Violin Acoustics

The physics of four strings, a wooden box, and horsehair.

Wave Equation

A vibrating string obeys the one-dimensional wave equation:

\[\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}\]

Where \(c = \sqrt{T/\mu}\) is the wave speed.

Standing Waves

With fixed endpoints at \(x = 0\) and \(x = L\), the solution yields standing waves:

\[y(x,t) = A \sin\left(\frac{n\pi x}{L}\right) \cos(\omega_n t)\]

The allowed frequencies are:

\[f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} = n \cdot f_1\]

Harmonic Series

The fundamental (\(n=1\)) and overtones form the harmonic series:

n Name Frequency Interval from fundamental

1

Fundamental

\(f_1\)

Unison

2

1st overtone

\(2f_1\)

Octave

3

2nd overtone

\(3f_1\)

Octave + fifth

4

3rd overtone

\(4f_1\)

2 octaves

5

4th overtone

\(5f_1\)

2 octaves + major 3rd

6

5th overtone

\(6f_1\)

2 octaves + fifth

7

6th overtone

\(7f_1\)

2 octaves + minor 7th (flat)

8

7th overtone

\(8f_1\)

3 octaves

Helmholtz Motion

Idealized bowed string motion discovered by Hermann von Helmholtz (1863):

The Kink

A sharp corner travels around the string:

\[\text{Period} = \frac{2L}{c} = \frac{1}{f_1}\]

Stick-Slip Cycle

  1. Stick phase: Bow friction pulls string

  2. Slip phase: String releases, corner propagates

  3. Return: Corner reflects at nut, returns to bow

  4. Capture: Bow catches string again

The frequency of this cycle equals the fundamental pitch.

Bow Dynamics

Minimum Bow Force

To maintain Helmholtz motion:

\[F_{min} \propto \frac{v_b}{\beta^2}\]

Where:

  • \(v_b\) = bow velocity

  • \(\beta\) = relative bow position (distance from bridge / string length)

Maximum Bow Force

Beyond this, motion becomes chaotic (scratchy):

\[F_{max} \propto \frac{v_b}{\beta}\]

Playable Region

The ratio:

\[\frac{F_{max}}{F_{min}} \propto \frac{1}{\beta}\]

Playing closer to the bridge (smaller \(\beta\)) narrows the playable force range but increases potential volume and brightness.

The Body

Resonances

The violin body amplifies string vibrations. Key resonances:

Mode Frequency Character

A0 (air)

270-280 Hz

Helmholtz resonance of air cavity

CBR (body)

440-480 Hz

Main corpus resonance

B1-

460-490 Hz

Lower body mode

B1+

520-580 Hz

Upper body mode

Bridge Function

The bridge converts lateral string motion to vertical soundboard motion:

  • Acts as a mechanical filter

  • Shapes the harmonic spectrum

  • Transmits energy to the top plate

Radiation

Directivity

The violin radiates sound asymmetrically:

  • High frequencies: More directional, project forward

  • Low frequencies: More omnidirectional

  • Wolf notes: Occur when string frequency matches body resonance

Room Acoustics

The player hears differently from the audience. The "edge" heard under the ear becomes "warmth" at distance.

Strings

Construction

Type Construction Character

Gut

Sheep intestine, wound or plain

Warm, complex, sensitive to humidity

Synthetic

Nylon/Perlon core, metal winding

Stable, gut-like warmth

Steel

Steel core, various windings

Bright, powerful, stable

Tension and Gauge

Higher tension = brighter, louder, but harder to play

\[f = \frac{1}{2L}\sqrt{\frac{T}{\pi r^2 \rho}}\]

For a given pitch, heavier gauge requires more tension.

Related

  • Violin — Overview

  • Mathematics — Wave equations, Fourier analysis

  • Music — Theory, composition, production