GP 306 Linear Systems
John Scales
Office: Green Center B52
jscales@mines.edu
http://acoustics.mines.edu/~jscales

Course Description:

Beginning with simple linear systems of coupled elements (springs and masses or electrical circuits, for instance) we study linearity, superposition, damping, resonance and normal modes. As the number of elements increases we end up with the wave equation, which leads, via separation of variables, to the first signs of Fourier series. One of the unifying mathematical themes in this course is orthogonal decomposition, which we first encounter in the comfort of finite dimensional vector spaces associated with springs and masses. But the idea extends naturally to infinite dimensional spaces where it appears as a Fourier series. The course culminates in an exposition of Fourier series, integrals and transforms, both discrete and continuous. Throughout, these ideas are motivated by and applied to current geophysical problems such as normal mode seismology, acoustic wave propagation and spectral analysis of time series. In addition to the lectures, there will be classroom and laboratory demonstrations, and all students will complete a variety of computer exercises, using packages such as Mathematica and Matlab.


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Course Requirements:

There will be a quiz or computer exercise roughly every 2 weeks throughout the semester. Each of these will be weighted equally and will be averaged to form your grade. The quizzes are normally in-class and take about 50 minutes. There will be no mid-term of final exam. Click here for the course syllabus (pdf)

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The lecture notes one pdf file.

Roel Snieder's notes on Fourier series (click here for the postscript)

Click here to find out how to view postscript documents.

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News

12/04/03: Final Quiz

here is the pdf file for the final quiz. It due (by email or hardcopy in my mailbox) by the end of the day, December 11. These data are plotted on page 105 in your notes. So do not need any computer tools to solve this problems. Also, remember there is a typo in the bottom part of this plot (in the power spectrum). The numbers on the x-axis should be multiplied by two.

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Mathematica Notebooks




• Resonance in a forced, damped harmonic oscillators
• Animate the equations of motion for a couple spring/mass system. Evaluate the notebook and then click on the bracket that surrounds all the snapshots. Then you can "animate selected graphics".
• Random lattices If you let the spring constants become random, some remarkable things happen
• The plucked string. Here we use Mathematica's Fourier Series capability to compute the time variation of a string clamped at the endpoints and plucked at t=0. Simply download the notebook and then animate the snapshots after all 50 have been computed.
• The 2D drum. This notebook shows how to plot the normal modes of a rectangular drum. It also gives an animation of the motion of one of the modes. If you were to excite this mode in via the initial conditions, the animation is exactly what you would observe. For a real measurement of such a mode, see my WWW page.
• Matrices. A very short intro to Mathematica's matrix handling capability.
• Fourier series 1. Fourier series of a sawtooth function.
• Fourier series 2. Fourier series of the absolute function. Notice that in both cases the series is slowly convergent at those points where the function is not differentiable.
• lots of FFT examples
• A naive low-pass filter. This uses the a small data file containing a single ultrasonic trace.
• Sinc function interpolation

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Some Readings

Jensenius and Zocchi, "Measuring the spring constant of a single polymer chain", Physical Review Letters, December 1997, pages 5030--5033.

Kohl and Levine, "Measurement and interpretation of tidal tilts in a small array", Journal of Geophysical Research, March 10, 1995, pages 3929-3941.


Here is a short discussion of the simplest exactly solvable nonlinear generalization of the spring/mass system. If you're looking for an interesting research topic, let me know.

Isospectral drums. Can you hear the shape of a drum? On the mathematics from Peter Doyle's web page. Including some nifty Mathematica tools for making your own isospectral drums. On the physics from S. Sridhar's web page at Northeastern. Ivars Peterson's MathLand column on isospectrum drums.

Extremely cool measurement of a normal mode of a stadium-shaped plate made with the laser-Doppler vibrometer in the Physical Acoustics Laboratory. For a traditional view you can look at Chladni figures of the same plate.