integral transforms in applied mathematics

INTEGRALTRANSFORMSIN APPLIEDMATHEMATICS

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INTEGRALTRANSFORMSIN APPLIEDMATHEMATICS

John W. MilesUNIVERSITY OF CALIFORNIA, SAN DIEGO

CAMBRIDGEAT THE UNIVERSITY PRESS • 1971



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© Cambridge University Press 1971

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PREFACE

The following treatment of integral transforms in applied mathematicsis directed primarily toward senior and graduate students in engineeringand applied science. It assumes a basic knowledge of complex variablesand contour integration, gamma and Bessel functions, partial differentialequations, and continuum mechanics. Examples and exercises are drawnfrom the fields of electric circuits, mechanical vibration and wave motion,heat conduction, and fluid mechanics. It is not essential that the studenthave a detailed familiarity with all of these fields, but knowledge of atleast some of them is important for motivation (terms that may be un-familiar to the student are listed in the Glossary, p. 89). The unstarredexercises, including those posed parenthetically in the text, form anintegral part of the treatment; the starred exercises and sections arerather more difficult than those that are unstarred.

I have found that all of the material, plus supplementary material onasymptotic methods, can be covered in a single quarter by first-yeargraduate students (the minimum preparation of these students includesthe equivalent of one-quarter courses on each of complex variables andpartial differential equations); a semester allows either a separate treat-ment of contour integration or a more thorough treatment of asymptoticmethods. The material in Chapter 4 and Sections 5.5 through 5.7 couldbe omitted in an undergraduate course for students with an inadequateknowledge of Bessel functions.

The exercises and, with a few exceptions, the examples require onlythose transform pairs listed in the Tables in Appendix 2. It is scarcelynecessary to add, however, that the effective use of integral transformsin applied mathematics eventually requires familiarity with more extended


http://dx.doi.org/10.1017/CBO9780511897351.001

VI PREFACE

tables, such as those of Erdelyi, Magnus, Oberhettinger, and Tricomi(herein abbreviated EMOT, followed by the appropriate entry number).

Chapter 1 and minor portions of Chapters 2 through 5 are based on alecture originally given at various points in California in 1958 and sub-sequently published [Beckenbach (1961)] by McGraw-Hill. I am indebtedto the McGraw-Hill Publishing Company for permission to reuse portionsof the original lecture; to Cambridge University Press for permission toreproduce Figures 4.2 and 4.3; to Professors D. J. Benney and W. Pragerfor helpful criticism; to Mrs. Elaine Blackmore for preparation of thetypescript and to Mr. Y. J. Desaubies for his aid in reading the proofs.

J. W. M.

La Jolla, 1968



CONTENTS

1 INTEGRAL-TRANSFORM PAIRS1.1 Introduction 1

1.2 Fourier's integral formulas 6

1.3 Fourier-transform pairs 7

1.4 Laplace-transform pairs 10

Exercises 12

* THE LAPLACE TRANSFORM2.1 Introduction 13

2.2 Transforms of derivatives 14

2.3 Simple oscillator 15

2.4 Convolution theorem 17

2.5 Heaviside's shifting theorem 18

2.6 Periodic functions 20

2.7 The inversion integral 24

2.8 Wave propagation in a bar 30

2.9 Heat conduction in a semi-infinite solid 34

2.10 Oscillating airfoil in supersonic flow 36

Exercises 38



•J FOURIER TRANSFORMS

3.1 Introduction 43

3.2 Transforms of derivatives 43

3.3 Operational theorems 45

3.4 Initial-value problem for one-dimensional wave equation 46

3.5 Heat conduction in a semi-infinite solid 47

3.6 Two-dimensional surface-wave generation 48

3.7 The method of stationary phase 50

3.8 Fourier transforms in two or more dimensions 53

Exercises 54

T" HANKEL TRANSFORMS

4.1 Introduction 57

4.2 Oscillating piston 59

4.3 Axisymmetric surface-wave generation 61

Exercises 65

FINITE FOURIER TRANSFORMS5.1 Introduction 67

5.2 Finite cosine and sine transforms 68

5.3 Wave propagation in a bar 70

5.4 Heat conduction in a slab 71

5.5 Finite Hankel transforms 72

5.6 Cooling of a circular bar 73

5.7 Viscous diffusion in a rotating cylinder 74

5.8 Conclusion 75

Exercises 76



A P P E N D I X 1. P A R T I A L - F R A C T I O N E X P A N S I O N S 79

A P P E N D I X 2. TABLES 83

2.1 Laplace-transform pairs 83

2.2 Operational theorems 84

2.3 Infinite integral transforms 85

2.4 Finite integral transforms 86

A P P E N D I X 3. LIST OF N O T A T I O N S 87

G L O S S A R Y 89

BIBLIOGRAPHY 90

Texts and treatises 90

Tables and handbooks 92

INDEX 95



To Oliver Heaviside



1 INTEGRAL-TRANSFORM PAIRS

1.1 Introduction

We define

F(p)= f K(p,x)f(x)dx (1.1.1)

to be an integral transform of the function /(x); K(p,x), a prescribedfunction of p and x, is the kernel of the transform. The introduction ofsuch a transform in a particular problem may be advantageous if thedetermination or manipulation of F(p) is simpler than that of /(x), muchas the introduction of log x in place of x is advantageous in certainarithmetical operations. The representation off(x) by F(p) is, in manyapplications, merely a way of organizing a solution more efficiently, as inthe introduction of logarithms for multiplication, but in some instancesit affords solutions to otherwise apparently intractable problems, just asin the introduction of logarithms for the extraction of the 137th root ofa given number.

Today, the Laplace transform [K = e~px, a = 0, and b = oo in(1.1.1)],

fF(p)= e-pxf(x)dx,

Jo

may be claimed as a working tool for the solution of ordinary differentialequations by every well-trained engineer. We consider here its application

fin Heaviside's form of operational calculus, the right-hand side of (1.1.2) appears multi-plied by p, but the form (1.1.2) is now almost universal.



2 / INTEGRAL-TRANSFORM PAIRS / CH. 1

to both ordinary and partial differential equations. We also considerapplications to partial differential equations of the Fourier transform(also called the exponential-Fourier transform or the complex Fouriertransform),

F(p)= I e'ipxf(x)dx9 (1.1.3)J-00

the Fourier-cosine and Fourier-sine transforms,

F(P) = f(x) cos pxdx (1.1.4)Jo

and

JoF(p)= fix) sin pxdx, (1.1.5)Jo

and the Hankel transform (also called the Bessel or Fourier-Besseltransform),

F(p)= f(x)Jn(px)xdx. (1.1.6)Jo

Our definitions are those of Erdelyi, Magnus, Oberhettinger, and Tricomi(1954, hereinafter abbreviated as EMOT) except for (1.1.6); other defini-tions of the Fourier transforms, differing from those of (1.1.3) to (1.1.5)by constant factors, are not uncommon. Notations for the transformsthemselves vary widely, and symbols other than p are not uncommon forthe arguments. In particular, engineers typically use 5, rather than p, inthe Laplace transforms of time-dependent functions [in which case tappears in place of x in (1.1.2)]. The properties of the foregoing infinitetransforms are summarized in Table 2.3 (Appendix 2, p. 85). We con-sider these and other properties, together with applications, in the follow-ing chapters.

Consider the following elementary example, in which we anticipatecertain results that are derived in Chapter 2. We require the charge q(t) ona capacitor C, in series with a resistor R, following the application of aconstant voltage v at t = 0. The differential equation obtained by equatingthe sum of the voltages across each of C and R to the applied voltage is

Let

d^-+C~1q = v. (1.1.7)dt

Q(P)= e~ptq(t)dtJo



SEC. 1.1 / INTRODUCTION / 3

be the Laplace transform of q(t). Taking the Laplace transform of (1.1.7),transforming the derivative through integration by parts,

Jo *and invoking the fact that v is a constant, we obtain

(Rp + C-l)Q(p) - Rq(0) = vp-1

Solving for Q(p), we obtain

<7(0) , (v/R)Qip) = p + (RCri

which we rewrite in the form

Q(p) = q(0)(p + a)"1 + Cv[p~l - (p + a)"1], (1.1.8)

where a = 1/RC. Referring to entry 2.1.2, in Table 2.1 (Appendix 2,p. 83), we find that the functions whose transforms are l/(p + a) and1/p are exp( — otf) and 1, the latter being a special case of the former.Invoking these results in (1.1.8), we obtain

q(t) = q(0)e-a1 + Cv{\ - e'"). (1.1.9)

This example is too straightforward to illustrate the real power of theLaplace transform, but it does serve to illustrate such basic advantagesas the reduction of differential to algebraic operations and the automaticincorporation of initial conditions. [Bracewell (1965) and Gardner-Barnes (1942) give extensive applications of Fourier and Laplace trans-forms, respectively, to circuit analysis.]

The transforms defined by (1.1.2) to (1.1.6) are the only infinite ones inwidespread use at this time, but many others have been studied andtabulated (see EMOT, Vol. 2), and still others may be introduced in thefuture. If one's goal is merely to produce formal solutions, it suffices toknow the inversion formula that determines /(x) from F(p), but extensivetabulations of f(x) versus F(p) are essential in applied mathematics.Returning to our analog of the logarithm, we note that formal analysisrequires only the knowledge that the inverse of y = loga x is x = ay,whereas a table of x versus y is indispensable for numerical compu-tation.

In addition to the possibility of defining new transforms through newkernels, there is also the possibility of adopting finite limits in (1.1.1),thereby obtaining so-called finite transforms. If, for example, we replace




the upper limits in (1.1.4) and (1.1.5) by n, the inversion formulas areordinary Fourier-cosine and -sine series summed over integral valuesof p. More generally, if the kernel K(p, x) in (1.1.1) yields a set of functionsorthogonal, with suitable weighting function, over the interval a, b for aninfinite discrete set of values p, then the inversion formula defines aFourier-type series.

The result of introducing a finite Fourier transform in a given prob-lem is merely to mechanize the classical technique of Fourier series;however, it is generally true that the more tedious solution of the prob-lem by the classical technique is straightforward, albeit sometimescalling for greater ingenuity (compare the use of Lagrange's equationsin mechanics). This is to be contrasted with the applications of infinitetransforms, which frequently offer entirely new insight and reducetranscendental to algebraic operations, thereby affording solutionsto problems that might have required far greater ingenuity for theirsolution by classical techniques. Nevertheless, the student must bearin mind that, with few exceptions, integral transforms are applicableonly to linear differential and/or integral equations with either constantcoefficients or (as with the Hankel transform) very special nonconstantcoefficients.

Integral transforms in applied mathematics find their antecedents inthe classical methods of Fourier and in the operational methods ofHeaviside, antecedents that had rather different receptions by contem-porary mathematicians. The eleventh edition (1910) of the EncyclopediaBritannica devotes five pages to Fourier series but does not mentionHeaviside's operational calculus; indeed, no direct entry appears forHeaviside in that edition, although his name is mentioned peripherally.The fourteenth edition (1942) does contain a brief biographical entry onHeaviside, but the only reference to his operational calculus is the ratheroblique statement that "he made use of unusual methods of his own insolving his problems."

Fourier's theorem has constituted one of the cornerstones of mathe-matical physics from the publication of his La Theorie Analytique de laChaleur (1822), and its importance was quickly appreciated by mathema-ticians and physicists alike. For example, Thomson and Tait remarked that

. . . Fourier's Theorem... is not only one of the most beautiful results ofmodern analysis, but may be said to furnish an indispensable instrument inthe treatment of nearly every recondite question in modern physics. To men-tion only sonorous vibrations, the propagation of electrical signals along atelegraph wire, and the conduction of heat by the earth's crust, as subjectsin their generality intractable without it, is to give but a feeble idea of its im-portance.



SEC. 1.1 / INTRODUCTION / 5

The concept of an integral transform follows directly from Fourier'stheorem (see Section 1.2 below), but the historical approach, at least tophysical applications, was largely through operational methods. Opera-tional methods in mathematical analysis, having been introduced origi-nally by Leibniz, are nearly as old as the calculus, but their widespreaduse in modern technology stems almost entirely from the solitary geniusof Oliver Heaviside (1850-1925). To be sure, the bases of Heaviside'smethod, as he recognized and stated, lay in the earlier work of Laplaceand Cauchy, but it was Heaviside who recognized and exposed the powerof operational methods not only in circuit analysis but also in partialdifferential equations. Unlike Fourier, Heaviside had no university train-ing and was not a recognized mathematician; indeed, he scorned notonly mathematical rigor ("Shall I refuse my dinner because I do not fullyunderstand the process of digestion?") but also, it sometimes appeared,mathematicians ("Even Cambridge mathematicians deserve justice.").This lack of rapport with mathematicians may have delayed the full appre-ciation of his work. Even some modern mathematicians have been reluc-tant to give Heaviside his due; thus, Van der Pol-Bremmer (1950) criticizeDoetsch (1943) for his description of Heaviside as merely "ein englischerElektroingenieur," using methods that were "mathematisch sehr un-zulanglich" and "allerdings mathematisch unzureichend." E. T. Whit-taker, on the other hand, offered the following evaluation (in Heaviside'sobituary): "Looking back..., we should now place the operationalcalculus with Poincare's discovery of automorphic functions and Ricci'sdiscovery of the tensor calculus as the three most important mathematicaladvances of the last quarter of the nineteenth century."

We conclude this introduction by contrasting the approaches of thepure mathematician and the physical scientist to transform theory. Atone extreme we have Titchmarsh's statement, in the preface to his treatise(1948), that "I have retained, as having a certain picturesqueness, somereferences to 'heat,' 'radiation,' and so forth; but the interest is purelyanalytical, and the reader need not know whether such things exist." Atthe other, we have Heaviside's cavalier statement, "The mathematicianssay this series diverges; therefore, it should be useful." In the followingpresentation of integral-transform theory, we attempt to follow the lineset down by Lord Rayleigh:

In the mathematical investigation I have usually employed such methodsas present themselves naturally to a physicist. The pure mathematician willcomplain, and (it must be confessed) sometimes with justice, of deficient rigour.But to this question there are two sides. For, however important it may be tomaintain a uniformly high standard in pure mathematics, the physicist may oc-casionally do well to rest content with arguments which are fairly satisfactory




and conclusive from his point of view. To his mind, exercised in a differentorder of ideas, the more severe procedure of the pure mathematician mayappear not more but less demonstrative. And further, in many cases of diffi-culty to insist upon the highest standard would mean the exclusion of the sub-ject altogether in view of the space that would be required.

1.2 Fourier's integral formulas

We first give a formal derivation of Fourier's theorem in complexform, following in all essential respects the argument offered by Fourierhimself.t Let/(x) be represented by the complex Fourier series

/(*)= £ cnexp(iknx) H A < x < i 4 (1.2.1)n = — oo

/•A/2

where cn = X~1 f(^)Qxp( — ikn^)d£, (1.2.2)J-A/2'

and kw = T " ( L 1 3 )

This representation is evidently periodic with a wavelength X. We nowallow X to tend to infinity, noting that the consecutive kn are separatedby the increment Afc = 2n/X; then, combining (1.2.1) and (1.2.2), we obtain

£ Afc fA/2

f(x) = lim £ exp(zfcnx)— f(^)exp(-ikn^)di. (1.2.4)A-»00 n=-oo 2ft J - A / 2

Replacing the sum by an integral in the limit, we obtain

f(x) = -*- f°° dk P M)exp[ik(x - 0 ] d£. (1.2.5)* n J-00 J-00

Expressing the exponential in terms of its trigonometric componentsand invoking the even and odd nature of cos k(x — £) and sin k(x — £),respectively, as functions of fc, we obtain Fourier's integral formula:

f(x) = - [^ dk f °° M) cos k(x - Q d£. (1.2.6)

Fourier's derivation differed from the above only in starting from thetrigonometric form of his series. We emphasize that the order of integra-tion in (1.2.5) and (1.2.6) must be preserved; its reversal would lead to

fThis derivation is included in order to emphasize the relation between Fourier series andFourier integrals. It is assumed that the student has had previous experience with bothFourier series and Fourier integrals and that he has been exposed to a more rigorous deriva-tion of Fourier's integral theorem [see, e.g., Churchill (1963)].



SEC. 1.3 / FOURIER-TRANSFORM PAIRS / 7

meaningless integrals. On the other hand, we assume that /(x) vanisheswith sufficient rapidity for large |x| to ensure the existence of the doubleintegrals as written. Actually,/vanishes exponentially in typical, not-too-idealized, physical problems.

Now let/(x) be either an even or an odd function (any function that isnot even or odd can be split into a sum of two such functions). Expandingthe cosine in (1.2.6), we obtain Fourier's cosine formula,

/ (x)=/(-x) = - cos kxdk I f({) cos k£di (1.2.7)71 Jo Jo '

for an even function, and Fourier's sine formula,

f{x) = -f ( - x) = - I sin kx dk I °° /"({) sin fc£ d{ (1.2.8)71 Jo Jo '

for an odd function.

13 Fourier-transform pairs

Equation (1.2.5) may be resolved into the transform pairfoo

F(k) = &{f} = f(x)e~ikxdx (1.3.1a)J-oo

and

^-i{F} = _L f00 F(k)eikxdk, (1.3.1b)2TC J

where ^ is the Fourier-transform operator and 3F~^ is its inverse (weomit the braces around the operand when this can be done without am-biguity). The location of the factor 1/(2TC) is essentially arbitrary; fromthe viewpoint of establishing an analogy between Fourier series andFourier integrals, it would appear preferable to place it in (1.3.1a) ratherthan (1.3.1b), whereas from an aesthetic viewpoint a symmetric disposalof the identical factors (2TT)~1/2 would be desirable. Each of these con-ventions has been adopted by various writers, but the form chosen in(1.3.1a, b) has two major advantages. First, it agrees with the notationadopted by Campbell-Foster (1948), in their very extensive table ofFourier transforms (see also EMOT, Vol. 2). Secondly, it affords a directtransition to the accepted definition of the Laplace-transform pair (seeSection 1.4). A third advantage, of special interest in electric-circuit orwave-motion problems, is that if x, implicitly a space variable in theforegoing discussion, is replaced by the time-variable t and k is replacedby 2TCV, where v is a frequency, (1.3.1a, b) go over to the symmetric pair




F(v) f(t)e - 2nivt dt

and

f(t)= F(v)e2nivt dv,

(1.3.2a)

(1.3.2b)

in which f(t) is represented as a spectral superposition of simple harmonicoscillations of frequency v and complex amplitude F(v). We remark thata similar form for the space-variable pair of (1.3.1a, b) follows from thesubstitution k = 2TZK, where K is a reciprocal wavelength.

