Guitar String Bending Physics

Physics of String Bending

Vibrating String Length

The vibrating length at fret \(k\) is calculated as:

\[ L_k = L_{\text{total}} \cdot 2^{-k/12} \]

Where \(L_{\text{total}}\) is the scale length (648mm for 25.5" guitars).

Scaling Factor

The geometric scaling factor that accounts for the position along the neck:

\[ \alpha_k = \frac{\sqrt{L_k \cdot (L_{\text{total}} - L_k)}}{324} \]

Normalized to the 12th fret where \(L_{12} = 324\)mm.

Displacement Calculation

The displacement \(y\) (in mm) required to bend the string by \(n\) semitones:

\[ y = (1.5 \cdot n) \cdot \alpha_k \cdot G \]

Where \(G\) is the gauge factor (1.0 for high E string).

Key Physics Principles

String bending increases tension, which raises the fundamental frequency
Frequency is proportional to the square root of tension (from wave equation)
Displacement relates to tension via Hooke's Law: \( F = k \Delta x \)
Pitch change is exponential: each semitone is a \(2^{1/12}\) frequency multiplier
Geometric factor \(\alpha_k\) accounts for the position-dependent mechanical advantage

Displacement Calculator

Fret Position

Semitones to Bend

String Gauge

Required Displacement

2.6 mm

This is the vertical distance you need to bend the string to achieve the desired pitch change.

Displacement Reference Table

Fret	Length (mm)	1 Semitone (mm)	2 Semitones (mm)	3 Semitones (mm)
1	611.5	0.7	1.4	2.3
5	485.5	1.3	2.6	4.0
12	324.0	1.5	3.0	4.7
17	242.7	1.5	2.9	4.5
22	152.9	1.3	2.5	3.9

Technical Explanation

Physics of String Bending

When a guitarist bends a string, they are increasing the tension in the string, which raises its fundamental frequency. The relationship follows these physical principles:

\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]

Where \(f\) is frequency, \(L\) is vibrating length, \(T\) is tension, and \(\mu\) is linear density.

For a pitch increase of \(n\) semitones, the frequency ratio is:

\[ \frac{f_{\text{new}}}{f_{\text{original}}} = 2^{n/12} \]

The tension increase required for this frequency change is:

\[ \frac{T_{\text{new}}}{T_{\text{original}}} = \left(2^{n/12}\right)^2 = 2^{n/6} \]

Using Hooke's Law, the displacement \(y\) relates to tension change:

\[ \Delta T = k \cdot y \]

Where \(k\) is the effective spring constant of the string. The geometric factor \(\alpha_k\) accounts for how the mechanical advantage changes at different fret positions. Near the nut, the string is stiffer (smaller displacement needed), while near the middle of the neck, the displacement is maximized.