3: PLANE STANDING
WAVES
Spherical Standing Waves
Standard plane standing
waves
The above diagram shows two sets of plane
waves traveling in opposite directions. This produces standing waves,
a well known wave structure.
Note that amplitude between peaks is nil. Such places are called nodes. Peaks are called antinodes. Both are present two times per period and two times per wavelength, hence regularly spaced lambda / 2 node and antinode pattern. There is absolutely no energy transfer between antinodes. However, the medium pressure alternates between a minimum and a maximum inside antinodes.
From a mechanical point of view, this may be seen
as kinetic energy transformed into potential energy, and again into
kinetic. In 2008, Mr. Jocelyn Marcotte elaborated a method (Lagrangian
= kinetic
– potential)
in order to detect standing waves. Such an achievement deserves the
warmest congratulations. Standing waves are certainly among the most neglected domains in physics. One can identify many misunderstood characteristics. For example:
Scientists seldom mention such characteristics. They prefer to spread out their knowledge with a lot of complicated equations. This is sometimes quite useless. For instance, many diagrams from this site were simply displayed by computer using sine functions only. Others were made using Huygen's Principle, without the help of equations. Augustin Fresnel stated that energy equals the square of amplitude. Because amplitude is two times higher for two equal sets of waves adding constructively, energy is four times higher there. This explains why billions of waves mixing together do not cancel energy in accordance with the Great Numbers Law. The wave energy simply cannot be destroyed on condition that it is not transferred into heat because of the imperfect mechanism of the medium. Standing waves do not contain waves traveling in opposite directions. Two sets of plane parallel waves traveling in opposite directions will indeed produce plane standing waves. One may place a plane parallel screen in the way of just one set of plane traveling waves and also obtain standing waves. Other methods are possible. From a mathematical point of view, one then can add their amplitude and obtain the classical standing wave pattern. However, it should be emphasized that although this method works most of the time, it is only an artificial one. It is not what is really going on. I especially showed recently that, if amplitude differs, the central antinode for circular standing waves becomes elliptic. In such a case, waves undergo a hard reflection on the zero energy ellipse and the reflected waves are no longer circular. This had never been pointed out. There is no equation for this up to now. Such a phenomenon is important because it clearly indicates that any external wave can influence circular or spherical waves by modifying its unique central structure. Waves can influence other waves, hence matter can influence matter even if it is solely made of waves. So the door is open to a totally new action and reaction mechanism based on the Doppler effect. We finally can upgrade Newton's laws in accordance with the Lorentz transformations and Lorentz's Relativity. Inside standing waves, energy is alternately transferred from kinetic energy to pressure energy, in accordance with Hooke's law. The medium behaves like a spring section moving to and fro. Inside regular traveling waves both types are present simultaneously and constantly move at the same speed. On the contrary, there is no energy transfer inside standing waves. Finally, when lossless standing waves are present inside a finite space, such waves will theoretically continue to oscillate eternally. But because all standing waves actually lose some of their energy, they need replenishment, hence amplification. So the electron, which is a spherical standing wave system, does not need "inwaves" in order to be replenished. It needs an amplification process to prevent it from fading out. This is done by all external waves going through it, and undergoing a lens effect. This amplification process allows the system to go on oscillating permanently.
MOVING STANDING WAVES Waves emitted by a moving device undergo the Doppler effect. Let's suppose that its speed is one half of the speed of light. Then the beta normalized speed is given by: v / c = .5. The Doppler effect equals 1 – beta = .5 forward and 1 + beta = 1.5 backward. The wavelength ratio is: (1 + beta) / (1 – beta) = 1.5 / .5 = 3, hence 1 : 3. Surprisingly, using such waves, and as compared to the diagram shown above, the wave pattern contracts to 75% of its normal length. Moreover, it moves forward at the same beta speed. 
Standing waves motion and contraction.
This phenomenon may have been discovered by Mr. Yuri Ivanov in 1990. www.keelynet.com/spider/b104e.htm www.keelynet.com/spider/b111e.htm Mr. Ivanov is a genius because he applied this phenomenon to matter contraction, which was Lorentz's main (and correct) theory. So he was right on this. Unfortunately, he did not understand the Lorentz time transformation and he proposed his own transformation with t' = t similar to Michelson's calculations. According to Lorentz the wavelength increases because the overall frequency is slowed down. Finally there is no crosswise standing wave contraction: x' = x; y' = y. So Mr. Ivanov's system is obviously wrong. Additionally, his spider and levitation effects are rather weird. Anyway, let us examine standing wave contraction first. In order to observe the above diagram in a more easy and intuitive manner, here is the same diagram as shown above, except for the observer who now follows the apparently unmoving node and antinode pattern: 
Here, the observer moves towards the right. Now, nodes and antinodes seem to be immobile.
Three times longer waves seem to move three times faster, explaining the same wave rate or apparent frequency.
STANDING WAVE CONTRACTION
g = (1 – beta^{ 2 })^{ (1 / 2)} Please remember that beta = v / c. Then the wave pattern contracts like this: 
beta = 0 and g = 1. No contraction.
System at rest: 0 c. No Doppler effect.
beta = .5 and g = .866 Contraction to .75 according to g^{ 2}.
Doppler leftward: 1 + beta = 1.5 Doppler rightward = 1 – beta = .5
beta = .7071 and g = .7071 Contraction to .5 according to g^{ 2}.
Doppler leftward: 1 + beta = 1.7071 Doppler rightward = 1 – beta = .2929
Standing waves also undergo a less severe contraction according to the contraction factor g (not squared) on any transverse direction (see below). However, Lorentz also showed that the system frequency should slow down according to the same contraction factor g. This means that the electron should finally undergo a contraction on the displacement axis x only according to the g value not squared. There is no contraction at all on both y and z axes. This is Lorentz's well known distance contraction. This explains why matter, which is made of standing waves, should also undergo the Lorentz transformations. This explains why the Michelson interferometer did really undergo such a contraction and could not reveal the aether wind. 
PARTIALLY STANDING WAVES
(Unequal amplitude)
The animation below shows standing waves where compressed waves are two times stronger. Such standing waves evolve inside a very peculiar peapodlike envelope. 
Partially standing waves: E_{1} = 67 %, E_{2} = 33 %, beta = 0
PARTIALLY MOVING STANDING WAVES
(Unequal amplitude and wavelength)
This is the general case for electrons. These pages show that stronger aether waves can "push" an electron. This is the radiation pressure, which can be explained by fields of force, which are "partially moving standing waves". In the animations below amplitude differs but there is also a Doppler effect. Such waves still evolve inside the same peculiar envelope, whose speed remains exactly the same as for equal amplitude waves. Note that nodes are partially hustled sideways; any stronger wave will act in the same manner on any sort of standing wave systems. It will especially push the electron central antinode. 
Partially moving standing waves. Forward contracted waves are stronger.
E_{1} = 67 %, E_{2} = 33 %, beta = .5
Partially moving standing waves. Backward dilated waves are stronger.
E_{1} = 33 %, E_{2} = 67 %, beta = .5
TILTED WAVES TRAVELING ALONG A TRANSVERSE AXIS
Let's imagine two train cars at rest along parallel railways. Their flat sides can reflect sound waves so that one can produce standing waves between them. However those waves must travel according to an angle in order to follow the cars while they are moving. This is Lorentz's theta angle, which equals: arc sin beta. So two sets of waves traveling in opposite directions will undergo a scissor effect. Moreover, Michelson showed that such trains would not accept the same resonant frequency at high speed any more because the wind will decrease the apparent speed of sound . Such waves behave in a very strange way. So I first showed in the left animation below those trains traveling rather slowly : 10 % of the speed of sound (76 mph). The scissor effect is obvious. The Time Scanner Page explains that the intersection points follow the places where Lorentz's t' time does not change. Scanning this diagram right on those points would produce standard plane standing waves.