Consider, for example,

f(x) = e~a]xl (-00 < x < oo),

the substitution of which into (1.3.1a) yields(-0

=

J -poo pO

= I e~^a+ik^x dx + e^a~il

JO J-oo

= (a + ik)'1 + (a - ik)'1

= 2a(a2 + /C2)-1 .

ik)*dx

•2Y\ (1.3.3)

Invoking (1.3.1b), closing the path of integration (see Figure 1.1) by a semi-circle of infinite radius in kt > 0 | for x > 0 or in kt < 0 for x < 0, ob-serving that the integrand has poles at k = ± ia, and invoking Cauchy'sresidue theorem, we obtain

1

/ - -/

//

{ +1 w

\\ 1

\\

• ia \

1( JQ 1

/

FIGURE 1.1 Path(s) of integration and poles for the example of (1.3.4).The semicircles have radii that tend to infinity.

fWe use the subscripts r and i here and subsequently to designate the real and imaginaryparts of a complex variable.



SBC. 1.3 / FOURIER-TRANSFORM PAIRS / 9

^laia2 + k2)'1} = - (a2 + k2yleikx

71 J-00dk

= -(±2ni)R{(a2 + k2)'1 eik%=±ia (x £ 0)71

= ^(±2ni)(±2iay1 eTax (x ^ 0)

= «rflW, (1.3.4)

where the upper and lower signs correspond to x > 0 and x < 0, re-spectively, and R{ } is the residue of the bracketed quantity.

Fourier's cosine and sine formulas, (1.2.7) and (1.2.8), may be resolvedinto the transform pairs

and

F(k) = :

f(x) = •

F(k) = ;

fix) = .

f °°FAf) = f(x) cos kxdx,

Jo "2 f00

&-l{F} = - F(k) cos kxdkn Jo

f00^s{/"} = /"(x) sin fcx dx,

Jo2 f00

^ - ^ F } = - F(k)sin kxdk,n Jo

(1

(1

(1

(1

3.5a)

.3.5b)

.3.6a)

.3.6b)

thereby providing the inversions of (1.1.4) and (1.1.5), respectively. Again,the 2/7C factors may be resolved differently—in particular, symmetrically;the notation adopted here is that of EMOT. We remark that the Fourier-cosine (-sine) transform is suited either to a function that is defined only inx > 0 or to an even (odd) function of x; conversely, the function definedby (1.3.5b) or (1.3.6b) is an even or odd function, respectively, of x.

Consider, for example,

f(x) = e~ax {a > 0, 0 g x < GO),

the substitution of which into (1.3.5a) yieldspoo

&c{e~ax} = e~ax cos kxdxJo

e~ax(eikx + e~ikx)dxJo

= i[(fl - iky1 +(a + ifc)-1]

= a(a2 + fc2)"1. (1.3.7)




We remark that (1.3.7) follows almost directly from the example of (1.3.3)(or conversely) by virtue of the fact that exp(-a|x|) is an even functionof x. The student should show that

=k(a2 (1.3.8)

and observe that this result is not directly related to the example of (1.3.3)(see the last sentence in the preceding paragraph).

1,4 Laplace-transform pairs

The path of integration for (1.3.1b) may be deformed into a complex-feplane in any manner that ensures the convergence of the integrals for bothF(k) and f(x). Suppose, to cite the most important, special case, thatexp(-oc)/(x) vanishes appropriately at both limits; in particular,/(x)may vanish identically in x < 0. Then the modified transform

F(k) = &{f} = r f{x)e~ikxdx (kt = -c) (1.4.1a)J-oo

exists, where kt denotes the imaginary part of fc, and/(x) is given by

F(k)eikxdk. (1.4.1b)2?r J-

Thus, Fourier's integral formula extends to functions for which (1.2.5)might not be valid. In the most important, physical applications, c ispositive and the path of integration appears as in Figure 1.2a.

—m

(a)

— Ml (>

(b)

FIGURE 1.2 Paths of integration for the inverse transforms of (1.4.1b)and (1.4.3b) and the poles for the corresponding transforms of sin mx.



SEC. 1.4 / LAPLACE-TRANSFORM PAIRS / 11

If we now rotate the path of integration through a right angle (seeFigure 1.2b), introduce p = ik, and at the same time replace F(k) byF{p\ (1.4.1a) and (1.4.1b) go over to the two-sided Laplace-transform pair^

J-ooF(P)=\ f(x)e~pxdx (Pr = c) (1.4.2a)

and

2ni

'c + ico

F(p)epxdp. (1.4.2b)

Finally, we suppose that f(x) vanishes for x < 0, so that the lowerlimit in (1.4.2a) may be replaced by zero. The resulting integral convergesif the real part of p exceeds some minimum value, say c*, not necessarilynonnegative, such that all singularities of F{p) lie in pr < c^, where pr

denotes the real part of p; the inverse transform (1.4.2b) exists for allc > c*, and we have the Laplace-transform pair

F(p)

and

Consider,

= <e{f(x)} = PJo

for example,

e~pxf(x)dx (Pr>

-^-. F(p)epxdp2ni Jc-icK

( C > c*).

(1.4.3a)

(1.4.3b)

(x^O), (1.4.4)

for which the Fourier transform defined by (1.3.1a) does not exist. Invok-ing (1.4.1a), we obtain (the student should fill in the details)

JF{sin mx] = m(m2 - k2)~x {kt < 0), (1.4.5)

which has poles at k = ±m on the real axis (see Figure 1.2a) but is ananalytic function of k in kt < 0. Similarly,

J£?{sin mx} = m(m2 + p2)~x (pr > 0). (1.4.6)

The student should evaluate the corresponding inverse transforms, withthe aid of the calculus of residues, to recover the original function.

t Applications of the two-sided Laplace transform are considered by Van der Pol-Bremmer(1950).




EXERCISES1.1 Derive the Mellin-transform pair from the Laplace-transform pair of

(1.4.2a, b):

=Jo

F(p)

and

f(x) = ^- I F(p)x-pdp (ocj.

1.2 Derive (1.3.8).



2 THE LAPLACETRANSFORM

2.1 Introduction

The Laplace transform, it can be fairly said, stands first in importanceamong all integral transforms; for, while there are many specific examplesin which other transforms prove more expedient, the Laplace transformis both the most powerful in dealing with initial-value problems and themost extensively tabulated. We consider in this chapter some of thefundamental properties that give it this flexibility and illustrate theseproperties by application to typical problems in vibrations, wave propa-gation, heat conduction, and aerodynamics.

The Laplace-transform pairs required in this chapter are tabulated inTable 2.1 (Appendix 2, p. 83). These results may be extended in variousways with the aid of the operational formulas of Table 2.2 (Appendix 2,p. 84). We give derivations of the more important formulas in the textand set the derivations of others as exercises. We refer to entries in thesetables by the prefix T, followed by the appropriate entry number. Moreextensive tables are listed under Bibliography. We regard Erdelyi et al.(EMOT) as a standard reference, although the most extensive table ofLaplace transforms now available is that of Roberts-Kaufman (1966).

Many of the Laplace transforms that arise in the analysis of a dynamicalsystem with a finite number of degrees of freedom have the form

where G and H are polynomials, of degree M and N,N > M. The inver-sion of F(p) may be inferred from a suitable partial-fraction expansion,together with the basic entries of Table 2.1. The basic techniques for

13



14 / THE LAPLACE TRANSFORM / CH. 2

partial-fraction expansions are developed in Appendix 1. A general ex-pansion, subject to the restriction that the zeros of H(p) be simple, isgiven by (2.7.7) (see also Exercise 2.8). It may be more direct, in manyapplications, to obtain appropriate, partial-fraction expansions by in-spection, as in the examples of (2.3.7), (2.6.13), and (2.6.16).

The general procedure for the application of an integral transform toboundary- and/or initial-value problems comprises the following steps:

(a) Select and apply a transform that is appropriate to the partial dif-ferential equation! and its boundary and/or initial conditions in thesense that it reduces the differential operations with respect to a givenindependent variable to algebraic operations. Different transformationsmay be applied simultaneously to different variables—e.g., a Laplacetransformation with respect to time and a Hankel transform with respectto a radial variable, as in Section 4.3.

(b) Solve the transformed problem for the transform.(c) Invert the transform through any or all of the following proce-

dures: (i) preliminary simplification through operational theorems,(ii) direct use of tables, (iii) direct use of the inversion integral, (iv) approxi-mate inversions through appropriate expansions of the transform or modi-fications of the contour integral. Asymptotic approximations such asmay be obtained through the invocation of Watson's lemma (Section2.7) or the method of stationary phase (Section 3.7) are especially im-portant.

1,2 Transforms of derivatives

The Laplace transformation is typically applied to initial-value prob-lems in order to reduce the derivatives of some function, say /(£), toalgebraic form. Let

F(p) = X{f{t)} EE f °° e-"f(t) dt (2.2.1)

denote the Laplace transform of f{t).% We obtain the transforms of thefirst and second derivatives,/'^) and/"(f), through integration by parts(the student should fill in the details of these integrations):

t Integral transforms also may be useful for the solution of integral equations, but we donot consider such applications here. See Sneddon (1951), Tranter (1956), and Van der Pol-Bremmer (1950).| We use t as the independent variable in most of this chapter in recognition of the fact thatusually that variable is time; nevertheless, initial-value problems may be encountered inwhich t is a space variable—e.g., those in linearized, supersonic flow, as in Section 2.10



SEC. 2.3 / SIMPLE OSCILLATOR / 15

Se{f'(t)} =pF(p)-/(0) (2.2.2a)

and 2>{f"(t)} = p2F(p)-pf(O)-f'(O), (2.2.2b)

where/'(0) denotes the initial (f -» 0 + ) value off'(t). Similarly, we obtainthe transform of the nth derivative, ftn\t), by integrating n times by parts:

" l P"-""1/""^), (2.2.3)

where /(m)(0) s lim ^ g l (2.2.4)

One of the virtues of the Laplace transformation for initial-valueproblems is the systematic manner in which the initial values, /(0), . . . ,/(lI~1)(0), are incorporated in the calculation. If, on the other hand, one(or more) of these initial values is not specified, it must be regarded as anunknown parameter, to be determined by some other condition; forexample, /'(0) might have to be determined in such a way as to ensurethat f(t) has a prescribed value at t = tl5 and the Laplace transformthen will be less efficient than in the solution of the corresponding initial-value problem.

There exists an important class of problems, dealing with systemsthat are initially at rest, in which each of the initial values, /(0), . . . ,j(n -1 ) QJ j s z e r o . t j i e n ^ a n c j onjy the i l ) w e h a v e t h e Heaviside operational rule

= f{n\t) [/(O) = - " = /(n"1)(0) = 0]. (2.2.5)

A related but generally valid, result is

1 f(t)(dt)\ (2.2.6)Jo

23 Simple oscillator

We consider (Figure 2.1) a one-dimensional oscillator of mass m andspring constant k (such that a displacement x is opposed by a restoringforce — kx) that is subjected to a constant force w0 in the positive-x direc-tion at t = 0. Given the initial displacement and velocity of m, x0 and

FIGURE 2.1 The oscillator of Section 2.3.




v0, we seek x(t). [The student presumably will be familiar with the non-operational treatment of this elementary example. We introduce it, to-gether with its subsequent modifications, in order to illustrate basicoperational procedures. Gardner-Barnes (1942) and Thomson (1960) giveextensive applications of the Laplace transform to more sophisticatedmechanical-vibration problems.]

The equation of motion, obtained by applying Newton's second lawto the mass m under the action of the forces w0 and — kx, is

mx"(t) + kx(t) = w0. (2.3.1)

The initial conditions are

x(0) = x0 and x'(0) = VQ. (2.3.2)

Let X(p) denote the Laplace transform of x(t). Transforming the left-handside of (2.3.1) through (2.2.2b) and w0 through T2.1.1, we obtain

m(p2X - px0 - v0) + kX = wop~K (2.3.3)

Solving (2.3.3) for X and introducing the natural frequency of the oscillator,

we obtain

X(p) = (P2 + P2Y 'Ixop + v o + ^ p - 1 \ . (2.3.5)

The inverse transforms of the terms in x0 and v0 on the right-handside of (2.3.5) are given by T2.1.3 and T2.1.4. The remaining term doesnot appear in Table 2.1, but we may invert it either by applying T2.2.4to T2.1.4 according to

&~l{p~\p2 + P2)"1} = P"1] sin ptdt = P~2{\ - cos fit) (2.3.6)Jo

or by invoking the partial-fraction expansion f

_ i ! f1 P ^ (237)

which yields the same result. Invoking T2.1.3, T2.1.4, and (2.3.6) in (2.3.5)and setting m/?2 = /c, we obtain

x(t) = x0 cos Pt + -%• sin pt + - ~ (1 - cos /ft)- (2.3.8)

tThe student will find, with some experience, that simple partial-fraction expansions maybe constructed by inspection. Formal rules for partial-fraction expansions are given inAppendix 1, where the derivation of (2.3.7) is given as an example.



SEC. 2.4 / CONVOLUTION THEOREM / 17

The term wo//c on the right-hand side of (2.3.8) represents the staticdisplacement of the oscillator under the force w0; the remaining termsrepresent free oscillations, which would have been transient (decaying ast -> GO) if the oscillator had been damped. The first two terms on theright-hand side of (2.3.8) represent the response of the oscillator to theinitial displacement and velocity and would have the same form for anyapplied force. The third term represents the response to the applied forcefrom an initial state of rest (both the displacement and the velocity givenby the third term vanish at t = 0). Replacing w0 on the right-hand sideof (2.3.1) by some other force, w(t), alters the third term but not the firstand second; accordingly, we take x0 = v0 = 0 in the subsequent varia-tions on this example.

2.4 Convolution theorem

It frequently proves expedient to resolve a Laplace transform into aproduct of two transforms, either because the inversions of the lattertransforms are known or because one of them represents an arbitraryfunction—typically an input to some physical system. Let F^p) andF2(p) be the Laplace transforms of/x(r) and/2(0; the convolution theoremstates that

= [h(t - T)f2(T)dT. (2.4.1)Jo

The convolution integral on the right-hand side of (2.4.1) is often denotedby/i(0 *f2(t).

To prove (2.4.1), we form the product of the defining integrals forFx and F2 to obtain

1 P 2 P ~ J o J o • ' •Introducing the change of variable G = t — T and invoking the require-ment that f\ must vanish for negative values of its argument, we obtain

Fl(P)F2(p) = r e~pt I P/i(* - T)/2(T) d r l dt9

whence the transform of the quantity in brackets is F1F2, the inversion ofwhich yields (2.4.1).

We remark that, in typical applications, the right-hand side of (2.4.1)represents a superposition of effects of magnitude /2(T), arising at t = T,

for which f^t — T) is the influence function, i.e., the response to a unitimpulse. Indeed, it constitutes the extension to impulsive inputs ofDuhamel's superposition theorem for step inputs.




The unit impulse is known as the Dirac delta function and has the formalproperties c/ ,F F 3(t - T) = 0 ( M i ) (2.4.2a)and

f(T)S(t -x)dx = f{t). (2.4.2b)0

Setting f(t) = S(t) in (2.2.1) and extending the integration to t = 0 - ,W e ° b t a i n - = 1 . (2.4.3)

5(x-a) = -_ lim ,._ ; , , , (2.4.4)

The delta function, as defined by (2.4.2), is improper, but it can be definedas the limit of a proper function and was so introduced by both Cauchyand Poisson. The function used by Cauchy was

1

7i y"o+ (* - a)2 + y2

the right-hand side of which may be identified as a solution to Laplace'sequation for a doublet source located at (x, y) = (a, 0). Perhaps the mostsatisfactory way of incorporating such functions as S(t) in the applicationof integral transforms is provided by the theory of generalized functions[Erdelyi (1962) and Lighthill (1959)], but we rest content with the in-tuitive support provided by such physical idealizations as that of a con-centrated force (which may be regarded as the limit of a large pressuredistributed over a small area).

We illustrate the convolution theorem by replacing the constant forcew0 in (2.3.1) by an arbitrary force w(f), and setting x0 = v0 = 0. Replacingthe transform of w0, wo/p in (2.3.5), by W{p\ we obtain

X(p) = m-\p2 + p2ylW(p). (2.4.5)

Setting Fx = m~\p2 + jff2)"1 and F2 = ^ in (2.4.1) and inverting F1

through T2.1.4, we obtain

x(t) = (mjS)"1 W(T) sin [j3(f - T)] rfr. (2.4.6)Jo

Setting w = w0 (as a particular example) in (2.4.6), we recover the thirdterm on the right-hand side of (2.3.8).

2.5 Heaviside's shifting theorem

Disturbances often arise at times other than zero (we may chooset = 0 to correspond to any particular event and include the effects ofprior events in the initial conditions). Let f(t — a) be a function thatvanishes identically for t < a9 where a > 0; then,



SEC. 2.5 / HEAVISIDE'S SHIFTING THEOREM / 19

- a)} =\ e~ptf(t - a) dt (2.5.1a)

e~pxf{x) dx (2.5.1b)Jo

= e~apF(p\ (2.5.1c)

where (2.5.1b) follows from (2.5.1a) after the change of variable! = t — a,and (2.5.1c) follows from (2.5.1b) by virtue of (2.2.1), in which we maywrite T as the variable of integration in place of t. Inverting (2.5.1c), weobtain

' (t ^ a, a > 0). (2.5.2)

This is Heaviside's shifting theorem.It often proves convenient, in dealing with discontinuities that arise at

other than t = 0, to introduce the Heaviside step function,

H(t) = 1} ( t ^ 0). (2.5.3)

This introduction permits a result like (2.5.2) to be exhibited in the morecompact form

J?-l{e-apF(p)} = f{t - a)H(t - a). (2.5.4)

A related theorem, which is complementary to (2,5.1), is

2-l{Fip + *)} =e-"tf(t). (2.5.5)

The proof follows directly from the inversion integral

rc+iooF(p)eptdp, (2.5.6)i

together with an appropriate change of variable (the student should fillin the details). The parameter a may be complex provided that theparameter c in the corresponding inversion integral is appropriatelychosen.

Suppose, for example, that the constant force in the problem of Sec-tion 2.3 is removed at t = T. Then,

w(t) = wo[H(t) - H(t - T)-] (2.5.7)

describes the applied force as a function of time. Replacing wo/p in (2.3.3)and (2.3.5) by

\ \ -e~pT), (2.5.8)




we obtain

x(t) = Y {(1 - cos pt)H(t) - [1 - cos P(t - T)]H(t - T)} (2.5.9)

as the response of the oscillator to the applied force. We emphasize thatthis example is intended only to illustrate the mechanics of the shiftingtheorem, since we could have obtained (2.5.9) directly from the super-position theorem for the oscillator. We use the shifting theorem to greateradvantage in the following section.

