Komentary on W W Sawyer's: A Path To Modern Mathematics.

( 3 ) Kupl'd osilator and wav equation.

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( Kapital-i, in 'I, myself', now spels Il as in isle or aisle.
Leter y spels sym for seem or seam and partys for parties.
Leter w spels swn for soon. )


( Notation: Symbols normaly in sub-skript ar presyded by a koma.
Powers, normaly in super-skript, ar presyded by a singl quotation mark, tho x-squar'd, for instans, is riten the normal way, x. )


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Weited vibrating string.

An exampl of a kupl'd osilator is tw weits osilating on the saim stretch'd pys of string. If they osilat with ych other, they wil shaip the string into an arch wen they go up together, and into a dip or bowl's out-lIn, wen they go down together. Ther osilations ar said to be in feiz.
The weits ar exaktly out of feiz, wen they ar moving in exaktly the oposit direktions. So, wen uon weit had jump'd up to its furthest, the other weit has jump'd down its furthest. If the tw objekts ar the saim weit, they hav gon equal distanses in the oposit direktion. ( That asums gravity dosnt exert an important downward pul, in this kais. )

The out-of-feiz motion of the weits givs the string a zig-zag shaip of uon wav-lenth. The in-feiz motion, over the saim lenth of string, forms haf a wav-lenth. Therfor, the in-feiz haf wav-lenth is twIs the lenth of a haf wav-lenth for the out-of-feiz string motion.

The mor weits ther ar to kupl'd osilators, the mor ther osilations aproximat to smwth wav motions. So, kupl'd osilators wer orijinaly study'd as an aid to understanding the propertys of wavs. In partikular, they aproximat to the wav equation, as a partial diferential equation, wen the weits ar asum'd to be kontinuus along a lenth of string, in efekt given a sertain mas density, forming smwth ( not zig-zag or angular ) wavs.

The tw previus web pajes konsider'd ordinary diferential wav equations ( damp'd or undamp'd ) wer the displasment, from equilibrium, of a weited spring, or of a tens string, was the dependent variabl on uon independent variabl, the tIm. But osilations kan tak plas in spas, as wel as tIm, for instans, if the weited spring akted on the surfas of a liquid, making wavs.

LIkwIs, visibl wavs ar form'd along a lenth of tens'd string. In a partial diferential equation, the string displasment is a funktion of at lyst tw independent variabls, tIm and spas in uon dimension, if not tw or thry dimensions, as fysikal wavs kan be linear, planar or sferikal.

Ther ar tw basik kinds of wavs, longitudinal wavs and transvers wavs. If yu fasten a pys of string to a post and hold the other end, yu kan mak longitudinal wavs with a quik bak and forward mov of the hand. Transvers or perpendikular wavs ar mayd by an up and down jerk of the hand. The strenth and direktion of the wavs depends on the enerjy and direktion of the motion yu impart to them.
LIkwIs, a hamer tap, to the end of a metal bar, along its lenth, sets up longitudinal vibrations. Wer-as, a tap downward, nyr uon end, sets up transvers vibrations in the bar.

In simpl terms, the mathematikal form of the partial diferential equation of wav motion, for instans in the metal rod, is much the saim. The klasikal wav equation of the vibrating string is for transvers motion.

In the tw previus web pajes, diferens equations wer konsider'd as mor or les aproximat substituts of aljebra for the kalkulus of diferential equations. The kupl'd osilator servs as a fysikal model for kalkulating the diferens equation's aproximation to a diferential equation, bekaus the kupl'd osilator's angular wav forms ar a fysikal aproximation to smwth wav motion.

In fakt, the equation of motion, for the midl of thry mases on a vibrating string, found as a diferens equation, is also equal to the diferential equation ( for smal displasments of the string ).
Sy graf:

Displasments of thry smal mases on a string under tension.

The midl mas, or nth mas, with the hIest displasment from the string's equilibrium, is pul'd down on both sIds by the string tension, T.

That is fors, from the left, T sIn P = ( a,n - a,n-1 )/ ^x,
and fors, from the rIt, T sIn Q = ( a,n - a,n+1 )/ ^x.

