How the Plucked String Dynamics Simulator Works

This tool visualizes the vibration of a plucked guitar string and how its energy is transferred to the guitar body through the bridge.

The simulation starts from an ideal triangular pluck and separates the string motion into its harmonic components. It then calculates the forces acting at the saddle and uses a simplified 4-DOF model of the guitar body to show how bridge mobility affects energy transfer and string decay. Pasted text

The string controls allow you to compare different vibrating lengths, tensions and masses per unit length, including the same note played on different strings.

How to Use the Simulator

String and Pluck Controls

String selects one of six ideal reference strings in standard tuning, from the sixth string, E2, to the first string, E4. These are not presets for commercial string sets: their mass per unit length is calculated from the selected tension, open-string tuning and reference scale length of 650 mm.

Reference string — free note retains the original full-length reference model. In this mode, you can change the fundamental independently of a numbered string, and the fret control is disabled.

String Fundamental selects the played note using the equal-tempered scale. On a numbered string, selecting a note automatically sets the corresponding fret. Changing strings preserves the played note when it is available within frets 0–20; otherwise, the nearest playable note is selected. The resonances of the guitar body remain in place.

Fret Position changes the vibrating length from the open string to fret 20. In this idealized model, fretting shortens the vibrating length and raises the pitch without changing the tension, mass per unit length or characteristic string impedance. UNSW Sites

String Tension sets the tension used in the force and impedance calculations. The simulator keeps the selected tuning fixed and recalculates the required mass per unit length. This represents comparing differently specified strings at the same tuning, not tightening or loosening one physical string of fixed mass per unit length.

The String Properties readouts update automatically:

• Vibrating length: the distance between the nut or pressed fret and the saddle, in millimetres.
• Tension: the selected string tension, in newtons.
• Linear mass: the mass per unit length, in grams per metre, not the total mass of the vibrating portion.
• String impedance: the characteristic transverse wave impedance, in N·s/m. This is a property of the string, distinct from the frequency-dependent impedance of the bridge.

Pluck Displacement sets how far the string is displaced before release. Keeping this value unchanged compares plucks with the same displacement, not necessarily the same finger force or stored energy.

Pluck Position changes where the string is displaced before release. This changes the relative strength of the harmonics and therefore the initial spectrum of the pluck. The position is expressed relative to the vibrating length; an additional readout gives its distance from the saddle as both a percentage of that length and a distance in millimetres.

Keeping the same position setting while changing frets preserves the relative plucking position, not the physical distance from the bridge.

Displayed Harmonics controls how many individual harmonic components are shown. The black curve represents the complete string motion using a higher-resolution internal model, while the colored curves show the selected harmonics separately. Pasted text

Bridge Mechanical Impedance

Bridge Mechanical Impedance changes the boundary condition at the saddle:

• Fixed Bridge represents an ideally rigid termination. No energy is transferred to the guitar body.
• Real Guitar introduces the frequency-dependent bridge mobility calculated from the 4-DOF guitar-body model.
• Intermediate positions allow you to explore the transition between these two conditions. Pasted text

Coupled Body Modes

Activate Edit Body Modes to change the main resonances of the guitar body: Air, Top and Back.

These are final coupled resonances of the complete guitar system, not isolated plate frequencies. The names indicate which component contributes most strongly to each mode.

Moving these resonances changes the mechanical impedance seen by the string and therefore changes which harmonics transfer energy most efficiently to the body.

Press Pluck to start the slow-motion simulation. The vibration is deliberately slowed down so that the motion of the string and the evolution of the forces can be observed over complete cycles. The oscillation phase and decay envelope use different time scales: one displayed cycle advances the decay envelope by one second. This is a teaching visualization, not a literal real-time waveform. Pasted text

What the Graphs Show

1. String Motion

The first graph shows the instantaneous shape of the string between the nut or pressed fret and the saddle. When a fretted note is selected, the left endpoint is labelled with the fret number and the length axis shows the shortened vibrating portion.

The black line is the complete string motion. The colored lines show its individual harmonic components.

As the vibration decays, different harmonics lose energy at different rates. The shape of the string therefore changes progressively with time.

2. Spectrum or Bridge Forces

The second graph can show either the Spectrum or the Bridge Forces.

Spectrum

The Spectrum view offers two different quantities:

String Displacement shows the harmonic content of the motion of the string itself: how much each harmonic contributes to the string displacement.

Bridge F⊥ shows the harmonic content of the transverse force applied by the string to the saddle. Higher harmonics can contribute more strongly to bridge force than their displacement alone would suggest. Pasted text

In simple terms:

String Displacement shows what is vibrating in the string.

Bridge F⊥ shows the force spectrum driving the guitar.

