# Lea Kenigsberg's REU 2015 Web Page

Name: Lea Kenigsberg lkenig (at) dimax.rutgers.edu Hill 323 Stony Brook Univesity Cymatics-Chlandi plates

Given a surface and certain boundary conditions (i.e a square metal plate clamped in the center), vibration of the surface at certain frequencies will result in regions of minimum and maximum displacement.These regions can be visualized by sprinkling a thin layer of particles such as sand or dust on the surface. Pending on the type of the particles and the acceleration of the surface these particles will accumulate in the nodal (regions of zero displacement) or antinodal (regions of maximum displacement) domains and form various patterns. This is the essence of cymatics and Chlandi plates, the topics of my project.

• To a full description of the project:
• I will be studying the motion of various surfaces and particles. Specifically, given a certain pattern of nodal domains, I will attemps to find the surface, boundary conditions, frequencies and type of particles that will generate the pattern.

## Weekly summary

### Week 1

This week I looked into diffenet ways of modeling the motion of the surfaces and the particles. I have read about Kirchoff plate tehory and Mindlin plate theory.

### Week 2

I looked into different MATLAB programs that model solutions to Laplace equation and simulate nodal regions and I started writing one for a very simple case.

### Week 3

I read about the finite difference method and the finite element method; I familiarized myself with Sobolev spaces and such. Also, I read MATLAB scripts for programs that are relevant to my project in an attempt to learn the language.

### Week 4

I wrote a simple MATLAB program that simulates the motion of a string. It took a long time, but it somehow works. I need to figure out how to change this program so that it is a better description of a physical system. Also, I will edit it so that it can be applied to 2-D surfaces.

### Week 5

I read some more on aprroximation methods in solutions to PDE's.

### Week 6

I wrote a simple MATLAB program that simulates the motion of a vibrating memebrane with fixed boundary conditions!! it looks wonderful.

### Week 7

I extended the program to a vibrating memebrane with free boundary conditions. The method employs the finite difference method to propogate a sinosoidal forcing function that is applied at a single point. We can clearly observe nodal sets.

### Week 8

I found out that given any separating closed hypersurface S in a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is S. I spent time reading on Riemannian metrics so that I can understand these results.

### Week 9

I discovered that the answer to the problem I am investigating is not necessarily. In fact many patterns of nodal domains can not be realized by any surface given any arbitrary boundary conditions. I was looking into a MATLAB toolpack that maps conformally the unit disk to an arbitrary proper subset of R^2, I will use this to track changes to nodal sets.