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Our discussion of string instruments is limited by time. However, if you wish to pursue a more in-depth analysis of string instruments, perhaps for your class project, a unique collection of research materials is maintained here at CCRMA. The Musical Acoustics Research Library
(MARL) is a collection of independent archives or libraries assembled by distinguished groups or individuals in the field of musical acoustics research. Currently, MARL is comprised of the Catgut Acoustical Society Library, the Arthur H. Benade Archive, the John Backus Archive, and the John W. Coltman Archive. The Catgut files in particular are well oriented toward string instrument acoustic research.
As we will see in this and subsequent sections, most musical instruments can be analyzed in terms of three principle acoustic components: a waveguide, a resonator, and an excitation source. The corresponding elements of a general string instrument are its strings, body, and plucking, striking, or bowing mechanism, respectively.
- The uniform stretched string forms a one-dimensional waveguide along which waves will travel with constant speed of propagation.
- The motion of a string of length , fixed at each of its ends, is determined by standing waves at discrete frequencies given by
and its harmonics (and linear combinations of these standing wave patterns), where is the speed of propagation along the string.
- The particular frequency components sustainable on the string are determined by the excitation mechanism and the boundary conditions at the ends of the string.
- The time evolution of the string's shape can be analyzed in terms of two traveling-wave components propagating in opposite directions along the string.
- 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 th the distance from one end will not have energy at the th partial and its integer multiples.
- The strength of excitation of the th vibrational mode is inversely proportional to the square of the mode number.
- The Struck String:
- A struck string is given an initial velocity distribution (kinetic energy).
- A string struck at th the distance from one end will not have energy at the th partial and its integer multiples.
- The strength of excitation of the th vibrational mode is inversely proportional to the mode number.
- 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.
- 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 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).
- Tuning Top and Back Plates:
- Various techniques exist to measure the modal frequencies of the various components.
- Complete systems, however, will have different mode structures than the individual plates.
- The Bridge:
- The bridge transforms the motion of the vibrating string into a driving force on the top plate of the instrument.
- The bridge must move a small amount in order to transfer energy from the string to the body. However, greatest losses on string due to air and internal damping.
- At low frequencies, sound radiation occurs from both plates. At high frequencies, however, most of the radiated sound comes from the top plate.
- Electric Guitars:
- Little to no body effects. Electrical circuits replace the body filtering function.