en>David Eppstein: unstub

2013-12-05T04:02:12Z

unstub

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	The ~~name of~~ the ~~writer~~ is ~~Figures~~. ~~Supervising~~ is ~~my profession~~. ~~North Dakota~~ is ~~our birth location~~. The ~~factor she adores most~~ is ~~physique building~~ and ~~now she~~ is ~~attempting~~ to ~~earn cash~~ with it.<br><br>~~My web page~~ ... at ~~home std test~~ ([http://~~ngonama~~.~~org~~/~~members~~/~~therestephens~~/~~activity~~/~~208304~~/ ~~inquiry~~])		{{Refimprove\|date=October 2010}}
			[[Image:Standing waves on a string.gif\|thumb\|250px\|Vibration, [[standing wave]]s in a string. The [[fundamental frequency\|fundamental]] and the first 5 [[overtone]]s in the [[Harmonic series (music)\|harmonic series]].]]

			A [[vibration]] in a [[strings (music)\|string]] is a [[wave]]. Usually a '''vibrating string''' produces a [[sound]] whose [[frequency]] in most cases is constant. Therefore, since frequency characterizes the [[pitch (music)\|pitch]], the sound produced is a constant [[note]].
			Vibrating strings are the basis of any [[string instrument]] like [[guitar]], [[cello]], or [[piano]].

			== Wave ==

			The speed of propagation of a wave in a string (<math>v</math>) is proportional to the square root of the tension of the string (<math>T</math>) (discovered by [[Vincenzo Galilei]] in the late 1500s) and inversely proportional to the square root of the linear density (<math>\mu</math>) of the string:

			<math>v = \sqrt{T \over \mu}.</math>

			===Derivation===

			[[Image:StringParameters.png\|right\|Illustration for a vibrating string]]
			Let <math>\Delta x</math> be the [[length]] of a piece of string, <math>m</math> its [[mass]], and <math>\mu</math> its [[linear density]]. If the horizontal component of [[Tension (mechanics)\|tension]] in the string is a constant, <math>T</math>, then the tension acting on each side of the string segment is given by

			:<math>T_{1x}=T_1 \cos(\alpha) \approx T.</math>
			:<math>T_{2x}=T_2 \cos(\beta)\approx T.</math>

			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, <math>a</math>, will be equal to the net force on the piece:

			:<math>\Sigma F_y=-T_{2y}-T_{1y}=-T_2 \sin(\beta)-T_1 \sin(\alpha)=\Delta m a\approx\mu\Delta x \frac{\partial^2 y}{\partial t^2}.</math>

			Dividing this expression by <math>T</math> and substituting the first and second equations obtains

			:<math>-\frac{\mu\Delta x}{T}\frac{\partial^2 y}{\partial t^2}=\frac{T_2 \sin(\beta)}{T_2 \cos(\beta)}+\frac{T_1 \sin(\alpha)}{T_1 \cos(\alpha)}=\tan(\beta)+\tan(\alpha)</math>

			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

			:<math>\frac{1}{\Delta x}\left(\left.\frac{\partial y}{\partial x}\right\|^{x+\Delta x}-\left.\frac{\partial y}{\partial x}\right\|^x\right)=\frac{\mu}{T}\frac{\partial^2 y}{\partial t^2}</math>

			In the limit that <math>\Delta x</math> approaches zero, the left hand side is the definition of the second derivative of <math>y</math>:

			:<math>\frac{\partial^2 y}{\partial x^2}=\frac{\mu}{T}\frac{\partial^2 y}{\partial t^2}.</math>

			This is the wave equation for <math>y(x,t)</math>, and the coefficient of the second time derivative term is equal to <math>v^{-2}</math>; thus

			:<math>v=\sqrt{T\over\mu},</math>

			where <math>v</math> is the [[speed]] of propagation of the wave in the string. (See the article on the [[wave equation]] for more about this). However, this derivation is only valid for vibrations of small amplitude; for those of large amplitude, <math>\Delta x</math> 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 <math>T</math>.

