is the q between P firing q and P resultant).

O_v - Target velocity along target trajectory.

P_ov - Projectile lab. Velocity.

O_vx - Target velocity on the x component.

P_v res - Projectile velocity on the resultant.

P (res) q- Projectile resultant angle.

h - Angle between S to O reference line and S Y axis.

⁺e - Angle between S forward X axis and P firing angle

^-e - Angle between S rear X axis and P firing angle.

Pfq - Projectile firing angle.

Since the projectile is fired from a moving source, the geometry and mathematical equations both

give the P i/c tf, which is the arithmetic factor representing the P distance to i/c. The i/c t must be obtained

from the t = d / s formula.

The previous equations used to obtain the results of Fig. 52 require certain modifications and are given here along with Fig. 53 to demonstrate more clearly the use of the symbols.

4 (y) O_vx = O_v x sin (qa ± qh). Add qh if fwd. of S Y axis.

4 (z) - O_vy = O_v x cos (qa ± qh). Add qh if fwd. of S Y axis.

4 (za) - P_vx = S_v + (P_ov x cos ⁺e) – (P_ov x cos ^-e).

4 (zb) - P_vy = P_ov x sin qe. Note- the max for P_vy is P_ov.

4 (zc) - Pfq = P_vy / P_vx ± S_v) = tan q. Add S_v if res q is off S rear X axis.

4 (zd) - P (res) q = P_vy / P_vx = tan q.

4 (ze) - P_v res = [P_ov x cos (qb ± qd)] + (S_v x cos g) – (S_v x cos ^-e). Subtract qd when rear of S Y axis.

4 (zf) - P i/c tf (res) =

Note – use T.E. #3 or #4. (Basic conditions for #1 or #2 will not lead to an i/c.

These equations are designed to describe the geometric process, not predict the process without

the geometry. The velocities on the Y component are used in T.E.#3 or #4 as under normal conditions. . The equations 4 (y) to 4 (zf) adjust the velocities on the Y component to be perpendicular to the S X axis. When forces due to fields or convection are introduced, the i/c t will be the same as the i/c t when no forces are applied, providing the force acts equally on both objects. The i/c point would be automatically relocated.

Refer to Fig. 34

Just one example now to show how angular ballistic intercepts are verified (not predicted) with the two main transformation equations.

Instead of the head-on i/c in Fig. 34, if the approach was at an angle, we use the cos of the target angle and the cos of the i/c missile firing angle, which is fired from a moving source.

Fig. 53

Fig. 54

The basic equation, which only covers two speeds, would be as –

But since Fig. 54 includes 3 speeds, we must alter the equation to include a S_v as this changes the P_ov, this way (P_ov is 240) –

(50.714 is amount of S_v to be subtracted from P_ov + S_v which would be P_v on the forward X axis).

The basic equations for the i/c conditions and separation conditions for angular motions including only 2 speeds each are –

For Fig. 32

For Fig. 33

For Fig. 34

For Fig. 35

Note – Two of these are distance of final separation when one projectile travels 120 km. The other two are i/c distances.

REFRACTION AND CONVECTION

Most of the information on these two topics is very well known and understood now and formulae exists which describes the operations. They are included here to record the method of using the index of refraction and convection coefficient with the new transforms under various conditions. Only in-line examples are given.

The symbol Æ represents Fresnel’s convection coefficient as determined by his formula –

I feel it should be mentioned here that it seems hard to believe the idea that different wavelengths of light are refracted by a different index of refraction, which is determined by individual wave phase velocities. It seems more logical that the frequency alone (according to the structure of the substance) determine the refractive index, because in other but related fields, the frequency determines the reaction. Such examples are found in the photoelectric effect, the formula for quantum energy, in sound the resonant vibrations set up in sympathetic volumes, the transversal Doppler effect, and the Mossbauer effect. However, since the frequency before, during, and after, refraction is always the same, it can not be used in equations to determine the refractive index.

We should not be able to measure a C' between two beams of light using wavelength interference where one goes directly from S to O through an air space, and the other from S through air and through a piece of refractive substance then through air to the O. Example –

Fig. 55

If the wavelength from the S was 24 mm and moved at C for beam #1, we should get x number of 24 mm waves/sec. on the screen. At the same time, if 24 mm wavelengths were also introduced into beam #2, h1 = .5 C, the refracted wavelength would be 12 mm inside the refractive substance and it would propagate at 1/2 C. We would get twice as many 12 mm wavelengths moving through the distance of the refractive substance propagating at 1/2 C in 2 seconds as we would get 24 mm wavelengths moving through the same distance of air at C in 1 second. The screen would receive the same number of 24 mm wavelengths/sec. from beam #1 as it received 24 mm wavelengths/sec. from beam #2, therefore, there would be no interference. The fact that the frequency is the same before, during, and after, refraction tells us this also. When a convection coefficient is introduced into one beam, the time and distance relationship is altered which would (Fizeau’s experiment) cause wavelength interference on the screen as the included equations for this condition demonstrate. When the light was first turned on, beam #1 would arrive at the screen first, but as soon as beam #2 got there both would appear identical excluding any special effects.

