Tuning Fork Simulator

This free tuning fork simulator treats each prong as a clamped-free cantilever beam and solves its vibration from the geometry and the material — nothing is looked up in a table of fork frequencies. Four rigs cover the whole curriculum: a fork on a resonance box with a live waveform and frequency spectrum, a two-fork beats bench, a resonance air column for measuring the speed of sound, and a design and tuning bench where you load, file and heat the fork.

The tool synthesises the real tone through the Web Audio API at the frequency it has just computed, including the clang overtone at 6.27× the fundamental, so what you hear is what the equations say. Modes are Simulate, Explore, Practice and Quiz.

The simulator opens on the Fork & Resonance Box rig with a steel fork of prong length 105 mm and thickness 6 mm — about 444 Hz, close to concert A. Click the fork on the canvas, press S, or press 🔨 Strike on the dock to set it ringing. Sound plays only after you strike, so nothing makes a noise on page load.

The controls you reach for while watching the fork sit on a dock along the bottom of the canvas: 🔨 Strike, ✋ Damp, 🔄 Reset (back to the default fork), and chips for 🔊 Sound, 📦 resonance box, 🔔 clang overtone and 🐢 slow motion, with a live level meter. A lit chip is on. Your choices are remembered for next time.

Pick a standard fork from the Preset list — the prong length is solved for the frequency on the label, not looked up, so “Concert A 440 Hz” really reads 440.00 Hz. Touch any control afterwards and the preset drops back to Custom.

Try the Concert A 440 Hz preset first, then type a new Prong Length or hold its − button down and watch the frequency fall as the square of the length. Every numeric box takes a typed value, steps with its − / + buttons, repeats if you hold one, and carries a fill bar showing where that variable sits between its own limits. Press the 🐢 chip on the canvas dock to see the prongs actually flex — at 440 Hz real time the motion is far too fast for the eye.

Fork & Resonance Box. Strike the fork and watch the prongs swing in antiphase, the stem drive the box, and the waveform decay. The spectrum panel shows the fundamental and the clang overtone. Switch off the 📦 Box chip to hear the bare fork: quieter but ringing far longer, because the box is what converts the fork’s energy into sound.

Beats (Two Forks). Fork B is fork A detuned by the Fork B Detune box, or loaded with wax. The canvas draws both waves and their sum with its beat envelope; the beat frequency is |f_A − f_B|. Adding wax always lowers f_B, which is how you find out whether an unknown fork is sharp or flat.

Resonance Air Column. A sounding fork is held over a tube closed by a water column. Lower the water with the Air Column box — or just drag inside the tube on the canvas — and the loudness meter peaks at each resonance. Press 📍 Mark at the first and second peaks; the tool then reports λ = 2(L₂ − L₁), v = fλ and the measured end correction, and compares that correction with 0.6 r.

Design & Tuning Bench. Every parameter at once: material, length, thickness, temperature, tip mass, and filing at the tips or the base. The note chip shows the nearest musical note and the error in cents.

Five categories — Basics, Formulas, Acoustics, Applications and Standards & Errors — each with concept cards carrying the equation, a worked example with real numbers, and a note on where students usually go wrong. Pick a category, then a concept from the grid.

Practice generates a fresh numerical problem each time — frequency from geometry, the shift after loading a prong, beat frequencies, the speed of sound from a resonance tube, end correction, temperature drift, and the prong length needed to hit a target note. Type your answer and press Check; a full worked solution appears either way, and your score is tracked.

Quiz is five multiple-choice questions drawn at random from a larger pool, with the four options shuffled on every sitting, finishing with a star rating and a question-by-question review. The readout cards are hidden in both modes so the answers are not on screen.

A prong is a beam built in at the yoke and free at the tip. Its first bending mode gives

f₁ = (β₁² / 2π) · (t / L²) · √(E / 12ρ),  β₁ = 1.8751

Worked example — steel, L = 105 mm, t = 6 mm, E = 200 GPa, ρ = 7850 kg/m³: √(E/12ρ) = 1457 m/s, t/L² = 0.544 m⁻¹, β₁²/2π = 0.5596, so f₁ = 444 Hz.

Three consequences worth checking on the controls: the width never appears (it cancels between the second moment of area and the mass per unit length); halving the length quadruples the frequency; and the second mode sits at (4.694/1.875)² = 6.27× the fundamental, not at 2× — a fork is not a harmonic oscillator like a string, which is exactly why it makes such a clean frequency standard once the clang has died away.

Internally the tool does not use the closed form. It evaluates the Rayleigh quotient over the real prong profile, so tip mass, tip filing and base filing all change the answer through the same integral. On a plain uniform prong that integral returns the closed form to machine precision.

File the tips → pitch rises. You remove mass where the prong moves most and shorten the vibrating length; L² in the denominator dominates.

File the base → pitch falls. You remove stiffness where the bending — and so the strain energy — is greatest. Both rules come out of the same integral rather than being bolted on.

Load the tips → pitch falls as f′ = f / √(1 + m/m_eff), with m_eff = 0.25 × prong mass for the first mode.

Warm the fork → pitch falls. Young’s modulus softens with temperature, and that term dominates: a steel fork drifts about −0.05 Hz per °C at 440 Hz, or roughly 120 parts per million per degree. Thermal expansion pushes the other way but only by ½α_LΔT — heating scales every length by g = 1 + α_LΔT while the mass stays put, so the density falls as g⁻³ and f ∝ (t/L²)√(E/ρ) gains a factor √g. Switch the material to Elinvar — the alloy Guillaume invented for exactly this problem — and the drift nearly vanishes.

• Do not forget the end correction. A quarter wavelength is not the first resonance length — the antinode sits about 0.6 r above the tube mouth. Taking two resonances and subtracting removes it exactly.
• Beats do not tell you the sign. Four beats per second means the unknown fork is 4 Hz away, sharp or flat. Load it with wax: if the beats slow, it was sharp.
• Thickness is measured in the direction of vibration, not across the prong face. The width really does not matter.
• The clang tone is not the octave. Rapping the fork hard on a bench excites it strongly; struck on rubber and held to a box it dies away in a fraction of a second.
• Keyboard: S strike, D damp, R reset the fork, M mark a resonance on the air-column rig.
• Right-click the canvas for Save as Image, Copy Readings, Download CSV and Reset. Sound feedback confirms strikes, marks and answers; the 🔊 Sound chip silences everything.