So far, we have learned many transformations and inverse transformation. Conceptually, we are just trying to decompose a given function $f(x)$ as a linear combination of $e^{ax}$ for various $a$. Because it is an eigenfunction of $(d/dx)$: $$ (d/dx) e^{ax} = a e^{ax}$$
Decomposition of a standing wave with Dirichlet condition. When we pluck a string on a violin, the string vibrate with the endpoints fixed.
Suppose the string's length is $L$, and you took a snapshot of the string and obtained the vertical displacement as a function $f: [0,L] \to \R$ (yes, it is $\R$ valued, not $\C$ valued).
Design a way to do Fourier decomposition of $f$.
If you like, pick $L = 2$ and $$ f(x) = 1 - |x-1|. $$ (just a linear peak).
Damped oscillation, reverse engineering.
Suppose you observe some damped oscillation that goes like $$ f(t) = A e^{-a t} sin(b t), \quad t > 0 $$ Can you find the second order equation that $f(t)$ satisfies? Namely, for what constant $A,B$ does $f(t)$ satisfies the equation?
$$ f“(t) + A f'(t) + B f(t) = 0. $$
If you like, pick $a=1, b= 2$.
Tuning the friction.
Consider the pure oscillation equation: $$ f”(t) + \omega^2 f(t) = 0 $$ it has solution like $f(t) = a \sin(\omega t) + b \cos(\omega t). $
What happens if we add some friction term? $$ f“(t) + a f'(t) + \omega^2 f(t) = 0 $$
Make a guess first. Then solve it explicitly, for $\omega = 1$, and try various $a$.
Hint: Is there a 'planewave' that solves the equation, i.e. somethign like $e^{ct}$ for some $c$?
You received the following a periodic sequence of numbers,
Use discrete Fourier transformation to analyse it.
$$ f(x) = \sum_{p=0}^8 F(p) e^{2\pi i (xp/9)} $$ Can you find $F(p)$?
If you have done the above exercise, try this one