# String Instrument Acoustics

As we will see in this and subsequent sections, most musical instruments can be analyzed in terms of three principle acoustic components: a waveguide, an output filtering system, and an energy input mechanism. The corresponding elements of a general string instrument are its strings, body, and plucking, striking, or bowing techniques, respectively.

## The String as a Waveguide

• The uniform stretched string forms a one-dimensional waveguide along which mechanical waves can travel with a constant speed of propagation (ignoring for the moment other dimensions of wave propagation).

• The time evolution of the string's shape is given by the linear superposition of traveling-wave components propagating in opposite directions along the string (Matlab example).

• The resonance frequencies (or normal modes) of the string are determined by its length, the wave speed of propagation, and the boundary conditions at each of its ends.

• For a string of length L, fixed at each of its ends, a discrete set of standing-wave patterns is possible with frequencies given by integer multiples of f0 = c/(2 L), where c is the speed of wave propagation on the string. These standing-wave frequencies correspond to the resonances of the system.

• The extent to which the resonance frequencies of the string are present in any particular string vibration pattern is determined by the way the system is driven or excited.

• The stiffness of the string produces a restoring force that was neglected in our derivation of the wave equation. When we account for this force, the string modes are ``stretched'' from perfect harmonic relationships (with greater stretching for higher modes).

• For thin metal strings, the decay time is determined mostly by air viscosity. For gut or nylon strings, internal damping is dominant for most modes.

## String Energy Input Mechanisms

1. The Plucked String:
• Plucking a string provides it with an initial energy displacement (potential energy).

• The shape of the string before its release completely defines the harmonic signature of the resulting motion.

• A string plucked at 1/nth the distance from one end will not have energy at the nth partial and its integer multiples.

• The strength of excitation of the nth vibrational mode is inversely proportional to the square of the mode number.

2. The Struck String:
• A struck string is given an initial velocity distribution (kinetic energy).

• A string struck at 1/nth the distance from one end will not have energy at the nth partial and its integer multiples.

• The harmonic amplitudes in the vibration spectrum of a struck string fall off less rapidly with frequency than those of plucked strings.

• Light ``hammers'' (mass much less than the mass of the string) result in little spectral drop-off with frequency. Heavier hammers produce a drop-off roughly proportional to the inverse of the mode number.

3. The Bowed String:
• ``Stick-slip'' mechanism: During the greater part of each vibration, the string is ``stuck'' to the bow and is carried with it in its motion. Then the string suddenly detaches itself and moves rapidly backward until it is caught again by the moving bow.

• Beginning and end of the slipping are triggered by the arrival of the propagating bend or ``kink''.

• The string's vertical motion at any one point is given by a sawtooth pattern.

• Round trip time depends only on the string length and the wave velocity.

• Bowing near the string end requires greater force and produces a louder, brighter sound than bowing farther from the end.

• Amplitude of vibration can be increased either by increasing the bow speed or by bowing closer to the bridge.

## The Violin Body

1. Air and Wood Modes:
• For a violin, there are typically three or four important body resonances below 1 kHz. These include the first air mode, the T1 top plate mode, and the third and fourth ``corpus'' modes.

• Above 1 kHz, the mode structure is usually difficult to decipher, though a concentration of resonances around 2 to 3 kHz may be important (related to perception).

2. Tuning Top and Back Plates:
• Various techniques exist to measure the modal frequencies of the various components (Chladni patterns, force hammers, hologram interferometry).

• Complete systems, however, will have different mode structures than the individual plates.

3. The Bridge:
• The bridge transforms the motion of the vibrating string into a driving force on the top plate of the instrument.

• The violin bridge typically has strong resonances around 3000 and 6000 Hz.

• The bridge must move a small amount in order to transfer energy from the string to the body. For most musical instruments, however, the rate of energy transfer from the string to the bridge and soundboard is quite small (energy decay in the string is most affected by air viscosity and internal string damping).

## Other Stringed Instruments

1. Guitars:
• At low frequencies, sound radiation occurs from both plates. At high frequencies, however, most of the radiated sound comes from the top plate.

2. Electric Guitars:
• The body has less effect on the sound. Electrical circuits replace the body filtering function.