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Akano's Blog



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Equation of the Day #8: Bessel functions

Posted by Akano Toa of Electricity , in Math/Physics Feb 15 2014 · 173 views
Equation of the Day
My very first Equation of the Day was about the wave equation, a differential equation that governs wave behavior. It doesn't matter whether you have linear waves (sine and cosine functions), cylindrical waves, or spherical waves, the wave equation governs them. Today I will focus on the second, the so-called cylindrical harmonics, or Bessel functions.

A harmonic function is defined as one that satisfies Laplace's equation,

 

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For cylindrical symmetry, the Laplacian (the operator represented by the top-heavy triangle squared) takes the following form:
 

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This is where a neat trick is used. We make an assumption that the amplitude of the wave, denoted here by ψ, can be represented as a product of three separate functions which each only depend on one coordinate. To be more explicit,
 

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This technique is known as "separation of variables."  We claim that the function, ψ, can be separated into a product of functions each with their own unique variable.  The results of this mathematical magic are astounding, since it greatly simplifies the problem at hand.  When you go through the rigamarole of plugging this separated function back in, you get three simpler equations, each with its own variable.
 

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Notice that the partial derivatives have become total derivatives, since these functions only depend on one variable.  These are well-known differential equations in the mathematical world; the Φ function is a linear combination of sin() and cos() (this azimuthal angle, ϕ, goes from 0 to 2π and cycles, so this isn't terribly surprising) with n being an integer, and the Z function is a linear combination of cosh(kz) and sinh(kz), which are the hyperbolic functions.  These equations are not what I want to focus on; what we've really been working so hard to get is the radial equation:
 

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This is Bessel's differential equation. The solutions to this equation are transcendental (meaning that you can't write them as a finite sum of polynomials; the sine and cosine functions are also transcendental). We write them as
 

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The Jn are finite at the origin (J0 is 1 at the origin, all other Jn are 0), and the Yn are singular (undefined) at the origin. They look something like this:
 

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The Jn are much more common to work with because they don't have infinities going on, but the Yn are used when the origin is inaccessible (like a drum head that has a hole cut in the middle). These harmonic functions are used to model (but are not limited to)
  • Vibrational resonances of a circular drum head
  • Radial wave functions for potentials with cylindrical symmetry in quantum mechanics
  • Heat conduction in a cylindrical object
  • Light traveling in a cylindrical waveguide
Note that, while they kinda look sinusoidal, they don't have a set period, so the places where they cross the x-axis are have different intervals and are irrational; thus, they must be computed. This results in some weird harmonic series for instruments like xylophones, drums, timpani, and so on. I got into them because I'm a trumpet player, and the resonances of the surface of the bell of a trumpet are related to the Bessel functions.

There are some cool videos (this one has a strobe effect during it) showing them in action.  There are also some cool Mathematica Demonstrations related to them as well.  There are also orthogonality relationships with them, but I'll save that for another day.

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Happy birthday, little bro

Posted by Akano Toa of Electricity , in Life Feb 14 2014 · 87 views
Birthday, Siblings, Aww, Shut up
So, today is Tekulo's birthday. Since his birthday also happens to be Valentine's Day, and his job is baking, he's quite busy making pastries and other flour and sugar-based products for happy couples and families. And for himself, by the looks of it.

Happy birthday, Tekky. I hope you have some part of the weekend off to make up for your busy day today.

P.S. Go over here to wish Tekulo a happy birthday yourself! 8D

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Snow Days

Posted by Akano Toa of Electricity , in Life Feb 05 2014 · 82 views

So, today was the second day this week where classes have been cancelled for snow/winter weather. Today the cancellation occurred due to power outages and falling frozen tree branches.

So, I'm now at home doing work since there's actually power here. So that's been fun.

Outside looks especially pretty, though.

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I made some things

Posted by Akano Toa of Electricity , in Math/Physics Jan 31 2014 · 108 views

I made these two images in Mathematica and tidied them up in Photoshop.

They're graphs in the complex plane. The color indicates the phase, or argument, of the complex number, and for this function, curves of equal phase are hyperbolas. To animate it, all I did was let the phase vary linearly in time.