By Newton's laws of motion, the joint fors of the string tension is equal and oposit to the fors, wich equals the mas tIms akseleration, responsibl for the displasment, a,n, of the midl weit:

m.da,n/dt = - T( sIn P + sIn Q )
Therfor, da,n/dt = ( a,n-1 - 2a,n + a,n+1 )T/m.^x.

The rIt hand sId of the equation relats to sekond order diferens equations ( as in web paj on the Fibonacci serys ). And ( as relats to the previus paj on sirkular funktion and the ordinary wav equation ) the displasment, a,n kan be konsider'd simpl harmoni, in tIm, t, with a frequensy of osilation, w, about its rest position or equilibrium.

That is to say a,n = A,n exp(iwt), wer A,n is the maximum displasment.

Exaktly the saim aplIs to the displasments of the other tw weits, hws displasments, a,n-1 and a,n+1 hav ther own maximum displasments, A,n-1 and A,n+1, respektivly.

The sekond derivativ of a,n kan now be equated to its sekond derivativ in exponential form ( as a sirkular funktion ) and the thry weits' displasments replas'd by ther maximum displasments:

da,n/dt = - wA,n.exp(iwt)
= ( A,n-1 - 2A,n + A,n+1 )exp(iwt)T/m.^x.

This reduses to the standard form:

- A,n-1 + ( 2 - ^x.wm/T )A,n - A,n+1 = 0.

This diferens equation of motion for n mases is solv'd with the equation repyted with substitutions of n = 1, 2, 3, etc up to the number of mases on the string. This kalkulation is only simpl for a fw mases.


Tw-weited string.

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The folowing figur is of a string with tw mases.

Tw mases on the string requir the standard form to be solv'd for n = 1 and n = 2.

To simplify maters further, the tension, T, mas, m, and distans, ^x, betwyn the weits on the string, ar al set at uon unit.

Substituting n = 1 into the standard form:

( 2 - w )A,1 - A,2 = 0.

Substituting n = 2 into the standard form:

- A,1 + ( 2 - w ) A,2 = 0.

A,n-1 in the former substitution equation and A,n+1, in the later equation both ar zero, bekaus they represent the zero-amplitud fix'd ends of the string, next to uon or other of the tw weits. ( A,0 = A,3 = 0. )

By the method of solving simultanus equations, substitut, say, the valu of A,1 in the sekond equation, into the first equation, so:

( 2 - w )A,2 - A,2 = 0.

A,2 kan be eliminated and the equation faktoris'd:

( 2 - w - 1 )( 2 - w + 1 ) = 0.

Hens, w is either 1 or 3. The tw valus for w ar the slow and the fast frequensys of osilation. In the former kais, the weits ar lwsly kupl'd in feiz. In the later kais, the tw weits ar tItly kupl'd out of feiz. ( Sy abov figur. )

The kupl'd weits hav byn konsider'd in terms of ther maximum displasments A,1 and A,2. But equations of motion kan also be given in terms of the tw weits' variabl displasments: kal them X and Y, respektivly. A sketch of the prosedur folows.

As with the sekond derivativ for a,n, ( wer T, m and ^x, ar set at uon ) ther kan be equations for the sekond derivativs of X and Y:

dX/dt = - 2X + Y.

dY/dt = X - 2Y.

Thys ar the equations of motion for the displasments of the tw weits. But equaly and opositly, they ar the equations, given oposit sIns, of the forses puling bak the stretch'd string from thos displasments.
It turns out that thys equations betwyn string displasments and forses ar the ky to a simpl expresion of ther relationship, thru a transformation of ko-ordinats from terms of X and Y to an other ko-ordinat system, naimly of eigen-vektors ( as diskus'd in the tw previus web pajes ).


Equivalent konserv'd systems.

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The ryson for this requirs som bak-ground explanation. Diferent fysikal systems may hav the saim mathematikal form, so that ther terms are analjus, tho they may not apyr alIk. The systems ar asum'd to be klos'd, with no signifikant los of enerjy thru friktion. They both obey konservation of enerjy, wich konsists of potential enerjy and kinetik enerjy.

Wen a vibrating string ryches its maximum displasment, al its enery is potential enerjy. Ryching its furthest extent, its kinetik enerjy or enerjy of motion syses a moment, until it starts to mov bak in the oposit direktion. The string's kinetik enerjy inkryses to a maximum at its equilibrium position, wen it has no potential enerjy. Pasing this mid-position, it starts to lws kinetik enerjy again, as the string's displasment inkryses in the other direktion.