2.6 Periodic functions

Let f(t) be a periodic function of period T for t > 0, such that

f(t + T) = f(t) (t > 0) (2.6.1)

and f(i) is piecewise continuous in 0 ^ t ^ T. We then may calculateF(p) as follows:

F(P) = \ e-"f(t)dt= £ fn+ e-»f(t)dt (2.6.2a)Jo n = O JnT

Y, e-npT\ e-pJ{x)dt« = o Jo

= (1 - -p'f(t) dt,

(2.6.2b)

(2.6.2c)

where (2.6.2b) follows from (2.6.2a) after the change of variable t = nT + T,and (2.6.2c) follows from the known result for the sum of a geometricseries and the change of variable T = t.

fit)

1 T2 7

FIGURE 2.2 The triangular wave of Section 2.6.



SEC. 2.6 / PERIODIC FUNCTIONS / 21

Consider, for example, the triangular wave sketched in Figure 2.2 anddescribed either by

It

v (2.6.3)-f) (iT^t^T)

in conjunction with (2.6.1) or by

2f 1fit) = = tH(t) + 2 £ (-)n(t - \nT)H(t - %nT) . (2.6.4)

Substituting (2.6.3) into (2.6.2c), carrying out the integration, and can-celling the common factor 1 — exp(—jpT) in the numerator and de-nominator of the result, we obtain

(2.6.5)

Transforming (2.6.4) with the aid of T2.1.1 and the shifting theorem,T2.2.5, we obtain

\ + 2 £ (-)nexp(-inpT)] ^-^S(p\ (2.6.6)J P l» = i

where S(p) denotes the bracketed series. The student may establish theequivalence of (2.6.5) and (2.6.6) by expanding the hyperbolic tangent in(2.6.5) in powers of exp(—jnpT).

The inverse transform of (2.6.6) obviously yields (2.6.4). An alternativerepresentation is provided by the expansion [Jolley (1961)]

tanh y = 2y £ [$sn)2 + y2y\ (2.6.7)

the substitution of which into (2.6.5) yields

,-i(p2 + ^ - 1 ^ (2.6.8a)

where

& = ^ , (2.6.8b)

and, here and throughout this section, s i? summed over 1, 3, 5, . . . , oo.Remarking that each of the individual terms in (2.6.8a) is similar to thetransform of (2.3.6), we obtain the inverse transform




fit) = 4S^^" 2 (1 - cos pst) (2.6.9a)

where (2.6.9b) follows from (2.6.9a) after invoking (2.6.8b) and the knownresult

yL c2 8

s S o

[which may be derived by letting y -• 0 in (2.6.7)]. We may identify(2.6.9b) as the Fourier-series representation of the periodic function /(r),as defined by (2.6.1) and (2.6.3).

The response, say x(t), of a linear system to a periodic input, say f(t\may be determined by either of two, more or less distinct, procedures:(i) expanding f(t) in a Fourier series, determining x(t) for a particularFourier component of /(£), and summing over all components (which ispermissible by virtue of linearity); (ii) determining x(t) for nT < t <(n + 1)T, say xn(t), and then determining the constants of integration inxn(t) by invoking the appropriate continuity requirements. The Laplace-transform determination of x(t) effectively combines these two procedures.Let us suppose, for example, that the constant force in the example ofSection 2.3 is replaced by the periodic force

(2.6.10)

where f(t) is given by (2.6.3), and wx is the peak force (w = vvx at t = %Replacing wo/p by w^ip), setting x0 = v0 = 0 in (2.3.5), and introducing

*i = -r = —&> (2.6.11)

we obtainX(p) = /?2*i(p2 + PT'Fip)- (2.6.12)

The parameter Xi is the static displacement of the oscillator under theaction of the peak force wx.

Considering first the representation (2.6.6) for F(p) in (2.6.12), weobtain

X[p) = 2~Lp-2(p2 + PT'Sip) (2.6.13a)

= 1J~\JP'2 ' (P2 + P2Y W (2.6.13b)



SEC. 2.6 / PERIODIC FUNCTIONS / 23

Inverting (2.6.13b) through T2.1.1, T2.1.4, and the shifting theorem,T2.2.5, we obtain

where

^) = ^1(t-r1sin^) (2.6.15)

is the inverse transform of (2.6.13b) for S = 1, corresponding to theleading term in the bracketed series of (2.6.6). We remark that the sumof the terms in t and t — \riT in (2.6.14) is equal to w1/(t)/fe, the staticdisplacement of the oscillator under the force w(t).

The representation (2.6.14) is especially advantageous for relativelysmall values of t, but it is not satisfactory for t > T in consequence of thelarge number of terms that must be retained in the series;! moreover,(2.6.14) obscures the resonances that occur if pT/2n is an odd integer.We therefore turn to the representation (2.6.8) for F(p\ the substitutionof which into (2.6.12) yields

X(p) = 1^f± X p~ V + &T V + P2r' (2.6.16a)

^ I P - U 2 - PT'liP2 + P2)'1 - (P2 + &T1] (2.6.16b)i s

after a partial-fraction expansion. Inverting the individual terms in(2.6.16b) with the aid of (2.3.6), we obtain

x{t) = l ^ J i X(ft2 - P2)'1^-2^ - cos jSt) - &-2(l - cos

(2.6.17)Invoking the result

^ZG»* - P2)'1 = I [ ( W - ( i / m 2 r ' = 203T)-1 tan(i/3T),

(2.6.18)

which follows from (2.6.7) with y = lifIT therein, we may place (2.6.17)in the alternative form

x{t) = ^tan(i/?T)(l - cos fit) - f± X ft 2(j52 - P2Y\\ - cosftr).

(2.6.19)

fit is, in fact, possible to sum the series in (2.6.14), but this reflects the simplicity of the par-ticular example rather than any general property of such problems.




Invoking the result

i , (2.6.20)

which follows from (2.6.18), and substituting fis from (2.6.8b) into theremaining series, we obtain the additional representation

\ - 2(jST)-1 tan$pT) cos pt

- i c o s ^ J | . (2.6.21)

The terms in cos pt in each of (2.6.17), (2.6.19), and (2.6.21) represent thefree oscillation that must be added to the forced oscillation to satisfy theinitial conditions; the remaining terms represent the forced oscillationproduced by the periodic force. Resonance occurs if PT/2n is an oddinteger, say s*, in which case both tan(i/?T) and the s^th term in each ofthe series are infinite. Applying L'Hospital's rule to the s^th term in(2.6.17), we obtain

**(0 = l^fi I1 ~ cos P* ~ IP* sin P^ (PT = 2ns*)- (2-6-22)

We could have dealt with the problem of resonance directly in (2.6.16a),for which the s^th term is

= 2ns*). (2.6.23)i

Differentiating (2.3.6) with respect to the parameter /?, we find that theinverse transform of (2.6.23) is given by (2.6.22).

2.7 The inversion integral

The most expedient procedure for the inversion of a given transformis through a suitable table of transform pairs, such as (but typically muchmore extensive than) that of Table 2.1, augmented by operational theorems,such as those of Table 2.2. This procedure is illustrated by the examplesin the preceding sections and in several of the exercises at the end of thischapter. It is clearly limited by the extent of the available tables, althoughit typically suffices for the solution of ordinary differential equations with



SEC. 2.7 / THE INVERSION INTEGRAL / 25

constant coefficients, such as those that arise in electric-circuit andmechanical-vibration problems.

The most general procedure is through the inversion integral,

= _i_ [C+i°2ni )c-iaD

F(p)ept dp, (2.7.1)

in which F(p) is a prescribed function of the complex variable p that hasno singularities in c ^ pr < oo. We assume that t > 0; the path of in-tegration could be closed in pr > c if t < 0, with the result that f(t) = 0by virtue of the fact that F(p) would have no singularities within the re-sulting, closed contour.

We consider first that class of problems in which F(p) is a meromorphicfunction—that is, a function having no singularities other than poles for\p\ < oo—and is bounded at infinity according to

lira |p|»F(p) < oo (b > 0, p * pn), (2.7.2)

where px, p2, . . . are the poles of F(p). Poles rarely occur in pr > 0, al-though they frequently occur on pr = 0; accordingly, it usually suffices tolet c -> 0+. Typical Laplace transforms (of the type considered herein)decay at least like l/\p\ as |p| -• oo, corresponding to b ^ 1 in (2.7.2), butit is useful to have the greater generality implied by the weaker restrictionb > 0.

Consider the contour integral

2ni ] c— \F(p)e» dp, (2.7.3)

** Pr

FIGURE 2.3 Contour of integration for meromorphic Laplace transform.




where (see Figure 2.3) C consists of a vertical segment, pr = c,—P ^p^P,and a semicircular arc of radius P in pr < c that does not intersect any ofthe poles of F(p). The integral over this arc vanishes in the limit P -> ooprovided that F(p) satisfies the conditions assumed above—in particular(2.7.2); the proof of this statement, which constitutes an extension ofJordan's lemma [which would be directly applicable for c = 0 + in (2.7.1)and b ^ 1 in (2.7.2)] is given by Carslaw-Jaeger (1953, Section 31). Theintegral over the vertical segment in this same limit tends to f(t\ as definedby (2.7.1); accordingly, / -• f(t) as P -^ oo. It then follows from Cauchy'sresidue theorem that f(t) is given by 2ni times the sum of the residues ofF(p)ept at its poles, say pi, p2,..., which, by hypothesis, lie in pr < c. If,as is typically true, these poles are all simple, the result is

f{t) = £ R(pn) expOv)* (2.7.4a)

where

R(pn) = lim (p - pn)F(p\ (2.7.4b)

the summation is over all of the poles, and R(pn) is the residue of F(p) atP = Pn.

It frequently proves convenient to place F(p) in the form

°(P)™-m-

where G(p) has no singularities in \p\ < oo [but see sentence following(2.7.6)], and H(p) has simple zeros at p = Pi,p2, Substituting (2.7.5)into (2.7.4b) and invoking L'Hospital's rule, we obtain

« ~ & ~ Pn)G(p)

where H(pn) == 0, and H'{pH) is the derivative of H(p) at p = pn. The result(2.7.6) may be generalized to allow any convenient factoring of the form(2.7.5) in the neighborhood of a given pole, p = pn, subject only to therequirement that G(p) be regular in that neighborhood. Substituting(2.7.5) and (2.7.6) into (2.7.4a), we obtain

xp(pn0, where H(pn) = 0. (2.7.7)

This last result is due essentially to Heaviside (who gave an equivalentform) and is generally known as the Heaviside expansion theorem.




The zeros of H(p) are typically either real or complex-conjugate pairs.The net contribution of such a pair to the summation of either (2.7.4a)or (2.7.7) is equal to twice the real part of either complex zero takenseparately.

It occasionally happens that Hip) has a double zero. The requiredmodification of (2.7.7) is given in Exercise 2.8. Zeros of third or higherorder rarely occur at other than p = 0 and are best handled on an ad hocbasis, rather than by further generalizations of (2.7.7).

We illustrate (2.7.7) by returning to the transform given by (2.6.5) and(2.6.12),

X(p) = 1 ^ - p " V + P2Yl tanh(ipT), (2.7.8)

which is a meromorphic function that has simple poles at

P = 0, ±ifi, ± ^ = ±ips (s = 1, 3, 5 , . . . , oo) (2.7.9)

and vanishes like |p |"4 as |p| -• oo; accordingly, it satisfies the requiredconditions for the validity of (2.7.7).t We may place it in the form of(2.7.5) by choosing

G(P) = ±E—± sinh(ipT), Hip) = p(p2 + p2) coshiipT). (2.7.10)pT

We emphasize that G is regular at p = 0. Substituting (2.7.10) into (2.7.6),we obtain

R(0) = ixl9 Ri±iP) = -XtipT)-1 tan(ij3T),

and

s) = ^pr2(ti - p2rl. (2-7.il)

Substituting (2.7.11) into (2.7.4a) and collecting the contributions of thecomplex poles in conjugate pairs, we obtain (2.6.21).

We turn now to that more general class of problems in which Fip)may have not only poles, but also branch points, in pr < c.X The contourC for the integral of (2.7.3) then must be deformed around the appropriate

flf the derivation of (2.7.4a) had been given explicitly for X(p), it would have been expedientto have assumed P = (N + j)(n/T\ where N is an integer, and let P -• GO by letting N -• ooin order to avoid the difficulty of having C pass through a pole.JThe remaining material in this section is not required in Section 2.8, and the student mayfind it expedient to read Sections 2.8 and 2.9 through (2.9.7) before studying this materialin detail.




+-pr

FIGURE 2.4 Contour of integration for Laplace transform having branchpoint at the origin.

branch cuts, as illustrated in Fig. 2.4 for the important special case of abranch point at p = 0. More generally, branch points of Laplace trans-forms are likely to be on the imaginary axis, but only occasionally else-where. The contributions of the poles, if any, may be evaluated as before,in particular from (2.7.4), but it also is necessary to include the integralaround the branch cut.

Considering the contour of Figure 2.4, we let p = ue~in on AB, p = sew

on BC, and p = uein on CD to obtain the contributions

/,« =

lBCl Cn

= — Qxp(etew)F(sei0)(iseie d9),

and

ICD = ^-.\ e-"F{ue*X-du)Zm ,

to the contour integral / [note that p = — u on both sides of the cut,whereas F(uein) and F(ue~in) have different values if, as assumed, p = 0is a branch point]. The integral IBc vanishes in the limit e -+ 0 if pF(p) -> 0as p -> 0; otherwise it must be evaluated by expanding the integrandabout 6 = 0 on a more or less ad hoc basis [note that eptF(p) does nothave a Laurent-series representation in the neighborhood of p = 0]. If,as in the example of Section 2.9 below, F(p) is of the form p~ 1F1(p), where




Fi{p) has a branch point at p = 0 but is finite there, the limit of IBC ase -» 0 is simply F^O). If F(p) has an infinity of higher order, the limitingforms of each of IAB, IBC, and ICD may be improper, in which case thelimit as e -> 0 may be taken only after the three integrals have beencombined.

Proceeding on the assumption that IBC exists, we combine IAB and ICDand write

J = l im (IAB + ICD) = e-ut<t>(u) du9 (2.7.12a)

£^o Jo

wherein

<Ku) = VmT^Fiue-**) - F{uein)\ (2.7.12b)It may be possible to regard J(t) as the Laplace transform of 0(w), with tas the transform variable, and to evaluate it from a table of Laplacetransforms; however, whether or not this is true, it often suffices to obtainan asymptotic approximation for large t (such approximations sometimesprove satisfactory for surprisingly small values of t).

The asymptotic evaluation of J is especially simple if (j)(u) has the repre-sentation (in u < R)

(j)(u) = £ anu{nlr)-1 (r > 0, u <R\ (2.7.13a)

wherein r = 1 or 2 in typical applications. If positive real numbers Cand a exist such that \(f)\ < Ceau, u > R, then Watsons lemma statesthat J has the asymptotic expansion

J - 5 "nr(-) t~nlr (t - QO), (2.7.13b)n=l VJ

where F denotes the gamma function. See Copson (1965, Section 22) forthe proof.

Consider, for example, F(p) = p 1 2 ; then

Setting r = 2, n = 3, and a2 = - l/n in (2.7.13), we obtain

J ~ - j r 1 ^ ) * - 3 ' 2 = -^Trf3)"1/2.

More generally (see Exercise 2.19),

F(P)= I M " / r h l ( r > 0 ) (2.7.14a)

implies




J ~ 71" 1 £ bnY (-) sin — r "/r (t -> oo), (2.7.14b)

to which terms for which n/r is an integer make no contribution.We obtain the complementary result that

F(p) - | CdT"" (|p| -+ ex), r > 0) (2.7.15a)

implies

by applying Watson's lemma directly to the Laplace-transform integral.The expansion of <f)(x) about x = 0 may contain logarithmic terms, in

which case a formal asymptotic expansion may be obtained by integratingterm by term and invoking the result (see Exercise 2.13)

i e~xtxm log x dx = m!(l + 7 + 7 + ••• + m"1 - y - log t ) r m ~ 1 ,

(2.7.16)

where y = 0.577215... is Euler's constant and 1 + ••• + m"1 = 0 form = 0.

Other methods of evaluating branch-cut integrals, of which J is onlyone—albeit the most important—form, are discussed in the monographsof Erdelyi (1956) and Copson (1965). Finally, we note that the numericalevaluation of integrals such as J may be entirely practical by virtue of theexponential convergence, although it may be necessary first to separatethe contribution of the singularity at the origin.

2.8 Wave propagation in a bar f

A uniform bar of cross section A is at rest and unstressed for t < 0,with one end fixed at x = 0 (see Figure 2.5). At t = 0, a force of mag-nitude PA is applied to the free end, x = /, in the direction of the positivex axis. We require the subsequent motion on the hypothesis of smalldisplacements.

Let y(x, t) denote the longitudinal displacement from equilibrium ofthe section at x. Hooke's law implies that the stress associated with thestrain yx at any section is Eyx9 so that the net force on a differentialelement of length dx and area A between x and x + dx (see Figure 2.6) is

t Carslaw-Jaeger (1953), Morse (1948), and Sneddon (1951) give many applications of theLaplace transform to problems of mechanical wave propagation.



SEC. 2.8 / WAVE PROPAGATION IN A BAR / 31

PA

FIGURE 2.5 A uniform bar to which a load PA is applied abruptly attime t = 0.

AE[yx + yxx dx] - AEyx = AEyxx dx,

where E is Young's modulus, and subscripts denote partial differentiation.The mass of that element is pA dx, where p is its density. Equating theproduct of this mass and the acceleration to the net force on the sectionand dividing the result by pA dx, we obtain the wave equation

c2yxx = ytt, where fE\'2

The initial conditions are

y = yt = 0 at t = 0 and 0 < x < I.

The boundary conditions are

y = 0 at x = 0and

Eyx = P at x = I (t > 0).

(2.8.1)

(2.8.2)

(2.8.3a)

(2.8.3b)

AEv

FIGURE 2.6 An element of the bar, showing the forces on the faces at xand x + dx and the acceleration vector, ytt.




Taking the Laplace transform of (2.8.1) with respect to t and invoking(2.8.2), we obtain

Yxx - (^ Y = 0. (2.8.4)

Transforming (2.8.3), we obtain

y = 0 at x = 0 (2.8.5a)and

Yx = ^- at x = I. (2.8.5b)Ep

The most general solution of the ordinary differential equation (2.8.4) is

A sinh — + B cosh —.c c

We choose B = 0 in order to satisfy (2.8.5a) and then determine A tosatisfy (2.8.5b). The result is

_ Pcsinh(px/c)Y ( Z 8 ' 6 )

as may be verified by direct substitution. Remarking that Y(p) has simplepoles at

± P " + ^ (n = 0,1,2,...), (2.8.7)P = 0,

we may invert (2.8.6) through (2.7.7) by choosing

G(p) = ^ sinh —, H(p) = p cosh - . (2.8.8)Ep c c

The end result is (the student should fill in the details)

y(x9 t) = ^ | y ~ ^ £ (-T(2n + I)"2 sin(knx) cos(fcnc01, (2.8.9)

where

The first term in (2.8.9), Px/E, represents the ultimate (static) displacementof the bar; the remaining terms represent standing waves that would dieout gradually if friction were admitted. In the absence of friction, how-ever, the displacement continues to oscillate about the static displacement.



