Acoustic Guitars and How They Make Sound
Acoustic Guitars and How They Make Sound
The acoustic guitar (which includes classical, flamenco and steel string acoustic guitars among others) needs no electrical amplification. The sound is initiated when a string is plucked, or excited. The strings, body and the air within the body vibrate when excited, causing the surrounding air molecules to be periodically pushed together (compression), and moved apart (rarefaction). This movement forms a longitudinal wave that is interpreted as a sound.
The vibration of the strings dominates that of the other parts, making them vibrate at the natural (resonance) frequencies of the string. The strings are thus known as the primary vibrators, and determine the pitch of the note. The body and air are secondary vibrators, and play an important role influencing the timbre and the loudness.
The interaction between the various vibrating parts is known as ‘coupling’, and the effectiveness of this is determined by instrument’s construction.
So, sound is produced when the string is plucked with a fingertip or nail. This inputs energy to the string and causes it to vibrate, producing a transient sound. The secondary vibrators are then excited, and cause the sound to radiate.
The initial sound is the result of one or more standing waves being set up on the string. A standing wave is formed from travelling waves moving in opposite directions. Standing waves feature two or more nodes, or areas where there is zero displacement of the air, with a half-wavelength distance between adjacent nodes. The plucking motion sends a pulse along the string, which is displaced perpendicularly to the direction in which the pulse is travelling. The string is fixed at both ends (the bridge and the nut or finger), and when the wave reaches these fixed points, it is reflected and the pulse inverted.
Standing waves are set up at the string’s natural frequencies, because the multiple waves travelling in each direction are in phase with each other, and behave as two single waves, one moving in each direction. This wave produces a constant pitch – an important feature for a musical instrument.
Since the length of the string must be measured in a whole number of half wavelength increments, the natural frequencies can be determined mathematically, with the following equation:
f = nv / 2L
f – frequency
n – an integer number
v – wave speed
L – string length
Modes of vibration
Various standing wave patterns, known as normal modes of vibration, can be set up at the string’s natural frequencies. In the first (fundamental) mode, the nodes occur only at the ends of the string, and the frequency at which the mode occurs is known as the fundamental frequency. In the second, there is also a node in the middle of the string, and the frequency at which this pattern occurs is twice the fundamental frequency, and so on.
These modes and their corresponding frequencies form a harmonic series, enabling the string to produce a pitched note. The string vibrates in several of its modes at once, and the resulting sound combines the harmonics from all the natural frequencies. The ear interprets these as a single pitch, based on the fundamental frequency of the group. The higher modes fade quickly, so the first mode soon dominates, but the other harmonics make an important contribution to the timbre of the sound.
If the string tension, length and mass per unit length are known, the fundamental frequency of the string can be calculated using the following equation:
f1 = 1/2L
f1 – fundamental frequency
L – length
T – tension
μ – mass per unit length
The pitch of the note can be changed by altering the string’s length, its tension, or the mass per unit length.
The string length is changed by fretting the notes. A hand position closer to the bridge equates to a shorter vibrating length and a higher pitch. The frets are arranged in semitone increments, in accordance with equal tempered tuning. The widths of successive frets are related by a ratio of 1.0594, (twelfth root of two), so they become closer together towards the sound hole.
The tension is changed when the instrument is tuned by using the machine heads to tighten or loosen the string, and thus to sharpen or flatten the pitch.
The highest three strings consist of a single nylon filament, whereas the bass strings are strands that are overwound with thin metal wire, which increases the mass per unit length. These vibrate more slowly, lowering the pitch.
The above description refers to an ‘ideal’ string, with a uniform mass and no resistance to bending. In practice, strings are slightly stiff and do not bend completely easily at the ends, so the effective length is less than the actual length. The resulting harmonics may produce a pitch that is not perfectly in tune – this is known as inharmonicity, and can be a particular problem on the thick single strand 3rd string, which tends to be sharp.
Note that the string’s normal mode of vibration has a frequency response curve; it can be excited even if it is driven at a frequency is close to, rather than exactly at, its resonance frequency. However, the amplitude is higher when closer to the resonance frequency.
The string is stretched when plucked, causing potential kinetic energy to be stored in it. Upon release, this is transformed into kinetic energy, which causes oscillations to be set up as a result of the string’s elasticity and inertia. A string plucked in isolation cannot move much air, and is very quiet, so the strings are coupled via the bridge to secondary vibrators. These radiate sound more efficiently, and are responsible for most of the instrument’s radiativity. The secondary vibrators add their own harmonics to the mix, strongly influencing the timbre.