Motion on a land-skaip ofers a simpl piktur, to kompar with mor komplikated-lwking systems. On a roler-koster, potential enerjy is greitest on the krests, wich the vehikl has slow'd almost to a stop in ryching. Kinetik enerjy is greitest in the dip, wen maximum akseleration has byn rych'd, and befor gravity gradualy slows down the charj up the next sloup.

Konsider a simpl system lIk an osilating weit on a spring. Potential enerjy is given by kx/2, wer k mIt be the elastik konstant of a spring. Kinetik enerjy is mv/2, wer m is for mas and v for velosity or rait of chanj of x.
The kupl'd osilator mIt also be konsider'd as a spring, insted of a string, with tw inter-mediat mases.
Given that the behaviors of konserv'd enerjy systems is determin'd by ther potential enerjy and their kinetik enerjy, thys systems kan mov in step. It depends if ther respektiv enerjy konstants k and m ar the saim.
The potential enerjy of the kupl'd osilator, tw weits on a stretch'd string, is found to be:

X - XY + Y.

This is determin'd by finding out how much the string is stretch'd by the weits. Noing the orijinal lenth of the string ( previusly tryted as thry unit lenths markd-of by the tw weits' positions ) the stretch'd lenths ar found by Pythagoras' theorem and by making yus of the fakt that for smal k, ( 1 + k ) ~ 1 + k/2.

( The kinetik enerjy of the kupl'd system is: m((dX/dt)) + (dY/dt))/2. )

The motion of a weit on a spring is mathematikly analjus to the motion of a bal in a bowl. LIkwIs, the motion of tw weits on a string is analjus to a bal's motion in an oval bowl, a bit lIk a boat. This can be shown by drawing the graf of the kupl'd system's potential enerjy, yusing the X and Y axes of the tw weits' displasments, with the potential enerjy as a third dimension of hIt.

For any given konstant level of potential enerjy, an oval kontur is map'd onto the graf, forming a serys of nested ovals depending on the 'hIt' of the potential enerjy. This is just lIk the jeografikal map of a valy, hws konturs wud show a perfektly oval holow.
But the lenth and the bredth of this holow or bowt ar not in lIn with the X and Y ko-ordinats. Insted, they ar at forty-five degrys to them. To mesur the lenth and bredth of the potential enerjy holow direktly wud requir nw axes, kal them U and V, at forty-fIv degrys to the old axes.

Asuming no los of enerjy thru friktion, the motion of a bal, from top to top, the lenth of the holow, is analjus, in mathematikal form, to the in-feiz osilation of the kupl'd system. This is along the U-axis, at Y = X. The bal roling from bredth to bredth of the holow is analjus to the out-of-feiz osilation. This is along the V-axis, wer Y = - X. It koresponds to the equal and oposit displasments of the tw weits on the string.

Thys tw osilations ( shown in the abov figur ) korespond to the fundamental and first harmonik in musik. Musicians, abl to hyr both nouts at uons, gav mathematicians the idea of ading such basik solutions to get mor komplikated solutions.
In terms of a bal roling down from the rim of the holow, this myns konsidering the mor komplikated motions wen the bal rols other than from a long end or a brod end. The nyrer the bal starts from the long end, the shalower wil be the path of its evolution; the nyrer the bal's start from the brod sId, the styper its rol.
To visualis how this adition of simpl solutions kan reprodus al posibl motions of the bal in the holow, uon may imajin the bal's motion mimik'd by the spot wer tw wIrs kros at rIt angls. Thys wIrs' motions ar govern'd by ther respektiv atachments to a long spring osilating at a lower frequensy than a shorter spring. The springs, paralel to the U and V axes, mimik respektivly the long shalow sloup and the narow styp sloup of the holow.
Indyd, this is a mekanikal model of the paralelogram law of vektor adition.

So, it turns out that thys nw axes ar the so-kal'd eigen-vektors or 'proper' vektors, in wich to formulat the motion of the system.

Re-formulated relation of displasments to forses.

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Returning to the end of the sektion on the tw-weited string, the eigen-vektors show an especialy simpl relation betwyn displasments and forses.

