This answer is a simplification and modern electronics now use different ways of driving fluorescent lights but I will explain in terms of how the original light works which will keep the answer as simple and understandable as possible. A fluorescent bulb (FB) contains a coating of phosphorescent powder on the inside of the bulb. This is what makes it look white when off. More on this later. The tube of the bulb is then evacuated of air and a small amount of mercury is injected inside of the bulb. Some manufactures also inject some amount of noble gas to aid in starting and maintaining the electrical properties needed for the FB to work and help in the useful life of the bulb. While the bulb is still under full or partial vacuum, the ends are sealed around two pins which act as the conductors to apply electricity to the inside of the bulb. Between the two pins on each side is a tungsten filament similar to a regular incandescent light bulb.When you flip the switch to turn the bulb on, two things happen. A transformer, called the “ballast” is energized. A ballast transformer works a little different than most transformers. Normally transformer step up or step down the AC voltage applied to them according to the ratio of the number of turns between the primary and secondary winding. A ballast transformer at first acts like a high voltage transformer, meaning there are many more turns of wire on the output (secondary) winding and a much higher ac voltage is produced on its output. The exact voltage depends on how big and how many bulbs the ballast transformer is designed to operate, but the voltage is in the range of thousands of volts. So during the first few seconds of turning on a florescent light the ballast transformer applies high voltage to one pin on each side of the FB. At the same time, a separate circuit controlled by a small relay called “The Starter” closes contacts which applies primary voltage (120 vac or 240 vac depending on the design) to the second pin on each side of the bulb providing power to the tungsten filaments mentions earlier. These heat up and start to glow dimly. The purpose of the filaments are heaters. Mercury evaporates, so after a bulb is manufactured, the mercury injected into the bulb slowly evaporates and replaces the vacuum inside the bulb with mercury vapor. This vapor will conduct electricity but initially has a high resistance which is too high for a spark to jump the several feet between the high voltage applied to each end of the bulb. The filaments heat the mercury vapor increasing the number of mercury atoms floating around in the tube. This lowers the resistance of the total vapor between the high voltage applied to each end until a spark is produced down the length of the tube. This process is similar to initial conditions in a bolt of lightning. In a small fraction of a second, this spark heats the mercury vapor into a plasma which conduct electricity very well.In milliseconds, the entire length and diameter of the mercury vapor in the tube is converted into a plasma which has completely different properties. The resistance to electron flow drops by several orders of magnitude. This conditions is very similar to the return stroke of a lightning bolt, but contained and maintained inside a glass tube. The ballast transformer that was outputting several thousand volts goes into a condition called saturation because it now has a near dead short across it secondary leads. Its output voltage drops to something in the range of 20 vac but the current flowing increases until it reaches a balance determined by the internal resistance of the copper wire in the secondary winding plus whatever the resistance of the mercury plasma in the tube happens to be. The plasma resistance depends on the temperature of the plasma, its pressure and the purity. The properties of mercury plasma is unique. Instead of heating up because of the electricity flowing through it like most conductors, the plasma ions vibrate at the frequency that gives off ultraviolet light (UV).UV light in general is very harmful to living things. It is commonly used to kill off microscopic organisms in water or to generally sterilize anything. Here is where the phosphorescent powder coating inside the FB comes in. When the UV light given off by the mercury plasma hits the coating inside the FB the coating adsorbs the UV light and vibrates in the range of visible light completely blocking and converting all the harmful UV into a nice cool white light. By varying the blend of phosphorescent powders, manufactures can control the color range of the FB’s output. Oh, and the starter circuit that lit the tungsten heaters on end of the bulb; once the mercury plasma starts conducting, the starter turns off. There are several ways manufactures control this, but for brevity, just accept that they do.All this explanation of how a florescent light works has been necessary to finally give and answer to the original question. What causes the wandering bright spots in fluorescent lamps? To answer this I need to explain what makes a FB go bad.Bulbs of any type that are filled with a gas go bad for one single reason. The gas leaks out and air leaks in. This happens because we have never been able to produce a perfect seal around the metal connections that penetrate a glass bulb. Glass and metal expand and contract differently as their temperature changes. Each time a light bulb is turn on or off the temperature changes dramatically. Even the change in ambient temperature produce this effect. There are ways this can be eliminated, for instance install the secondary coil inside the bulb and have the primary coil wound around the outside, but all these schemes are difficult and expensive, so you only see these types of things in lights that require extreme reliability. Today things like that are being replace with solid state lighting devices like LEDs. It’s the loss or introduction of contaminates that cause the effects you showed in the videos.FB produce these wave features because of several random effects mixed together. When the bulb looses some mercury vapor, the plasma channel does not fully expand inside the tube, so what you are seeing is a plasma arc that never expands to fill the tube and is subject of changing conditions around it that either attract or repel.First, the waves the plasma arc produces is the result of a resonate frequency that develops because the arc is a product of an alternating current. In the US all power is transmitted at 60 cycles per second. This means the current reverses 120 times per second. (A cycle is a positive current and the opposite negative current. In Europe it is 50 cycles per second. The ballast transformer will also have some natural resonate frequency depending on its shape and windings. Resonates happens all across the spectrum of electromagnetics and sound. For instance the tone a bell rings at is it’s resonate frequency. In the case of the first video, the length of the bulb and the shape of the ballast transformer produce two resonate frequencies that were nearly exactly divisible with no fraction left over. The two frequencies combine and produce a standing wave that was some multiple between the two and they produced light and dark spots that stood still. In the second video, the standing wave moved slowly down the tube, indicating that the transformer driving the tube and the tube length was not and exact even division of the other, leaving a fraction between the two that manifest in the wave appearing to slowly move in one direction.There are other things that affect how the plasma arc moves. One is gas circulation inside the tube. The plasma will heat the gas around it. The hotter gas will rise to the top of the tube and try to carry the plasma with it. You can see this effect on many YouTube videos showing a power line arc. A plasma arc jumps between two lines and then rises up in the center, following the lighter hot air it produces until the arc becomes too long to maintain and breaks, only to form again as a new straight arc between the two lines.Another variable that affects the plasma in the tube is electromagnetic and electrostatic fields. If you would bring a magnet near the tube, the plasma wave would be attracted to it. The same thing would happen if you touch the tube with your hand. Your body acts as a capacitor and the plasma wave will the attracted to your hand. This is the principle behind most modern touch screens today. Your finger changes the capacitance at the point where you touch the screen and this is detected by a fine grid of tiny wires in the screen, pinpointing where your finger is.Another variable that would alter the plasma wave would be to heat the bulb with a lighter. The heat transfers into the gas in the bulb and produces an area of lower conductivity making the plasma arc move toward the heat. A trick you can do with a florescent light that will not turn on is to take a lighter and run the flame from one side to the other, heating the glass as you go. The bulb will most likely light up and be fine until you cut it off and you have to repeat the process.As a FB get older, you will notice dark areas near each end of the bulb. This is the tungsten filament slowly vaporizing into gas and then condensing on the relatively cold glass near it.This was a good question. I did not expect I would have to write so much about how a FB works, but without the explanation, the short answer makes little sense. I have never seen a good explanation on florescent lights elsewhere, so hopefully my time will be helpful to someone. Sorry it was so long. Perhaps I will come back and clean it up, making it shorter and clearer.