Physical Modeling

What Physical Modeling Is

Every acoustic instrument produces sound through the same basic mechanism: something excites a resonant body, and the body vibrates in a pattern determined by its physical properties. A guitar pick strikes a string. A mallet hits a drumhead. Breath passes through a reed into a tube. The excitation starts the vibration. The body shapes and sustains it.

Physical modeling synthesis replaces the real object with a mathematical model of that object. Instead of recording a guitar string and playing back the sample, a physical modeling engine calculates what a string of a given length, tension, and material would sound like when plucked with a given force. The sound is generated in real time from equations, not retrieved from memory.

The advantage over sampling is expressiveness. A sample is a photograph — it captures one moment of one performance. A physical model is a living system. Change the pluck force and the timbre changes. Change the string tension and the pitch bends naturally. Change the damping and the decay shortens or lengthens. Every parameter maps to a physical property, so the interaction feels intuitive in a way that adjusting filter cutoff on a sample never quite does.

Exciters and Resonators

Physical modeling breaks instrument sounds into two components: the exciter and the resonator.

The exciter is the initial energy. For a plucked string, the exciter is a short burst of noise — a click that contains all frequencies. For a bowed string, the exciter is continuous friction — a sawtooth-like signal fed steadily into the resonator. For a blown tube, the exciter is turbulent air — noise shaped by the player’s embouchure.

The resonator is what gives the sound its pitch and character. A string resonates at a frequency determined by its length and tension. A tube resonates at a frequency determined by its length and whether it is open or closed at the ends. A drumhead resonates in a complex pattern that depends on its diameter, tension, and material.

Try pinging a highly resonant bandpass filter with a trigger. Set the bandpass filter to a high Q (narrow, sharp resonance) and send a single impulse into it. The filter rings at its center frequency — a decaying tone that sounds like something was struck. That ringing filter is the simplest possible resonator.

Chain multiple resonators together and you get more complex behavior. A network of resonant filters at different frequencies can simulate the multiple resonant modes of a guitar body, a room, or a bell. These “resonator networks” connect directly to reverb — a reverb is, at its core, a dense network of resonators simulating the reflective surfaces of a room.

The Karplus-Strong Algorithm

Karplus-Strong is the gateway to physical modeling. The algorithm is so simple it almost feels like a trick.

Start with a short burst of white noise. Feed it into a delay line. Set the delay time to match the period of the pitch you want — for 440 Hz (A4), the delay time is about 2.27 milliseconds (1/440 of a second). Feed the output of the delay line back into its input, but pass it through a lowpass filter first.

What happens: the noise burst enters the delay line and starts looping. Each time it passes through the lowpass filter, the high frequencies are attenuated slightly. The fundamental and lower harmonics survive longest. After a few cycles, the noise has been shaped into a pitched tone that decays naturally — the highs die first, then the mids, then the fundamental fades out last. It sounds like a plucked string.

Build this in VCV Rack with three modules: a noise source (the exciter), a delay (the resonator), and a filter in the feedback path (the damping). A VCA controlled by an envelope triggers the noise burst. The delay time sets the pitch. The filter cutoff controls the brightness and decay character.

Changing the filter in the feedback loop changes the instrument. A gentle lowpass filter gives you a nylon guitar string — warm, with a slow decay. A sharper lowpass filter gives you a steel string — brighter attack, quicker decay of the highs. Replace the lowpass with a bandpass and you get something more bell-like or metallic. Replace the noise burst with a sawtooth impulse and the pluck character changes — less random, more tonal from the start.

Waveguide Synthesis

Karplus-Strong models one direction of travel along a string. Waveguide synthesis models both. In a real string, a vibration travels from the pluck point to one end, reflects, travels back, reflects again at the other end, and continues bouncing. A waveguide uses two delay lines running in opposite directions to simulate this back-and-forth propagation.

The practical difference is subtle for plucked sounds but significant for sustained sounds. Bowing a string requires continuous excitation — the bow feeds energy into the string at a specific point, and the waveguide model tracks how that energy propagates in both directions from the bow position. You cannot model a convincing bowed string with basic Karplus-Strong. You can with a waveguide.

Waveguides also model tubes well. A clarinet or flute is essentially a tube with a vibrating column of air inside it. The waveguide represents the tube — one delay line for the air traveling one direction, another for the reflection. The exciter is the reed or embouchure. The tone holes along the tube change the effective length of the delay lines, which changes the pitch.

The distinction between bowing, striking, and plucking matters. Each excitation method produces a different interaction with the resonator. Striking is a single impulse (Karplus-Strong handles this well). Plucking is a single impulse with a specific spectral shape (the pluck position on the string affects which harmonics are present). Bowing is continuous friction that requires the bidirectional model to capture correctly.