8D

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Select Quotes From a Brilliant Man

Posted by Akano Toa of Electricity , in Math/Physics Jan 29 2014 · 136 views
David Griffiths, quotes
I have posted before about the genius of physicist David J. Griffiths. I thought I'd post a few quotes by him to share why I think he's awesome.

"…You can always tell the particles apart, in principle—just paint one of them red and the other one blue, or stamp identification numbers on them, or hire private detectives to follow them around."

"...And, of course, if you’re in a really bad mood you can create a state for which neither position nor momentum is well defined..."

"It is traditional to write the Bohr radius with a subscript: a0. But this is cumbersome and unnecessary, so I prefer to leave the subscript off."

"If you think this is starting to sound like a mystical numerology, I don’t blame you. We will not be using Clebsch-Gordan tables much in the rest of the book, but I wanted you to know where they fit into the scheme of things, in case you encounter them later on. In a mathematical sense this is all applied group theory—what we are talking about is the decomposition of the direct product of two irreducible representations of the rotation group into a direct sum of irreducible representation (you can quote that, to impress your friends)."

"I’m not at all sure what I’m supposed to say today. Maybe you’re expecting a grand philosophy of education. But I learned very early as a parent that almost any philosophy of childrearing is worse than no philosophy at all, and I am inclined to think the same applies to teaching."

"Personally, I never bring notes to a lecture unless I am egregiously ill-prepared, for they break a very delicate and important bond of trust with the listener: If B really follows from A, how come he has to refer to his notes?"

"There are a thousand ways to get a problem wrong—not all of them bad—and many ways to get a problem right—not all of them good."

"Above all, I think studying science—and especially physics—is a tremendously liberating experience. I don’t happen to know how a carburetor works; I’m not even sure what a carburetor does; let me be frank: I don’t know what a carburetor looks like. But I do know that the behavior of carburetors is perfectly rational; somebody understands them, and if I really wanted to I’m sure I could understand them too. For I have confidence, grounded in the study of physics, that the world is rationally intelligible, and this, to me, is the most important—and most profoundly liberating—idea in human experience. The universe is comprehensible..."

"A colleague of mine in Chemistry likes to boast that ‘‘anyone can teach; the important thing is to attract good researchers.’’ I think it’s exactly the reverse: competent research physicists are a dime a dozen, but good teachers are few and far between. Please don’t misunderstand: I’ve got nothing against research—I do a certain amount of it myself, and I think it goes hand in hand with good teaching. But I regard myself as a professional teacher, and an amateur researcher, whereas most physicists are professional researchers but amateur teachers, and it shows. In my opinion by far the most effective thing we can do to improve the quality of physics instruction—much more important than modifications in teaching technique—is to hire, honor, and promote good teachers."

There are many more wonderful quotes, but I don't remember them/don't have the sources on me. Perhaps I'll add to this in another blog entry.

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Physics Jokes!

Posted by Akano Toa of Electricity , in Math/Physics Jan 20 2014 · 207 views

In no particular order, some fun physics jokes. Ready? GO!
  • My friend was trying to talk to me about atoms, but I got Bohr'd.
  • Did you hear that Albert Einstein developed a theory about space? It was about time, too.
  • Never trust an atom; they make up everything.
  • The oddly pleasant feeling of looking down on a physicist as they finish the last of their drink. The strange charm of a top-down bottoms-up.
  • Why does hamburger have less energy than steak? It's in the ground state.
  • Why are physics books always unhappy? Because they're full of problems.
  • Neutrinos make the worst friends; they rarely interact with anyone.
  • In a quantum finish!
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Allons-y!

Posted by Akano Toa of Electricity , Jan 13 2014 · 146 views
Doctor Who, Geronimo
So, I may have just finished watching all six series of Doctor Who available on Netflix.

And I'm quite glad I did. Awesome show. David Tennant is best pony Doctor.

I want a sonic screwdriver.