The equations for the displasments, X and Y, of the tw weits, ar now expres'd in terms of the equal and oposit forses, label'd X* and Y*, puling them bak towards the string's equilibrium position:

X* = 2X - Y.

Y* = - X + 2Y.

X and Y, as rektilinear ko-ordinats, ar, by konvention, drawn on a graf pointing to the yst and north, respektivly. The eigen-vektors V and U ar the ko-ordinats to the north-west and the north-yst, respektivly. As such, the ordinat lIn U is positiv with respekt to both the X and Y axes. An X,Y ko-ordinat position, lIk 1,1, wich is on the U ordinat, kan show wat hapens to the displasment-fors equation, in terms of that eigen-vektor:

Wen X* = 2.1 - 1 = 1, and Y* = -1 + 2.1 = 1, the ko-ordinats remain unchanj'd. As diskus'd on the tw previus web pajes, the 'eigen-valu', of eigen-vektor, U, is uon.

The situation difers for eigen-vektor V, bekaus its position is represented by X,Y ko-ordinats, -1,1. Substituting thys: X* = -2 - 1 = -3. Y* = - -1 + 2 = 3. In terms of the V axis, the transformation of displasment into fors stretches the V axis thry tIms. In other words, the eigen-valu of V is thry.

The equations of motion, in terms of U and V, ar found by substituting X = U + V and Y = U - V into the equations of motion for X and Y. Thys kom out at dU/dt = - U and dV/dt = - 3V, again showing the simpl relation betwyn the oposing forses of string displasing mas and string elastisity, wen in terms of the eigen-vektors U and V. The solutions of U and V ar the usual form for the simpl harmonik osilator:

U = A kos t + B sIn t.

V = C Kos t3 + D sIn t3.
Wer A, B, C, D ar konstants.

Thus, an under-lIing simplisity to the dynamiks of the kupl'd osilator system is shown up by finding the proper ko-ordinats.


The ( partial diferential ) wav equation.

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The first sektion of this paj mention'd that the kupl'd osilator was lIk a mekanikal model of a finit diferens equation. This is a diskryt version of a diferential equation. The mases at periodik intervals on the string ar imajin'd to get smaler and kloser until they merj into a kontinuusly dens string, hws vibrations sys to be angular, or zig-zag from the separat aktion of the weits, but bekom a kontinuusly smwth wav form.

The equation of motion of the nth mas was given as:

da,n/dt = ( a,n-1 - 2a,n + a,n+1 )T/m.^x.

For the mases to merj into a kontinuus hevy string, the distanses betwyn them must aproch zero, or, ^x --> 0.

The braketed terms relat to a sekond order finit diferens. As shown on the Fibonacci serys paj, that myns they involv a diferens betwyn tw diferenses, or a chanj in a chanj. So, the equation kan be re-riten:

da,n/dt = ( ( a,n+1 - a,n )/^x - ( a,n - a,n-1 )/^x )T/m

= ( (^a/^x),n+1 - (^a/^x),n )T/m,

wer the ^ sIn is ment to serv as a delta sIn, yus'd in finit diferens equations as the equivalent term to the d for diferentiation.

Indyd, ( da/dx ),x+dx - ( da/dx ),x = ( da/dx )dx.

The konditional terms after the komas ( usualy denoted as sub-skripts ) nyd not be konsider'd as ^x --> 0. Therfor:

da/dt = ( da/dx )dx.T/m.

This is the wav equation as a partial diferential equation, wer the wav displasment, a, is at position, x, on the string, and at tIm, t.
The konstant, mas per unit lenth, m/^x, is kal'd the linear density, p. And T/p = v, wer v, in this kais, is the feiz velosity, or the velosity of the wav, defin'd as a 'feiz of osilation'.


Richard Lung.


Referenses:

W W Sawyer: A Path To Modern Mathematics. ( 1966. )

H J Pain: The Physics Of Vibrations And Waves. ( 1980 )

Marcelo Alonso and Edward J Finn: Fundamentals Of University Physics. ( 1967 )
( Exampl 18.5 shows the relation betwyn ordinary and partial differential wav equations for the longitudinal wavs in a spring. )
A later edition has Paul Davies as jeneral editor and is just kal'd: Physics.



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