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3190 results in Mathematical Methods

Page 49 of 160

Spectral Analysis for Univariate Time Series
  • Donald B. Percival, Andrew T. Walden
  • Published online:
    17 March 2020
    Print publication:
    19 March 2020
  • Spectral analysis is widely used to interpret time series collected in diverse areas. This book covers the statistical theory behind spectral analysis and provides data analysts with the tools needed to transition theory into practice. Actual time series from oceanography, metrology, atmospheric science and other areas are used in running examples throughout, to allow clear comparison of how the various methods address questions of interest. All major nonparametric and parametric spectral analysis techniques are discussed, with emphasis on the multitaper method, both in its original formulation involving Slepian tapers and in a popular alternative using sinusoidal tapers. The authors take a unified approach to quantifying the bandwidth of different nonparametric spectral estimates. An extensive set of exercises allows readers to test their understanding of theory and practical analysis. The time series used as examples and R language code for recreating the analyses of the series are available from the book's website.

  • 2 - Damped Harmonic Oscillator

  • Joel Franklin, Reed College, Oregon
  • Book:
    Mathematical Methods for Oscillations and Waves
    Published online:
    02 April 2020
    Print publication:
    05 March 2020, pp 31-64

  • Summary

    Here, we add damping to the harmonic oscillator, and explore the role of the resulting new time scale in the solutions to the equations of motion.Specifically, the ratio of damping to oscillatory time scale can be used to identify very different regimes of motion: under-, critically-, and over-damped.Then driving forces are added, we consider the effect those have on the different flavors of forcing already in place.The main physical example (beyond springs attached to masses in dashpots) is electrical, sinusoidally driven RLC circuits provide a nice, experimentally accessible test case.On the mathematical side, the chapter serves as a thinly-veiled introduction to Fourier series and the Fourier transform.

  • 4 - The Wave Equation

  • Joel Franklin, Reed College, Oregon
  • Book:
    Mathematical Methods for Oscillations and Waves
    Published online:
    02 April 2020
    Print publication:
    05 March 2020, pp 90-119

  • Summary

    Taking the continuum limit of the chains of masses from the previous chapter, we arrive at the wave equation, the physical subject of this chapter.The connection to approximate string motion is an additional motivation.Viewed as a manifestation of a conservation law, the wave equation can be extended to other conservative, but nonlinear cases, like traffic flow.Mathematically, we are interested in turning partial differential equations (PDEs) into ODEs, making contact with some familiar examples.Making PDEs into ODEs occurs in a couple of ways --- the method of characteristics, and additive/multiplicative separation of variables are the primary tools.

  • 3 - Coupled Oscillators

  • Joel Franklin, Reed College, Oregon
  • Book:
    Mathematical Methods for Oscillations and Waves
    Published online:
    02 April 2020
    Print publication:
    05 March 2020, pp 65-89

  • Summary

    We turn next to the case of additional masses.In one dimension, we can attach masses by springs to achieve collective motions that occur at a single frequency, the normal modes.Building general solutions, using superposition, from this ``basis" of solutions is physically relevant and requires a relatively formal treatment of linear algebra, the mathematical topic of the chapter.

  • 8 - Numerical Methods

  • Joel Franklin, Reed College, Oregon
  • Book:
    Mathematical Methods for Oscillations and Waves
    Published online:
    02 April 2020
    Print publication:
    05 March 2020, pp 207-233

  • Summary

    This chapter can be used as a six week “lab" component to a mathematical methods course, one section each week.The chapter is relatively self-contained, and consists of numerical methods that complement the analytic solutions found in the rest of the book.There are methods for solving ODE problems (both in initial and boundary value form) approximating integrals, and finding roots.There is also a discussion of the eigenvalue problem in the context of approximate solutions in quantum mechanics and a section on the discrete Fourier transform.

Page 49 of 160