Math Blues I
Gaaah.
So I'm trying to figure out a bit about wave dynamics - namely how a wave can move in a collimated fashion, such as in a light ray or laser beam.
I was playing around with the wave equation a bit. I set up field sims for a 3d wave and advanced time, reproducing all sorts of interesting effects. Diffraction, reflection, interference, refraction ect.
Then something started bugging me - if you have a travelling wave, how do you get it so that it retains a collimated shape? It seems to me that the divergence of the gradient on a point outside the beam is going to be nonzero due to the difference between the zero and nonzero amplitudes inside and outside the beam, and so the region outside the beam should be sucking up energy and spreading the wave as it travels.
At first I thought it was just that I was looking at a scalar wave, and light is a more complicated vector wave operating off of different rules. But in my optics book, they eventually transform maxwells equations into a set of 6 scalar equations for the electric and magnetic field, and the same laplacian(field) = acceleration(field) behavior results.
Okay, so then I did some reading and discovered that beams usually have gaussian distribution of amplitude along the beam radius.
I want to be able to figure out how the amplitude's radial profile changes as you move along the wave. So I wanted to transform the wave equation into a different coordinate system moving with the beam.
laplacian(field(x,y,z)) = acceleration(field(x,y,z)) -> acceleration(field(x-vt,y,z)) = ???
and here's where I have to quit for today, I need to make dinner.
This better not be something that some old math god fart like Newton or Euler solved in 5 minutes while waiting for the coffee to brew.
So I'm trying to figure out a bit about wave dynamics - namely how a wave can move in a collimated fashion, such as in a light ray or laser beam.
I was playing around with the wave equation a bit. I set up field sims for a 3d wave and advanced time, reproducing all sorts of interesting effects. Diffraction, reflection, interference, refraction ect.
Then something started bugging me - if you have a travelling wave, how do you get it so that it retains a collimated shape? It seems to me that the divergence of the gradient on a point outside the beam is going to be nonzero due to the difference between the zero and nonzero amplitudes inside and outside the beam, and so the region outside the beam should be sucking up energy and spreading the wave as it travels.
At first I thought it was just that I was looking at a scalar wave, and light is a more complicated vector wave operating off of different rules. But in my optics book, they eventually transform maxwells equations into a set of 6 scalar equations for the electric and magnetic field, and the same laplacian(field) = acceleration(field) behavior results.
Okay, so then I did some reading and discovered that beams usually have gaussian distribution of amplitude along the beam radius.
I want to be able to figure out how the amplitude's radial profile changes as you move along the wave. So I wanted to transform the wave equation into a different coordinate system moving with the beam.
laplacian(field(x,y,z)) = acceleration(field(x,y,z)) -> acceleration(field(x-vt,y,z)) = ???
and here's where I have to quit for today, I need to make dinner.
This better not be something that some old math god fart like Newton or Euler solved in 5 minutes while waiting for the coffee to brew.
posted by qwerty182764 at 4:21 PM
4 Comments:
I think you're talking about diffraction limit and coherence length. Any beam will only stay collimated (to a certain beam width) and coherent (phase-wise) so far out.
Well, it's been a long time since I messed with optics, which is a pretty fascinating subject but kind of like orbital mechanics, in that both are often rather non-intuitive.
Thanks for your comment on AeroGo, I left a reply there. From what I've heard, Purdue is a pretty good aero school, with a strong undergrad program, not just research kudos.
Nice post,i lick it.
: )
Ours is an envionment where evil is perceived to be rewarded while good is punished. As with everything the Gods have a reason for creating this perception::::
People who fall on the good side of the good/evil scale have more favor, and when they do something wrong the Gods punish them BECAUSE THEY WANT THEM TO LEARN. The Gods want them to receive this feedback in hope they make corrections and begin to behave appropriately. The Gods DON'T like evil and refuse to grant this feedback.
EVERYBODY pays for what they do wrong, only evil people must wait until their next life before they will experience the wrath of the Gods, manifested in their reincarnation as a lower form of life into environments with increased/enhanced temptations.
Sadly, this allows the Gods to position this perception of evil rewarded as temptation, one which they use as an EXTREMELY effective corruptor.
Both Africa and the Medittereanean are regions which have sexual issues. This is a sign of morbid disfavor once you understand that females are the God's favored gender. Muhammad's (Mohammed's) polygamy halfway through his life as a prophet was preditory. Now a huge percentage of Muslims believes in male superiority and that the abuse of women is God's will. Female genital mutilation is still practiced in Africa. Black misogyny is the most eggregious example in the recent past.
Black member size is temptation to a predisposed population.
The patriarchal cancer spread throughout Europe because of Christianity, of which the majority of policy makers were Italian men. Expect the largest landowner in Europe and the continent's original superpower also played a major role in African slavery.
Militancy in Africa is consistant with the Iraqi example, as was slavery and the KKK here in America:::Fear enforces proper behavior. Without it we see what happens as a result of gross/morbid disfavor:::::AIDS, crack babies, dead young men in gangland retaliation killings. This is the purpose behind many black's historical tendancy towards resistance.
The same principle was true in Europe and throughout the world for centuries:::People whom lived under iron fists were conditioned to think the right way. As a result they experienced higher numbers of children accend into heaven because they were taught to think and behave appropriately. Our preditory envionment of "freedom" was the primary purpose the Gods had when implimenting this strategy that is the United States, one which they used to spred the cancer of democracy and westernization throughout the world. And the Gods use this tool that is America to prey on the disfavored both at home and abroad.
Even the Old Testiment is not to be taken literally, but the Gods do offer clues throughout to help the disfavored:::The apple is a tool of temptation used to corrupt Adam and Eve and cast them out of the Garden of Eden.
There is another lesson to be learned from this passage, and it is quite similar to the vailing issue and the discourse over women's attire which ultimately died in the 70s:::Women are responsible for and control the fate of mankind.
Think about what I say. Consider what I teach. Society is going to become disturbingly ugly as we approach the Apocalypse due to spiralling, runaway disfavor.
I do not know when this will occurr, but it is the God's way to grant some time before they end on Planet Earth.
Make the decision to always be good and never look back. Until you do this technology will employ tactics to test your resolve:::Ridicule, beligerance, doubt and refusal to abandon what people perceive to be their "investment".
Pray daily. Think appropriately. Too many are confident, unaware of the God's awesome powers or their status as antients. Others may fall prey to their positioning.
Be humbled, God-fearing and beware of the God's temptations, for everyone is tested to evaluate their worthiness.
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