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? $$ f“(t) + a f'(t) + \omega^2 f(t) = 0 $$

• does it matter if $a$ is positive or negative?
• does it matter if $a$ is small or large?

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,

• sequence of period 9: 0 0 1 0 0 2 0 0 3

Use discrete Fourier transformation to analyse it.

• You can use this input number to as values of $f(x)$ for $x=0,1, \cdots, 8$.
• you know $f(x)$ can be written as

$$ f(x) = \sum_{p=0}^8 F(p) e^{2\pi i (xp/9)} $$ Can you find $F(p)$?

• These $f(x)$ are all real valued, what does that say about $F(p)$? (try to take complex conjugate of the equation)
• Can you guess, which $|F(p)|$s are largest?

If you have done the above exercise, try this one

• sequence of period 10: 2 5 3 8 2 7 1 9 0 4