Coupled Mesh + String — User Guide

What this model is

This is a physical-model synthesizer: instead of starting from an oscillator or recording, it numerically simulates a small mechanical network. The network contains two square meshes and a short string. Every neighboring pair of nodes is connected by a spring with damping. The outer rows and columns of both meshes are fixed, while the interior mesh nodes and all 12 string nodes can move.

The model contains 140 nodes in total but only 84 moving nodes: 36 interior nodes in Mesh 1, 12 string nodes, and 36 interior nodes in Mesh 2. There are 235 internal springs — 112 in each mesh plus 11 along the string — and two additional distributed mesh-to-string couplings.

The simulation uses a central-difference / Verlet-style update. You do not need to work with the equation directly to use the instrument: musically, stiffness controls how strongly neighboring nodes pull against displacement, damping controls energy loss, coupling controls how strongly the string and meshes exchange motion, and the pickup controls which part of that motion becomes the output.

Quick start

1. Run Coupled_Mesh_String.praat. No input Sound is required.
2. Choose one of the five network presets.
3. Leave Model rate at 2205 Hz for a useful starting point.
4. Use Edit all model parameters only when you want direct access to stiffness, damping, geometry, excitation, and pickup settings.
5. Click OK. If advanced editing is enabled, two compact parameter pages open after the preset has been applied.
6. The final mono Sound remains selected and is named coupled_mesh_string_<preset>.

Network structure

The mesh attachment coordinates are continuous values from 0 to 1. The script does not simply snap them to the nearest node: it computes a bilinear interpolation over the surrounding 2×2 mesh nodes. The coupling force is then distributed back to those same four nodes using the same weights. This lets the virtual connection sit between grid nodes.

Presets

The presets describe network configurations, not claims that the model literally reproduces a particular acoustic instrument. They set duration plus the detailed physical and geometric parameters. Sample rate, model rate, edge fade, normalization, visualization, and playback remain at the values in the compact launcher.

Excitation amplitude is not changed by the presets; its baseline value is 1.0. Presets 5 and 6 also leave both mesh attachment coordinates at the baseline center position, (0.5, 0.5).

Controls

Compact launcher

Advanced page 1: physics

Advanced page 2: geometry & pickup

Model rate and stability

Model rate is the physics update rate, not the audio sample rate. The mechanical system is first calculated at Model rate and only afterwards converted to the requested Sample rate. Raising Sample rate alone cannot create physical modes above the bandwidth represented by the model-rate simulation.

This scaling follows the time-step dependence of the explicit central-difference update. It keeps tuning and decay approximately invariant across model rates; it does not make different numerical resolutions mathematically identical.

The script also calculates a conservative stability bound from the scaled string, mesh, and coupling stiffnesses. If the bound reaches 3.80, the run stops and asks for a higher Model rate or lower stiffness/coupling. Excessive scaled damping is also rejected.

Excitation and virtual pickup

Excitation

A 5 ms half-sine force pulse is applied to the string. The normalized excitation position is converted to the two neighboring string nodes, and the force is divided between them according to the fractional position. This gives continuous placement along the discrete 12-node string.

Pickup

The output is not hard-wired to one string node. The script measures three actual structural trajectories on every physics step:

• the interpolated string pickup,
• the actual Mesh 1 attachment displacement,
• the actual Mesh 2 attachment displacement.

At Mesh pickup mix = 0, the output is only the selected point on the string. At 1, the string term disappears and the output becomes the average of the two mesh attachment probes. Intermediate values expose the coupled exchange between the three structures.

Output, resampling, and level

The physics produces one mono displacement trajectory at Model rate. If Model rate differs from Sample rate, Praat's Resample command converts the model-rate Sound to the requested audio rate. This replaces the old integer interpolation approach and supports arbitrary rate ratios with proper resampling filtering.

A short linear edge fade is then applied at both ends. It is not the old long musical fade-out: the network's natural decay remains audible through almost the whole requested duration.

If Normalize output is enabled and the result is non-zero, Scale peak: 0.90 performs target peak normalization. The Info window reports pre-normalization peak/RMS as well as final peak/RMS, so normalization does not hide the raw physical level from QC.

Visualization and QC

The current figure is built around the mechanism rather than a generic waveform display:

The QC box reports the 84 moving nodes, 235 internal springs plus two couplings, model/reference rates, coefficient scaling, stability bound, RMS of the three structural probes, and pre/post-normalization output metrics.