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Life Update: Winter 2014

Posted by Akano Toa of Electricity , in Life Jan 03 2014 · 71 views

When I visit home I tend to not do anything online. So, here's a basic rundown of my winter holidays:
  • KK, Tekulo, and I are in the same house again. This changes tomorrow, as KK and I are going to a friend's house, then KK returns to school.
  • Had a New Year's Eve celebration with friends from college. Had a blast.
  • My Christmas gifts include Super Mario Bros. 3D World, the Winter Market LEGO set (which I may trade with KK for his Winter Village Cottage, which I like a lot), a Fluttershy (my third so far), the mini-VW camper LEGO set, and a ceramic Snoopy sitting on his doghouse. All in all, a good haul.
I hope everyone has had a good winter holiday season. May 2014 bring you more joy and happiness. :)

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Equation of the Day # 7: The Virial Theorem

Posted by Akano Toa of Electricity , in Math/Physics Dec 10 2013 · 143 views
Equation, Virial
This is one that I didn't really know much about until recently, so I thought I'd share it. :) Today's equation is known as the Virial theorem,
 

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or, in component form,
 

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The word "virial" comes from the Latin vis, which means "force" or "energy," and looking at the equation, it makes sense why it's called that. Here the big Σ means sum, the "k" index denotes the kth particle of a system of N particles, V is the potential energy function affecting the kth particle, T is the potential energy of all the particles in the system, and rk is the position of the kth particle. This essentially relates the kinetic energy of all the particles to the positions and forces exerted on each particle (since -grad V is the force when energy is conserved, which is an assumption we are making). The brackets 〈 〉 denote that we're taking an average, so 〈T〉 is the average kinetic energy, etc.

Now, you may be thinking, "okay, that's a cute equation, I guess, but I don't see how it's particularly useful." Okay, here's where the usefulness comes in. Let's say I want to know the mass of some distant galaxy, but I don't have a good galaxy-weighing device on hand. We know that the gravitational potential energy of an object is given by

 

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where m is the mass of the star, M is the mass of the center of the galaxy, and r is the distance from the center of the galaxy.  Taking the distance r and multiplying by the gradient of the potential yields...the potential again, with a negative sign out front. So, for gravity,
 

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Plugging this into the Virial theorem above and noting that 2T = mv^2 (where v is speed), we get that, for an object in the gravitational pull of an object of mass M,
 

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Thus, we have at our disposal a way of measuring the mass of something like a galaxy by measuring only the speeds of stars and their distance away from the center. That's pretty incredible.

This actually is one of the ways scientists support the idea that there is dark matter in the universe; the Virial theorem gives an average of what speeds the stars in our galaxy should have based on their distance away from the center of the Milky Way, but what we actually observe is startlingly different. Thus, we can conclude that something is wrong with our knowledge of how gravity within a galaxy works. Based on this and other observations, the idea that there's extra stuff that can't be seen that adds to the gravitational force of a galaxy seems to be a reasonable idea.

In my research on diatomic hydrogen (H2), the Virial theorem is used in a different capacity. When figuring out the potential energy of an electron (or two) around the two positively charged protons, the virial has the Coulomb force term (which is just -V, just like gravity) and an additional term that pops up from assuming that the electrons are keeping the protons at equilibrium. I won't go too much into the physics, but the final product is

 

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where E, T, and V are the total energy, kinetic energy, and potential energy of the electron(s), respectively, and R is the distance between the nuclei. This tells us something useful about the energy of the electrons; more specifically, it tells us about how the energy changes as you move the nuclei farther apart or closer together. In other words, since E = T + V,
 

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which is very useful when constructing potential energy curves for hydrogen.

On a slightly related note, our lab's paper got published! Akano is now a for reals, published scientist! 8D

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I Caught Some Shinies!

Posted by Akano Toa of Electricity , in Pokémon Dec 01 2013 · 146 views

Over the past week I caught a shiny Remoraid (evolved it and gave it to Tekulo), shiny Octillery, and shiny Clauncher (traded from Tekulo's X version).

I have become a consecutive fishing master! I want to get a shiny Corsola next.

Also, I had a very enjoyable Thanksgiving with my family and our friends. Tomorrow I head back to my apartment, and then Monday is school as usual.

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About Me

Akano Toa of Electricity
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Name: Akano
Real Name: Forever Shrouded in Mystery :P
Age: 25
Gender: Male
Likes: Science, Math, LEGO, Bionicle, Ponies, Comics, Yellow, Voice Acting
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Misses the 90's. A lot.
Twitter: @akanotoe

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