CHAPTER V MULTIPLE SCALES..? # w. 5?œ% 0 a?ß?ß%.?.? # %?œ!.>#.>
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1 CHAPTER V MULTIPLE SCALES This chapter and the next concern initial value prolems of oscillatory type on long intervals of time. Until Chapter VII e ill study autonomous oscillatory second order initial value prolems of the form.?.> 5?œ% 0 a?ß?ß%?! a œ!.? a.>! œ ". It is clear from the results of the damped linear oscillator in Chapter IV that to otain a regular expansion hich is uniformly valid, e must pull a part of the effect of % into the first approximation of?>ß a %. The solution to the reduced prolem of a3.19.?.? %?œ!.>.>?! a œ!.? a.>! œ " can e ritten in polar form a5.1?œ 3 cosa> < or Cartesian form a5.2?œ! cosa > " sina >. The results of Chapter III sho us that the zeroth approximation has an error Sa% uniformly for > in a finite interval!ÿ>ÿx. The first task of the method of multiple scales is to improve this zeroth approximation to produce a first approximation having error S a % uniformly for > in an expanding interval of the form! > PÎ%. This is a different prolem from that of improving the accuracy to Sa% on a finite interval, a prolem handled satisfactorily y regular perturation methods. Up to no, the solution of the reduced prolem as also the leading term in the asymptotic series eing 51
2 52 constructed. Hoever, in the methods of multiple scales, the leading term of the asymptotic solution is not the solution of the reduced prolem ut is already an improvement on it. So there is no zeroth term in this method, the first term is the first approximation. The first approximation ill no take the form of a5.1 or a5.2 except that the quantities!, ", 3, and < hich ere constants of integration no ecome sloly varying functions of time. They are of the form or?œ!% a > cos > "% a > sin >?œ 3% a > cosa> <% a > here the functions of %> are found as the solutions of differential equations developed in the method of multiple scales. It is customary to rite 7 œ % > or use the notation X! œ > and X" œ % >. We ill refer to 7 and X " as slo time and > and X! as fast time. It may e seen upon examination of the exact solution of the damped linear oscillator that the damping half life is of the order X ", or slo time, and the period of the oscillations are of the order X!, or fast time. X! and X" are also referred to as to time scales. One of the distinctive properties of most perturation prolems is that no single scale can e used to characterize the complete solution of the prolem. In most applications of the method of multiple scales, the first approximation is sufficient. There are several different methods of multiple scales and they all give the same first approximation. Beyond the first approximation the different methods do not necessarily give the same result. This is possile ecause generalized asymptotic expansions are not a unique approximation of the exact result. Murdock[6], pp. 230, groups higher order multiple scale methods for oscillatory prolems into three types: 1. To scale methods using > and 7. The solutions are ritten in the form?! a>ß7 %?" a>ß7 â 2. To scale methods using slo time, 7 œ % >, and strained time X³>A a % A % A â. The solutions appear as! "?! axß7 %?" axß7 â 3. Multiple scale methods using Q scales here X7 œ % 7 >, 7 œ "ß ß á ß QÞ The solutions are ritten?! ax! ßX" ßáßX7 %?" ax! ßX" ßáßX7 â. A variation of this method omits one time scale for each successive term so that a three scale three term solution ould e ritten? ax ßX ßX %? ax ßX %? ax.!! " "! "!
