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Physics 103

Fall 2021

Spherical Harmonics Lab Project

We’ve studied standing waves on strings in detail, and we’ve investigated standing waves in two and three dimensions. In this lab project you will be investigating three-dimensional standing waves in more detail. In particular, you will be studying the solutions of three-dimensional waves on a round ball (like a ball of Jell-O!). In the case of a cube of Jell-O the solutions are simple sines and cosines, as we’ve seen. However, in the spherical case the solutions will involve new functions called Spherical Harmonic functions, and are similar to sines and cosines, but with some important differences.

For this lab project you should do the following:

• Discuss the 3D wave equation in spherical coordinates, and then reduce it to a 2D wave equation by taking the case that the radius of the ball is fixed, such that all the radial derivatives vanish.

• Solve the 2D wave equation on the surface of the ball, subject the boundary conditions that the solutions are symmetric when the polar angle, θ (as measured from the z-axis), and the azimuthal angle, φ (lying in the xy-plane), advance by a factor of 2π. This leads to the Spherical Harmonic functions (obviously, you should look up this derivation and reproduce it).

• Discuss some of the solutions (list some) - compare and contrast them with sines and cosines in the rectangular case.

• Discuss what we mean by multipole moments.

• Find pictures (or better yet - make plots!) of some of the standing wave patterns (the multipole moments) on the ball. Be sure that the standing wave patterns look how you expect.

• Comment on some applications of Spherical Harmonics in physics, engineering, or mathematics.

This problem has, of course, been solved many times, and the solutions are online - you’re welcome to use those as references. Learn from that material, but write things for yourself! Come up with your own way of thinking about things, and don’t just copy their solutions. The point of this project is to get you used to doing a little research on something, and learning to write it up in a professional way. You’ll be graded on the effort that you put into the writeup, and the quality of the writeup, itself (I assume that you actually get the correct solutions!). Some of the things that I am looking for:

• Did you follow the correct format for the formal lab write up - is it in LATEX? Are all the group members listed (alphabetically)? Did you write in complete sentences, with good grammar? Did you write things in paragraphs, instead of making lists, and so on? In other words, does the write up look professional - would it be something that you would proudly submit for publication in a real journal?

• Did you actually discuss things, instead of just showing a page of calculations? Imagine that you’re giving a lecture on this stuff, and type that up! Look at real papers, and how much writing there is, compared to the amount of math!

• Did you only write things that you can understand, and write at a level at which any classmate could understand? It’s tempting to include scary-looking equations in your paper because they look impressive - and you can! However, if you’re including these scary equations, be sure that you understand them, and can explain them!

• When including equations and formulae, did you define your variables? Don’t display equations without telling the reader what the variables are! Also, are the important equations numbered? Do the equations look nice on the page (as they would look in a textbook, or when you’re writing them down), or are they messily just stuck there (i.e., do you write 2/1at2, or 1/2 at∧2 - see how much better one looks than the other?)?

• Did you cite your sources, and list them in the references section? Were you sure to not simply copy and paste stuff from the internet (that’s plagiarism, and gets you kicked out of science!)?

• Look at real papers to see how it’s done!

This list of stuff isn’t just for this write-up, of course - it applies to all the labs that we’re performing, as well as to your research paper! The whole point of all this is to learn to write technical documents, which is much more important to your future careers than any particular lab that we’re going to do!