My proposed project will address musical instrument modeling, namely for the instruments I play, the harpsichord and organ. Computational Physics, by Nicholas J. Giordano and Hisao Nakanishi, addresses the piano, among other instruments, in chapter 11, which will serve as a starting point for modeling other musical instruments that have their own unique physical characteristics. For example, the harpsichord plucks, rather than strikes, its strings, and modeling a more accurate plucking force along with other features of the instrument would form part of the project. Another example is the variety of sounds produced by the pipe organ due to different pipe materials and shapes. I intend to model at least one pipe type (more if the harpsichord part of the project proves too simple). In all cases, I intend to compare the “bridge force spectrum” (basically, the force waves of certain frequencies exert at the boundary) that I calculate to an actual signal from the instrument itself, which I will record and analyze. This will verify, or nullify, the models that I will build.
- Introduction to Electrodynamics, 4th ed. by David J. Griffiths
- Physics of the Galaxy and Interstellar Matter by H. Scheffler and H. Elsässer
- Interstellar Grains by N.C. Wickramasinghe
- Physical Processes in the Interstellar Medium by Lyman Spitzer, Jr.
- The scattering of light, and other electromagnetic radiation by Milton Kerker
- Mathematica 9 Help Documentation Center (the most valuable resource!)
I will be exploring Mie theory and modeling this type of scattering for particles present in the interstellar medium (ISM). There are a number of parameters that can be easily derived from applying Mie theory to the ISM, including an asymmetry parameter (g), extinction coefficients, albedo, and phase functions. I will focus first on modeling the asymmetry parameter (g), but time permitting, will continue onto other properties as I move into Week 3 – as stated in my timeline (below).
While the space between stars and galaxies appears quite vast and barren given only the access of our eyes fixed quaintly at ground level, the ISM is teeming with a variety of matter and electromagnetic waves. These regions are rich with gas (atomic and molecular), dust, and are permeated by electromagnetic waves, or radiation, from starlight (and occasionally other sources). Observations of various astrophysical phenomena show that along a given line of sight, their is “extinction” of this radiation. Scattering and absorption account for these observations and occur due to the presence of various dust grains within the ISM. This is where Mie scattering fits into modeling the asymmetry parameter (g), a measure of the fraction of light scattered in the forward direction. Our efforts towards modeling this relationship as well as the values of other dust grain properties such as size, composition, etc, begins with a preliminary comprehension of Mie’s work.
In 1908, Mie was working well before the substantially assistive mechanisms of modern computational modeling. Although his successors refined the theory of scattering (for spherical particles) over subsequent years, Mie’s initial work is fitting for the relatively fundamental level of analysis in this course. As Kerker explains (5), the basic scattering functions can be derived from a process whereby the proper Ricati-Bessel function is chosen and scattering coefficients are derived:
where m indicates an index of refraction, and (and their respective primes) are the Ricati-Bessel functions chosen, and and are constants given by wave parameters (k, etc). These scattering coefficients are then combined with a mathematical tool called a “Legendre polynomial” to give amplitude functions for the scattering. Other parameters can then be derived once these functions are modeled.
Week 1 (4/7 – 4/13)
I will refresh and further extend my knowledge of Mathematica and continue my research into the computational and theoretical aspects of Mie scattering in the context of the interstellar medium (ISM). As my project is computationally based, I will focus heavily on bridging the divide between my theoretical knowledge of scattering in the ISM and my work in Mathematica.
Week 2 (4/14 – 4/20)
For Week 2 I plan on completing a working Mathematica model for the asymmetry parameter, g, and complimenting this work as needed with research into Bessel and Ricati-Bessel functions. Mie theory is founded in the rigors of these functions and gives solutions to Maxwell’s equations that illustrates the scattering of waves by spherical particles. Specifically useful with the Bessel and Ricati-Bessel functions are the eponymous differential equations. If this work is completed I will move on to Week 3 work.
Week 3 (4/21 – 4/27)
I will continue modeling. My initial research into the theory will be added to as needed. As time permits, I will continue on to explore the utility of Mie theory and scattering within the ISM as it relates to the derivation of other interstellar grain properties.
Week 4 (4/28 -5/4)
I will finalize any lingering modeling. My results will be nearly all in and I will begin to wrap up my blog – focusing on visualizations (plots and animations) and ensuring that they function properly within the blog space. This week will be mainly devoted to polishing off my blog posts (data and results especially), i.e. making all aspects of the blog look aesthetically pleasing.
Week 5 (5/5 – 5/11)
This week concludes my project as I finish commenting on my peer’s blog posts.
