Dynamics of mechanical and electrical oscillations using Fourier series and integrals, time and frequency reprentations for driven damped oscillator, resonance, vector spaces.

PREREQUISITE: Introductory Classical Physics with Calculus, PH 211-213 or equivalent calculus-based introductory course.

GOALS: This course is intended to give students both conceptual and practical tools for understanding the dynamics of systems close to equilibrium from different viewpoints.

The central concept is the duality of time and frequency representations; the corresponding tool is the Fourier transform. In a mathematical introduction with stylized exercises, the students will see examples of series of functions converging toward a limit. They will experience their mastery of the concept and technique of Fourier transformation by predicting the time response of a simple electronic circuit from its measured frequency response, then comparing their prediction with observations.

Another important concept is the use of truncated series as successive approximations. Introduced with the example of Taylor power series, the concept will be developed and reinforced with a case study of Fourier harmonic series applied to anharmonic oscillations.

A third important concept is the use of superposition to represent the state of a physical system as a point in a vector space. This higher-level abstraction generalizes the time/frequency duality by viewing it as an example of a change of basis in a vector space of functions. This concept will be introduced in the context of the one-dimensional anharmonic oscillator. By the end of the course, the students should perceive the complementarity of time and frequency measurements as alternate projections onto different bases of an infinite-dimensional vector space.

The students will become familiar with two different, simple systems. In these examples they will learn to connect the mathematical descriptions with physically realized examples. The example of a rigid pendulum is a time-honored illustration of many important principles; and is capable of chaotic behavior when excited beyond the oscillitory regime studied here. The example of response functions in an electrical circuit has its own importance as a widely used application of the central concept in linear circuit theory.

The practical skills to be learned include not only mathematical techniques, but also problem-solving methodology which forms the foundation for a physicist's approach to the investigation of a physical phenomenon.

Mathematical techniques include skills at computing and inverting Fourier series and integrals, but also the use of complex exponentials and their connection to the physical quantities describing the systems.

Problem-solving methodology includes learning to record and document observations and calculations, to investigate the realm of validity of simple models and approximations, to form hypotheses explaining unfamiliar observations, and to perform and interpret simple qualitative and quantitative tests of hypotheses. These techniques will be exercised both in the laboratory and on the computer. Equally important will be the skills of working with other scientists: oral and written communication, cooperating on complex activities, organizing a division of tasks when appropriate, gleaning knowledge from authoritative sources such as experts and books, and translating the notation and language to apply to the problem at hand.

TEXTBOOKS:

J.B.Marion and S.T. Thornton, *Classical Dynamics of Particles and
Systems*, 4th edition (Saunders, Fort
Worth, 1995). Required.

P. J. Siemens, *Notes on Oscillations*,
Course notes for sale in OSU Bookstore. Required.

K.F. Riley, M.P. Hobson, and S.J. Bence,
*Mathematical Methods for Physics and
Engineering*,* *(Cambridge University
Press, 1998). Recommended supplement to class notes.

I. G. Main, *Vibrations and Waves in Physics*, 3rd edition, (Cambridge University Press, 1993).
Recommended supplement to class notes.

D. V. Bugg, *Electronics: Circuits, Amplifiers and
Gates*, (Institute of Physics
Publishing, Bristol, 1991). Recommended supplement. Same text as PH
411.

R. A. Serway and R. J. Beichner,
*Physics for Scientists
andEngineers*, 5th edition (Saunders,
Fort Worth, 2000). Not required. Another calculus-based introductory
book will do fine.

GRADES: Based on homework (30% = 3 x 10%), a portfolio of the two laboratory projects (60% = 2 x 30%), and a final exam (10%). A complete portfolio will include records of data and computations, as well as brief essays giving summaries and conclusions for each of the experimental systems studied. Laboratory and computer exercises will be performed in teams, but each student will be responsible for his/her own report. Class assignments will be collected on Wednesdays or Fridays. The portfolio will be assessed using an Inventory of Achievement.

ADD/DROPS AND WITHDRAWALS: Special add/drop and withdrawal procedures are in place for the Paradigms, because of their unusual structure. The last day to add/drop a Paradigm is the third day it meets (not counting the Preface). The last day to withdraw is at the end of the second week it meets (not counting the Preface). See the instructor if you are having trouble getting this accomplished at the Registrar's Office.

STUDENTS WITH DISABILITIES: Students with documented disabilities who may need special accommodations in the lecture, lab, or examinations should make an appointment with the instructor no later than the first week of classes to discuss those accommodations.