			== Frequency of the wave ==
			Once the speed of propagation is known, the [[frequency]] of the [[sound]] produced by the string can be calculated. The [[speed]] of propagation of a wave is equal to the [[wavelength]] <math>\lambda</math> divided by the [[Wave period\|period]] <math>\tau</math>, or multiplied by the [[frequency]] <math>f</math> :

			:<math>v = \frac{\lambda}{\tau} = \lambda f.</math>

			If the length of the string is <math>L</math>, the [[Fundamental frequency\|fundamental harmonic]] is the one produced by the vibration whose [[node (physics)\|node]]s are the two ends of the string, so <math>L</math> is half of the wavelength of the fundamental harmonic. Hence one obtains [[Mersenne's laws]]:

			:<math>f = \frac{v}{2L} = { 1 \over 2L } \sqrt{T \over \mu} </math>

			where <math>T</math> is the [[Tension (mechanics)\|tension]], <math>\mu</math> is the [[linear density]] (that is, the [[mass]] per unit length), and <math>L</math> is the [[length]] 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 <math>\lambda_n = 2L/n</math>, then we easily get an expression for the frequency of the nth harmonic:

			:<math>f_n = \frac{nv}{2L}</math>

			And for a string under a tension T with density <math>\mu</math>, then

			:<math>f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}</math>

			== 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 [[Cathode ray tube\|CRT]] 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 lamp]], at a rate that 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 and other non-oscillating light sources, this effect does not occur and the string appears still but thicker, and lighter or blurred, due to [[persistence of vision]].

			A similar but more controllable effect can be obtained using a [[stroboscope]]. This device allows matching the frequency of the [[xenon flash lamp]] to the frequency of vibration of the string. In a dark room, this clearly shows the waveform. Otherwise, one can use [[bend (guitar)\|bend]]ing or, perhaps more easily, by adjusting the machine heads, to obtain the same, or a multiple, of the AC frequency to achieve the same effect. For example, in the case of a guitar, the 6th (lowest pitched) string pressed to the third fret gives a G at 97.999 Hz. A slight adjustment can alter it to 100 Hz, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia, 50 Hz. In most countries of the Americas—where the AC frequency is 60 Hz—altering A# on the fifth string, first fret from 116.54 Hz to 120 Hz produces a similar effect.

			== See also ==
			* [[Fret\|Fretted instrument]]s
			* [[Musical acoustics]]
			* [[Vibrations of a circular drum]]
			* [[Melde's experiment]]
			* [[3rd bridge]] (harmonic resonance based on equal string divisions)

			== References ==
			*{{cite journal \| last=Molteno \| first=T. C. A. \| coauthors=N. B. Tufillaro \| title=An experimental investigation into the dynamics of a string \| journal=American Journal of Physics \|date=September 2004 \| volume=72 \| issue=9 \| pages=1157–1169 \| doi=10.1119/1.1764557\|bibcode = 2004AmJPh..72.1157M }}
			*{{cite journal \| last=Tufillaro \| first=N. B. \| title=Nonlinear and chaotic string vibrations \| journal=American Journal of Physics \| year=1989 \| volume=57 \| issue=5 \| page=408 \| doi=10.1119/1.16011\|bibcode = 1989AmJPh..57..408T }}

			== External links ==
			* [http://www.falstad.com/loadedstring/ Java simulation of waves on a string]
			* [http://www.johnsankey.ca/string.html Physics of a harpsichord string]
			* [http://www.acoustics.salford.ac.uk/feschools/waves/string.htm#fullGuitar A friendly explanation of standing waves and fundamental frequency]
			* "[http://demonstrations.wolfram.com/TheVibratingString/ The Vibrating String]" by Alain Goriely and Mark Robertson-Tessi, [[The Wolfram Demonstrations Project]].

			{{Acoustics}}

			[[Category:Mechanical vibrations]]
			[[Category:Sound]]
			[[Category:String instrument construction]]

			[[da:Snorbølger]]
			[[fr:Corde vibrante]]
			[[de:Saitenschwingung]]
			[[it:Corda vibrante]]
			[[ko:끈의 진동]]
			[[hu:Húr]]

en>David Eppstein: not unreferenced (OEIS is a ref) but not well referenced either

2012-08-29T05:23:24Z

not unreferenced (OEIS is a ref) but not well referenced either

New page

The name of the writer is Figures. Supervising is my profession. North Dakota is our birth location. The factor she adores most is physique building and now she is attempting to earn cash with it.<br><br>My web page ... at home std test ([http://ngonama.org/members/therestephens/activity/208304/ inquiry])

Centered cube number - Revision history

en>David Eppstein: unstub

en>David Eppstein: not unreferenced (OEIS is a ref) but not well referenced either