SEC. 2.8 / WAVE PROPAGATION IN A BAR / 33

We obtain an alternative solution by expressing the hyperbolic func-tions in terms of exponentials and substituting the expansion

1 2exp(-p//c) „ ^ , „,____, , _ . . , . , , . , ( 2

cosh(p//c) 1+exp(-2p/ /c ) Mtti

into (2.8.6) to obtain

x jexp|-^((2n + 1)/ - x)1 - exp|-^((2n + 1)/ + x)11. (2.8.12)

Now the inverse transform of p~ 2 is t, whence the shifting theorem, T2.2.5,yields

y(x>t) = ? I (-nict - ((2n + 1)/ - x)] - [ct - ({In + 1)/ + *)]},

(2.8.13)

where, by definition, the square brackets vanish identically if their contentsare negative.

Equation (2.8.13) exhibits the solution as a series of traveling waves,the first and second sets moving respectively toward and away from x = 0.Such a representation is valuable not only because it presents the solutionin a finite number of terms (since each of only a finite number of the squarebrackets is positive at any finite time), thereby rendering numerical com-putation simpler for small ct/l, but also because it provides additional in-sight into the physical problem. It is, indeed, one of the great virtues ofthe Laplace-transform solution that it comprises both the standing- andtraveling-wave representations (cf. Section 2.6).

At x = /, where the load acts, the displacement given by (2.8.13)reduces to

yd,t) = Y

which is simply the triangular wave of Figure 2.2, with a mean value equalto the static displacement, Pl/E (so that the maximum value is 2P1/E,rather than unity as in Figure 2.2), and a period equal to the time requiredfor a wave to travel four times the length of the bar, T = 41/c.




2.9 Heat conduction in a semi-infinite solidf

We consider now the classical problem of a semi-infinite solid, x > 0,that is initially at temperature v = 0 and for which the boundary, x = 0,is maintained at temperature v = v0 for t > 0. The rate at which heatis transferred across a plane section, x = const, in the direction of in-creasing x is -Kvx, where K is the thermal conductivity; accordingly thenet rate at which heat is transferred into a slab bounded by x and x + dxis Kvxx dx per unit area. This must be equal to the rate at which the slabis gaining heat, namely pcvt dx, where p is the density, c is the specific heat,and dx is the volume (per unit of transverse area). Equating these tworates, we obtain the diffusion equation

vt = KVxx, where K = - (2.9.1)J

pc

is the diffusivity. The initial condition is

v = 0 at t = 0 ( JC> 0). (2.9.2)

The boundary conditions are

v = v0 at x = 0 (2.9.3a)and

\v\ < oo as x -• oo. (2.9.3b)

Transforming (2.9.1)-(2.9.3), we obtain

Vxx - V-V = 0, (2.9.4)

V = — at x = 0, (2.9.5a)

and

|F| < oo asx -> oo. (2.9.5b)

Choosing that exponential solution of (2.9.4) which satisfies (2.9.5b) andthen choosing its amplitude to satisfy (2.9.5a), we obtain

- - x\, (2.9.6)\K/ J

where (p1/2)r > 0 for pr > 0. Invoking T2.1.10, we obtain

tCarslaw-Jaeger (1949) give extensive applications of the Laplace transform to heat-conduction problems in one, two, or three dimensions.JThe right-hand side of (2.9.1) becomes KV2V for three-dimensional (isotropic) heat con-duction.



SEC. 2.9 / HEAT CONDUCTION IN A SEMI-INFINITE SOLID / 35

= 2n m e~u2 du, (2.9.7)

where erfc denotes the complementary error function.The error function, erf z = 1 - erfc z, is tabulated, and its expansions

for both small and large z are known; however, we use the transform of(2.9.6) to illustrate the general procedure for multivalued transformsdescribed in Section 2.7. We choose the branch cut for p1/2 along thenegative-real axis to satisfy (p1/2\ > 0 [the student should verify this by set-ting p = uew and showing (p1/2)r = w1/2cos(0/2) > 0 in |0| < re] and closethe contour of integration in pr < 0, as shown in Figure 2.4. The functionV(p) has no singularities inside the closed contour, whence the contourintegral of V(p) exp(pt) around that contour is zero. Moreover, F(p)satisfies (2.7.2) with b = 1 therein, whence the integrals over the arcs atinfinity also are zero. It follows that the inverse transform of V is equal tothe integral over the top and bottom of the branch cut plus the integralaround the small circle at the origin taken in a counterclockwise direction,

where the contour ABCD is as in Figure 2.4.In the neighborhood of the origin, V tends to infinity like vo/p, and the

contribution of the path BC as its radius tends to zero is simply v0 [asmay be proved by setting p = e exp(i#) in the integrand, letting £ -> 0,and integrating between — n and n; cf. IBC in Section 2.7]. LettingF(p) — V(p)/v0 in the development of Section 2.7, we obtain

v = vo(l + J), (2.9.9)

where J is given by (2.7.12a) with

- ( - M )- i expT- ^ Y ' 2 xe"

i r/wY isin - x\ (2.9.10a)

nu 1\KJ J- V _Jz£_ K -»-<l /2> x 2» .+ lum-(l/2) (2.9.10b)n m% (2m + 1 ) !




Invoking (2.7.13), with r = 2, and substituting the result into (2.9.9), weobtain

v_ 1% (2m

(t-+ oo). (2.9.11)

Finally, anticipating the alternative solution of Section 3.5 below, wesubstitute (2.9.10a) into (2.7.12a) and (2.9.9) and introduce the change ofvariable u = /cfc2 to obtain the representation

— = 1 - - I V 1 Qxp(-k2Kt) sin kx dk. (2.9.12)^o n Jo

2.10 Oscillating airfoil in supersonic flow t

We consider (see Figure 2.7) the disturbances produced by a thin(two-dimensional) airfoil that is traveling through a perfect (inviscid, non-heat-conducting) fluid with the supersonic speed U and that executessmall, transverse oscillations. The disturbances, being small by hypothesis,are sound waves (the flow is isentropic). The velocity potential for thesedisturbances, defined such that the perturbation velocity is given by

v = V(/>, (2.10.1)

satisfies the two-dimensional wave equation in a fixed reference frame (withrespect to which the fluid at infinity is at rest),

^(^XX + fiyy) = &TT) (2.10.2)where X and y are Cartesian coordinates, and Tis time. This referenceframe is not convenient for the description of the motion of the airfoil,

u u- • x

iv(x)ev

- • x

FIGURE 2.7 Schematic of oscillating airfoil in fixed (X, y, T) and movingreference frames.

fThis section may be omitted without loss of continuity.



SEC. 2.10 / OSCILLATING AIRFOIL IN SUPERSONIC FLOW / 37

which is more conveniently described in a reference frame that moveswith the mean velocity U; accordingly, we introduce the Galilean trans-formation

x = X + UT, t = T, (2.10.3)

under which (2.10.2) goes over to (the student should carry out this trans-formation in detail and observe that, whereas t = T, <j>t # (j)T\ con-versely, x ^ I , but (f)x = (j)x)

C2(<t>XX + tyy) = ( | + U 04>. (2.10.4)

We assume that the transverse velocity of the airfoil is harmonic in time,with angular frequency co, and has the complex amplitude i?(x), such thatit is given by the real part of v(x) exp(icot). We may evaluate the corre-sponding velocity in the air, </>y9 at y = 0+ (0 —) for points just above(below) the airfoil by virtue of the assumption that the airfoil is thin. Wethen may pose the boundary conditions for $ in the form

cj)y = v(x)ei(Ot on y = 0+ and x > 0 and |</>| < oo as \y\ -> oo,(2.10.5)

with the implicit understanding that the required solution is given by thereal part of (j>.

Invoking the fact that the airfoil is moving with supersonic speed, weinfer that no disturbances can appear upstream of the airfoil, in conse-quence of which (j> = 0 in x < 0. It then follows (since neither 0 nor V$can change discontinuously) that

0 = ^ = 0 at x = 0. (2.10.6)

These are, in effect, initial conditions with respect to the variable x, whichnow plays the same role as time in a conventional initial-value problem.

Invoking the linearity of (2.10.4) and (2.10.5) in 0, we assume that <j>exhibits the time dependence exp(icot), so that we may replace (d/dt) byico in (2.10.4). Taking the Laplace transform of the resulting equationand of the boundary conditions (2.10.5) with respect to x and invokingthe initial conditions (2.10.6), we obtain

) (2.10.7)\ c /

and

®y = V{p)ei(ot at y = 0 ± . (2.10.8)




That solution of (2.10.7) which is bounded as y -» oo is proportionalto exp( - ky\ where

= B[(p + ivM)2 + v2]1 /2 , (2.10.9b)

M = —, B = (M2 - 1)1/2, v = - ^ - , (2.10.10)c B e

and that branch of the radical which renders kr > 0 for pr > 0 is implied.Invoking (2.10.8), we obtain

<J> = - / l " 1 ^ ) ^ - ^ [y > 0), (2.10.11)

wherein the sign of X must be reversed if y < 0.We may invert O with the aid of the convolution theorem. Considering

first the factor X~x exp(-A^), we apply the shifting theorem to obtain

J & r 1 ^ - V } = B-xe-ixMx<£-x{($2 + v2)"1 / 2exp[~(p2 + v2)1/2By~]}(2.10.12a)

= B-le~ixMxJ0[y{x2 - B2y2)1/2']H(x - By), (2.10.12b)

where (2.10.12b) follows from (2.10.12a) through T2.1.12. Invoking theconvolution theorem, we obtain

By) TJo

K(x - I y)v(0 di, (2.10.13)

Jowhere

K(x,y) = eii(at-xMx)J0[v(x2 - B2y2)lt2~\. (2.10.14)This last example, which was originally solved by various workers

with the aid of more difficult techniques [see Miles (1959), Section 5.2 forcomplete references and for other applications of transform methods toaerodynamic problems], illustrates the power of the Laplace transformwhen applied to problems of some complexity.

EXERCISES t2.1 Derive T2.1.5 and T2.1.6 by: (a) considering a to be complex in T2.1.2 and

taking the real and imaginary parts of the result; (b) applying the shifting theorem,T2.2.6, to T2.1.3 and T2.1.4.

"fThe starred exercises are of greater difficulty.



EXERCISES / 39

2.2 Derive T2.1.7 by: (a) differentiating T2.1.2 with respect to the parameter a;(b) applying the shifting theorem to T2.1.1.

2.3 Derive T2.1.1 for v = n by differentiating T2.1.2 with respect to a.2.4 Derive T2.1.2 by expanding the exponential in a power series, invoking

T2.1.1, and summing the resulting series in p.2.5 Derive T2.1.8 by: (a) transforming the derivative of the Heaviside step

function, H(t); (b) transforming the function

= f 1/e (0 < t < e)

" l0 (t > e)

and considering the limit e -> 0 + .2.6 Derive the scaling theorem, T2.2.7.2.7 Derive T2.2.10 directly from the definition of F(p).2.8 Suppose that one of the zeros of H(p) in (2.7.5), say pl9 is a double zero,

such that //(pj) = //'(Pi) = 0. Show that the first term in the series of (2.7.7) mustbe replaced by

where

(P ~ Pi)2G(p)<D(p) =

Hip)

Use this result to invert (2.6.23).2.9 Show that

lim pF(p) = lim f(t)

provided that both limits exist.2.10 Use the convolution theorem, T2.2.9, to obtain a general solution of the

second-order differential equation

f"(x) + af(x) + bf(x) = g(x).

2.11 The rectified sine-wave voltage v(t) = fo|sin cot\ is applied to a series cir-cuit consisting of a resistance R in series with an inductance L at t = 0. Use themethods of Section 2.6 to determine two different representations of the resultingcurrent.

2.12 Suppose that a viscous damper is placed across the spring in Figure 2.1,such that the equation of motion becomes

mx" + dx' + kx — w0.

Show that the resulting motion is given by

wor / « Ylx = e-at[x0 cos fit + (ax0 + vo)p~l sin pt] + — 1 - e~at (cos pt + - sin pt)

where a = d/2m and p2 = (k/m) — a2 > 0. Find the motion in the special case ofcritical damping, for which d is adjusted to give P = 0.




2.13 Use the known result

f°°y = - \ e~u log udu (y = 0.577...)

Jo

to derive (2.7.16) for m = 0 and then generalize the result through integration byparts.

2.14 Derive T2.1.11 by choosing the branch cut for F(p) = (p2 + a2)'1'2 toconnect the branch points at p — ± ia along the imaginary axis, closing the contourfor the inversion integral with an infinite semicircle in pr < 0, then deforming thecontour to a dumbbell-shaped figure consisting of the two vertical sides of the cutplus small circles around the branch points, and invoking the integral representation(4.1.6) for the Bessel function Jo.

2.15 Show that

fJo

dx ~ £ (-)"(2n)\rn = 0

2.16 Use the integral representation

K0(t) = I e-tco*he dO= I e-tcJo

for the modified Bessel function Ko to obtain the asymptotic approximation

V'2

2.17 A uniform bar of unit cross section is at rest and unstressed for t < 0and has free ends at both x = 0 and x = I. When t = 0, a force P is applied atx = I. Show that the subsequent displacement of any section initially at x is given by

Pt2 2PI " , / nnct\ nnx

where c2 = E/p, m = pi, and p and E denote density and Young's modulus. (Note:The Laplace transform of y has a triple pole at the origin.) Obtain also an expressionfor the motion in terms of traveling waves.

2.18 Let v0 be a function of t in the problem of Section 2.9. Use the convolu-tion theorem to obtain the solution

t;(x, t) = -(nK)-1'^ \vo(x)(t - x)-3'2 exp | ~ *L Jo \J\K\J, —

2.19 Derive (2.7.14) and use the result to obtain the asymptotic solution (2.9.11)by expanding (2.9.6) in ascending powers of p.

2.20 A slab of infinite area extends from x = 0 to x = I and is initially at zerotemperature. The temperature of the face at x = 0 is raised to v = v0 when t = 0;radiation takes place at the other face in accordance with

v + hv = 0 at x = /.



EXERCISES / 41

Show that the subsequent temperature is given by

v(x, t) = 2v0 £ k-'[h + (k2 + h2)iy\k2 + H2)[\ - exp(-/c2K*)] sin /ex,k

where the k are determined by

k cot kl = -h.

• 2.21 A solid is bounded internally by the cylinder r = a and extends to in-finity. The initial temperature is zero, and the surface is kept at a constant tempera-ture v0. The temperature in the solid at t > 0 is denoted by v and satisfies

vt = K(VFF + r~lvr).

Show that the Laplace transform of v is given by

v0K0(qr) pw h e r e « = ~ >

and Xo is a modified Bessel function. Show that v is given by

v i 2 f00 / 2— = 1 H— exp( —/cw2

Note:K0(xe±in/2) = ( + in/2)[J0(x) + iT0(x)] (x > 0).

*2.22 A uniform bar of length /, cross-sectional area A, total weight w0, andcompression-wave speed c is hanging vertically from a fixed point and stretched (instatic equilibrium) under its own weight. A concentrated weight w is suddenlyattached to the lower end of the bar at t = 0. Show that the stress of the fixed endis given by

T / pi w pi . , p / V 1 !p M cosh — h sinh —I

L \ c w o c c / J

0 ,a = — H— « p co h

A A L \ c w o

If w = vv0, show that the time at which the stress achieves its first maximum isgiven by

ct 1_ = 3 + - ( l + e-2).

*2.23 A uniform string of length 2/ and line density p has a particle of mass mattached to its midpoint and is stretched to tension pc2 between the fixed pointsx = ± /. At t = 0, when the string is straight and at rest, the particle is set in motionby a transverse impulse /. Show that its subsequent displacement is given by

211 £ - i , _ i <*nCt— > an

l(l + k cscz a_) 1 sin ,men=i /

where k = 2p//m and otn, n = 1, 2, . . . , are the positive roots of k cot a = a.




*2.24 A constant, transverse force of w0 per unit length is applied suddenly to acircular wire in a perfect (but compressible) fluid at t = 0. Prove that the subsequentvelocity of the wire has the following representations

npca f°° e~uldu- v(t) = t + *vv0 - ' ' ' Jo u\K\(u) + n2l\(u)\

- r + r"1 + r~3[ln(4r) - 7 - i ] + ••• (r -> 00),

where c = velocity of sound, a = cylinder radius, t = (c/a)(time). Sound waves inthe fluid are governed by the wave equation,

1 1firr + - <t>r + "T 060 = 4>n '

where v = V0, and ar and 0 are cylindrical polar coordinates.Notes: (a) X^s) has no zeros.

(b) Kl(ue±in)= -KMT MM

2.25 Derive the results for the Mellin transform tabulated in the last two col-umns of Table 2.3 (cf. Exercise 1.1).



3 FOURIERTRANSFORMS

3.1 Introduction

We consider in this chapter the Fourier-transform pairs defined by(1.3.1), (1.3.5), and (1.3.6); see also Table 2.3. We designate the complexFourier transform of (1.3.1) simply as the Fourier transform, and thetransforms of (1.3.5) and (1.3.6) as the Fourier-cosine and Fourier-sinetransforms, respectively.

We have seen that the Laplace transformation is especially suited toinitial-value problems in that the transform of the nth derivative incorpo-rates the initial values of the first n — 1 derivatives. The Fourier trans-formation, on the other hand, appears to best advantage in boundary-value problems associated with semi-infinite or infinite domains, with theappropriate selection depending on the boundary conditions and/orsymmetry considerations.

3.2 Transforms of derivatives

We assume, in applying the Fourier transform to the solution of annth-order differential equation, that f(x) and its first n — 1 derivativesvanish at x = + oo. Replacing/(x) by its nth derivative,/(n)(x), in

55 fF(k) = ?{f(x)} = f(x)e-ikxdx (3.2.1)J-00

and integrating n times by parts, we obtain

x)} = (ik)nF(k). (3.2.2)

43



44 / FOURIER TRANSFORMS / CH. 3

An analogous result holds for the nth integral of f(x) from either x = + ooor x = — oo; in particular,

-fJ ± x>

(3.2.3)

ifandonlyifj»00/(fld£ = 0.Similar results hold for the Fourier-cosine and Fourier-sine transforms,

Fc(k) = ^c

and

In particular,

3Fc{f'\x

and

Jo

Jo

)} = ~k2Fc

:)} = -k2F

f(x) cos kx dx

f(x) sin kx dx.