Left : 10 % of the speed of sound. Right : 50 %.
The animation on the right shows the trains traveling at 50 % of the speed of sound (380 mph). Then beta = .5 and waves are tilted to : arc sin beta or 30°. Surprisingly, this system seems to move sideways. A "checkerboard effect" moving sideways is even more visible in the more sophisticated animation below, which nevertheless shows the same situation:

Transverse standing waves : v = .5 c.
Same frequency: transverse contraction to .866 according to Lorentz's contraction factor g.
However, no contraction actually occurs because frequency also slows down according to g.
It should be pointed out that such transverse standing waves still do not contain two sets of traveling waves moving in opposite directions. This is a completely new and different wave system where the medium pressure alternates from one point to another. Now one may speak about energy transfer, whose speed is that of the system. This was also the case for longitudinal "moving" or "lively" standing waves. The time shift cancels the checkerboard effect. Now let's suppose that you are a bat. Instead of seeing objects by means of light, you can "hear" objects so well that you actually see them. What will you see if you are moving on the top of those trains ? The answer is : you will "hear" regular standing waves. They will not appear tilted any more. You will not see a checkerboard moving sideways because the sound traveling forward is three times slower because of the wind. So the information traveling from the front of the train is much faster. This also produces a "time shift" if you are observing with your ears a clock placed in front of the train and compare it to another one at the rear. Henri Poincaré discovered well before Einstein that clocks could not indicate the same hour along a moving system in order to preserve the apparent simultaneity. This happens with light too because we see things with our eyes, by means of light. Such information is not simultaneous when we are moving through the aether, and so we encounter the same effect as we would as a bat. According to the Lorentz transformations the time shift equals simply –beta for each one lightsecond uncontracted distance. For example, a .5 beta speed produces a –.5 second time shift for one light second distance forward contracted to g = .866. This is the true time shift displayed by clocks, whose time rate is however slowed down to .866 times their original rate at rest. One light second is about the distance to the Moon, so the true distance to the moon would be actually .866 light second if we were moving at one half of the speed of light, the Earth being at the rear. Then the clock on the Moon would be .5 second late according to –beta. However, this situation would not be detectable either by radar or GPS firstly because of the contraction, secondly because of the slower clocks, and finally because of the Doppler effect. This is easily verifiable because the calculus shown above is amazingly simple. The pages on Relativity propose a more dramatic example because of a faster speed: 86.6 % of that of the light. Then objects and distances are contracted to 50% of their original length. The animation below shows that inside such a frame of reference moving to the right, a plane wave emitted very far away on the CD axis will be tilted to as much as 60°. However, any observer placed on this axis will record that observers A and B did receive this wave at the same time because of the time shift. This will "prove" that this wave is not tilted : 

A tilted wave seems parallel to the displacement x axis in a moving frame of reference.
Bradley's star aberration is not noticeable if both the Earth and a distant star move at the same speed.
Any observer on Earth will only record the speed difference, so Bradley's aberration is symmetrical.
Next page is on Spherical Standing Waves.
Gabriel LaFreniere, 