10.1 String instruments and resonance

String Vibration and Resonance

String vibrations are the foundation of how instruments like violins, guitars, and pianos produce sound. The physics governing a vibrating string determines pitch, while the instrument's body shapes the timbre and volume we actually hear. This section covers how strings vibrate, how instrument bodies amplify those vibrations, and what makes each string instrument sound distinct.

Principles of string vibration

The fundamental frequency of a vibrating string depends on three physical properties: length, tension, and linear density. The relationship is captured by this equation:

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

where $f$ is frequency (Hz), $L$ is the vibrating length of the string, $T$ is tension (in Newtons), and $\mu$ is linear density (mass per unit length, in kg/m).

Each variable pulls its weight here. Increasing tension raises the pitch. Increasing length or density lowers it. Every time you turn a tuning peg or press a finger down on a fret, you're changing one of these variables.

When a string vibrates, it doesn't just move at the fundamental frequency. It also vibrates at integer multiples of that frequency, called harmonics (or overtones). The second harmonic is twice the fundamental, the third is three times, and so on. The relative strength of these harmonics is what gives an instrument its unique timbre, which is why a guitar and a violin playing the same note still sound different.

Standing waves form on the string when waves traveling in opposite directions interfere with each other. This creates:

• Nodes: points along the string that don't move at all
• Antinodes: points of maximum displacement

The fundamental has one antinode in the middle and nodes at each fixed end. Higher harmonics have progressively more nodes and antinodes along the string's length.

Resonance occurs when a driving frequency matches one of the string's natural frequencies. At resonance, energy transfers efficiently into the vibrating system, amplifying the vibration and increasing sound output.

Role of soundboard and body

A vibrating string on its own barely moves any air. The string's surface area is too small to radiate sound effectively. That's where the soundboard and body come in.

The soundboard (the top plate of a guitar or violin, for example) acts as a transducer: it converts the string's mechanical vibrations into air pressure waves. Because the soundboard has a much larger surface area than the string, it pushes far more air and produces a much louder sound. The choice of wood matters here. Spruce and cedar are common soundboard materials because they combine stiffness with low density, allowing them to vibrate responsively.

The bridge is the physical link between strings and soundboard. It transmits vibrational energy from the strings into the top plate. Bridge design and placement affect how efficiently that energy transfers, which in turn influences the instrument's responsiveness and tonal balance.

The body as a whole functions as a resonating chamber. The enclosed air volume reinforces certain frequencies, particularly in the lower range. Body shape and size have a direct effect on tonal character. A large dreadnought guitar body emphasizes bass frequencies, while a smaller parlor body produces a more midrange-focused sound.

The coupling between strings and soundboard also determines two important performance qualities: projection (how far the sound carries) and sustain (how long a note rings).

Factors influencing string sound

Pitch varies with three factors, all visible in the fundamental frequency equation:

• String length: shorter vibrating length means higher pitch. A violin's strings are much shorter than a cello's, which is why the violin plays higher.
• Tension: higher tension raises pitch. This is what you adjust when you turn tuning pegs.
• Linear density: heavier (thicker or denser) strings vibrate more slowly, producing lower pitches. Bass strings are wound with metal to increase their mass without making them impractically long.

Timbre is shaped by several interacting factors:

• Playing technique and position change the harmonic content dramatically. Bowing near the bridge (sul ponticello) emphasizes higher harmonics and produces a glassy, metallic tone. Bowing over the fingerboard (sul tasto) suppresses upper harmonics for a warmer, softer sound. Plucking position on a guitar has the same kind of effect.
• Body resonances emphasize certain frequency ranges. One important example is Helmholtz resonance, where the air inside the body vibrates as a mass through the soundhole, boosting a specific low frequency.
• String material affects overtone structure. Steel strings produce bright, strong overtones. Nylon strings sound warmer and more mellow. Gut strings (historically common) have a complex, rich overtone profile.

Volume depends on:

• The amplitude of the string's vibration (how hard you pluck, bow, or strike)
• How efficiently the bridge and soundboard convert that vibration into sound
• The size and design of the resonating body

Acoustic properties across instruments

Violin family instruments are played with a bow (arco) or plucked (pizzicato). The family includes violin, viola, cello, and double bass, with graduated sizes that shift the pitch range lower as the instrument gets larger. F-shaped soundholes (f-holes) are a defining feature; their shape and placement contribute to sound radiation and influence the instrument's Helmholtz resonance frequency.

Guitar family instruments are typically plucked or strummed. Acoustic guitars rely entirely on the body for amplification, with the soundhole and internal air volume playing a central role. Electric guitars, by contrast, use electromagnetic pickups to convert string vibration into electrical signals, making the acoustic body far less important for volume.

Harps have multiple strings of varying lengths, each tuned to a specific pitch. There's no fingerboard, so each string produces only one note (or a limited set with pedal adjustments). The soundboard is oriented differently than in guitars or violins, running roughly parallel to the strings.

Pianos use felt-covered hammers to strike strings. Most notes have two or three strings tuned in unison to increase volume and richness. The piano's large soundboard and cast iron frame work together: the frame withstands the enormous combined string tension (which can exceed 20 tons), while the soundboard efficiently radiates sound across a wide frequency range.