Komentary On W W Sawyer's A Path To Modern Mathematics.

( 2 )Sirkular funktions and the wav equation.

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


( Notation:
subskripts to a term ar shown by a koma in betwyn the term and the symbols usualy subskripted: x,n myns the n-th term of x insted of the konvention that n is a sub-skript.
Super-skripts, such as powers, ar shown by a quotation mark betwyn the term and its power: x'n myns x to the power of n. But the usual notation x is yus'd for x squar'd.
)

Links to sektions:


A simpl periodik serys.

We kan yus the teknyk of the karakteristik equation for the much simpler wav equation, that is the undamp'd wav equation. If the damp'd wav equation, in its krud Fibonacci serys version, wer undamp'd, then the wav wud maintain its orijinal amplitud of ( plus or minus ) 0.618, at every subsequent krest and trof of its zig-zag sIkl.

For simplisity's saik, mak the maximum amplitud unity. A sirkl, of unit radius, kan be kros'd thru its senter by a vertikal y-axis, of hIt, 1, and depth, - 1, and a horizontal x-axis, mesuring 1 to the rIt, and - 1 to the left of sirkl senter. If yu rol'd out the sirkl sirkumferens, it wud mesur tw pi tIms the radius, as a horizontal ordinat in a Cartesian x-y ko-ordinat system, insted of the sirkular or polar ko-ordinats of radius, r, and angl ( say, angl Q ) swept out by the radius.

Wen x equals uon or minus uon, y equals zero, and vice versa. Tak uon komplyt anti-klokwIs swyp of the radius, mesur'd in terms of y = 1, at the top of the sirkl ( wer x = 0 ) down to y = 0 ( wer x = - 1 ) and down to y = - 1, at the botom ( wer x = 0 ) up to y = 0 again ( at x = 1 ) and rIt round to the top.

In terms of the equivalent Cartesian or rektilinear ko-ordinats, a lIn from y = 1, on the y-axis, wud slant down thru the x-axis, at uon quarter of the lenth, 2πr, ryching a trof at haf that lenth, and zig-zaging bak up, in a similar maner. ( Ych quarter point koresponds to the radius swyp moving thru a quadrant of the sirkl, a 90 degrys angl. ) This wud form a v-shaip or simplest aproximation to uon wav-lenth, from krest to krest.
For every komplyt swyp of the sirkl, an other wav-lenth wud folow on along the x-axis in the rektilinear system, lIk a serys of ripls in a pond.

The aljebra of thes repyted swyps or repyted wav-lenths is a periodik serys: 0, -1, 0, 1, 0, - 1, 0, 1, 0, - 1, 0, 1,..

In the anti-klokwIs swyp of the sirkl, divided into quadrants by x-y axes, x mesur'd this serys, from the first term, wer-as the y-ordinat pair'd off, from the sekond term in the serys, with the x-ordinat. This was also the kais with the previus web-paj trytment of the Fibonacci serys, wich we now repyt, yusing the karakteristik equation.

We tak for the first tw x-y pairs ( the initial konditions ) of this sirkular serys, x = 0 and y = 1, x = - 1 and y = 0. Thys ar a, b, c, d, respektivly, in the karakteristik equation:

T + ( a + d )T + ( ad - bc )I = 0.

Therfor, T + 1 = 0. And T = +i or -i. ( Wer i equals the squar rwt of minus uon. )

( Notis that the mid-term, the first order term in T is mising. Konsider'd as an auxiliary equation to a wav equation, this mising term disipats the enerjy of the wav. This indikats a simpl wav equation without a damping efekt on the amplitud. )

Plus or minus i ar the eigen-valus of tw nw axes, X and Y, wich eigen-funtions under-go no expansion or kontraktion, bekaus both valus ar unitary. The equation of Y is: iy = - x. The equation of X is: iy = x. Therfor, the nw axes apyr at 45 degres to the old, so the sirkl is split into eit equal segments.

As for finding the n-th term of the Fibonacci serys, the eigen-valus kan be yus'd to find the n-th term of the periodik serys, x,n:

x,n = C( i )'n + D( - i )'n.

Taking x,n = 0, C + D = 0. Therfor, C = - D.

Taking x,n = - 1,
- 1 = iC - iD. Therfor, - 1 = 2iC. Or, C = i/2. And D = - i/2.

Substituting:

x,n = ( i/2 )( i )'n - ( i/2 )( - i )'n.

Tabl 3 shows how this formula works for the first fIv terms. Tho, simpl arithmetik for this 4-term periodik serys, mor ysily gIds us to the anser, rather as we no wich yer is a lyp yer:
Let n be any od number. Al the terms, hws order in the serys is od, equal zero: x,n = 0.
Let n+1 be al the even order'd terms. Al thys terms, from the sekond term onwards, hws number order is divisibl by tw but not by for, equal minus uon: x,(n+1)/2 = - 1.
Al thys terms, from the forth term onwards, hws number order is divisibl by for, equal plus uon: x,(n+1)/4 = 1. Therfor, x,n = x,(n+1)/2 + x,(n+1)/4.