While I plan on working alone, Professor Magnes will be assisting and advising as this project progresses. Both of my predecessors focused on the theory and modeling of Rayleigh scattering – a related phenomena but still outside of the purview of my project. I intend to move in a direction much different than my predecessors who explored scattering of the Rayleigh domain. I do, however, anticipate that as my project develops, I will be reflecting on my results and the possibility exists to compare and contrast them in the context of the work done by these individuals. I cannot predict what these comparisons may entail, but as I conclude my project, these ties may reflect elements shared between background theory, computational processes, and my results.
This project is based on an initial interest in the RLC circuit lab in 114 where students put together parts of a radio and demonstrated how inductance and capacitance was used to tune a radio. My end goal is to compare measurements across components of a real RLC circuit (radio) to the ideal values one might expect to find based on equations from our textbook that model these circuits. I would begin with a simple LC circuit, maybe constructed out of components that would later be found in the radio circuit (or whatever components are available to make the simplest LC circuit I can make measurements on). I would collect data such as potential across various components. An important part of the analysis would be proposing possible/probable causes of any discrepancy between expected values and experimental values. I would use an oscilloscope to make the measurements. Ideally I would use a computer oscilloscope and a laptop and set up in room 203A.
Earth’s Van Allen Radiation Belts are areas of concentrated charged particles surrounding the planet due in part to its naturally generated magnetic field. These particles come from the Sun’s solar wind and are divided into two layers whose borders are the magnetic field lines. While these radiation belts are not unique to Earth, they are of utmost importance to us when considering safe space travel and the placement of satellites. The goal of my project is to use Mathematica and publicly supplied data to model the Earth’s magnetic field. Next, that model will be used to define the location of Earth’s Van Allen Belts. These experimental values will then be compared to the observed values of the belts in order to determine the accuracy of the model which can then hopefully be altered to determine the equivalent belts on other planets.
While this class has covered the electric and magnetic fields of stationary objects, it has not addressed the concept of objects in motion. In this project, I will first derive the relativistic transformation equations for electric and magnetic fields. Then, I will apply these equations to a number of scenarios that have already arisen in the stationary form, such as a parallel plate capacitor, a bar magnet, a solenoid, and a current-carrying wire. These scenarios will then be assessed at varying speeds.
When analyzing the brightness of a star, or a mode of a laser beam we observe the effects of that object. In this case we gather photons, and we use tools to gather as much radiation as possible. This data is then transmitted onto a screen to see a representation of what is actually going on. When an electric signal is sent to an oscilloscope, are we actually seeing the signal? A spectrum of light must enter through a series of optical and electrical things before being displayed, and those things can and do distort the image. These are optical filters. Sometimes this is done intentionally to block out certain frequencies, but other times the distortion is unavoidable. By understanding the convolution of electromagnetic waves one can isolate the desired data from the signal presented. I will model using Mathematica different spectra and examine how convolution and deconvolution work as means of setting up usable data.
For my project, I will work with Peter Florio to model the electric and magnetic fields of a bar magnet, sphere and cylinder. I will be looking at the magnetic fields of these distributions in 3-space and modeling their vector fields using Mathematica and Maxwell’s Equations. Problems given in David Griffiths’ Introduction to Electrodynamics will be used as specific examples of these kinds of charge distributions. My results will then be compared to the electric fields of the same distributions found by Peter.
For my project, I will be studying Mie scattering and it’s relevance to the study of the interstellar medium. The ISM, the space between stars and galaxies, is filled with gas (atomic and molecular), dust, and is permeated by radiation – starlight. Observations of various astrophysical phenomena show that along a given line of sight, their is “extinction” of this radiation. Scattering and absorption account for these observations and occur due to the presence of various dust grains within the ISM. I will model Mie scattering and look at the asymmetry parameter g, a measure of the fraction of light scattered in the forward direction, in an effort to model the relationship between this value and dust grain properties such as size, composition, etc.
The C. elegans nematode is a common subject of biological studies, and has become more and more popular in physics research. I intend to find the diffraction patterns generated by the worms’ shape and log the findings into a Symmetries Library (with the eventual goal of using Group Theory to get the worm shape directly from a diffraction image).
The shape of the worm (photos to be taken with a microscope) will correspond to a particular diffraction pattern. I will model the Fraunhoufer diffraction patterns (Far-Field diffraction) of the electromagnetic waves (light waves) by generating images with Mathematica using the Fourier Transforms. The idea is that $\left | Fourier Transform | \right ^2 $ = the diffraction pattern. This project is a study of the behaviors of light waves.
I will eventually be keeping a log of my findings on the already existing website, the Diffraction Symmetries Library.
For my project, I will study RLC circuits. RLC circuits have three main components; resistors, inductors, and capacitors. These circuits are important because of their prevalence in radios and televison (among other things). Specifically, I will develop a few different configurations of these circuits and measure characteristics such as voltage, current, resonance, and most importantly, damping. The physics that governs RLC circuits relies heavily on the mathematics of differential equations. For this reason, Mathematica, due to its strong capabilities in regards to differential equations, is the ideal program to use to model and study these systems.