Random Questions in Applied Mathematics

Can you answer any of these questions?

  • "It is better that a question be unanswered than unasked." - old proverb
  • "The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill" - Einstein

Written 2001

Formatted 2010

  Focused Waves vs. Planet Formation

The speed of sound will vary with temperature, pressure, and the matter through which it passes. If these things vary in space it can have the consequence of refocusing the wave and preventing it from dispersing. (This approach is used in fiber optics.) If the right conditions were to occur in an atmosphere sound waves could be concentrated into a single turbulent layer, or rapidly dispersed out of another layer. Could this refocusing of waves account for planet formation in the gas cloud around a young star?

focused waves

Are there such layers in our atmosphere, or the atmosphere of the other planets? If so, how is the wave concentration manifest?

Galaxies vs. Computer Spiral Simulations

Some scientist have claimed that the existence of spiral galaxies suggest there is something odd in the universe beyond our explanation. The outer part of the spiral arms and the inner parts should rotate at different speeds causing the spiral arms to fly apart.

chaos spiral

Years back, I wrote a computer program using a simple formula to study chaos. That program produced mostly spirals. When I varied the "seed" (starting number) by a small amount the number of spiral arms would change as well as their arcing. This suggests that galaxy arms do fly apart only to regroup with a different number and a pattern. The non-spiral galaxies might actually be between phases in the number of spiral arms. A more focused computer simulation would solve this.

 

Spirals and Chaos

spiral 3

My chaos program that produced spirals, as mentioned above, produced various other patterns including some arrangements that were random on one side and ordered on the other, the yin-yang, and the swastika. I was surprised that a simple equation with different seeds produced two well known symbols along with other patterns. I lost my chaos program long ago. But I do recall it used a very simple reiterative formula using complex (square root of -1) numbers and plotted the points produced at each iteration. As so often happens in chaos mathematics, the same thing shows up from different sources. Will other reiterative equations produce these same patterns? Pick an equation and try. Report your results. Where do theses results show up in nature?

 
 

Lift and Quantum Theory

According to quantum theory, every wave carries energy E=hv and momentum P=h/k. This suggests that lift may be produced by the emission of waves. What grouping of waves would have the highest efficiency: lift to energy ratio? What would the wave front producing the most lift look like? Assuming acoustic waves, what would it sound like? What would the vehicle that rode on these waves look like?

 

lift

 
 

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