Physics Model

The core simulation solves the wave equation:

$u_{tt} = c^2 u_{xx} - 2\sigma u_t$

Where:

• $u(x,t)$ is the displacement of the string.
• $c$ is the wave speed.
• $\sigma$ is the damping coefficient.

Discretization

The string is discretized into $N$ segments. The update equation for each point $i$ at time step $n+1$ is:

$u_i^{n+1} = (2 - 2C^2)u_i^n + C^2(u_{i+1}^n + u_{i-1}^n) - u_i^{n-1}$

Where $C$ is the Courant number ($C = c \Delta t / \Delta x$).

This method accurately reproduces physical phenomena such as:

• Traveling Waves: Excitation creates two pulses traveling in opposite directions.
• Reflections: Waves invert upon reflection at fixed boundaries (bridge/nut).
• Interference: Standing waves emerge from the superposition of traveling waves.

Features

1. Damping Profile Editor

The interface includes a “Damping Profile” editor in the HUD. This allows you to draw a curve representing the damping factor along the length of the string.

• X-Axis: Represents the position along the string (Left = Bridge, Right = Nut).
• Y-Axis: Represents the damping intensity.
  • Bottom (0.0): Standard sustain (Base Damping).
  • Top (1.0): Maximum damping (Mute).

This allows for simulating techniques like Palm Muting (drawing a hump near the left side) or preparing the string with specific decay characteristics.

2. Reverb

A global reverb is applied using a ConvolverNode. The impulse response is generated procedurally by creating a buffer of white noise with an exponential decay, simulating a large, diffuse space. The mix is controlled by the reverbMix parameter in AUDIO_CONFIG.

3. Interaction

The instrument supports multi-touch and mouse interaction:

• Pluck: Click or tap to excite the string.
• Drag: Move across strings to continuously excite them (bowing/scraping effect).
• Velocity: The speed of interaction determines the amplitude.
• Acceleration: Sudden changes in velocity affect the spectral brightness.

4. Composition

• Strings: Adjustable from 1 to 24 strings.
• Layout: Strings are arranged randomly across the viewport (“Mikado” style) when count > 1.
• Tuning: Uses a 12-TET ratio system relative to the base frequency.

Architecture

The codebase is structured into three main modules:

1. VirtualString: Encapsulates the FDTD physics engine and Web Audio ScriptProcessorNode for sound generation. It maintains the state arrays ($u, u_{prev}, u_{next}$) and the damping profile.
2. StringComposer: A Factory/Builder that manages the ensemble, generating string layouts and tuning ratios.
3. InteractionManager: Handles input events (Mouse/Touch), calculates velocity/acceleration vectors, and dispatches excitation commands to the strings.

Usage

1. Start: Click anywhere to unlock the AudioContext.
2. Play: Swipe or click on the strings.
3. Modify: Use the HUD sliders to change physical properties.
4. Damp: Draw on the “Damping Profile” canvas to create custom decay envelopes.
5. Reset: Click “Reset Instrument” to generate a new random layout.

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Based on the “The Vibrating String” model by Francisco Esquembre et al. and “StringWave” by Paul Falstad.