2.6 Fourier preview — motion as a sum of sinusoids

The oscillator equation is linear. That means: if $x_1(t)$ is a solution for drive $F_1(t)$, and $x_2(t)$ is a solution for drive $F_2(t)$, then $a x_1(t) + b x_2(t)$ is a solution for drive $a F_1(t) + b F_2(t)$. Superposition holds.

Combined with the fact that any reasonable function can be written as a sum (or integral) of sinusoids, linearity means we can solve the driven oscillator for an arbitrary drive by:

1. Decomposing the drive into sinusoids: $F(t) = \int \tilde F(\omega)\, e^{i\omega t}\, d\omega$.
2. Solving each sinusoidal piece individually: $\tilde X(\omega) = \dfrac{\tilde F(\omega)/m}{(\omega_0^2 - \omega^2) + 2 i \gamma \omega}$.
3. Summing the responses: $x(t) = \int \tilde X(\omega)\, e^{i\omega t}\, d\omega$.

This is the entire program of Fourier analysis applied to linear systems. We will spend chapter 8 making it rigorous and operational. For now, just preview the basic claim — that sinusoids are a basis — with a sound or two you can build.

One sinusoid

The simplest sound is a pure tone, $y(t) = A \sin(2\pi f t)$.

Slide the frequency, hear the pitch change. The amplitude controls loudness; the frequency controls pitch.

Two sinusoids

Add two sinusoids at different frequencies, and the result depends sharply on the frequency ratio:

Two close frequencies produce beats — a slow amplitude envelope at the difference frequency. Two harmonically related frequencies (ratios like 2:1, 3:2) produce a stable shape — the building block of every voiced musical tone. Two unrelated frequencies produce a pattern that never quite repeats.

What this preview promises

By the end of chapter 8 we will have made three things precise:

1. Any periodic signal is a sum of sinusoids at harmonics of the fundamental — the Fourier series.
2. Any transient signal is an integral over a continuum of frequencies — the Fourier transform.
3. Any linear system’s response is determined by how it acts on each frequency component independently — the transfer function.

For now we have the oscillator down. Time to couple oscillators together and watch a wave emerge.