(k) - /'(0)

,(fe) + k/(0)

(3.2.4)

(3.2.5)

(3.2.6)

(3.2.7)

if and only if both f(x) and f\x) vanish at x = oo.Analogous results may be established for the cosine and sine trans-

forms of higher derivatives of even order, but the cosine (sine) transformof an odd derivative involves the sine (cosine) transform of the originalfunction. Thus, as implied directly by their trigonometric kernels, thesetransforms are intrinsically suited to differential equations having onlyeven derivatives with respect to the variable in question. Moreover, thecosine (sine) transform of such a differential equation incorporates onlythe values of the odd (even) derivatives at x = 0; the values of otherderivatives at x = 0 could be incorporated as constants to be determined,but the most satisfactory applications are those in which the unincorpo-rated boundary conditions are null conditions at x = oo.

The complex Fourier transform, on the other hand, may be appliedto all derivatives, but it incorporates no boundary values and thereforearises naturally only for infinite domains; to be sure, it may be appliedto semi-infinite domains, but then it becomes essentially a Laplace trans-form.

The cosine or sine transform also may be advantageously applied toan infinite domain if /(x) is an even or odd function of x, respectively, inwhich case /'(0) or /(0) vanishes in consequence of symmetry.



SEC. 3.3 / OPERATIONAL THEOREMS / 45

3.3 Operational theorems

Many of the operational theorems for the Laplace transform (seeTable 2.2) have analogous counterparts for the Fourier transform. Wenote here the shifting theorem,

&{f(x - a)} = e~ikaF(k) (3.3.1a)

or

^~l{e~ikaF(k)} =f(x - a), (3.3.1b)

where a is real (but not necessarily positive), and the convolution theorem,

J-

x-Z)dt. (3.3.2)

Setting x — 0 in (3.3.2), we obtain

MZ)f2(-Z) K = ^ P F1(k)F2(k) dk (3.3.3)

or, equivalently,

f°° fid)fid) d£ = ^- T F fcJFKfc) dfe, (3.3.4)J-oo 2TTJ_0 0

where

F*(k) = ^T f(x)eikx = F(-k) (3.3.5)27t J

is the complex conjugate of F(/c), and fc is real. Lett ing/2 =j\ = j in(3.3.4), we obtain ParsevaFs theorem

=± f (3.3.6)

The quantity / 2(x) in (3.3.6) is typically proportional to the energy inphysical applications, and |F(fc)|2—or \F(k)\2/(2n)—is the power spectrumof/(x).

Many of the operational theorems for the Laplace and Fourier trans-forms depend on the exponential kernel that characterizes these trans-forms; accordingly, these theorems typically have no simple counterpartsfor the Fourier-cosine and Fourier-sine transforms. The most importantexception is the convolution theorem for the cosine transform, namely,




- - rn Jo

F;l{Fc{k)Gc{k)} = - Fc(k)Gc(k) cos kx dk

Setting x = 0 in (3.3.7), we obtain

rc Jodk (3.3.8)

JO ?t JO

and

(3.3.9)f 72(£) = - f "Jo 7C J o

as the analogs of (3.3.4) and (3.3.6). No such analogs exist for the Fourier-sine transform, although there are related theorems that involve both theFourier-cosine and Fourier-sine transforms [Sneddon (1951), Section 3.6].

34 Initial-value problem for one-dimensional wave equation^

A classical problem in wave motion requires the solution to the waveequation

c2ct>xx = 4>tt (3.4.1)

for the initial values

(j) = f(x) and <t>t = g(x) on — oo < x < oo. (3.4.2)

We may suppose f(x) and g(x) to be the initial displacement and velocityof an infinitely long string.

Taking the Fourier transform of (3.4.1,2) with respect to x and in-voking (3.2.2), we obtain

<£„ + (fec)2O = 0, (3.4.3)

<J> = F(k), and O, = G{k) at t = 0. (3.4.4)

The solution of (3.4.3) is a linear combination of either cos kct and sin kctor Qxp(ikct) and exp(-ikct). Determining the coefficients in these al-ternative solutions to satisfy (3.4.4), we obtain

= F(k) cos{kct) + (kc)'1^) sin(fecr) (3.4.5a)

' t o . (3A5b)

fSee Morse (1948) for further examples of this type.



SEC. 3.5 / HEAT CONDUCTION IN A SEMI-INFINITE SOLID / 47

Invoking the shifting theorem, (3.3.1b) with a = ±ct9 and (3.2.3), weinvert (3.4.5b) to obtain the required solution in the form

<t> = i L f l * + ct) + f(x - ct)-] + \ c - ' [ g i O d t ( 3 . 4 . 6 )Jx-ct

5.5 Heat conduction in a semi-infinite solid

The Fourier-cosine and Fourier-sine transforms are less flexible thanthe Laplace transform when applied to a semi-infinite domain; neverthe-less, they may offer distinct advantages. We illustrate this last assertionfor the heat-conduction problem of Section 2.9, although it should beemphasized that we are comparing the application of the Laplace trans-form relative to t with the Fourier transform relative to x; the Laplacetransform is not well suited to x, nor is the Fourier transform well suitedtot.

It is evident from (3.2.7) and the discussion in Section 3.2 that theFourier-sine transform is well suited to the second-order differentialequation (2.9.1) and the boundary conditions of (2.9.3). Designating theFourier-sine transform of v(x, t) by Fand transforming (2.9.1)-(2.9.3), weobtain

Vt + Kk2V = Kkv0 (3.5.1)and

V = 0 at t = 0. (3.5.2)

The solution of (3.5.1) is a linear combination of the particular solutionvo/k and the complementary solution exp( — k2Kt). Determining the co-efficient of the exponential to satisfy (3.5.2), we obtain

V = ~ [1 - exp(-/c2/cr)]. (3.5.3)

(This last solution could have been obtained through a Laplace trans-formation with respect to r.) Substituting (3.5.3) into the inversionformula (1.3.6b), we obtain

— = - fe-1[l - exp(-/c2Kf)] sin kx dk. (3.5.4)^o ft Jo

Invoking the known result

1k ~ * sin kx dk = - n (x > 0), (3.5.5)




we find that (3.5.4) is equivalent to (2.9.12), and hence also to (2.9.7).Alternatively, we may invert (3.5.4) with the aid of the transform pair inEMOT Section 2.4(21),

•Ff1^-1 Qxp(-ak2} = erf(ia"1/2x). (3.5.6)

3.6 Two-dimensional surface-wave generation f

We illustrate the simultaneous application of integral transformationswith respect to space and time variables by considering the developmentof two-dimensional gravity waves on a semi-infinite body of water froman initial displacement of the free surface. We regard the water as in-compressible, frictionless, and initially at rest; it then follows from theknown laws of hydrodynamics that the velocity and gauge pressure, sayv(x, z, t) and p(x, z, t\ at a given point in the fluid may be derived from avelocity potential 0(x, z, t) according to

v = V(/> (3.6.1)and

p= -p{(j)t + ^z + iv2), (3.6.2)

where (/> satisfies Laplace's equation in two dimensions,

<t>xx + 0Zz = 0, (3.6.3)

p is the density of the water, g is the gravitational acceleration, and z ismeasured vertically upwards from the free surface.

Let C(x, t) be the elevation of the free surface relative to its equilibriumposition, z = 0. We assume that this displacement is sufficiently small tojustify the neglect of all terms of second order in its amplitude—that isto say, we linearize the equations of motion. This assumption permitsthe neglect of the second-order term \pv2 in (3.6.2) and the evaluation of 0and its derivatives at z = 0, rather than z = £ in the free-surface boundaryconditions. % The kinematical boundary condition on the vertical velocityat this surface then is, from the definitions of <j> and £

(j>z = Ct a t z = 0. (3.6.4a)

fSee Lamb (1932). Sections 238-241 or Sneddon (1951), Sections 32.1-32.4 for a morecomplete exposition and references (note that Lamb's velocity potential is opposite in signto that used here). This problem and that of Section 4.3 were originally posed and solvedindependently by Cauchy and Poisson, whence it is known as the (two-dimensional) Cauchy-Poisson problem.{This approximation assumes that <j> and its first derivatives can be expanded in powersof C about z = 0.



SEC. 3.6 / TWO-DIMENSIONAL SURFACE-WAVE GENERATION / 49

The dynamical boundary condition, corresponding to the requirementp = 0 at the free surface (we neglect the aerodynamic reaction as smallcompared with the hydrodynamic forces), is

p = -p((t>t + gQ = 0 at z = 0. (3.6.4b)

We complete the statement of the boundary conditions by invoking thefiniteness conditions

|0| < oo as x -+ ± oo or z -> - oo (3.6.5a)and

|C| < oo asx-> ±00. (3.6.5b)

Prescribing the initial displacement C0(x) and invoking the assumptionthat the water is initially at rest, we obtain the initial conditions

4> = 0 and C = C0(x) at t = 0. (3.6.6)

We attack the mathematical problem posed by (3.6.3)-(3.6.6) by in-voking a Fourier transformation with respect to x and a Laplace trans-formation with respect to t. Let

O(fc,z,/>) = J2?J^, Z(k,p) = JPtFt, and Z0(k) = J Co (3.6.7)

denote the required transforms, where & implies Fourier transformationwith respect t o x ( - o o < x < o o ) , and if implies Laplace transformationwith respect to t (0 < t < oo). Transforming (3.6.3) with the aid of (3.2.2),we obtain

(bzz - fe2O = 0. (3.6.8)

Transforming (3.6.4a, b) and invoking (3.6.6), we obtain

- O z + pZ = Zo and pQ> + gZ = 0 at z = 0. (3.6.9)

The solutions of (3.6.8) are proportional to exp(±/cz), where the upper(lower) sign must be chosen for k > (<) 0 in order to satisfy the finitenesscondition (3.6.5a) as z -• — oo; accordingly, the required solution has theform

<D = F(k, p)e^z. (3.6.10)

Substituting (3.6.10) into (3.6.9), eliminating Z between the resultingequations, and solving for F, we obtain

* = -diP2 + glkty^ZoikyeW*. (3.6.11)

Taking the inverse-Laplace transform of (3.6.11) with the aid of T2.1.4 andwriting out the inverse-Fourier transform with the aid of (1.3.1b), weobtain




4>= - (27i) -^1 / 2 \k\-1/2Z0(k)eWz+ikxsm[co(k)t]dk, (3.6.12)J — oo

wheream = co(-k) = (g\k\^2 (3.6.13)

is the angular frequency of a gravity wave of wave number k.Substituting (3.6.12) into (3.6.4b), we obtain

1 f00

C = —g 14)t\z=o — — Z0(k)elkx cos cot dk (3.6.14a)^ ^ J-oo

0

Z0(k)eikx cos cot dk (3.6.14b)

1 f00

= —0t\ Z0(k)[emx-a*) + eiikx+(Ot)'] dk, (3.6.14c)

2?r Jo

where (3.6.14b, c) follow from (3.6.14a) by virtue of the identities Zo( — k) =ZJ(fe), co(-fc) = co(k), and the exponential representation of the cosine.The first and second terms in the integrand of (3.6.14c) represent wavestraveling in the directions of increasing and decreasing x, respectively,with the phase velocity co/k; the corresponding integrals represent super-positions of such waves over the wave-number spectrum, 0 < k < oo.

The free-surface displacement can be expressed in terms of tabulatedfunctions (Fresnel integrals) in the special case of a delta-function dis-placement (this development is given by Lamb), and the result can begeneralized with the aid of the convolution theorem. However, it suf-fices for many purposes and is, in any event, more instructive to considerthe asymptotic representation of (3.6.14c) by the method of stationaryphase, which we now proceed to develop.

3.7 The method of stationary phase

We consider the asymptotic behavior, as t -• oo, of the integral

•Ib

A(k)e^k) dk, (3.7.1a)

where4m = kx - co(k)t (x > 0) (3.7.1b)

is the phase and A(k) the amplitude density (either real or complex) ofplane waves in the wave-number spectrum a < k < b. The form of <j>(k)



SEC. 3.7 / THE METHOD OF STATIONARY PHASE / 51

displayed in (3.7.1b) is especially convenient for problems in wave propa-gation, but it would be equally general to let <j> = tf(k), which reduces to(3.7.1b)for/ = k(x/t) - co(k).

The integrand of (3.7.1), regarded as a function of k, oscillates with in-creasing rapidity as t -• oo, in consequence of which the contributions to /of adjacent portions of the integrand cancel one another except in theneighborhoods of the end points (there are no contributions from justbelow k = a to cancel those from just above k = a and conversely fork near b) and those points, if any, at which (f>(k) is stationary, say

</>'(U = 0 (a < ks < b). (3.7.2)

The points determined by (3.7.2) are known as points of stationary phase,and here and subsequently the subscript s implies evaluation at k = ks.We consider separately the special cases ks -+ a or ks -• b.

Let us suppose that / has one and only one point of stationary phase;if it has n such points, it may be subdivided into n integrals, each of whichhas only one such point. Expanding A(k) and 0(/c) in Taylor series aboutk = fes, we obtain

!\ A'JLk - ks) + ±A'J(k - ks)2 + . - ]

' exp{f[0s + i#'(fc - ks)2 + W(k - fes)

3 + ...]} dk. (3.7.3)

Introducing the change of variable k — ks + eu, where2 Y / 2

we obtainr(b~ks)/e

\ eA'sur(b~ks)/e

I = e\ [AsJ-(ks-a)/E

• exp jf ^ s + u2 sgn # ' + ie ( ^ ) u3 + ...11 dii, (3.7.4)

where sgn implies the sign of (sgn x is known as the signum function of x).Letting s -• 0 (t -> oo) in (3.7.4) and invoking the known integral

r°°exp(±iu2) du = n1/2 exp(±Jin),

J-oowe obtain

iJ 4S exp[#s + \n sgn O ] (t - oo). (3.7.5)




Repeating the argument on the supposition that either ks = a or ks = b,we infer from (3.7.4) that the asymptotic limits of integration for u becomeeither (0, oo) or ( - oo, 0), rather than (— oo, oo), in consequence of whichthe right-hand side of (3.7.5) must be divided by 2. [It is evident from thisresult that the asymptotic approximation (3.7.5) is not uniformly valid asks approaches either a or b.~\

The approximation (3.7.5) obviously fails if < ' = 0. This importantspecial case and others of less importance are discussed in the mono-graphs of Copson (1965) and Erdelyi (1956). The approximation may beextended to complex (j>(k) [such as would arise in approximating theintegral of (3.6.12) if z, as well as x and t, is large], in which case it appearsas Riemann's saddle-point approximation [Copson (1965, Section 36)], aspecial case of Debye's method of steepest descent [Copson (1965, Sec-tion 29)].

The asymptotic behavior of (3.7.1) may be determined through in-tegration by parts if (t>'(k) does not vanish in a ^ k ^ b. The result is

(t -> oo). (3.7.6)

The right-hand sides of (3.7.5) and (3.7.6) may be superimposed to obtainthe leading terms in the complete asymptotic expansion of / providedthat ks # a or b. More general cases are discussed by Erdelyi (1956).

Returning now to the problem of Section 3.6, we observe that only thefirst of the two exponentials in (3.6.14c) has a point of stationary phase forx > 0; accordingly, this exponential dominates the asymptotic approxi-mation to the integral, such that

C - (2n)'lSt \Z0(k)e^k) dk (x > 0, t -> oo), (3.7.7)Jo

where <t> is given by (3.7.1b). Invoking (3.6.13) and (3.7.2), we obtain

4>\ks) = x- co'(ks)t = x - i (g/kf'h = 0, (3.7.8)

which implies

fcs = £? , ^ = - £ ? ' and # ' - ^ F (3J-9)

Substituting (3.7.9) into (3.7.5), setting A(k) = Z0(k)/(2n)9 and taking thereal part of the result, we obtain



SEC. 3.8 / FOURIER TRANSFORMS IN TWO OR MORE DIMENSIONS / 53

(x > 0, t -> oo). (3.7.10)

A consideration of the neglected terms in the stationary-phase approx-imation implies that (3.7.10) is valid for gt2 > 4x. If x < 0, only thesecond exponential in (3.6.14c) has a point of stationary phase, and arepetition of the preceding argument with (/) = — fc|x| + co(k)t yields

(x <0 , t-> oo). (3.7.11)

We may render (3.7.11) valid for x > 0 simply by multiplying the ex-ponent by — sgn x.

3.8 Fourier transforms in two or more dimensions

We may extend the Fourier-transform pair of (1.3.1a, b) to a function oftwo variables to obtain

F(kl9k2) = I P f(x, y) exp[-i(*i* + k2y)] dx dy (3.8.1a)J — oo J — oo

and

*> y) = Ti r P *•(*!>™ J-ooJ-oo

f(*> y) = Ti r P *•(*!> M exp[i(k!X + k2yj] dkx dk2. (3.8.1b)JJ

More generally, letting r denote a vector having the Cartesian componentsxx, x 2 , . . . , xn in an n-dimensional space andk a similar vector in the wave-number space kt, k2,..., fcw, we obtain

F(k) = f°° .- f°° f(r)e-*r dx, .- dxn (3.8.2a)J — oo J — oo

and

/(r) = (271)"" f" - . f°° F ( k y k r dfcx - . d/cn. (3.8.2b)J — oo J — oo

Further transform pairs may be obtained from (3.8.1a, b) and (3.8.2a, b)by coordinate transformations. In particular, the transformation of(3.8.1a, b) to polar coordinates leads to the Hankel transform (1.1.6) andits inverse. We give the derivation in Section 4.1.




The Laplace-transform pair of (1.4.3) may be extended to functions oftwo variables through transformation of the Fourier-transform pair of(3.8.1); see Ditkin-Prudnikov (1962) and Voelker-Doetsch (1950).

EXERCISES

3.1 Determine the Fourier transform of Heaviside's step function, H(x), by(a) allowing k to be complex (state the necessary restriction on k{) and (b) transform-ing e~axH(x) and considering the limit a -• 0 + .Answer: (a) &{H{x)} = (iky1 (kt < 0).

(b) &{e-axH(x)} = (a + iky1 (a -+ 0 + ).3.2 Determine the Fourier transform of the signum function,

sgnx = ± 1 (x^O) ,

by transforming e~a^ sgnx and considering the limit a -• 0 + . Compare the re-sult with those of Exercise 3.1.a, b and discuss the seeming paradox. Can the diffi-culty be resolved by permitting k to be complex?Answer: ^{e~aM sgnx} = -2ik(a2 + k2)'1. The limiting result, 20k)'1, ap-pears to be the Fourier transform of 2H(x) and cannot be rendered valid by permittingk to be complex.

3.3 Show that the Fourier transform of f(x) = x" 1 sinmx is F(k) =nH(m - \k\), where m > 0 and kt = 0.

3.4 Show that the inverse Fourier transform of

is

by an appropriate deformation of the path of the inversion integral in the complex-/cplane.

3.5 Use the Fourier-sine transform to solve the following problem in potentialtheory:

<t>XX + <l>yy = 0 (X>O, y> 0),

(j) = 1 on x = 0, y > 0, </> = 0 on x > 0, y = 0,

and

V0 - • 0 as x 2 + y2 - • oo.

