Ron Lebar. Music Technology & Theory

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Starburst Vibration in music strings, resonance

Music
Technology

Formulae &
Equations

Sine waves &
Frequency

DeciBels &
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Active & Passive
Components

Strings etc., Modes of Vibration & Resonance .

A plain steel string, stretched between two rigid terminations, has three main modes of vibration and resonance.
1: Transverse. Controlled by the longitudinal wave propagation velocity on the string (wv) and the length of the string (L). The fundamental frequency is given by wv divided by 2L.
This frequency is usually lower than the pitch of a blown pipe of the same length. So transverse waves (deflections) in strings tend to propagate slower than compressive waves in air.
This mode is the one made use of in most stringed instruments, there are exceptions. The other modes are generally more of a nuisance, needing to be taken into account, rather than directly used.
2: Longitudinal (Compressive). Controlled by the propagation velocity of a compressive disturbance along the string (speed of sound in steel) (vs) and the length of the string (L). The fundamental frequency is given by vs divided by 2L.
The speed of sound in rigid materials is higher than in air. By a factor of 17 times in the case of steel.
So this frequency is higher than that due to mode 1. In a practical string it is more than ten times higher & only slightly affected by tension. It is particularly significant for struck strings, and to a lesser degree for plucked strings.
It is more noticeable for thicker strings, i.e. the bass end.
3: Torsional. This mode is usually of limited significance when struck or plucked strings are considered. It is however a more significant factor in the performance of bowed strings.
Calculation of the fundamental resonant frequency is more complex than with the other modes. It is proportional to the square root of { radial mass (m) divided by the torsional spring constant (t) }. It is usually higher than the resonance of mode 1 and lower than mode 2.
Torsional vibration has comparatively little effect on bridge movement and so does not play a major part in acoustic output. Its effect on bowing is however quite marked.
The string is bowed tangentially at its radius and is rotated. This initially detracts from bowing efficiency. At a certain point the combined lateral tension and radial tension in the string reach a critical level (the slip point).
The string is released, moving laterally and radially against the bow's influence. When its increasing tension overcomes momentum, the string sticks again, and the cycle repeats.
The torsional movement is repeated at the repetition rate of the lateral movement, but introduces its own resonance as a 'formant'. This is, in the main, only heard directly from the string.
So the player will be more aware of it than the audience. Steel strings may produce a noticeable 'whistle', due to torsional resonance, at certain bowing speeds. As with mode 2, mode 3 is only weakly affected by string tension.
The more significant effect, under certain conditions, relates to the absence of a direct harmonic relationship with mode 1. This may result in a periodic 'jitter' of the moment of slip. This effect can be made use of, with sufficient skill, to produce a 'growl', or it can simply sound un-musical.
More on the Modes of Vibration.
Mode 1: Frequently misunderstood, it is often assumed to be a simple lateral vibration of the string. With greatest amplitude at the antinodes & zero amplitude at the nodes. For the fundamental, the nodes are at the two terminations (bridges or frets), with its antinode at the centre.
This assumption is not strictly correct, there is no mechanism allowing such a vibration to be sustained & to be resonant. The lateral only movement of the string, tracing two arcs for the fundamental, is an optical illusion.
Certain Web sites show animations of simple transverse vibrations of a string. Some show the fundamental and a number of harmonics. Although some work has been put into these animations, they are wrong.
One site I found has an animation, copied on left, showing initial lateral deformations, due to plucking at one third of length, followed by release. These travel along the string both ways, reflecting from each termination. A stylised but more accurate depiction of real string wave motion.
This animation is copyright & by courtesy of Dr. Dan Russell, at Kettering University. Clicking on it links to the site page (not always active). The site's other pages also have much useful information, well worth a visit.

More will be written as time permits...
In the meantime - an example, from my work on an unconventional electric piano..
A string, tuned to A440, is plucked sharply downwards, as though by a fast glancing blow from a plectrum. This is the nearest conventional equivalent to the 'pulse hammers' on my prototype electric piano. The strike point is one tenth of the string's speaking length from the left bridge. This is the first antinode for the 5th harmonic, which will be strongly excited. A magnetic pickup is placed under the string, a similar distance from the right bridge.
The fundamental period is 2.2727 milliseconds. The entire strike action is over in 10% of this time, about 227 microseconds. Equal to a half period of the 5th harmonic, which will consequentially be further enhanced. A short length of the string is deflected downwards. This deflection pulse will propagate both left and right at the velocity of wave travel in the string.
One tenth of a fundamental half period later, the left-bound pulse (pulse A) hits the bridge. It is reflected towards the right & inverted. The now upwards pulse is chasing the downwards right bound pulse (pulse B) along the string.
Eight tenths of a half period after the strike, pulse B crosses the pickup, producing a negative pulse output. It then hits the right bridge, reflecting left as an upwards pulse.
Two tenths of a half period after crossing the pickup, it crosses it again. Meeting the upwards pulse A as it does so. This results in a double strength positive output pulse. Pulse A then hits the right bridge, returning left as a downward pulse. Two tenths of a half period after crossing the pickup, it crosses it again, another negative pulse output.
This to & fro movement continues. The output is initially a short burst of pulses, repeated at the fundamental frequency. The fifth harmonic & many higher ones are present. Propagation starts from the strike point, where maximum deflection occurs. Harmonics with a node at that position ( 10th, 20th etc. ) are absent, as they must be.
The string is an imperfect wave guide, pulses are distorted by finite stiffnes, inter-molecular friction & incomplete reflection from the bridges. After a number of cycles they become increasingly sinusoidal in shape. They also affect one another's shape, as they cross.
Upper harmonics are affected more rapidly by this process, they require more energy & sharper bending. All the time, overall amplitude is decaying. Gradually the pulses blur into one another, until the fundamental, not present at the start, is almost all that remains. The rate of decay is proportional to level, thus constantly reduces. So a low level sustained output continues for some time, eventually being lost in background noise.
With conventional plucking, the entire string is first deformed & stretched. So upon release, the fundamental is excited as the whole length is in motion. Release transients will travel from end to end. Due to the high strike speed involved in the example above, the results are appreciably different. The principle remains the same however.
A simple sideways motion alone can't determine a particular frequency, the length determines resonance. Take a road, no matter how many times a chicken crosses it, she will never know how long it is.
More soon...
Author; Ron Lebar.
Dot

