10.1 String instruments and resonance

String vibrations create the mesmerizing sounds of many instruments. From violins to guitars, the interplay of tension, length, and density determines pitch, while harmonics shape unique timbres. Soundboards and bodies amplify these vibrations, transforming them into rich, resonant tones.

Different instruments harness string physics in unique ways. Violins rely on bowing and body shape, guitars on plucking and soundhole design, and pianos on hammers and large soundboards. Understanding these principles unlocks the secrets behind each instrument's distinctive voice.

String Vibration and Resonance

Principles of string vibration

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• Fundamental frequency of vibrating string determined by length, tension, and linear density expressed as $f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$ where $f$ is frequency, $L$ is string length, $T$ is tension, $\mu$ is linear density
• Harmonic series consists of integer multiples of fundamental frequency shaping instrument's unique timbre (overtones)
• Standing waves form on string from interference of incident and reflected waves creating nodes (points of no motion) and antinodes (maximum displacement)
• Resonance amplifies vibration and sound production when driving frequency matches string's natural frequency enhancing overall output

Role of soundboard and body

• Soundboard acts as transducer converting string vibrations into air pressure waves increasing effective radiating area with material properties affecting tonal quality (spruce, cedar)
• Body functions as resonating chamber enhancing lower frequencies with shape and size influencing tonal characteristics (dreadnought, parlor)
• Bridge transmits vibrations from strings to soundboard affecting efficiency of energy transfer and overall instrument responsiveness
• Coupling between strings and soundboard determines instrument's projection and sustain capabilities

Factors influencing string sound

• Pitch varies with
  • String length: shorter strings produce higher pitches (violin vs. cello)
  • Tension: higher tension increases pitch (tuning pegs)
  • Linear density: heavier strings produce lower pitches (bass vs. treble strings)
• Timbre shaped by
  • Harmonic content influenced by plucking/bowing technique and position (sul ponticello, sul tasto)
  • Body resonances emphasizing certain frequencies (Helmholtz resonance)
  • String material affecting overtone structure (nylon, steel, gut)
• Volume determined by
  • Amplitude of string vibration
  • Efficiency of energy transfer to soundboard
  • Size and design of resonating body (concert vs. parlor guitar)

Acoustic properties across instruments

• Violin family played with bow (arco) or plucked (pizzicato) with graduated sizes affecting pitch range and F-holes contributing to sound radiation
• Guitar family typically plucked or strummed with acoustic guitars relying on body for amplification while electric guitars use electromagnetic pickups
• Harp features multiple strings of varying length without fingerboard producing single pitch per string with unique soundboard orientation
• Piano uses struck strings with multiple strings per note for increased volume employing large soundboard and cast iron frame for structural support and tonal stability