## Introduction to Fourier analysis ### Simple harmonic motion Simple harmonic motion could be characterized by the [[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=20&selection=57,0,69,1|differential equation]] of the form $u''(t)+c^{2}u(t)=0$ and the solution is given by $u(t)=a \cos ct + b \sin ct$ One may notice that the coefficient in the ODE, $c$ gets into the frequency part of the solution. So if the solution must follows some type of periodicity, then there could be some restriction on $c$. e.g., for $u(t)$ with the period being $2\pi$, we may say this $c$ must be an integer. (Since $\frac{2\pi}{c}$ is the period) ### Standing and traveling waves For a function corresponds to wave motions, we may use two approaches to express it, a travelling wave equation, or superposition of standing waves. - Standing waves should be separatable, i.e., $u(x, t)=\phi(x)\psi(t)$. - Travelling waves are non-separatable, with the form $u(x,t)=F(x-ct)$, its *position* travels with time. ### Solving wave equations by superposition of standing waves The wave equation (1D) has [[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=25&selection=127,0,142,1|the form]] $\frac{\partial^{2}U}{\partial T^{2}} = \frac{\partial^{2}U}{\partial X^{2}}$ and if write $U$ as the standing waves, then we have $u(x, t)=\phi(x)\psi(t)$, so $\phi(x)\psi''(t)=\phi''(x)\psi(t)$ $\frac{\psi''(t)}{\psi(t)}=\frac{\phi''(x)}{\phi(x)}$ Notice that we separated $x$ and $t$ at left and right side, so both could only be a constant, say $\lambda$, then $\begin{cases} \psi''(t)-\lambda \psi(t) = 0 \\ \phi''(x)-\lambda \phi(x) = 0 \end{cases}$ Notice this is the form of [[Introduction to Fourier analysis#[Simple harmonic motion](https //en.wikipedia.org/wiki/Simple_harmonic_motion)|simple harmonic motion]], so we may let $\lambda = -m^{2}$ to make the equation reasonable. And we may get the [[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=29&selection=173,2,214,3|solutions]], with BCs we may ensure $m$ being only integers, so $u_{m}$ may have the expression as $u_{m}(x,t)=(A_{m}\cos mt + B_{m} \sin mt) \sin mx$ >[!Note] >If here set $m=2$, we may have the **[[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=30&selection=180,1,181,15|second harmonic]]**, or first overtone, as $u(x,t) = \cos 2t \sin 2x$. This frequency doubling is the origin of the name of SHG. >For $m=1$, it is called fundamental tone or first harmonic of the vibrating string. Since the wave equation is linear, the superposition of $u_{m}$ could give the solution to the equation, here we neglect the problem of convergence first and have that discussed later, we should have $u(x,t)=\sum_{m=1}^{\infty} (A_{m} \cos mt + B_{m}\sin mt)\sin mt$ If we set $t=0$, and let $f(x)=u(x,0)$, then we get $\sum_{m=1}^{\infty} A_{m} \sin mt=f(x)$ This is our initial condition (initial value) for the wave function, which should be preassigned. The problem is then, **Given a (any) function $f$ on $[0,\pi]$ and BCs, is it possible to find coefficients $A_{m}$ satisfy above expression?** The precise answer could only be given in later chapters, but here we may have the expression for $A_{n}$ by multiply both sides with $\sin nx$ and integrate with respect to $x$. Here $n$ is an integer (selected from the same range of $m$) and called the $n^{th}$ Fourier coefficient, $A_{n} = \frac{2}{\pi} \int_{0}^{\pi}f(x)\sin nx \ dx$ One may also write above expression [[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=32&selection=315,2,338,1|in complex form]] using Euler formula, i.e., for $F(x)$ $F(x)=\sum_{m=-\infty}^{\infty} a_{m} e^{imx}$ with $a_{n} = \frac{1}{2\pi} \int_{-\pi}^{\pi} F(x) e^{{-inx}}\ dx$ >[!Notice] >Here we sum from $-\infty$ so the problem statement becomes: >Given any reasonable function $F$ on $[-\pi, \pi]$, with Fourier coefficients defined above, is it true that $F(x)=\sum_{m=-\infty}^{\infty} a_{m} e^{imx}$? ### Examples - [[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=34&selection=67,0,67,31|1.3 Example: the plucked string]] - [[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=35&selection=172,0,172,19|2 The heat equation]] The heat equation considers the 2D time independent heat transfer problem, with mathematical model called [Dirichlet problem](https://en.wikipedia.org/wiki/Dirichlet_problem). The differential equation itself is $\nabla^{2}u=0 $ for $D={(x,y) \in \mathbb{R}^{2}: x^{2}+y^{2} < 1}$. >[!Notice] >Here we use $\nabla ^2$ for Laplacian, we may also use $\Delta$ in the future. >The expression is just >$\frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} = 0$ This problem would be easier stated and solved under polar coordinates. >[!Info] >See also: https://en.wikipedia.org/wiki/Simple_harmonic_motion >[[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=21&selection=56,0,56,28|Standing and traveling waves]] in the book. >[[Stein-Shakarchi-1-Fourier_Analysis.pdf#page=28&selection=311,0,311,31|Solving wave equations by superposition of standing waves]] in the book.