3 53 The theory of higher order approximations y the first form is ell understood. This form is applicale to a ide variety of prolems and gives approximations to any order of accuracy hich are valid on expanding intervals of length Sa % ". The second and third forms are intended to give approximations on expanding intervals longer than Sa % " ut they are not alays successful and there is no adequate theory to explain the success they do have. We return no to the damped linear oscillator. The method of strained parameters failed to improve the solution over the regular method due to its inaility to handle transient characteristics. Both methods use only one time scale in their solution. The exponential decay of the oscillator due to the damping introduces a characteristic time ( the half life of the decay) that occurs on a longer scale than the oscillations. We ill then introduce a longer time scale right from the eginning and see if a to scale method can resolve this difficulty. To Scale Method. E XAMPLE 5.1: Damped Linear Oscillator. We seek an approximate solution of the IVP a5.3? %?? œ!?! a œ!? a! œ " in the form 8 a? a>ß 7 ß % œ? a>ß 7 %? a>ß 7 %? a>ß 7 â %? a>ß 7 Sˆ 8 " 5.4 % here! " 8 a5.5 7 œ % >Þ. Ð œ.> Ñ. In this example, only the first term of a5.4 ill e determined. Using a5.5 and the chain rule e have a5.6? œ? %? > 7 a5.7 >> > 7 77? œ? %? %? here partial derivatives are indicated y suscripted variales. Sustituting a5.6 and a5.7 into the P.D.E. of a5.3 and sustituting a5.4 into the initial conditions e otain
4 54 a5.8? >>? % a? > 7? > % a? 77? 7 œ!! "? a!ß! %? a!ß! %? a!ß! â œ!?!> a!ß! % c?! 7a!ß!? "> a!ß! d % c? a!ß!? a!ß! d â œ " " 7 >. No sustitute a5.4 into the P.D.E. of a5.8 and use the fundamental theorem to get the folloing sequence of partial differential equations and initial conditions: a5.9?? œ!!>>!?!? a!ß! œ " a5.10?? œ a? 7?!> ">> "!>!>?"? a!ß!? a!ß! œ! ">! 7 a5.11?>>? œ a? "> 7? ">?! 77?! 7?? a!ß!? a!ß! œ! > " 7 We note that a5.4 is a generalized asymptotic expansion since % enters oth through the gauges and also through the coefficients? 8 y ay of 7. Although there is no general theorem alloing the differentiation of a generalized asymptotic expansion term y term it is nevertheless reasonale to construct the coefficients of a5.4 on the assumption that such differentiation is possile, and then to justify the resulting series y direct error estimation afterards. It is also important to note that even the step of equating coefficients of equal poers of % (the fundamental theorem) is not justified y any theorem aout generalized asymptotic expansions since they have no uniqueness properties. a5.9 is a partial differential equation. The general solution is otained as usual except e let the aritrary constants ecome aritrary functions of 7. a5.12?!a>ß7 œ E a7cos > F a7sin > œ 3a7 cosa> < a7 The initial conditions impose the folloing restrictions on the aritrary functions in (5.12) À
5 55 a5.13 E! a œ! F! a œ" 3 a! œ È! " tanc< a! d œ " Î! Þ At this point e see that e get no information on Ea7ß Fa7ß 3a7 ß or < a7 except their initial values. Thus, the initial value prolem for does not completely determine?!. The polar form of a5.12 ill expedite calculations ut the computational advantage for higher order terms lies ith the Cartesian form. Therefore, e ill compute the first order approximation in Cartesian form ecause e ill need the functions Ea7 and Fa7 to compute the second order term. We rite the general solution to a5.9 as here?!?!a>ß7 œ E a7cos > F a7sin > a5.15 E! a œ! The equation for? " is F! a œ". a5.16?? œ cae a7 E a7sin > af a7 F a7cos > d. ">> " Since 7 is ounded on expanding intervals of S a%, it is permissile for?" to contain so called 7-secular terms hich involve factors of 7, ut not > -secular terms hich involve factors of >. The terms resonant in > hich ill produce > -secular terms are the sin > and cos > terms. The coefficients of these terms must e set to zero. Thus Ea7 and Fa7 must satisfy the O.D.E.s " E a7 œ Ea7 F a7 œ Fa7Þ The solutions of these equations satisfying a5.15 are Ea7 œ!/ Fa7 œ "/ Þ
6 56 So, in Cartesian form, the first approximation is computed to e a5.17? a>ß7 œ / a! cos > " sin >.! We can convert a5.17 directly to polar form y taking a5.18! œ 3 cos < " œ 3 sin < 3 œ È! " tana< œ " Î! Þ then? a>ß7 œ / 3 cosa> < a7þ! We found that the entire right side of a5.10 as resonant so e eliminated it y our choice of coefficients. a5.10 then ecomes hose solution is?? œ! ">> "?"? a!ß! œ? a!ß! œ ">! 7!?"a>ß7 œ G a7cos > H a7sin > G! a œ! H! a œ! Þ As ith the first order approximation, e do not have enough information to determine Ga7 and H a7 unless e use the expression for a5.11 and suppress all terms in? hich ould produce > -secular terms in? ". We can no rite a5.11 as?? œ a???? >> "> 7 ">! 77! 7 œ cg G "/ dsin > c H H!/ dcos >. Elimination of resonant terms requires that
7 57 " Ga7 œ 7/! Ha7 œ 7 /!/ 7 so that?" a>ß7 œ " 7 7/ > Š! 7 7 cos 7 /!/ sin >Þ Sustituted into a5.4, the complete second approximation ecomes?>ß a % œ!/ cos > "/ sin > "! 7 % 7/ > Š 7 /!/ > Sˆ cos sin % or in polar form?>ß a % œ 3 / cos a> < " 7 % 37 / a> <!/ > Sˆ sin sin % Þ Adapted from Murdock[6], pp and Simmonds and Mann[10], pp u 1 2 Pi 4 Pi 6 Pi t Figure 6. Exact Linear Solution vs. To Scale Expansion