Answer:2 f00 2 v

0 = - k~\l - e-ky)sinkxdk = - t a n " 1 - .n Jo n x

3.6 The transverse oscillations of a semi-infinite string, x > 0, are governedby the wave equation, (3.4.1). The string is originally at rest, and the end at x = 0



EXERCISES / 55

is subjected to the displacement f(t) for t > 0. Obtain the classical solution

by invoking a Fourier-sine transformation with respect to x and a Laplace trans-formation with respect to t.

*3.7 The preceding result may be inferred directly from the known propertiesof the wave equation—in particular, the fact that the wave propagation is non-dispersive. In contrast, wave propagation along an ideal beam, which is governedby

a2yxxxx + yu = 0 (a2 =

where El is the bending stiffness and m is the mass per unit length, is dispersive.Consider a semi-infinite beam that is initially at rest and for which the freely hingedend has the prescribed motion/(r), so that

y =f(t), yxx = 0 at x = 0, t > 0.

Show that the Laplace transform of y(x, t) is given by

2>y(x, t) = [i?/(f)] exp [ - ( l ; ) - * ] cos [ ( l ;

and invert this result to obtain

y(x9t) = (87ra)"1/2x f(x)(t - i rJ

3.8 The initial temperature in an infinite medium is given by v0 = f(x). Use aFourier transformation to obtain the solution of (2.9.1) in the form

v(x,t) = (4mcty112 I"" /({)exp[-(x - O^fcr)"1]^J-00

fix + 2(Kt)ll2tfexp(-r,2)dv.

3.9 Obtain the counterpart of (3.6.12) for £0(x) = S(x) by invoking symmetryconsiderations in the original statement of the problem and using a Fourier-cosinetransform. [N.B. £fd(t) = 1, but ^cS(x) = £ Justify the latter result.]

3.10 Repeat the solution of Section 3.6 for £0(x) = 0 and

(/>(*, 0,0) = -<5(x),P

corresponding to the application of a concentrated impulse / at t = 0.




Answer :

ekzcos(kx)cos[(gk)ll2tr\dk.=- \n Jo

3.11 Use the method of stationary phase to obtain the asymptotic form of thefree-surface displacement in Exercise 3.10.



4 HANKELTRANSFORMS

4.1 Introduction

The Hankel-transform pair,

Fn(k) = Jtn{f(r)} = \f(r)Jn(kr)r dr (4.1.1a)Jo

and

f(r) = ^-l{Fn{k)} = [^Fn(k)Jn(kr)k dk, (4.1.1b)

arises naturally in connection with the differential operator2 1 8 fn\2 1 8 r8 fn\2

which is derived from the Laplacian operator

after separation of variables in cylindrical polar coordinates (r, 9, z).We derive (4.1.1a, b) by introducing the polar-coordinate transfor-

mations x = rcos0, y = rsinfl, kx = fccosa, and k2 = /csina in thetwo-dimensional, Fourier-transform pair of (3.8.1a, b) to obtain (afterappropriate changes in functional notation)

poo rinF(k, a) = /(r, 9) exp[ - ikr cos(0 - a)> rfr d9 (4.1.4a)

Jo Joand

57



58 / HANKEL TRANSFORMS / CH. 4

/(r, 0) = _ F(/c, a) exppfcr cos(0 - a)]/c d/c da. (4.1.4b)4TC J O J O

[We remark that if /(r, 0) is multiplied by exp( —icot), (4.1.4b) representsa packet of plane waves having the amplitude distribution F(/c, a), thewave speeds co/k, and wave-front normals inclined at the angles oc to thex axis.] Now let us suppose that

f(r,9) = f(r)eine; (4.1.5)

this is not a restrictive assumption, for we may expand /(r, 9) in a complexFourier series, in which each term has the form (4.1.5), and then considerthe series term by term. Substituting (4.1.5) into (4.1.4a) and introducingthe change of variable cp = 9 — oc + ?n, we obtain

/•GO r(3/2)n-a

F(k, oc) = exp[m(a - ^7r)] f(r)r dr Qxp[i(ncp - kr sin cp)] dcp.Jo J-(l/2)7t-a

Invoking the representation [Watson (1945), Section 2.2(5)]

2nJn{kr) = exp\_i(ncp — kr sin cp)~] dcp (4.1.6)%)(p0

for the Bessel function /„ , choosing cp0 = \n - oc, and identifying theintegral over r with the Hankel transform of/(r), as defined by (4.1.1a),we obtain

F(k, oc) = In exp[m(a - ?n)~]Fn(k). (4.1.7)

Substituting (4.1.7) into (4.1.4b), introducing the change of variablecp = oc — 9 — |TT, and comparing the result to (4.1.5), we obtain

I roo r(3/2)n-0f(r)eine = — Fn(k)k dk Qxp[i(ncp — kr sin cp)\ dcp ein0.

27C JO J-(l/2)7T-0

Invoking (4.1.6) with cp0 = -\n - 9 and cancelling exp(in0), we obtain(4.1.1b), thereby establishing the inverse relation between (4.1.1a, b).

Another form of the Hankel transform, which is used in EMOT, isgiven by

Gn(k) = [™gn{r)Jn(kr){kryi2 dr (4.1.8a)Jo

and

gn(r) = [°°Gn(k)Jn(kr)(kr)1/2 dk. (4.1.8b)Jo

This evidently can be reconciled with (4.1.1a, b) by setting gn = r1/2fn and



SEC. 4.2 / OSCILLATING PISTON / 59

Gn = k1/2Fn. As it stands, it reduces, except for the constant factor (2/TT)1/2,

to the Fourier-sine or -cosine transform for n = \ or - 7 , respectively.Returning now to the operator An of (4.1.2), we take the nth-order

Hankel transform of An0 and integrate twice by parts to obtain

= lJo

- n2r-2(j>}Jn{kr)rdr

{AJJLkr)}4>r dr.{"Assuming that the partially integrated terms vanish at both limits (notethat Jn vanishes like rn as r -• 0 and like r~1/2 as r -+ 00) and invokingBessel's equation,

(AH + k2)Jn(kr) = 0, (4.1.9)

we obtain

^n{K(j>} = -k2^n(j>. (4.1.10)

The most important, special cases of the Hankel transform correspondto n = 0 and n = 1, but we emphasize that, subject to appropriate re-strictions on (/>, the reciprocal relation implied by (4.1.1a, b) and the result(4.1.10) are valid for 0ln > - \ [see Sneddon (1951), Chapter 2].

4.2 Oscillating piston

We consider an oscillating piston of radius a mounted in an infiniteplane, z = 0, and radiating sound into a half-space, as shown in Figure 4.1.This configuration serves as a simple model of a loudspeaker.!

Let the displacement of the piston from its equilibrium position, z = 0,be given by

z = 0l{Ae™% (4.2.1)

where A is a complex amplitude, and co is the angular frequency. We as-sume that \A\ is small compared with the other characteristic lengths de-fined by the problem, namely a and c/a>, where c is the velocity of sound(the wavelength corresponding to the angular frequency co is 2nc/a>). Werequire a solution to the wave equation,

c2V2(t> = 4>u, (4.2.2)

tThis problem, which was solved originally by Lord Rayleigh, is discussed in considerabledetail by Morse (1948, pp. 326-338).




FIGURE 4.1 Oscillating piston in an infinite baffle.

for the complex velocity potential (j) (only the real part of which is to beretained in the calculation of the actual velocity). The boundary conditionis (we prescribe the piston velocity at z = 0, rather than the displacedposition, by virtue of our assumption that \A\ is small)

(j)z = i(oAei(OtH{a - r) at z = 0, (4.2.3)

where H is Heaviside's step function. We also require <f> to be boundedand to yield an outgoing wave at infinity (radiation condition).

We begin our construction of the solution by remarking that 0 must beindependent of 6 by virtue of the fact that the boundary conditions aresimilarly independent. We also may replace (f)tt by — co2(/> by virtue of theprescribed time dependence. Invoking these results in (4.2.2), we obtain

<t>z1- <t>r ~J 0=0.

Applying the operator J«f 0 to (4.2.3) and (4.2.4) and defining

we obtain

where

and

= 0,

kr)r dr = ioiAak'lJl(ka)e'imt

(4.2.4)

(4.2.5)

(4.2.6)

(4.2.7)

(4.2.8)



SEC. 4.3 / AXISYMMETRIC SURFACE-WAVE GENERATION / 61

The required solution of (4.2.6) and (4.2.8) is

O = -ioiAa{kX)-lJ ^kay"1-^, (4.2.9)

where X must be positive real for k > co/c and positive imaginary fork < co/c in consequence of the requirements that the solution be boundedand behave as an outgoing wave, as x -• oo [let X = ifi for k < co/c; then(4.2.9) represents a disturbance moving in the positive-x direction with thephase speed co/n]. Applying the operator JFQ \ as defined by (4.1.1b),to O, we obtain

f00

0 = -icoAa k~lJ1(ka)JQ{kr)^at'^ dk. (4.2.10)Jo

The gauge pressure on the piston is given by

p= -pcj)t\z = 0 = -pco2Aaei(Ot X~'J\(ka)Jr0(kr) dk. (4.2.11)

Jo

The corresponding power (which goes into acoustic radiation) is given by

2TC prdr\[ia)AeUur\*\, (4.2.12)

Jo J Jwhere the first and second terms in square brackets are the complex forceand the complex conjugate of the velocity, respectively. Substituting(4.2.11) into (4.2.12), carrying out the integration with respect to r, andtaking the real part of the result (to which only that portion of the kspectrum in which X is imaginary contributes), we obtain

rto/c r / r A 2 ~|~1/2

P = na2\A\2pco3 k'1 - - k2\ J\(ka) dk. (4.2.13)J° LW J

Introducing the change of variable k = (co/c) sin a, we find that the in-tegral in (4.2.13) can be evaluated [Luke (1962), Section 13.3.2(24)] toobtain

2

a result due originally to Lord Rayleigh.

43 Axisymmetric surface-wave generation f

We consider the axisymmetric analog of the hydrodynamical problemof Section 3.6, namely, the development of gravity waves on a semi-infinitebody of water from the initial displacement £ = Co(r)- The mathematical

fSee Lamb (1932), Section 255 and Sneddon (1951), Section 32.6 for more detailed dis-cussions of this problem.




development closely parallels that of Section 3.6, with x replaced by thecylindrical radius r and the Fourier-transform operator & replaced bythe Hankel-transform operator Jf0.

Let ${r, z, t) be the velocity potential. The velocity and gauge pressure,v(r, z, t) and p(r, z, t\ then are given by (3.6.1) and (3.6.2), and <j> satisfiesLaplace's equation for axisymmetric motion,

c^ + r - 1 ^ + <£„ = (), (4.3.1)

in place of (3.6.3). The boundary conditions on the assumption of smalldisplacements are given by (3.6.4) and (3.6.5). The initial conditions aregiven by (3.6.6). Carrying out a zero-order Hankel transformation withrespect to r and a Laplace transformation with respect to t, we write

O(fe, z, p) = jSPjr0^, Z(fe, p) = <£Jf0C, Z0(k) = ^oCo (4.3.2)

in place of (3.6.7). Transforming (4.3.1), (3.6.4), and (3.6.6), we obtain theboundary-value problem posed by (3.6.8) and (3.6.9). The required solu-tion is given by (3.6.11) or, since k is nonnegative for the Hankel transform,

<*> = ~g(P2 + gk^Zoiky. (4.3.3)

The solution departs from that of Section 3.6 at this point in conse^quence of the geometrical differences. Inverting (4.3.3) with the aid ofT2.1.4 and (4.1.1b), we obtain

# • , z, t) = -g112 I °°fc1/2Zo(fc)Jo(fer)^ sin cot dk, (4.3.4)

where co(fe) = (gk)1/29 as in Section 3.6.

We now obtain the asymptotic approximation to the free-surface dis-placement with the aid of the stationary-phase approximation of Section3.7. Setting z = 0 in (4.3.4) on the assumption that Z0(k) vanishes withsufficient rapidity as k -> oo to ensure the convergence of the integral andinvoking the boundary condition (3.6.4b) to determine £, we obtain

r, t) = -g-'fair, 0, t) = |Jo

{(r, t) = -g-'fair, 0, t) = | kZ0(k)J0(kr) cos cot dk. (4.3.5)Jo

Letting r -* oo and replacing Jo by the asymptotic approximation (cf.Exercise 4.4)

/ 2 \ 1 / 2

Mkr) - l w ) cos(fcr - &U (43.6)

we obtain



SEC. 4.3 / AXISYMMETRIC SURFACE-WAVE GENERATION / 63

2y/2 p£(r, r) ~ (— I k1/2Z0(k) cos(kr - fa) cos a* dk. (4.3.7)

W JResolving the product of the cosines into the cosines of the sum and dif-ference of the arguments, we obtain a representation similar to that of(3.6.14c),

c°°C(r, t) ~ (2nr)-1/2 @ k1/2Z0(k) exp[i(cot - kr + fa)] dk, '4.3.8)

Jowhere we have retained only that exponential which has a point of sta-tionary phase, namely [cf. (3.7.9)],

* . - £ • (4-3-9)

Carrying out the stationary-phase approximation (the student should fillin the details, following the example in Section 3.7), we obtain

(4.3.10)

The classical solution for the Cauchy-Poisson problem is based on aninitial displacement of unit volume that is concentrated at the origin,which yields Z0(fc) = (27c)"1. However, actual displacements of an incom-pressible fluid must satisfy the constraint of zero net volume,

V =

which implies

In Co(& dr = 0,Jo

Zo(0) = 0.

(4.3.11)

(4.3.12)

A displacement that does satisfy (4.3.11) and resembles the initial dis-placement produced by a surface explosion f or a dropped pebble is givenby

[ ( ) 1 ] [ ( ) 1 ] (43'13)which represents a cavity of depth d with a concentric lip, as shown inFigure 4.2. Invoking EMOT 8.3(5),

| I t need scarcely be added that the motion near an explosion is too violent to justify theassumption of small displacements. Nevertheless, this assumption permits an adequatedescription of the motion at a sufficient distance from the explosion.




FIGURE 4.2 The cavity and lip described by (4.3.13) [from / . Fluid Mech.34, 368 (1968) by courtesy of Cambridge University Press].

we obtain

(4.3.15)

04 -

FIGURE 4.3 The asymptotic displacement described by (4.3.16) [from / .Fluid Mech. 34, 368 (1968) by courtesy of Cambridge University Press].



EXERCISES / 65

Substituting (4.3.15) into (4.3.10), we obtain

(igt2 > rp a). (4.3.16)

This displacement is plotted in Figure 4.3.

EXERCISES4.1 Replace J0(kr) in (4.2.10) by the asymptotic approximation

v l / 2

and use the method of stationary phase to obtain the asymptotic approximation

<t> ~ - iAac{R sin 0)"xJt (— sin 0 J exp lico (t - - J 1 ,

where R and 0(0 = 0 on axis) are spherical polar coordinates. Calculate the radiatedpower by integrating the mean product of pressure and radial velocity over a hemi-sphere of radius R -• oo and show that the result agrees with (4.2.13).

*4.2 The piston in Section 4.2 is given the displacement AH(t). Show that theresulting force on the piston is

F = -pc2A 4 - ( - ) H(2a - ct).I V a ) I

Note:

c°°J2Ax)JAux)dx = i7c~1(4 - n2)1/2H(2 - fi).

Jo4.3 If V(s) and F(s) are the Laplace transforms of the velocity of, and force on,

the piston in Section 4.2, then

Z(s) = —— and Y(s) = ——

are the Laplace transforms of the impulsive impedance and impulsive admittance ofthe piston. Use the result of Exercise 4.2 to show that

Z(s) = 1 - - I V2 s u(l - u2)112 du71 Joafter rendering all variables appropriately nondimensional. Use this result toshow that

lim y(t) = -£.




4.4 The asymptotic approximation of (4.3.6) requires kr P 1, which leaves somequestion as to the validity of (4.3.7), wherein k goes to zero. Repeat the derivationof (4.3.10) by substituting the integral representation

2 f"/2

J0(kr) = - cos(kr cos a) dec

into (4.3.5) and carrying out stationary-phase approximations with respect to eachof k and a in that order.

4.5 The impulse of heat Q is supplied uniformly over the circle r < a to the semi-infinite solid z > 0 at t = 0, after which the boundary z = 0 is insulated (no heat istransferred across it). Show that the subsequent temperature of the boundary isgiven by

0 / \ 1/2 poov = •£- — J0(kr)J,{ka) exp(-

Ka \n3tj Jowhere K and K are the conductivity and diffusivity of the solid.Hint: The rate at which heat is transferred across r < a is (Q/



5FINITEFOURIERTRANSFORMS

5.7 Introduction

The transforms considered thus far are applicable to semi-infinite orinfinite domains and have a common antecedent in Fourier's integralformula (1.2.6). It is natural to inquire whether transforms can be definedby (1.1.1) and their inverses derived from the theory of Fourier series. Theessential result of this theory is that if the infinite sequence ^ (x ) , ^2(

x)> • • •constitutes a complete orthogonal set of functions for the intervala < x < b and the weighting function w(x), corresponding to a discreteset of eigenvalues p x , p2,...—that is, if

fb

^m(x)^B(x)w(x) dx = SnmN(pn) (m, n = 1,2,.. . , oo), (5.1.1)fwhere 8nm is the Kronecker delta, defined by

Snm = 0, m # n, Snm = 1, m = n, (5.1.2)

and

N(pn) = fVnWw(x) dx, (5.1.3)

then it may be shown that

F(p) = \bf(x)il/(x, p)w(x) dx (5.1.4a)Ja

and

^ 4 ^W(X)' where Ux) = (x'Pw)> (5-L4b)

67



68 / FINITE FOURIER TRANSFORMS / CH. 5

corresponding to K(p,x) = ij/(x,p)w(x) in (1.1.1). Equations (5.1.4a, b)constitute a generalized, finite Fourier-transform pair.

The choice of the orthogonal functions \j/(x9 p) depends both on thedifferential equation and on the boundary conditions to be satisfied by/(x),just as with the infinite transforms of the preceding chapters. We considerhere finite sine and cosine transforms, which are appropriate to differentialequations containing only even derivatives with respect to x and for whichw = 1, and finite Hankel transforms, which are appropriate to the dif-ferential operator An of (4.1.2) and for which w = r, corresponding to thatin the element of area r dr dO in polar coordinates. The technique isgenerally applicable to all functions and boundary conditions of theSturm-Liouville type and serves to mechanize much of the time-consum-ing detail associated with the determination of the unknown coefficientsin the classical procedure that begins with separation of variables. Theproperties of several such transforms are tabulated in Table 2.4 (Appendix2, p. 86).