The spectrum is referenced to the strongest initial displayed harmonic for the selected quantity, so you can watch each harmonic decrease as the string decays. Because each setup has its own reference, equal heights in this normalized view do not necessarily mean equal forces when comparing different strings. Use the Bridge Forces view to compare force values in newtons. Pasted text

Optional references allow you to compare:

• Initial Spectrum: the harmonic content at the instant of release.
• Fixed-Bridge Reference: how the same string would decay without transferring energy to the body.
• 4-DOF Bridge Mobility: where the guitar body is most easily driven. This overlay is also normalized; it shows the shape of the mobility response, not its absolute magnitude. Pasted text

This makes it possible to see directly why some harmonics decay faster than others in the model.

Bridge Forces

The Bridge Forces view shows the forces generated at the saddle:

• Transverse force: the main oscillating force driving the soundboard.
• Longitudinal force: the variation in string tension produced by the changing geometry of the vibrating string, usually smaller than the transverse force.

Both are calculated at the saddle, corresponding to the right-hand endpoint of the first graph.

The force calculations use the selected tension and vibrating length. The vertical scale expands when necessary, so compare the numerical values and axis scales rather than the apparent height of the curves alone.

3. Energy Transfer to the Guitar

The third graph shows the estimated instantaneous mechanical power exchanged between the string and the guitar body within the modelled frequency range.

It is calculated from the transverse force components included in the body-coupling model and the resulting bridge velocity:

Mechanical power = transverse bridge force × bridge velocity. Pasted text

The power curve does not necessarily have the same shape as the force curve. A plucked string contains several harmonics, and the guitar body responds differently to each frequency. Their different amplitudes and phases can therefore produce several power peaks within one string cycle.

• Positive values: energy is transferred from the string to the guitar body.
• Negative values: some energy is temporarily returned from the body to the string.
• Mean absorbed power: represents the net energy removed from the string by the body per unit time, averaged over an oscillation cycle at the current decay envelope. Pasted text

This back-and-forth exchange is normal in a coupled vibrating system. What matters for string decay is the net energy transferred over time, not the number of individual peaks in the instantaneous power curve.

The String as a Mechanical Source

A string’s vibrating length, tension and mass per unit length determine its fundamental frequency. Tension and mass per unit length also determine its characteristic mechanical impedance: the relationship between transverse force and velocity in a travelling wave along the string. This should not be confused with the resistance felt when slowly displacing the string with a finger. UNSW Sites

The transverse force applied to the bridge depends on the tension and the slope of the string near the saddle. Increasing the pluck displacement increases that slope. For the same displacement and relative plucking position, a shorter vibrating length also produces a steeper slope. Euphonics

The resulting energy transfer depends on both the string and the guitar body. Force alone does not determine how much energy leaves the string: the bridge must move, and the phase relationship between force and velocity determines the net transfer. Pasted text

Same Note, Different String

Select the fifth string at fret 0, then change to the sixth string. The simulator keeps the played note, A2, and sets the sixth string to fret 5.

Keep the tension, pluck displacement, relative plucking position and body settings unchanged. The two strings now produce the same fundamental, with the same initial relative displacement spectrum, but their vibrating lengths, masses per unit length and characteristic impedances differ.

Compare the force values, mean absorbed power and harmonic decay. This isolates differences in the mechanical source without introducing a change in the initial relative plucking position.

On a real guitar, keeping the picking hand at the same physical distance from the bridge does not preserve the relative plucking position when the vibrating length changes. To explore that separate effect, adjust Pluck Position until the distance-from-saddle readout matches between the two setups.

String and Body Damping

Even with a fixed bridge, the simulated string has a small intrinsic damping. The model uses a fixed intrinsic Q of 3000, which gives faster amplitude decay at higher harmonic frequencies. Pasted text Pasted text

Connecting the string to the guitar adds another loss mechanism.

The mechanical impedance of the bridge changes with frequency, so every harmonic sees a different boundary condition. Its decay depends on both the string’s characteristic impedance and the bridge mobility at that frequency. For net energy transfer, the relevant part of the mobility is the component associated with energy absorption, not simply the magnitude of bridge motion. Pasted text Pasted text

Changing the String Fundamental or Fret Position moves the harmonic frequencies relative to the guitar resonances. Changing the Coupled Body Modes moves the guitar resonances relative to the string harmonics. Comparing the same note on different strings changes the mechanical source while leaving those frequency relationships unchanged.

This makes it possible to explore one of the central ideas behind string–body interaction:

The guitar body is not simply driven by the strings. Its mechanical behavior also changes the way the strings vibrate, evolve, and decay.

Model Scope

The body-coupling and power calculations cover 60–500 Hz. Harmonics outside that range retain intrinsic string damping but do not transfer energy to the body in this model. This is a modelling limit, not a claim that a real guitar stops responding above 500 Hz. Pasted text

The simulator retains a common reference axial stiffness, effective EA = 8.5 kN, for its longitudinal-force estimate. It does not model string-specific bending stiffness, winding construction, fret-contact losses or differences in internal damping between commercial strings. It illustrates how string properties affect force, coupling and decay, but does not resolve the beating or pitch instability of a complete wolf-note interaction.