8 58 To Scales ith Strained Time. The term proportional to 7/ hich at first gives the appearance of a 7-secular term turns out not to e a prolem. This is ecause the exponential damping acts to reduce not only the size of the solution ut also the error as >p_. Because of this, the " error, rather than eing Sa% on expanding intervals of Sa%, is actually Sa% for all time. Although the term does no harm in this case, its appearance is due to the fact that fast time in the exact solution a3.33 is not > ut the strained time > È" %. Recall that damping alters the period of oscillation. Failure to introduce a strained time results 7 in the appearance of such terms as / in? Þ Just as secular terms ere suppressed y 7/ strained time in the Lindstedt expansion, strained time can e used to suppress 7-secular terms in multiple scale expansions. We ill no examine this approach even though the only apparent reason for suppressing these terms is that doing so ill simplify the equations. The regular method has only one time scale and could not account for the damping nor for the frequency change in the period of oscillation caused y the damping. We have just seen ho the to scale method successfully handled the exponential decay ith the introduction of a second longer time scale. No, folloing Lindstedt, e ill see if e can more closely match the actual frequency of the pertured solution y introducing a strained time. The term proportional to 7/ gives the appearance of a 7-secular term. Its appearance is due to the fact that fast time in the exact solution a3.33 is not > ut the strained time > È " %. Failure to introduce a strained time results in the appearance of 7 such terms as / in? Þ Just as secular terms ere suppressed y strained time in the Lindstedt expansion, strained time can e used to suppress 7-secular terms in multiple scale expansions. We ill no examine this approach E XAMPLE 5.2: Damped Linear Oscillator. We ill use the damped linear oscillator prolem once again so that e may compare our results ith previous methods and to simplify calculations. Instead of fast time, >, and slo time, 7, e ill no use strained time, X, and slo time, 7, defined y a5.19 X ³> ˆ A! A" % A % We seek a solution of the form 7 ³ % >Þ
9 59 a5.20? axß7 ß% œ? axß7 %? axß7 â! " To save some time and space, e report that the end result of using a5.19 in its current form is A! œ " and A" œ!. So instead of a5. "* e ill use strained time to e X³>" A ˆ %. Sustituting a5.20 into a5.3, using the chain rule and the fundamental theorem results in the folloing sequence of partial differential equations: a5.21?? œ!!x X!?!? a!ß! œ "!X a5.22?? œ?? "X X "!X 7!X?"? a!ß! œ? a!ß! "X! 7 a5.23?? œ A????? X X!X X! 77 "X 7! 7 "X The solution to a5.21 is?? a!ß! œ A? a!ß!? a!ß! X!X " 7 a5.24?! œ Ea7cos X Fa7sin X a5.22 ecomes requiring E! a œ! F! a œ".?? œ c E a7sin X F a7cos X d "X X " cea7sin X Fa7cos X d E a7 Ea7 œ! F a7 Fa7 œ!
10 60 to suppress resonant terms. With initial conditions a5.24 the solutions are Ea7 œ!/ Fa7 œ "/ Þ a5.22 no ecomes?? œ! "X X "?"? "X hose solution is?" œ Ga7cos X Ha7sin X G! a œ! H! a œ!. a5.23 ecomes?? œ A E E E H H cos X X X ca F F F G Gdsin X. To suppress resonant terms e require " G G œ ŒA "! / " H HœŒA " " / Þ We see here another important principle of perturation theory. There is no unique ay to distriute the original independent variale, >, eteen the to ne variales X and 7. The rule of thum is to use any aritrariness to simplify the equations as much as possile. In this case e choose then A œ "
11 61 Ga7 œ! Ha7 œ!/ With this information e may no rite our second approximation 7 a5.25?xß a 7 ß% œ/! cos > Œ" " % " " " sin > Œ" % %! sin > Œ" % hich is free of > -secular and 7-secular terms according to our design. We may convert a5.25 to polar form y using the equations a5.18 and get 7 a5.26?xß a ß œ/ > Œ" " > Œ" " 7 % 3 cos % < %! sin % Þ Adapted from Simmonds and Mann[10], pp u 2 Pi 4 Pi 6 Pi t Figure 7. Exact Linear Solution vs. To Scales ith Strained Time
12 62 Multiple Scales. The method of multiple scales has een employed to solve many applied prolems and is used extensively in engineering applications. Hoever, it has incomplete mathematical foundations. The folloing example from Murdockc6 d, pp , is intended to illustrate the method and to allo examination of some of its irregularities. Anyone ishing to learn the method for practical use ould have to study many different examples to see ho the difficulties are dealt ith in each specific situation. EXAMPLE 5.3: Using the example of the damped linear oscillator, e ill no find an approximation of the form! " a5.27?µ? a>ß7ß5 %? a>ß7 %? a> here a œ % >. 5 œ % > Using the chain rule, a5.27, a5.28, and a5.3 e have the folloing sequence of PDEs: a5.29?? œ!!>>!?! a!ß!ß! œ!? a!ß!ß! œ"!> a5.30?? œ?? ">> "!> 7!>?"? a!ß! œ? a!ß!ß! ">! 7 a5.31?? œ????? >>!> 5! 77 "> 7! 7 "> The solution to a5.29 is ritten? a! œ!? a! œ? a!ß!ß!? a!ß!. >! 5 " 7?! œ Ea7ß 5cos > Fa7ß 5sin > E!ß! a œ! F!ß! a œ".