Tabl 3: finding the n-th term of a simpl periodik serys.
n ( i/2 )( i )'n - ( i/2 )( - i )'n x,n
0 i/2 - i/2 0
1 - 1/2 - 1/2 - 1
2 - i/2 - - i/2 0
3 1/2 - - 1/2 1
4 i/2 - i/2 0

Noing the n-th term of x, x,n, also determins its y partner, wich equals the folowing term in the periodik serys: y,n = x,n+1. Taking x and y together as a komplex variabl, we also no the n-th komplex number, z,n = x,n + iy,n.
Komplex numbers refer to order'd pairs of numbers, that kan lokat or ko-ordinat any point on a playn.

We noted that the nw X-Y ko-ordinats split the sirkl into eit segments. Ther is no ryson wI we shud not yus them to plot an eit term periodik serys, insted of for. The amplitud, at the for nw points, at forty fIv degres to wer the orijinal x-y ko-ordinats tuch'd the sirkumferens ( deriv'd from the sIn of that angl, given a radius of uon for the hypotenus ) is about 0.707.
The nw periodik serys wud be: 0, - .707, - 1, -.707, 0, .707, 1,..repyted.

Translating this mor detail'd serys, from sirkular or polar ko-ordinats to rektilinear ko-ordinats, the v-shayp'd wav wud under-go a sekond aproximaton, to the normal smwth kurv, with bends nyr either sId of the krest and trof of the wav-lenth, giving a first indikation that a wav kurv has shalower sloups ther.

Further sub-divisions of the sirkl wud translat into kloser aproximations to a smwth wav shaip. In this way, a diferens equation aproches a diferential equation, the ordinary wav equation.


The ryl and imajinary parts, of a komplex number and its konjugat, as ther eigen-vektors.

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Al the mathematiks jargon, of this sektion tItl, expreses som quIt simpl ideas. The ryl and imajinary parts ar simply the order'd pairs of numbers, label'd x and y, respektivly, from the previus sektion. The y ordinat is multiply'd by the leter i ( for imajinary ). Then the x and y ordinats ar pair'd with an adition sIn and kal'd a komplex number, denoted z, lIk so: z = x +iy.

This kan be represented on a graf of x and y, wer y is the vertikal and x is the horizontal axis. But this is not to be konfus'd with the traditional graf of y as a funktion of x, expresing a ratio of y over x, or y/x, lIk velosity equals distans travel'd over tIm.

The komplex number servs a diferent funktion, naimly to pin-point every posibl spot on a playn or squar, mark'd-off by the x and y axes. In partikular, z may represent every posibl point on the sirkumferens of a sirkl, hws senter is drawn from the orijin of the x and y ko-ordinats. ( See graf of the komplex number. )

On the graf, the x and y quadrants ar mark'd 1, i, - 1, - i. This is aktualy a short-hand for the for positions of the komplex number z, wer either x or y is zero. The ful deskription of z at thys for points is, respektivly: z = x + iy = 1 + i0, 0 + i1, - 1 + i0, 0 - i1.

The graf shows a partikular point for z on the sirkl rim, indikated by a radial lIn, r ( for radius ) at an angl of Q degrys betwyn the x-axis and the radial lIn. ( The usual symbol for the angl is not Q but Greek leter theta. )

The graf also shows a miror imaj or reflektion of point z, below the x-axis. It so hapens, that wen uon works out the aljebra, this miror imaj position koms to the invers of z, or 1/z, wich equals x - iy. This miror position has a special naim, kal'd the komplex konjugat. ( It also has a special symbolisation of the leter z given a tild, or wavy lIn, over it. )

The komplex number z and its komplex konjugat, 1/z, ar vektors, radius vektors, in this kais. That is, they hav both magnitud, in the lenth of the radius, r, and they hav direktion, in the angl Q of the radius with the x-axis. The vektor z is chanj'd or transform'd, by miror reflektion, into its konjugat vektor, 1/z. But in that proses, the ko-ordinats x and y ar not chanj'd in direktion. Konsider'd as vektors in ther own rIt, they ar so-kal'd eigen-vektors, or vektors that dont chanj ther direktion under a transformation, such as miror reflektion.

This is as it shud be, bekaus the problem of finding eigen-vektors, is the task of finding the ko-ordinats that stait a mathematikal situation most transparently, for the solving of problems.

How-ever, eigen-vektors hav eigen-valus. This simply myns that ther magnitud may be chanj'd: they may stretch or shrink. In the exampl of the graf, the x-axis remains unchanj'd by miror reflektion. In efekt, it is multiply'd by uon, or its eigen-valu is plus uon. The efekt of miror reflektion on the y-axis is to multiply it by minus uon, for an eigen-valu, bekaus the komplex konjugat is on the negativ y-axis.

The forgoing explanation of x and y as eigen-vektors kan be presented, lwsly, in terms of a theorem that every funktion kan be split into a symetrik funktion and an anti-symetrik funktion. Thys ar eigen-vektors, bekaus a symetrik funktion is uon wich is left unchanj'd by an operation upon it, just as the x-axis was left unchanj'd by miror reflektion of z into 1/z. Also, an anti-symetrik funktion myrly chanjs the sIn of a funktion, as the sIn of the y-axis chanj'd, thru miror reflektion of z into its komplex konjugat, 1/z.