5.2 Finite cosine and sine transforms

The simplest finite cosine and sine transforms are those for whicha = 0, b = 7T, and p = n in (5.1.4a, b), which then reduce to

-JX:F(n)= | f(x)\C°\xdx (5.2.1a)

and

f(x) = - J (2 - dn0)F(n)\ ~.~" nx, (5.2.1b)

where either upper or lower alternatives must be taken together. Thecorresponding transforms of/"(x) are given by

f"(x) cos nxdx= - n2F(n) + ( - ff'(n) - /'(0) (5.2.2)Jo

and

f"(x) sin nx dx = - n2F(n) + n[/(0) - ( - ff(n)l (5.2.3)Jo

whence these transforms are expedient for problems in which f\x) or/(x), respectively, is prescribed at the end points of the interval. Thegeneralization to an interval of lengtn / merely requires a scale transforma-tion, with x replaced by nx/l (as in the following paragraph).



SEC. 5.2 / FINITE COSINE AND SINE TRANSFORMS / 69

More general forms of these transforms, corresponding to Fourier'sown generalization of his series, are given by

and

F(k) = | f(x)\ C ° S kx dx, (5.2.4a)• s in

f cosk2+ h2)lY1F(k)\ . kx, (5.2.4b)

l s i n

where

j j t a n £/=+/,, (5.2.5)(cot

7"(x){COS kx dx = - k2F(k) + [/'(/) + fc/(0] j C O S kl " / " ^ , (5-2.6)

and either upper or lower alternatives must be taken together. Thesetransforms are applicable to problems in heat conduction in which radia-tion takes place at x = / and to problems involving lumped parametersat the boundaries of electrical or mechanical systems, such that /'(/) +hf(l) is prescribed.

We remark that h usually is nonnegative in physical problems, by vir-tue of which (5.2.5) has only real roots, which occur in pairs of equalmagnitude and opposite sign, with only the positive values being includedin the summation of (5.2.4b). But if h is negative, (5.2.5) has a pair of con-jugate imaginary roots, say ±iK, to which the corresponding term in(5.2.4b) is

Jo sinh(5.2.7)

We note, however, that/(x) depends only on k2 and therefore remains real.If h = 0, then the value k = 0 appears as a nontrivial root of (5.2.5) forthe cosine transform, and (5.2.4b) must be replaced by (5.2.1b).

The roots of (5.2.5) are given approximately by

nn + h{nnl)~l ~ Q,

(n — ^)n + h\(n — j;)nl]

with an accuracy of 10% (1%) for all n (n = 2) if W ^ 1 (the studentshould derive this result). These roots also are tabulated numerically[Abramowitz and Stogun, Table 4.20].




5.5 Wave propagation in a bar

We apply the finite Fourier transform to the problem of Section 2.8, asposed by

c2yxx = y t t 9 (5.3.1)

y = 0 at x = 0, (5.3.2a)

Eyx = P a t x = /, (5.3.2b)

and

y = yt = 0 at t = 0 and 0 < x < /. (5.3.3)

Referring to (5.2.4)-(5.2.6), with f(x) replaced by y(x, t\ we find thatthe boundary conditions of (5.3.2) may be accommodated by using afinite sine transform with h = 0 and kl equal to an odd multiple of jn:

k9t)= y(x, t) sirJo

Y(k, t) = | y(x, t) sin kx dx (5.3.4)

and

fc = /cB = ^ ^ (n = O,l , . . . ) . (5.3.5)

Transforming (5.3.1)—(5.3.3), we obtain

Ytt + (kc)2Y =^- sin kl (5.3.6)

and

Y = Yt = 0 at * = 0. (5.3.7)

A particular solution of (5.3.6) is given by (P/Ek2) sin kl so that the generalsolution is

pY = -—- sin kl + A cos kct + £ sin kct.

Ekr

Invoking (5.3.7) to determine A and B, we obtain

Y(/c, 0 = - | - [1 - cos(fccO] sin kl (5.3.8)

Substituting F = Y and kl from (5.3.5) and (5.3.8) into (5.2.4b) and settingh = 0, we obtain



SEC. 5.4 / HEAT CONDUCTION IN A SLAB / 71

o p / oo

><*. ') = - ! = ! (~r(2n + I)"2 sin knx\_l - cosset)]- (5-3.9)71 & n = 0

The terms in the series that are not time-dependent may be identified asthe Fourier-series representation of Px/E, whence (5.3.9) may be reducedto (2.8.9).

5.4 Heat conduction in a slab

We illustrate the generalized transform described by (5.2.4)-(5.2.6) withh # 0 by considering one-dimensional conduction of heat in a slab ofthickness / and initial temperature i?0, one face of which, x = 0, is in-sulated, and the other face of which, x = /, radiates into a medium atconstant temperature, say v = 0 (the temperature in the slab beingmeasured relative to that in x > I). The heat-conduction equation is,from (2.9.1),

vt = KVXX, (5.4.1)

where K is the diffusivity. Invoking the condition of zero heat flow atx = 0 and Newton's law of cooling (which assumes that the relativetemperature v is small compared with the absolute temperature), weobtain the boundary conditions

vx = 0 at x = 0 (5.4.2a)

and

vx = -hv a tx = / (5.4.2b)

and the initial condition

v = v0 at t = 0. (5.4.3)

Referring to (5.2.4)-(5.2.6), we find that the boundary conditions of(5.4.2) may be accommodated by using a finite cosine transform,

k, t) = v(x, i

JoV(k, t) = I v(x91) cos fex dx, (5.4.4)

with

k tan kl = h. (5.4.5)

Transforming (5.4.1)-(5.4.3) with the aid of (5.2.6), we obtain

Vt + Kk2V = 0 (5.4.6)

and




V = vo\ cos kx dx = vok * sin kl at t = 0. (5.4.7)

Determining the coefficient of the exponential solution of (5.4.6) to satisfy(5.4.7), we obtain

V = vok-1 Qxp(-Kk2t) sin kl. (5.4.8)

Inverting (5.4.8) with the aid of (5.2.4b), we obtain

(]A + h2) sin klv(x, t) = 2v0 I * (k2 h2)n

c o s kx exP( - K^2t\ (5.4.9)

where the summations are over the positive roots of (5.4.5).

5.5 Finite Hankel transforms

The Bessel functions Jv(/cr), v ^ — , form a complete, orthogonal setfor the interval 0 < r < a with the weighting function w = r if the se-quence /q, / c 2 , . . . is determined by

Jv(ka) = 0 (0 < kx < k2 < . . . ) . (5.5.1)

The corresponding, series representation of the function/(r) is known asa Fourier-Bessel series. Setting x = r, \jjn = Jv(kr), and w = r in (5.1.3)and (5.1.4), we obtain the finite Hankel transform

Hk) = f(r)Jv(kr)r dr (5.5.2a)Jo

and its inverse

a2 k Jj-i(fca)

The evaluation of iV in passing from (5.1.4b) to (5.5.2b), and also (5.5.6)below, follows from the known integral

fJo

- v2)Jv3(x) + \x2J'2{x). (5.5.3)

We also remark that (5.5.1) implies J'x = Jv_1 = — J v + 1 through the re-cursion formulas for Jv and its derivative, J'v.

The finite Hankel transform, like its infinite counterpart of Section 4.1,arises naturally in connection with the Laplacian operator in cylindrical



SEC. 5.6 / COOLING OF A CIRCULAR BAR / 73

polar coordinates. Proceeding as in Section 4.1 and invoking (5.5.1) andJ'v = Jv_! in the partially integrated terms, we obtain

I (Avf)Jv(kr)rdr = -k2Fv(k) - kaJv_^M/H (5-5.4)

where the operator Av is defined by (4.1.2). We may replace Jv_i by— Jv+1; this is convenient for v = 0.

A more general form of the finite Hankel transform, corresponding tothe Dini-series representation of f(r) in 0 < r < a [Watson (1945),Chapter 18] and analogous to the transforms defined by (5.2.4H5.2.6), isdetermined by the roots of

kJ'x{ka) + hJv(ka) = 0, (5.5.5)

which leads to

/ ( r ) - 2 ? [(fc* + fc V - v2]in place of (5.5.2b).

The summation in (5.5.6) is over the positive roots of (5.5.5) if h > 0.If h < 0, (5.5.5) has a pair of conjugate imaginary roots, say k = ±i/c, forwhich the corresponding (single) term in (5.5.6) is

= 2 K2

where /v is a modified Bessel function. The counterpart of (5.5.4) is

(Av/)Jv(fcr)r dr = aJ,{ka\fr + hf)r=a - k2Fv(k) (5.5.8a)Jo

= -kaJ'x{ka){h-lfr + / ) r = a - k2Fv(k\ (5.5.8b)

where (5.5.8a) and (5.5.8b) are equivalent by virtue of (5.5.5).

5.6 Cooling of a circular bar

We now consider the axisymmetric, and physically more interesting,counterpart of the heat-conduction problem of Section 5.4. An infinitelylong, circular cylinder of radius a, diffusivity K, and initial temperature v0

radiates into a medium at zero (relative) temperature according to New-ton's law of cooling. Invoking axial symmetry in the three-dimensionalgeneralization of (2.9.1) [see footnote following (2.9.1)], we obtain theheat-conduction equation




vt = KAOV = ?c(i;rr + r~ 1i;r), (5.6.1)

the boundary condition

vr = -hv at r = a and t > 0, (5.6.2)

and the initial condition

v = v0 at t = 0 and 0 < r < a. (5.6.3)

The form of (5.6.1) and (5.6.2) suggests the finite Hankel transform

K t) = [av(r,Jo

V(k91) = v(r, t)J0(kr)r dr, (5.6.4)

where the k are determined by (5.5.5) with v = 0 or, equivalently,

kJ^ka) = hJ0(ka). (5.6.5)

Transforming (5.6.1)-(5.6.3) with the aid of (5.5.8a), we obtain

Vt + Kk2V= 0 (5.6.6)

and

JoJ0(kr)r dr = vok Wi(fca) at t = 0, (5.6.7)

the solution of which is [cf. the solution of (5.4.6) and (5.4.7)]

y = vok~laJ iika) exp( — Kk2t). (5.6.8)

Setting v = 0, replacing Fv by V in (5.5.6), and eliminating Jl through(5.6.5), we obtain

, fv 2i;0/iY,J0(/cr)exp(-/c/c20

where the summation is over the positive roots of (5.6.5).

5.7 Viscous diffusion in a rotating cylinder f

The axisymmetric motion of a viscous fluid in concentric circles aboutthe axis of rotation of an infinitely long cylinder is governed by [Lamb(1932), Section 328a(5)]

vt = vAxv = v(vrr + r~lvr - r~2v\ (5.7.1)

where v is the tangential velocity, and v is the kinematic viscosity (not to

t Sneddon (1951) gives other examples of this type.



SEC. 5.8 / CONCLUSION / 75

be confused with the order of the Bessel function in Section 5.5). Weconsider the spin-up of a cylinder of fluid of radius a, which is initially atrest, following the imposition of the angular velocity Q at the outerboundary. The corresponding boundary and initial conditions are

v = Qa at r = a and t > 0 (5.7.2)and

v = 0 at t = 0 and 0 < r < a. (5.7.3)

The form of (5.7.1) and (5.7.2) suggests the finite Hankel transform

k,t) = v(r9t)Jl

JoV(k,t)=\ vfatyjkrydr, (5.7.4)

where the k are determined by the zeros of

J^ka) = 0. (5.7.5)

Transforming (5.7.1)—(5.7.3) with the aid of (5.5.4), we obtain

Vt + vk2V= -vQa2kJ0(ka) (5.7.6)

andV=0 atf = 0. (5.7.7)

Remarking that a particular solution of (5.7.6) is given by -Qa2k~ 1J0(ka)and choosing the complementary solution to satisfy (5.7.7), we obtain

V(k, t) = - Q a 2 / T ^ M l " ! - exp(-vfe2t)]. (5.7.8)

Setting v = 1 in (5.5.2b), we obtain

v(r, t) = - 2Q X [feJo(M] " i(fer)[l - exp(- vk2t)\ (5.7.9)

where the summation is over the positive roots of (5.7.5).We remark that r has the corresponding representation

r = - 2 X [fcJo(M] " dkr), (5.7.10)k

so that v ~ Qr as t -+ oo, as also may be inferred from physical considera-tions.

5.# Conclusion

The superiority of the finite-transform method, over either the classicalprocedure of separation of variables or the Laplace transformation, inobtaining solutions as expansions of natural modes tends to increase with




the complexity of the problem. In particular, the finite-transform methodalways provides the modal expansion of the static solution, as in (5.3.9)above; however, this may not always be an advantage. We emphasize,nevertheless, that the Laplace transform is both more flexible and morepowerful. It both incorporates alternative interpretations, such as thetraveling-wave expansion of (2.8.13), and places less stringent conditionson the boundary conditions that may be accommodated.

EXERCISES5.1 A uniform string of length / and line density p is stretched between two fixed

points, x = 0 and x = /, to tension pc2. It is displaced a small distance a at a pointdistant b from the origin and released at t = 0. Given the equation of motion,

c2yxx = y«,

show that the subsequent displacement is

2al2 " „ nnb nnx nncty(x, t) = V n 2 sin sin cos

n2b(l

5.2 A string of line density p and length / is stretched to tension pc2. The endx = 0 is fixed, and the end x = I is attached to a massless ring that is free to slideon a smooth rod. At t = 0, when the system is at rest with the ring displaced a smalldistance a from equilibrium position, the ring is released. Show that the subsequentdisplacement of the string is

( (2n + l)ncty j X sin cos — .

n n = 0 21 11

5.3 A string of line density p and length / is stretched to tension pc2. The endx = I is fixed and at t = 0, when the string is at rest in its equilibrium position, theend x = 0 is given a small oscillation a sin cot. Show that the subsequent displace-ment of the point x is

a sin cot sin[co(l — x)/c] * 2lcaco . rnx . met+

r ^ co2/2 - 7 i 2 r 2 c 2 S m T S m T

5.4 (a) Show that (5.4.9) has the alternative representation

v = 2vohY^\h + (k2 + h2)iy1 sec kl cos /cxexp(-/c/c2Ok

by solving (5.4.5) for sin kl and cos kl. (b) Show that (5.4.9) reduces to

v = ^ £ (-)n(2n + I )" 1 cos/c,,xexp(-K/c20,n M = 0

where kn = (2n + l)(n/2l), for h = oo.



EXERCISES / 77

5.5 Determine the solution to the diffusion equation

vt = KVXX

with

x for 0 < x < \\ at t = 0

— x for \\ < x < I at t = 0

and

v = 0 at x = 0 and / and t > 0

by (a) Laplace transformation and (b) finite-Fourier transformation.Answer:

4/ » (-)" f (2n + I)2rc2x*l .t; = — > r exp si

n\%(2n + l)2 yl I2 J

: + l)7tXs in-

5.6 Consider the problem of Section 5.4 with the boundary condition v = f(t)in place of (5.4.2a) and the initial condition v = 0 in place of (5.4.3). Show that

v(x9t) = 2K5>( /C 2 + h2)\h + (k2

where

k cot kl = -h.

5.7 Solve Exercise 2.17 by using a finite Fourier transform.5.8 Solve Exercise 2.20 by using a finite Fourier transform.5.9 Use the Laplace transform to solve the problem of Section 5.6.

5.10 The top (0 < x ^ /) and bottom ( - / ^ x < 0) portions of a cylindricalbox bounded by x = ± / and r = a are insulated from one another and charged tothe potentials v0 and — v0, respectively. Show that the potential inside the box,which must satisfy Laplace's equation,

vxx + vrr + r ~ V = 0,

is given by either

2v0 [ sinh/c(/- |xl)1 J0(kr)

where the summation is over the positive roots of J0(ka) = 0, or

where / 0 is a modified Bessel function.




5.11 The functions

ZJLkr) = Jn{kr)Yn{ka) - Yn(kr)Jn(ka)

form a complete orthogonal set in a < r < b if the k are given by the positive roots of

ZJLkb) = 0.Obtain the corresponding finite-transform pair (see Table 2.4). Use the result todetermine the temperature distribution in a long annular tube of diffusivity K if theinner and outer surfaces, r = a and r = b, are held at zero temperature and theinitial temperature is v0.Answer:

v = nv0Y, W M + Jo(to)]-%(to)Zo(fcr) exp(-Kk2t).k

5.12 Suppose that the viscous fluid of Section 5.7 is bounded internally by astationary cylinder of radius a and externally by a cylinder of radius b that begins torotate with the angular velocity Q at t = 0. Show that

JAkaVAkbyZAkrfil - exp(-vfe2t)]

*M) = 7rQ*? VKka) -where

- Y,{kr)J\(ka),

and the k are determined by the positive zeros of Z^kb).



PARTIAL-FRACTIONEXPANSIONSAPPENDIX 1

LetF ( p ) =H' (AU)

where

and

H(p) = h0 + hlP + ••• + hNpN (N > M) (A1.3)

are polynomials with no common factors (common factors may be can-celled if originally present), and the degree of H is higher than that ofG (N > M).

We consider first the case where the zeros of H{p\ say p1, p2> • • • > PN>are all distinct. It then follows from known results in algebra that H(p)can be factored to obtain

H(p) = hN(p - Pl)(p - p2) - . (p - pN) (A1.4)

and that F(p) can be developed in a partial-fraction expansion of the form

F(p) = ^ ^ £ ()P - Pi P - Pi P - PN

The Cn, n = 1, 2, . . . , N, in the expansion (A1.5) may be determinedby one of the following procedures: (a) multiplying both sides of (A1.5)through by H(p), equating the coefficients of pn, n = l ,2 , M o n bothsides of the resulting equation, and requiring the coefficients of pn, n = M+ 1 , . . . , N on the right-hand side thereof to vanish identically; (b) multi-plying both sides of (A1.5) through by p - pn and letting p -> pn to obtain

79



80 / PARTIAL-FRACTION EXPANSIONS / APPENDIX 1

= lim {P ~J^P) s lim(p - pn)F(p) (n = 1,2, . . . , N); (A1.6)

(c) invoking the identity (in effect, L'Hospital's rule)

lim H{P)n ~ f P°] = H'(P)lP=Pn ^ H'(pn)

p-p n P Pn

in (A 1.6) to obtain

The procedure (a) is efficient only in relatively simple cases, although itoccasionally may be more direct than either (b) or (c). The latter pro-cedures are almost equivalent, but (c) is generally more efficient.

Substituting (A1.8) into (A1.5), we obtain the general result

= 0 ] - < A 1 9 >

The inverse transform of this expansion is given by (2.7.7), which, is, how-ever, of greater generality [see discussion preceding (2.7.7)].

