I am a retired mathematician living in San Diego, California. My wife and I have two daughters and five grandchildren. I obtained a PhD in mathematics from Carnegie-Mellon University in 1968 with an emphasis in Continuum Mechanics. I worked at SPAWAR (Space and Naval Warfare) center from 1959 to 2003. My principal job was the mathematical modeling of Sonar transducers and arrays. This involved, among other things, familiarity with elasticity theory, electrodynamics, acoustics, and numerical analysis. I was a co-developer of the widely distributed computer program CHIEF that numerically solves the Helmholtz integral equation to predict the acoustic radiation or scattering from arbitrary shaped bodies. As retirees, my wife and I spend as much time as possible with our grand kids, work part-time at our church, and attend a weekly Bible study. I have also been writing some tutorials on subjects that are of interest to me. Links to some of these are listed below. Additional papers on acoustics, the days of creation in Genesis, and the life and faith of Leonhard Euler can be reached through the navigation panel. I hope these tutorials and papers will be of interest and of use to you.

• Harmonic Analysis (pdf)

  Tutorial discussing some of the numerical aspects of practical harmonic analysis. Topics include Historical Background, Fourier Series and Integral Approximations, Convergence Improvement, Differentiation of Fourier Series and Sigma Factors, Chebyshev Polynomial Approximations, The Tau Method, Fast Fourier Transforms, and Fast Sine or Cosine Transforms.

• Numerical Linear Algebra (pdf)

  Tutorial describing many of the standard numerical methods used in Linear Algebra. Topics include Gaussian Elimination, LU and QR Factorizations, The Singular Value Decomposition, Eigenvalues and Eigenvectors via the QR Method with Shifts or the Divide-and-Conquer Method, and the Conjugate Gradient and Lanczos Iterative Methods.

• Digital Encoding (pdf)

  Tutorial describing methods for encoding digital information in an electric or optical signal. Topics include Manchester and 4B/5B Encoding, Amplitude Modulation, Frequency Modulation, Phase Modulation, Minimum Shift Keying (MSK), Quadrature Amplitude Modulation (QAM), Orthogonal Frequency Division Multiplexing (OFDM), Frequency Hopping (FHSS) and Direct Sequence Spread Spectrum (DSSS), and Code Division Multiple Access (CDMA).

• Modeling of Sonar Transducers and Arrays (pdf)

  Tutorial describing some of the mathematical tools used in the modeling of Sonar transducers and arrays. Topics include The Acoustic Wave Equation, Acoustic Array Interactions, The Helmholtz and Kirchhoff Integral Equations, Infinite Element Methods, The Wave Envelope Method, Doubly Asymptotic Methods, T-Matrix Methods, The Finite Element Method (structural, fluid, piezoelectric, and magnetostrictive ), Simple Variational Approximations, and Structure-Acoustic Coupling. This tutorial contains links to many reference documents. Among these is the CHIEF 2004 User Manual.

• Symmetry Reductions (pdf)

  The numerical solution of many physical problems can be reduced to the solution of a system of linear algebraic equations. This paper shows how to use geometric symmetry in order to reduce the solution of the original system of equations to the solution of a number of smaller systems of equations. The reduction technique employs the spectral decomposition of an appropriate symmetry operator. The use of this technique can significantly reduce computation time.

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Modeling of Sonar Transducers and Arrays File: Modeling.zip (104 MB)

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