A
vibrationVibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
in a
stringA string is the vibrating element that produces sound in string instruments, such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a flexible material kept under tension so that they may vibrate freely, but controllably. Strings may be "plain"...
is a
waveIn physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
. Usually a
vibrating string produces a
soundSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
whose
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
in most cases is constant. Therefore, since frequency characterizes the
pitchPitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
, the sound produced is a constant
noteIn music, the term note has two primary meanings:#A sign used in musical notation to represent the relative duration and pitch of a sound;#A pitched sound itself....
.
Vibrating strings are the basis of any
string instrumentA string instrument is a musical instrument that produces sound by means of vibrating strings. In the Hornbostel-Sachs scheme of musical instrument classification, used in organology, they are called chordophones...
like
guitarThe guitar is a plucked string instrument, usually played with fingers or a pick. The guitar consists of a body with a rigid neck to which the strings, generally six in number, are attached. Guitars are traditionally constructed of various woods and strung with animal gut or, more recently, with...
,
celloThe cello is a bowed string instrument with four strings tuned in perfect fifths. It is a member of the violin family of musical instruments, which also includes the violin, viola, and double bass. Old forms of the instrument in the Baroque era are baryton and viol .A person who plays a cello is...
, or
pianoThe piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...
.
Wave
The speed of propagation of a wave in a string (
) is proportional to the square root of the tension of the string (
) and inversely proportional to the square root of the linear density (
) of the string:
Derivation
Let
be the
lengthIn geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
of a piece of string,
its
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, and
its linear mass. If the horizontal component of
tensionIn physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...
in the string is a constant,
, then the tension acting on each side of the string segment is given by
If both angles are small, then the tensions on either side are equal and the net horizontal force is zero. From Newton's second law for the vertical component, the mass of this piece times its acceleration,
, will be equal to the net force on the piece:
Dividing this expression by
and substituting the first and second equations obtains
The tangents of the angles at the ends of the string piece are equal to the slopes at the ends, with an additional minus sign due to the definition of beta. Using this fact and rearranging provides
In the limit that
approaches zero, the left hand side is the definition of the second derivative of
:
This is the wave equation for
, and the coefficient of the second time derivative term is equal to
; thus
where
is the
speedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
of propagation of the wave in the string. (See the article on the
wave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
for more about this). However, this derivation is only valid for vibrations of small amplitude; for those of large amplitude,
is not a good approximation for the length of the string piece, the horizontal component of tension is not necessarily constant, and the horizontal tensions are not well approximated by
.
Frequency of the wave
Once the speed of propagation is known, the
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of the
soundSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
produced by the string can be calculated. The
speedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
of propagation of a wave is equal to the
wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
divided by the period
, or multiplied by the
frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
:
If the length of the string is
, the
fundamental harmonicThe fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
is the one produced by the vibration whose
nodeA node is a point along a standing wave where the wave has minimal amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the...
s are the two ends of the string, so
is half of the wavelength of the fundamental harmonic. Hence:
where
is the
tensionIn physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...
,
is the
linear densityLinear density, linear mass density or linear mass is a measure of mass per unit of length, and it is a characteristic of strings or other one-dimensional objects. The SI unit of linear density is the kilogram per metre...
, and
is the
lengthIn geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
of the vibrating part of the string. Therefore:
- the shorter the string, the higher the frequency of the fundamental
- the higher the tension, the higher the frequency of the fundamental
- the lighter the string, the higher the frequency of the fundamental
Moreover, if we take the nth harmonic as having a wavelength given by
, then we easily get an expression for the frequency of the nth harmonic:
And for a string under a tension T with density
, then
Observing string vibrations
One can see the waveforms on a vibrating string if the frequency is low enough and the vibrating string is held in front of a
CRTThe cathode ray tube is a vacuum tube containing an electron gun and a fluorescent screen used to view images. It has a means to accelerate and deflect the electron beam onto the fluorescent screen to create the images. The image may represent electrical waveforms , pictures , radar targets and...
screen such as one of a television or a computer (
not of an oscilloscope).
This effect is called the stroboscopic effect, and the rate at which the string seems to vibrate is the difference between the frequency of the string and the refresh rate of the screen. The same can happen with a
fluorescent lampA fluorescent lamp or fluorescent tube is a gas-discharge lamp that uses electricity to excite mercury vapor. The excited mercury atoms produce short-wave ultraviolet light that then causes a phosphor to fluoresce, producing visible light. A fluorescent lamp converts electrical power into useful...
, at a rate which is the difference between the frequency of the string and the frequency of the alternating current.
(If the refresh rate of the screen equals the frequency of the string or an integer multiple thereof, the string will appear still but deformed.)
In daylight, this effect does not occur and the string will appear to be still, but thicker and lighter, due to
persistence of visionPersistence of vision is the phenomenon of the eye by which an afterimage is thought to persist for approximately one twenty-fifth of a second on the retina....
.
A similar but more controllable effect can be obtained using a
stroboscopeA stroboscope, also known as a strobe, is an instrument used to make a cyclically moving object appear to be slow-moving, or stationary. The principle is used for the study of rotating, reciprocating, oscillating or vibrating objects...
. This device allows the frequency of the xenon flash lamp to be exactly matched to the frequency of vibration of the string; in a darkened room, this clearly shows the waveform. Otherwise, one can use bending or, perhaps more easily, by adjusting the machine heads, to obtain the same frequency, or a multiple of, the AC frequency to achieve the same effect. For example, in the case of a guitar, the bass string pressed to the third fret gives a G at 97.999 Hz; with a slight adjustment, a frequency of 100 Hz can be obtained, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia. In most countries of the Americas, where the AC frequency is 60 Hz, one can start from A# at 116.54 Hz, on the fifth string at the first fret, to obtain a frequency of 120 Hz.
See also
- String instrument
A string instrument is a musical instrument that produces sound by means of vibrating strings. In the Hornbostel-Sachs scheme of musical instrument classification, used in organology, they are called chordophones...
s
- Fretted instrument
A fret is a raised portion on the neck of a stringed instrument, that extends generally across the full width of the neck. On most modern western instruments, frets are metal strips inserted into the fingerboard...
s
- Musical acoustics
Musical acoustics or music acoustics is the branch of acoustics concerned with researching and describing the physics of music – how sounds employed as music work...
- Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
- Vibrations of a circular drum
The vibrations of an idealized circular drum, essentially an elastic membrane of uniform thickness attached to a rigid circular frame, are solutions of the wave equation with zero boundary conditions....
- Melde's experiment
Melde's experiment is a scientific experiment carried out by the German physicist Franz Melde on the standing waves produced in a tense cable originally set oscillating by a tuning fork, later improved with connection to an electric vibrator. This experiment attempted to demonstrate that...
- 3rd Bridge
The 3rd bridge is an extended playing technique used on some string instruments , that allows a musician to produce distinctive timbres and overtones that are unavailable on a conventional string instrument with two bridges...
(harmonic resonance based on equal string divisions)
External links