When light travels through a substance its propagation velocity inside the substance is altered according to the refraction index of the substance. Since the substance in motion receives the incident wavelengths and transmits the exit wavelengths, it would act as a relay, and the fifth law would apply, but the wavelengths altered by the Otf would be altered by the refraction tf for reception, and the reversed tf would be applied at the exit interface for transmission and would then be altered by the Stf.

To take one step at a time, consider the refractive substance stationary. It receives a 6 000 Å wavelength, slows it down according to the index of refraction (let h = .5), and retransmits it back into the air at the other interface.

Fig. 56

The 6 000 Å l moving at C is received by one glass surface which slows it to 1/2 C. The l is altered to –

C' / C = .5, 6 000 Å l (C' / C) = 3 000 Å l.

This 3 000 Å l travels across the glass to #2 interface and is again altered when it is transmitted to the air, but this time in the reverse.

C / C´ = 2, 3 000 Å l (C / C´) = 6 000 Å l.

Thus, it is lengthened to 6 000 Å once more.

Now consider a refractive substance (solid) in motion. We use the same glass piece and cause it to move in a straight line at 1/2 C. No convection is introduced at this stage.

Fig. 57

The Otf and Stf are found by the proper transforms and are described in previous pages so will be omitted here, and the choice of numbers make it easy to follow the 6 000 Å l through the various transformations. It is interesting to note that the Otf and Stf cancel out for any motion as long as the angle of incidence is equal to the angle of exit measured from the X axis, and the refraction factors h1 and h2 cancel out.

If we make a detour at this point to draw an analogy from the sound spectrum.

Consider an aircraft moving at 3/4 C (3/4 the speed of sound) with a rubber window on the front and one on the rear so sound can pass through.

Fig. 58

Inside the aircraft standard laboratory conditions are duplicated and sound travels up and down the cabin at C. This same sound though would be transmitted out the front altered by the 0° X axis Stf, and out the rear by the 180° X axis Stf, and the Doppler effect now shows an outside observer that the source is in motion.

Now consider the same aircraft receiving a wavelength through the rear rubber window altered by the Otf, and carrying it up the cabin (n0 refraction here), and transmitting it through the front rubber window altered by the Stf. S and O transform factors cancel out and the original wavelength is as though it had passed through nothing. The wavelength on exit from the cabin is identical to the one before entry.

Fig. 59

If the aircraft was filled with water so a refraction index would apply for sound the process would follow the same pattern as that for light, except sound travels faster in the denser medium so the refraction tf’s would be reversed. Both spectrums would exhibit identical basic operating principles for the Doppler phenomena so far.

This far the handling of the equations is pretty straightforward as long as the special factors are solved for separately and applied in proper sequence to effect the transformation being described. To illustrate, the process for Fig. 57 would follow this course. The Otf for the conditions described is worked out giving an Otf of 2. This alters the 6 000 Å wavelength as 6 000 x 2 = 12 000 Å. Next, the index of refraction from tables using velocities is converted to a factor for h1, in this case C¢ ¸ C it is .5. This alters the new wavelength as 12 000 x .5 = 6 000 Å. The reverse refraction tf C ¸ C¢ = 2 is obtained for h2, and applied to the latest wavelength as, 6 000 x 2 = 12 000 Å. Now the Stf for existing conditions gives Stf = .5, and this alters the wavelength to, 12 000 x .5 = 6 000 Å.

Now we look at convection.

The convection in a moving substance would parallel a wave carrier motion outside a moving substance or in vacuum. In a dispersive medium, if the carrier between h1 and h2 is stationary relative to both interfaces, only the relative index of refraction affects C. But if a carrier motion exists (as in Fizeau’s experiment) it would alter the C´ (which is due to the refraction index) as C´ + Æ or C´ - Æ, but it would not affect the Otf or Stf which are effects of conditions which are ²outside² the both interfaces. If the dispersive medium was removed and the convection took place over the entire distance between S and O, the Stf and Otf would be affected by the Æ adding to or subtracting from the conditions which cause real or apparent wavelength alterations.

A list of equations are selected which show how the Æ is used with this theory. In some cases, a simple drawing with arrows is given before the equation to describe the special conditions. When Stf and Otf are required the transformation equations (T.E. #s 1, 2, 3, and 4) are used.

TRANSFORMATION FORMULAE – REFRACTION AND CONVECTION

Eq. 9 l entering #1 interface –

Eq. 10 Refracted l at #1 interface –

Eq. 11 h2 tf at #2 interface –

Eq. 12 l´ at #2 interface –

Eq. 13 Moving refractory substance. At #1 interface –

l_r = (l_S x Otf) h1

Eq. 14 Moving refractory substance. At #2 interface –