13 63 a5.30 ecomes?">>? " œ c E7a7ß 5sin > F7a7ß 5cos > d c Ea7ß 5sin > Fa7ß 5cos >Þ d Here e see an apparent contradiction in that?"a>ß 7appears to e a function of >ß 7ß and 5. This prolem is resolved, hoever, hen e realize that the right side of a5.30 is set equal to zero to eliminate secular terms. Thus, the apparent dependence upon 5 is eliminated along ith the forcing term. Requiring that there e no > -secular terms in the final solution means that all > -resonant terms must vanish. Therefore hich leads to E a7ß5 œ Ea7ß5 7 F a7ß5 œ Fa7ß5 7 Ea7ß5 œ O a5/ Fa7ß5 œ K a5/ a5.32 O! a œ! K! a œ". No e see a ig difference eteen this method and the to scale method. At this point in the latter ould e completely specified, hile in the present method e see a?! dependence upon 5 hich is not yet determined. Returning to a5.30 ith all resonant terms eliminated e have hose solution is?? œ! ">> "?"? a5.33?" œ G a7cos > H a7sin > "> G! a œ! H! a œ!. Sustituting these results into a5.31 e find
14 64 a5.34? tt? œ c F5 E77 E7 H7 H dcos > c E F F G G dsin > here e require that the resonant terms vanish to suppress 7-secular terms in the final solution. Why do e ant to get rid of 7-secular terms? The aim of a three scale method is to achieve uniformity on an expanding interval of Sa%. On an interval of this length, a 7 -secular term ill produce the same kind of prolems that a > -secular term ill " produce on an interval of Sa%. We ill follo the rule that Oa5 and Ka5 should e chosen so that Ga7 and H a7 contain no 7-secular terms. Suppressing > -resonant terms in a5.34 e require that " a5.35 G7a7 Ga7 œ Œ O a5 K a5 / " H7a7 Ha7 œ Œ K a5 O a5 / 7. O and K are functions of 5 alone and may e treated as constants hen solving a5.35 for G and H as functions of 7. It is apparent that the forcing terms in a5.35 proportional to / produce a response proportional to 7/. In order to avoid these terms it is necessary to take a5.36 O a5 œ " Ka5 K a5 œ " Oa5. The solutions of a5.36 ith initial conditions a5.32 are Finally,?! is completely determined 5 5 Oa5 œ! cos " sin 5 5 Ka5 œ! sin " cos Þ 5 5 a5.37?! a>ß7ß5 œ! cos " sin / cos > 5 5! sin " cos / sin >. Using
15 65 a5.18! œ 3 cos < " œ 3 sin < 3 œ È! " tana< œ " Î! Þ e may convert a5.37 to its polar form and get 5?! a>ß7ß5 œ / 3 cosš > < Þ With the information from a5.35 and a5.36 e see that With a5.33 these are solved as and G 7 a7 œ Ga7 H a7 œ Ha7Þ 7 Ga7 œ! Ha7 œ!/? a>ß7 œ!/ 7 sin >Þ " Since the entire right hand side of a5.31 as eliminated y our choice of terms it ecomes and its solution is here the initial conditions require that?? œ! >>? a! œ! "?> a! œ?a> œ I cos > J sin >
16 66 Iœ! Jœ " so that "?a> œ sin >Þ The entire three term approximation may e given in the form 7 5 "?>ß a 7ß5 à% œ/ 3 cosš > < %! sin > % sin > Þ If e take 5 œ" e can see that this expansion is the same first order expansion as a5.26. The aove example as adapted from Murdock[6], pp and Nayfeh[8], pp The fundamental theorem does not hold for generalized asymptotic expansions so the theoretical foundations of the multiple scale method are very eak. Nevertheless it does provide correct and interesting results in many cases. In this example, for instance, e can see that the second scale, 7, "automatically" corrects the solution for the damping and the third scale, 5, "automatically" corrects for the frequency shift.
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