Modeling Strings, Tubes, Membranes, and Plates

Strings and tubes are one-dimensional — vibrations travel along a line. Membranes (drumheads) and plates (cymbals, gongs) are two-dimensional — vibrations spread across a surface. This makes them harder to model and produces more complex, often inharmonic spectra.

A drumhead vibrates in multiple modes simultaneously, and those modes are not harmonically related the way a string’s overtones are. The result is the characteristic “thud” or “ring” of a drum — pitch is present but diffuse, and the relationship between the fundamental and the upper partials is more complex than simple integer ratios.

Plates are even wilder. A cymbal or a gong has so many resonant modes, so closely spaced and so inharmonic, that the result is somewhere between pitched tone and noise. Physical modeling engines that handle plates use networks of coupled resonators or two-dimensional waveguide meshes — computationally expensive but capable of producing sounds that no other synthesis method can match.

In VCV Rack, the Audible Instruments Modal Synthesizer (a virtual version of the Mutable Instruments Rings module) offers multiple resonator models in one module: strings, bars, tubes, membranes, and plates. Each model responds differently to the same exciter input. Feed it a trigger and switch between models to hear how the resonator type transforms the same impulse into a guitar pluck, a marimba strike, a bell tone, or a metallic crash.

Formant Synthesis and the Voice

The human voice is a physical modeling challenge worth tackling as a sound design exercise. Try recreating a cat’s meow using synthesis. The progression of approaches is instructive.

Start with analog synthesis in VCV Rack: resonant bandpass filters tuned to formant frequencies, shaping a noise source into something vowel-like. It gets close but not close enough — the transitions between formant shapes are too abrupt.

Next, tune the bandpass filters and their modulation envelopes to create formants that shift over time, mimicking the way a mouth changes shape during speech. Better, but still mechanical.

Then try a soft synth with built-in formant filters and modulation. The dedicated formant engine handles the transitions more naturally.

Finally, try a vocal formant synthesizer (the Pink Trombone model in VCV Rack) that directly models the vocal tract — tongue position, throat opening, nasal cavity. This is physical modeling of the voice, and it produces the most convincing result because it simulates the right thing: not the sound of a voice, but the mechanism that produces a voice.

This progression — from generic filters to dedicated physical models — illustrates why physical modeling exists. General-purpose synthesis tools can approximate acoustic sounds. Physical modeling can nail them, because it simulates the source of the sound rather than the sound itself.

Physical Modeling in Practice

Physical modeling shows up in more places than you might expect. Many software instruments marketed as sample-based are actually hybrid: they use samples for the initial attack and physical modeling for the sustain and release, because the model handles expressive variation better than a static recording.

In VCV Rack, the key modules for physical modeling work are:

• Audible Instruments Modal Synthesizer (Rings): multiple resonator models, external exciter input, built-in exciter options
• Delay modules with feedback for building Karplus-Strong patches from scratch
• Resonant bandpass filters at high Q for simple resonator experiments
• Noise sources for excitation

Combining techniques is where physical modeling becomes powerful: Karplus-Strong for pitched string sounds, resonator networks for body simulation, and formant filters for vocal-like timbres. Think of physical modeling not as a separate synthesis method but as a way of thinking about sound — every sound in the real world has an exciter and a resonator, and once you see that structure, you can model anything.

What to Practice

• Build a Karplus-Strong plucked string in VCV Rack from scratch: noise source into a VCA (triggered by an envelope), into a delay line with feedback, with a lowpass filter in the feedback path. Set the delay time to produce a specific pitch (A = 2.27 ms, middle C = 3.82 ms). Adjust the filter cutoff and listen to how it changes the decay character.
• Experiment with different exciters on the same delay-line resonator. Try noise bursts (pluck), a sawtooth impulse (different pluck character), and a continuous noise source through a VCA controlled by pressure or a slow envelope (bowing approximation).
• Use the Audible Instruments Modal Synthesizer in VCV Rack. Feed it a trigger and switch between the string, membrane, and plate models. Notice how the same excitation produces radically different timbres depending on the resonator model.
• Build a resonator by setting a bandpass filter to maximum Q and feeding it a single trigger pulse. Tune the filter to different frequencies and listen to how the ringing tone changes. Chain two or three resonant filters at different frequencies for a more complex body simulation.
• Attempt the cat meow challenge: try to synthesize a recognizable animal vocalization using only filters, noise, and modulation. Start with formant frequencies for the target vowel sounds and modulate between them with envelopes.
• Compare a Karplus-Strong string to a sampled string. Load a guitar sample in one channel and play your Karplus-Strong patch in another. Listen for what the model captures well (decay shape, brightness variation with pitch) and what it misses (body resonance, fret noise, string-to-string interaction).