$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} \newcommand{\intd}[1]{\int\hspace{-0.7em}\int\limits_{\hspace{-0.7em}{#1}}} $

Lecture 31, 11/11/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

Chapter 7: Fourier Series and Transform

  1. Simple Harmonic Motion:
    $\m y = A \sin\theta = A \sin \omega t \mm (\theta = \omega t)$; $\mm y = A \cos \omega t; \mm y = A( \cos \omega t+i \sin \omega t) = A e^{i \omega t}$.
    $A =$ amplitude; $\m$ velocity amplitude $ = A \omega; \m$ period $ T = \frac{2\pi}\omega; \m $ frequency $ f = \frac\omega{2\pi}$;
  2. Application: typical fourier series:
    (a) Square wave; $\m$ (b) Sawtooth wave; $\m$ (c) Rectified half-wave (by diode);
    Try this: 
     
       % Velocity = 2,  time = 0.03 second per frame, Sine wave
       x = 0:.03:10;  for i=1:.1:10  plot(x, sin(x - 2* i)), pause(0.03), end  
       % Square Wave
       for i=0:.03:10  plot(x, sign(sin(x - 2* i))), ylim([-2 2]), pause(0.03), end