Waves & oscillations

Oscillations

Start

Use springs and pendulums to connect equilibrium, sinusoidal motion, and energy exchange.

Resonance

Use coupled oscillators to connect driven resonance, normal modes, and the first hints of wave motion.

Waves

Build intuition for wave propagation by comparing what the pattern does with what the medium does.

Superposition and Reflection

See how incident, reflected, and transmitted waves combine at fixed, free, closed, and open boundaries.

Standing Waves and Harmonics

Connect reflected waves, nodes, antinodes, harmonics, and musical pitch on strings and air columns.

Sound Synthesis

Here

Shape harmonic amplitudes, play a piano keyboard, and watch sound as a waveform or spectrogram.

Sound Synthesis

The overall sound quality of an instrument, or timbre, is shaped by a variety of factors. These include the mixture of frequencies present in a sound wave, as well as the wave’s intensity over time.

Additive synthesis builds this mixture by adding pure sine waves together:

x(t)=A_1\sin(2\pi f t)+A_2\sin(2\pi(2f)t)+A_3\sin(2\pi(3f)t)+\cdots

The amplitudes $A_1, A_2, A_3, \ldots$ shape the harmonic content of the sound. The pitch of a particular note is largely determined by the fundamental frequency $f$ , but the higher harmonics (aka overtones) $2f, 3f, ...$ also effect the sound quality.

Below, you also have access to the sound’s ADSR envelope (attack, decay, sustain, release). Together, these shape the volume as a function of time for each note played.

Additive Synthesizer

Shape harmonic amplitudes, play chords on the piano keyboard, and compare the live waveform with a spectrogram.

What the Graphs Mean

The composite waveform shows one repeating cycle of the recipe you are shaping. The live waveform uses a fixed scrolling speed and fixed display scale, so pitch shows up as spatial spacing. The spectrogram turns that same sound into a moving frequency picture: lower notes sit near the bottom, higher harmonics stack above them, and brighter colors mean stronger frequency components.

Problem Solving

Harmonic Frequencies

Problem

A note has fundamental frequency $f = 220\ \text{Hz}$ . Find the first four harmonic frequencies.

Solve

Harmonics occur at integer multiples of the fundamental:

f_n = nf.

The first four are $220\ \text{Hz}$ , $440\ \text{Hz}$ , $660\ \text{Hz}$ , and $880\ \text{Hz}$ .

Reading a Spectrogram

Problem

A spectrogram has bright bands at $180\ \text{Hz}$ , $360\ \text{Hz}$ , and $540\ \text{Hz}$ . What fundamental frequency do these bands suggest?

Solve

The bands are integer multiples of $180\ \text{Hz}$ :

180,\quad 2(180),\quad 3(180).

The likely fundamental is $180\ \text{Hz}$ .

Sound Synthesis Checkpoint

Question 1 of 3

What does the fundamental frequency mainly determine for a musical note?

Choose an answer to get instant feedback.