The string’s high pressure oscillations make the bridge push and pull on the body, transferring the vibrations to the flexible soundboard, which is the major secondary vibrator. This, and the rest of the body, oscillate at the string’s resonance frequencies, and radiate them to the air, creating a stronger pressure variation and thus a louder sound. In this way, the large surface area of the body acts as an efficient converter of the initial energy input.
The relative roles of the different parts varies; lower pitched sounds are more dependent on the vibrations of the body cavity, sound hole, back and ribs, whereas higher pitched sounds are more influenced by the bridge and soundboard. In all cases, careful construction is needed to ensure the correct degree of interaction between the coupled components, to produce a desirable volume and tone.
The secondary vibrators have resonance frequencies of their own, but the body is designed to vibrate quite strongly at all frequencies within the instrument’s range, so the natural frequencies of the secondary vibrators do not interfere with that of the primary vibrator.
Not all of the vibrations at the bridge are converted to corresponding sound pressure variations in the air; instead the body resonances ensure that some frequencies are amplified to a larger extent, and the radiated sound is equivalent to the frequency response of the body multiplied by the frequency spectrum of the string force at the bridge.
The body’s modes of vibration can be measured using various optical, mechanical, acoustic, electrical methods, including time-averaged holographic interferometry, which involves exciting all of the guitar’s modes of vibration simultaneously, and creating a hologram to record the positions the body takes over time. All classical guitars show the same general patterns, but exhibit individual differences based on their construction.
Guitar body resonances are strongest at about 100 Hz and 180 – 230 Hz. The first of these modes is occasionally called the air resonance, and it involves contractions and relaxations of the body, with the top and bottom moving in opposite directions. When combined with the compression and rarefaction of the air inside the body, this causes large volumes of air to be moved in and out of the sound hole. This movement (and not structural vibrations) is responsible for most of the sound radiation. The instrument also acts like a Helmholz resonator at air resonance (tuned to about G#2 – 103.8 Hz). Below air resonance the radiation from the sound hole is not in phase with that of the structural vibrations, so the response declines.
In the higher of these two modes, the top and bottom plates move in the same direction. This mode also radiates sound strongly, since the body motion is in phase with the radiation of the sound hole.
At higher frequencies, the body motion is more complex, with the top and bottom plate motion becoming more significant, and the sound hole radiation less so. The modes that radiate sound most effectively induce the greatest changes in the body. However, the weaker higher frequency modes still play a role in ‘colouring’ the sound.
Overall, the first few modes of vibration are most efficient at radiating sound, as these move more air and radiate evenly. At higher frequencies, the modes become more complex, and the radiation more directional and varied. This is why recordings don’t quite match the original sound, and the instrument sounds different from the performer’s perspective than further away.
Timbre refers to the characteristic ‘colour’ of musical notes, and enables instruments to be differentiated. It is determined by the note’s harmonic components, and influenced by the performance style.
The manner in which the string is plucked affects the timbre. Variations include:
The vibrational pattern of the bridge varies depending on where the string is plucked; if close to the bridge, the upper harmonics are louder, and even numbered harmonics more prominent, resulting in a harsher sound. If plucked close to the sound hole, the even numbered harmonics have lower amplitudes (closer to a square wave pattern), and the upper harmonics fade quickly, giving a softer sound.
Classical and flamenco guitars are played by plucking the strings with the fingernails. Force applied perpendicular to the top plate makes the strings vibrate vertically, producing the strongest body resonance. This can be facilitated by filing the nails into a ramp shape.
Angle of release
Quieter, more sustained sounds are produced when the string vibrates parallel to the soundboard, and there is less coupling between the string and the body.
Vibrato is a pitch fluctuation (usually 5-7 Hz) created by rocking the fingers on the fretboard, which alters the string length. The amplitude also varies slightly due to the changes in the response of the body.
Other factors relating to the Acoustic guitar’s sound:
A wolf note results from poor coupling between the string and the body, which causes the string to deliver its energy to the body too quickly. It has a high amplitude and decays rapidly.
These result from the deformation of the body caused when the string is pulled. When the string is released, the body relaxes and emits its tap tones, which play an important role in determining timbre.
Intonation (the degree to which the actual frequency of each note corresponds to what it theoretically should be) is affected by the distance between the saddle and the nut, which determines the positions of the strings’ harmonic nodes. In practice, the intonation is imperfect, due to the limitation of equal temperament. In addition, the intonation of the same note played on different strings varies.
In general, there is no optimal technique for producing a good tone, given the significant individual differences in each player’s hands and fingers. It is easy to get a sound of the acoustic guitar, but not so easy to produce a good one!