The following section refers to a subject not needed for success in basic electrical or electronic engineering. It is essential if a true understanding of the processes involved is desired.
Such understanding will not help with exams, curricula are set by old-worlders.
A Form of Words

Much of the technical information on this site is written in the language of classical electrical theory. Formulae & equations etc. are those in common usage. These are correct, accurate & have stood the test of time. Their use can solve problems & enable design projects.
Some 19th century scientists & a small number in the 20th century realised that many basic assumptions are wrong. 20th century mainstream physics became fossilized, particularly after the Second World War. The spirit of enquiry which got it all going has been replaced by complacency.
The word 'theory' has been largely replaced by 'description', in other words there is nothing left to learn. Any attempt to correct the record, or introduce new theory is actively resisted. Early & later dissenters have effectively been written out of the records. (e.g. Heaviside, Tesla, Fleming, Ivor Catt etc.)
It is now mostly a cosy way of earning a good living. In many ways there is a correlation with the 16th century campaign to halt science. This was conducted by the church, to protect their power & influence. Now it is physicists who, as a body, resist new ideas. These are seen as likely to show old ideas as mistaken, damaging reputations & endangering the huge funds currently being spent.
On various pages we refer to current flow, capacitors being charged etc. This is basically a convenient form of words, based on habits now spanning four centuries. It does not mean we accede to the dualist or particle theories. Nor do we accept quantum mechanics, the uncertainty principle & all their ramifications. The Theory V Reality page will develop to cover this.
Until we get our house in order, a new form of language, in keeping with reality, will not fully evolve. The nearest was Oliver Heaviside's use of 'energy current', James Fleming also touched on the subject. For the present, we are stuck with the archaic way of describing things.
Some branches of physics are genuinely in the business of moving forward. Some progress has been made, it is heavy going while incorrect concepts are taken as gospel. Radical new theories should not be suppressed, they should be afforded the full glare of publicity & open debate. If they are obviously wrong they will be shot down or simply wither away.
A good theory should fit the observed facts without paradoxes. It should be capable of future modification, if new discoveries demand it. If such necessary modification negates the original concept the theory should probably bow out gracefully. We must always accept the possibility of being wrong, a theory's age or pedigree is no guarantee of infallibility.
An obvious & old example is displacement current. This caused a paradox even in the eyes of James Maxwell, its proposer. It requires the polarised displacement of molecules in a capacitor's dielectric. Yet a capacitor works with a vacuum between its plates. He then proposed the Aether, an indetectible substance pervading all space.
We are now pretty sure this does not exist, yet it is still required for Maxwell's equations to work. These equations are a mainstay of current electromagnetic theory. We must therefore accept the improbable aether or look for an alternative theory explaining a capacitor's function.

One of many quotes attibuted to Filipo Bruno, my nomination as the patron saint of science & truth. Burnt at the stake by the Roman Catholic church in the 16th century, for refusing to recant the truth. The wording, from Latin, is modernised by me, but is fundamentally original:
"I make no personal claim to the truth, only the right to seek it, prove it in argument, and to be wrong many times in order to reach it."
Dot
Science has, over the last century, divided into more & narrower specialisations. One such branch, physics, has sub-divided into further specialisations. On the other hand, some workers generalise, becoming multi-disciplinary, or 'jack of all trades'.

Specialising means learning more and more about less & less.
Until ultimately everything is known about nothing.

Multi-disciplinary or generalising means learning less and less about more & more.
Until nothing is known about everything.

The T.W. Hammonds
tab links to the Tone Wheel Hammond Organ page. A series will develop on Major Classics, probably also some minor ones.

Included will be guides to routine maintenance, building on the listings in the Classic Instruments section. Some dos & don'ts together with tips etc. Plus sections on emergency repairs. Not just classics, a small amount of modern gear may achieve that status if looked after & maintained.

As we progress, feedback from readers will be welcome. Ron.

Our Music Industry connection goes back to the mid '60s independent since 1968.

All Brand & Model names are Trademarks and/or Copyright of their respective owners.

Statistical Post Script

A statistic quoted by someone writing about the irrelevance to potential pupils of much music taught in schools. Six percent take up music courses offered.

Back in the early '60s the only keyboard instruments available for general sale were acoustic pianos, electric & electronic organs. The question was asked of the industry. 'Why are almost no organs built for professional use?'

The answer given was that professionals were only one percent of the market. Multiplying those figures means that one percent of six percent of people are professional musicians. That is six people in every ten thousand.

The experience of professionals, as I see it, is that there is a one percent chance of making a living from music, the rest do it out of working hours. That leaves six in one million. Probably no more than one percent of those will make a fortune.

That makes six in a hundred million. There are about six thousand million people on Earth, fifty million in Britain. Three hundred and sixty will become wealthy from music, only three in this country.

Information given is generally brief & is based on our experience. If you spot any factual mistakes or 'typos' please feel free to let us know. We are not perfect & won't sulk over constructive criticism.

All Brand & Model names are Trademarks and/or Copyright of their respective owners.

Regards & thanks for reaching our Site, a developing project.
Watch this space.

E-Mail: organs@alphaentek.com