9.1 Fourier series

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §9.1. Rewritten in original wording — not a paste of textbook figures or listings. Continues 8.7 ODE solver sketches. Next: 9.1b Fourier series labs.

From anharmonic clocks to spectra

Chapter 8 gave periodic orbits that are not pure sinusoids: soft $\alpha$ wells and $|x|^p$ forces produce pyramids, asymmetric swings, and amplitude-dependent periods. Those $x(t)$ still satisfy $x(t+T)=x(t)$ once a true period $T$ is known. Fourier’s theorem then lets you write any such (nice enough) signal as a sum of harmonics of the fundamental

$$\omega\equiv\omega_1=\frac{2\pi}{T}.$$

The computational program of this chapter is: measure or impose $T$, project coefficients, and later replace the integrals by a DFT / FFT. Short-time spectra and the quantum Fourier transform appear in later chapters; here we stay classical and continuous-time.

Series for a known period

Any single-valued period-$T$ signal with finitely many jumps admits a real Fourier series in harmonics of $\omega$. Each pair $(a_n,b_n)$ is the amplitude of the pure tone at frequency $n\omega$; intensity scales as $a_n^2+b_n^2$. Walk the series form, the orthogonality projection for $(a_n,b_n)$, and the odd-sawtooth $b_n$ — pick the matching equation each step.

This is the same least-squares projection idea as Chapter 6: the truncated series is the best trigonometric fit of that order. At a jump the partial sums converge to the midpoint of the left and right limits; near a sharp corner they overshoot (Gibbs phenomenon, roughly $9\%$ of the jump).

Symmetry shortcuts

• $a_0=2\langle y\rangle$ (twice the mean).
• Odd $y(-t)=-y(t)$ $\Rightarrow$ all $a_n=0$; integrate sines on half a period if convenient.
• Even $y(-t)=y(t)$ $\Rightarrow$ all $b_n=0$; cosine half-range formulas.

Causal signals that vanish for t<0 are rarely perfectly odd/even, so small “forbidden” coefficients can appear numerically.

Why this still helps nonlinear oscillators

Linear systems enjoy superposition of normal modes. Nonlinear Newton laws do not. Fourier does not claim that each harmonic separately solves the ODE — only that the observed periodic $y(t)$ can be expanded. Traditional mechanics often finds the fundamental analytically and adds overtones by perturbation; here you start from the full numerical orbit (or an analytic toy signal) and read the spectrum off.

For spectral analysis of driven / dissipative systems, wait until transients die. Strongly nonlinear attractors can jump among configurations; then a single long FFT mixes regimes — motivation for short-time transforms later.

Worked toys: sawtooth and half-wave

Sawtooth (odd, discontinuous)

Shifted to be odd on $(-T/2,T/2)$,

$$y(t)=\frac{t}{T/2},\qquad -\frac{T}{2}\le t\le\frac{T}{2},$$

and extended periodically. Only sine terms survive (the $b_n$ you picked above), so

$$y(t)=\frac{2}{\pi}\Bigl(\sin\omega t-\tfrac12\sin 2\omega t+\tfrac13\sin 3\omega t-\cdots\Bigr).$$

A few terms capture the overall ramp; many terms are needed for the corners. At each jump the series averages to $0$; just beside the jump, partial sums overshoot (Gibbs).

Half-wave (continuous, kinked)

$$y(t)=\begin{cases} \sin\omega t,&0<t<T/2,\\ 0,&T/2<t<T. \end{cases}$$

Projection gives $b_1=\tfrac12$, other $b_n=0$, and

$$a_n=\begin{cases} \dfrac{-2}{\pi(n^2-1)},&n\text{ even or }0,\\ 0,&n\text{ odd}, \end{cases}$$

so

$$y(t)=\tfrac12\sin\omega t+\frac{1}{\pi}-\frac{2}{3\pi}\cos 2\omega t-\frac{2}{15\pi}\cos 4\omega t+\cdots.$$

No corners as harsh as the sawtooth — truncation converges faster.

Quiz

Fourier series

1 / 4

If y(t+T)=y(t), the fundamental frequency is…

Pick an answer to see if you are right.

Takeaway

Know $T$, project $(a_n,b_n)$, watch power $a_n^2+b_n^2$, and expect midpoint convergence plus Gibbs at jumps. Nonlinear orbits from Chapter 8 are fair game once you extract their period. Labs lock analytic $b_n$ and a few partial sums before DFT machinery arrives.

Next: 9.1b Fourier series labs.