( Symetry deriv'd from jeometrikal konsiderations. For instans, the human fais is rufly symetrikal. If the hair is parted down the midl of the hed, then the miror imaje's parting wil be, tu. The mirror imaj lyvs the hed's midl parting unchanj'd: ther partings ar symetrikal. But a left-parted hed of hair wil hav a rIt-parting for a miror imaj. A parting on uon sId and its reflekted imaj ar anti-symetrikal to ych other. )

The theorem of symetrik and anti-symetrik parts is staited in terms of the standard text-bwk formulas ( given hyr without prwf ) for z and 1/z in terms of rektilinear ko-ordinats x and y, and polar ko-ordinats, r and Q ( hyr standing in for angl theta ):

z = x + iy = r ( kos Q + i sin Q ) = r exp(iQ).

1/z = x - iy = r ( kos Q - i sin Q ) = r exp(-iQ).

Ading the tw sets of equations:

z + 1/z = 2x.

This kan be shown to be a symetrik expresion under the operation of miror reflektion, denoted M. This operator works similarly to the operator, i, wich aplI'd to y myns a turn thru nInty degrys with respekt to the x-axis. The diferens is that M reflekts z, thru the angl Q about the x-axis, from the positiv y-axis onto the negativ y-axis. An other operator yus'd is the Identity operator, I, wen aply'd to a funktion, myrly lyvs it as it is: Iz = z. ( This was yus'd, on the previus web paj, in the karakteristik equation. )

Hens, Mz = 1/z.

So, 2x = z ( I + M ).

But a miror operation aplI'd twIs ( denoted by M ) produses a situation identikal to the orijinal:

M = I or M - I = 0.

It folows that: z ( M - I ) = 0 = z ( M + I )( M - I ) = 2x ( M - I ).

Therfor, Mx = Ix. In other words, the operation of reflektion on x is the saim as the Identity operation on x. Or miror reflektion produses no chanj on x.

Uon kan also subtrakt 1/z from z, for 2iy. Folowing the saim resoning, as for x, this shows My = - Iy, or that the miror operation on y chanjes its sIn, so that it is anti-symetrik with the identity of y.


The ordinary wav equation.

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For exampl, the wav equation kud deskrib the regular osilations, over tIm, of a spring, of elastisity, k, with a bob, of mas, m, on the end. The mas puls the spring down and the elastisity puls it bak in a kontinuing jig. A simplify'd form of the equation has k and m equal, or k/m = 1. Normaly, k/m is set equal to w, the squar of the angular frequensy, wich apyrs in the equation as the ko-eficient of the dependent variabl.

The frequensy, f, is the number of osilations per unit tIm; w = 2πf. ( This is a relation w and f sher for the kais of sirkular motion, wen w is kal'd the angular velosity. )

The equation arises from Newton's law that fors is proportional to akselerating mas: F = ma. The fors, F, is in equal and oposit re-aktion to the elastik konstant, k, with regard to a displasment distans of osilation, x, so that F = - kx. The akseleration is a sekond order chanj of distans with tIm, involving a sekond derivativ ( d²x/dt² ).

The up and down of the displasment, x, about a spring's equilibrium position, kan be mimik'd by the chanjing amplitud of a radius swyp around a sirkl, lIk a klok fais that kyps tIm with it. Alternativly, in rektilinear ko-ordinats, the chanjing amplitud of the displasment x, mesur'd on the vertikal axis, under-gos a wav motion over tIm, mesur'd on the horizontal axis.

Kal the radius-swyp angl, Q. This equals angular frequensy, w, tIms time, t. The sIn of Q equals displasment, x. Or, x = sIn Q = sIn wt.

The akseleration, a, ( wich is the sekond derivativ of the distans, x, with respekt to tIm ) is: d²x/dt² = - w sIn wt = - wx.

( We hav left out a kupl of things. SIn wt is a multipl of the maximum amplitud, A, the hIt of the wav krest, or depth of the trof, from equilibrium level, wich for water is the surfas level. We asum'd, at the start that A = 1.
Also, a feiz angl, a, is jeneraly inkluded so x = sIn ( wt + a ). Making angl, a, equal nInty degrys wud be to start at the top of the vertikal axis, nInty degrys on from the positiv horizontal axis. This is equivalent to x = kos wt. And, in fakt, this is wer we did start the simpl periodik serys, deskrib'd abov. )

Hens, F = ma = - wxm = - kx.

So, md²x/dt² + kx = 0. Or, d²x/dt² + wx(t) = 0.

Under previus simplifikations, w = k/m = 1.

So, d²x/dt² + x = 0. This ordinary wav equation for the osilating spring, or analjus fysikal motions, deskribs the abov periodik serys, as the intervals betwyn the terms klos so much, that the funktion bekoms, in efekt, kontinuus.


Richard Lung.




Referenses:

W W Sawyer: A Path To Modern Mathematics.

G Stephenson: Mathematical Methods For Science Students. ( 1973 )



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