Consider, for example, the transform of (2.3.7),

Choosing G(p) = 1 and

H(p) = p{p2 + P2) = p(p - W)(P + Wl (Al.ll)

we have M = 0, N = 3, p1 = 0, p2 = i/J, p3 = —ifi, and

+ ^ + ^ . (A1.12)H(p) p p - i p p + ip

Considering first procedure (a), we multiply both sides of (A 1.12) throughby H(p) in its factored form to obtain the identity

1 = Ct(p - W(p + iff) + C2p{p + iP) + C3p(p - ip)

= CJ2 + (C2 - C3)ipp + (Ci + C2 + C3)p2. (A1.13)

Equating the coefficients of p° and requiring the coefficients of p and p2

on the right-hand side of (A 1.13) to vanish, we obtain

1 = CJ2, C2-C3= 0, and Ct + C2 + C3 = 0, (A1.14)

the solution of which yields



APPENDIX 1 / PARTIAL-FRACTION EXPANSIONS / 81

CY = j 2 and C2 = C3 = - ^ j . (A1.15)

Substituting (A 1.15) into (A 1.12), we obtain

(A1.16b)

the latter form being that of (2.3.7). Considering procedure (b), we sub-stitute G = 1 and H = p{p2 + p2) in (A1.6) to obtain

r -lim

r _ y (p - ip) . 1 1p-+tp p(p + P ) p->ip p(p + ip) 2/?2

andC3 = lim -—2 -r^ = lim — — = — —^. (A1.17)

Considering procedure (cj, we substitute G = 1 and / f = 3p2 -f p2 in(A1.8) to obtain

C l = ^ ' C l = 3(ij8)2 + p2 = " 2 ^ *

and

^ 3 (_ l i S )2 + ^2 2 j S2- v—/

It is evident, even in this rather simple example, that (a) is the most cum-bersome procedure, whereas (c) is the most efficient.

Various special cases, and generalizations, of (A 1.9) are given byGardner-Barnes (1942), pp. 153-63. The most important generalizationallows for a double zero of if (p), say px = p 2 , such that (A 1.4) is replaced by

H(p) = hN(p - ptfip - p3) •.. (p - pN). (A1.19)

The expansion (A1.5) then must be replaced by

11 . 12 , ^ 3 . . ^N / A 4 ^/Vw

— + 7 1 2 + + + • (AL2°)P - Pi (P - Pi) P ~ Pi P - PN

Procedure (a) above may be applied directly to this expansion to obtain



82 / PARTIAL-FRACTION EXPANSIONS / APPENDIX 1

C n , C12 , C3, . . . , CN. Procedures (b) and (c) may be applied to obtainC3, . . . , CN, but must be modified to obtain C u and C t 2 . Multiplyingboth sides of (A1.20) through by (p — px)2 and introducing the function

we obtain

O(p) = C12 + C n (p - Pl) + (p- P l )2 j z ^ ^ 3

(A1.22)Setting p = Pi in (A 1.22), we obtain

C12 = O(Pl). (A1.23)

Differentiating (A1.22) with respect to p and then setting p = p l 5 weobtain

Cn^fo). (A1.24)

Substituting (A1.23) and (A1.24), together with (A1.8) for n = 3, . . . , AT,into (A 1.20), we obtain

(P - Pl) P - Pl . -3 «(?„)(? - Pn)

where O(p) is given by (A1.21). See also Exercise 2.8.



TABLESAPPENDIX 2

Table 2.1 Laplace-transform pairs

(Greek letters in the following may be complex as noted.)

2.1.1

2.1.2

2.1.3

2.1.4

2.1.5

2.1.6

2.1.7

2.1.8

2.1.9

2.1.10

2.1.11

2.1.12

fit)

tv (0i\ > -

e-'â ^

cos bt

sin bt

e~at cos bt

e~at sin bt

te-îMtx :

d(t)

tm log t

-1)

0)

erfc(|flr1/2)

J0(at)

J0[a(t2 - (t-b)

F(p) = \e~ptf(t) dtJo

r(v + i)p—

(p + oc)"1

p(p2 + b2)-1

b(p2 + b2)-1

{p + a)[{p + a)2 + b2y

blip + af + b2]'1

(P + «)"2

1

m\p~m~l[\ + i + i + ••• + m"1 - y - logp]

pêxpC-ap 1 / 2 )

(p2 +a2yit2

(p2 + a2)-1/2exP[-Mp2 + tf2)1/2]

83



84 / TABLES / APPENDIX 2

Table 2.2 Operational formulas

fit) F(p)Jo

2.2.1 oif(t)

2.2.2 fx(t) + f2(t) F.ip) + F2(p)

2.2.3 /<">(*) p"F(p) - V

2.2.4 |/(t)dT

2.2.5 / ( t - a)H(t - a ) ( a ^ 0 )

2.2.6 e~"f(t) F(p + a)

2.2.7 f(at)

2.2.8 /(t + T) = /W (1 - e'")-1 \Te-"f(t) dtJo

2.2.9 fVi(t - T)/2(T) di F1{p)F2{p)Jo

2.2.10

22A1 i



Table 2.3 Infinite integral transforms

Transform

Laplace

Fourier

Fourier-cosine

Fourier-sine

Mellin

Hankel

F{p) = P{f(x)}

r°°f(x)e~px dx

Jo

f(x)e~ipxdxJ-oo

f00

f(x) cos px dxJo

foof(x) sin px dx

Jo

C oo

/Mx'^dxJo

1 f(x)Jn(px)xdxJo

i fc + iao

— F(p)epxdp

i r°° .pjc

27rJ_QO

2 f00

- F(p) cos px dpn Jo

2 f00

- F(p) sin px dp^ Jo

2^'Jc-ioo

Jo

Transforms of typical derivatives

3T{fM} = p"F(p) - X p"/^-"" 1 ^)m = 0

•^{/(n)} = (ip)nF(p)

3T{f^} = (-)"p2"F(p) - "E (_rp2«/ (2--2--i

m = l

^~{xn f^} = (~l)"p(p ~l" 1) "• (p + w — l)F(p)

r 1 /n\2 1^ " ] / " + - / ' - ( - ) / [ = ~P2F

Convolution theorem&~~1{F(p)G{p)}

Jo

J — oo

Jo

().ft. None available, but see Sneddon^ (1951), Section 3.6, for related

theorems

No simple result

ID

m

a><to



Table 2.4 Finite integral transforms

Transform

Cosine

Sine

F(k) = 3T{f(x)}

nf(x)coskxdx

Jo

pf(x) sin kxdx

Jo

f(x)= F-l{F(k)}

( 2 - dk0\k2 + h2)F(k) cos kx

£ ft + /(fe2 + ft2)

(fc2 + fc2)F(fc)sinfcx

^ t * + Kk2 + ft2)

Eigenvalue equation forK (K 0)

k tan /c/ - ft

kcotkl= ft

Transforms of typical derivatives

3T{f"(x)} = -fc2F(/c)-/ '(0)

•^{/"W} - - f e 2 ^ ) + fc/(0)

r

a

Hankel f(x)Jn(kx)xdxk2F(k)Jn(kx)

(ft > 0) kJ'n(ka) + ftJB(/ca) = 0 - g) } = -»]^B(M

AnnularHankel f

I = YJLkaVJLkx)- Jn(ka)Yn(kx)

1 2 k2Jl(kb)F(k)Z,(kx)

2% t = 0 (6 > a)

Legendre (2n + \)F(ri)PJ,x)



LIST OFNOTATIONSAPPENDIX 3

The numbers after each entry indicate either the equation or the section inwhich that entry is defined. Symbols used only in a local context, whereinthey are defined explicitly, may not be included in this list.

C closed contour of integration in complex planeE Young's modulus for elastic solid (Section 2.8)EMOT Erdelyi, Magnus, Oberhettinger, and TricomiF typically an integral transform of/(x)J% &~l Fourier-transform operator and its inverse (1.3.1)J*c, J%~1 Fourier-cosine-transform operator and its inverse (1.3.5)^ ^ ^7 1 Fourier-sine-transform operator and its inverse (1.3.6)H Heaviside step function (2.5.3)Jf, 3^ ~* Hankel-transform operator and its inverse/„ modified Bessel function of first kind and nth orderJn Bessel function of first kind and nth orderK transform kernel (1.1.1), thermal conductivityif, $£ ~ * Laplace-transform operator and its inverse (1.4.3)R spherical radius; also residue0t operator that implies real part of its operandT entry in table (e.g., T2.1.1 implies entry 1 in Table 2.1); period

of periodic function2T,3~~X integral-transform operator and its inverseYn Bessel function of second kind and nth orderc (1.4.1) and Figure 1.2; wave speed; specific heat (Section 2.9)e base of natural logarithms = 2.7183- ••erf z = 2TT" 1 /2JQ exp( - W2) du = error functionerfcz = 1 — erf z = complementary error function/ typically an arbitrary function to be transformed (1.1.1)

87



/ LIST OF NOTATIONS / APPENDIX 3

fn\x) nth derivative of /(x)/(n)(0) nth derivative of /(x) evaluated at x = 0

r - /(1)

r ^ /(2)

# gravitational acceleration/i (5.2.5)i imaginary unit; subscript i denotes imaginary part of corre-

sponding variablek Fourier-transform variable (1.3.1), (1.3.5), (1.3.6); spring con-

stant in Section 2.3kn discrete value of k in the monotonically increasing sequence

fci,/c2, •••/ lengthm, n integersp transform parameter (1.1.1)pn a pole of F(p) or a discrete value of p in the monotonically

increasing sequence pi, p2,r radius in polar coordinates; subscript r denotes real part of

corresponding variables index of summation; s takes only odd values (s = 1, 3, •••) in

Section 2.6; also subscript denoting point of stationaryphase

sgn signum function (3.7.4)x independent variable; typically, but not necessarily spatialx, y, z Cartesian coordinatesF gamma function; T(n + 1) = n\An (4.1.2)E summat ion signy Euler's constant = 0.577215-••5 Dirac delta function (2.4.2)dmn Kronecker delta (5.1.1)£ a small parameter that tends to zero ( through positive values

unless otherwise stated)£ water-wave displacement (Section 3.6)6 angle, typically a polar coordinateK diffusivity (2.9.1)p density (mass per unit volume or, for string, per unit length)co angular frequencyV vector-gradient operatorV2 Laplacian opera tor ; V2 = (d/dx)2 + (d/dy)2 + {d/dz)2 in Car-

tesian coordinates; see (4.1.3) for polar coordinates~ implies is asymptotic to= implies equal by definition= implies approximately equal



GLOSSARY

boundary condition a condition imposed on a bounding surface (in three dimen-sions) or line (in two dimensions) or at a bounding point (in one dimension)

convolution theorem see (2.4.1), (3.3.2), (3.3.7), and Table 2.3finiteness condition a degenerate form of a boundary condition, whereby a de-

pendent variable is required to be finite in some limitHeaviside's expansion theorem see (2.7.7)Heaviside's shifting theorem see (2.5.2)initial condition a condition imposed at t = 0 or, more generally, at the origin of a

timelike variable (as in Section 2.10)Jordan's lemma Let L denote the semicircle \p\ = P in pr < 0; then

lim eapf(p) dp = 0 (a > 0)P-+CK JL

provided that f(Peie) vanishes uniformly in \n ^ 9 ^ frr as P -* oo.L'Hospital's rule' Iff (a) = g(a) = 0, l im,^ [/(*)/</(*)] = / ' ( a ) M 4meromorphic a function of a complex variable for which every point in the finite,

complex plane is either a regular point (in the neighborhood of which the functionis analytic) or a pole

radiation condition a requirement that a disturbance appear as an outgoing wave;e.g. f(r — ct) is an outgoing wave as r -• oo, whereas f(r + ct) is an incomingwave.

residue

where p* is a pole of nth order.stationary phase see Section 3.7Watson's lemma see (2.7.13)

89



BIBLIOGRAPHY

Texts and treatises

BECKENBACH, E. F. (Ed.), Modern Mathematics for the Engineer, Second Series(New York: McGraw-Hill, 1961).

BRACEWELL, RON, The Fourier Transform and its Applications (New York: McGraw-Hill, 1965).A stimulating treatment from the viewpoint of modern electrical engineering.

CARSLAW, H. S., and J. C. JAEGER, Operational Methods in Applied Mathematics(New York: Oxford Univ. Press, 1953).Deals only with the Laplace transform. Contains extensive collection of workedand unworked problems involving both ordinary and partial differential equations.

CARSLAW, H. S., and J. C. JAEGER, Conduction of Heat in Solids (New York: OxfordUniversity Press, 1949).Standard treatise on mathematical theory of heat conduction in solids. Extensiveapplication of Laplace transform.

CHURCHILL, R. V., Fourier Series and Boundary Value Problems (New York:McGraw-Hill, 1963).

CHURCHILL, R. V., Operational Mathematics (New York: McGraw-Hill, 1958).Elementary text dealing with the Laplace transform, finite Fourier transforms,and, briefly, complex-variable theory. Reasonable mathematical rigor is main-tained, but the problems are rather elementary.

COPSON, E. T., Asymptotic Expansions (Cambridge: Cambridge Univ. Press, 1965).Excellent treatment of asymptotic evaluation of integrals.

DITKIN, V. A., and PRUDNIKOV, A. P., Operational Calculus in Two Variables andits Applications (New York: Pergamon Press, 1962).

DOETSCH, G., Theorie und Anwendung der Laplace-Transformation (New York:Dover, 1943; originally published in Berlin: Springer-Verlag, 1937).Standard mathematical treatise on the Laplace transform; extensive bibliographyof pre-1937 works.

90



BIBLIOGRAPHY / 91

DOETSCH, G., Handbuch der Laplace-Transformation (Basel: Birkhauser, 1950-1956).Three-volume treatment of theory and applications, both mathematical andphysical, of Laplace transform.

ERDELYI, A., Asymptotic Expansions (New York: Dover, 1956).Treats asymptotic development of both integrals and solutions to differentialequations.

ERDELYI, A., Operational Calculus and Generalized Functions (New York: Holt,Rinehart, and Winston, 1962).A brief but authoritative treatment of the modern, rigorous approach to opera-tional calculus; complements Lighthill's (1959) treatment of Fourier integrals.

FOURIER, JOSEPH, The Analytical Theory of Heat (Paris, 1822; translated by A.Freeman, Cambridge, 1878; reprinted New York: Dover, 1955).The fountainhead of Fourier methods.

GARDNER, M. F., and J. L. BARNES, Transients in Linear Systems (New York:Wiley, 1942).Application of Laplace transforms to analysis of lumped-constant, electrical andmechanical systems.

HEAVISIDE, OLIVER, Electromagnetic Theory (New York: Dover, 1950 reprint).Contains much of Heaviside's original use of operational calculus.

LAMB, H., Hydrodynamics (Cambridge: Cambridge Univ. Press, 1932; reprint NewYork: Dover, 1945).The classical treatise on hydrodynamics.

LIGHTHILL, M. S., Introduction to Fourier Analysis and Generalised Functions(Cambridge: Cambridge Univ. Press, 1959).Deals with the Fourier-transform and -series representations of functions thatwould conventionally be regarded as improper, and with the asymptotic propertiesof these representations.

MIKUSINSKI, JAN, Operational Calculus (New York: Macmillan, 1959).Original and fundamental treatment of operators in operational calculus [seeErdelyi (1962) for more concise treatment].

MILES, J. W., The Potential Theory of Unsteady Supersonic Flow (Cambridge:Cambridge Univ. Press, 1959).Extensive application of Fourier and Laplace transforms to aerodynamicboundary-value problems.

MORSE, P. M., Vibration and Sound (New York: McGraw-Hill, 1948).SNEDDON, I. N., Fourier Transforms (New York: McGraw-Hill, 1951).

Extensive applications of various transforms to physical problems at research-paper level.

THOMSON, W. T., Laplace Transformation (Englewood Cliffs, N.J.: Prentice-Hall,1960).Similar in scope to Churchill (1958); less rigorous mathematics but more elaborateengineering applications.

TITCHMARSH, E. C , Introduction to the Theory of Fourier Integrals (New York:Oxford Univ. Press, 1948).Standard treatise on Fourier, including Fourier-Bessel or Hankel, integrals and



92 / BIBLIOGRAPHY

transforms; largely complementary to Doetsch (1937); extensive bibliography ofpre-1948 works.

TRANTER, C. J., Integral Transforms in Mathematical Physics (London: Methuen,1956).A brief, but not elementary, coverage of all the commonly used transforms.

VAN DER POL, B., and N. BREMMER, Operational Calculus Based on the Two-sidedLaplace Integral (New York: Cambridge Univ. Press, 1950).A modern, rigorous presentation of Heaviside's operational calculus as opera-tional calculus. Advanced and stimulating applications in such diverse fields aselectric circuits and number theory.

VOELKER, D., and G. DOETSCH, Die Zweidimensionale Laplace-Transformation

(Basel: Birkhauser, 1950).WATSON, G. N., Bessel Functions (Cambridge: Cambridge Univ. Press, 1945).

The classical treatise on Bessel functions.

Tables and Handbooks

ABRAMOWITZ, M., and I. A. STEGUN, Handbook of Mathematical Functions (Wash-ington: National Bureau of Standards, 1964; reprint New York: Dover, 1965).An extensive tabulation of analytical properties and numerical values of tran-scendental functions. Highly recommended.

CAMPBELL, G. A., and R. M. FOSTER, Fourier Integrals for Practical Application(New York: Wiley, 1948).The most extensive table of (exponential) Fourier integrals; many entries areeffectively Laplace-transform pairs and are presented as such.

ERDELYI, A., and J. COSSAR, Dictionary of Laplace Transforms (London: Depart-ment of Scientific Research and Experiment, Admiralty Computing Service,1944).Most of the material from these tables has been included in EMOT.

ERDELYI, A., (Ed.), with W. MAGNUS, F. OBERHETTINGER, and F. TRICOMI [EMOT].Tables of Integral Transforms, 2 vols. (New York: McGraw-Hill, 1954).The most comprehensive tables of integral transforms presently available.Volume 1 contains Fourier-exponential, -cosine, and -sine, Laplace, and Mellintransforms: Vol. 2 contains Hankel transforms, along with many transformsnot introduced in the foregoing treatment.

JOLLEY, L. B. W., Summation of Series (New York: Dover, 1961).An extensive tabulation of summable series.

LUKE, Y. L., Integrals of Bessel Functions (New York: McGraw-Hill, 1962).The most extensive table of integrals that contain Bessel functions.

MAGNUS, W., and F. OBERHETTINGER, Formulas and Theorems for the SpecialFunctions of Mathematical Physics (New York: Chelsea, 1949).This valuable (for the applied mathematician) compendium contains short butwell-selected tables of both Fourier- and Laplace-transform pairs.

MANGULIS, V., Handbook of Series for Scientists and Engineers (New York: Aca-demic Press, 1961).A tabulation that is both more extensive and more expensive than that ofJolley.



BIBLIOGRAPHY / 93

ROBERTS, G. E., and H. KAUFMAN, Tables of Laplace Transforms (Philadelphia,

Pa.:W. B. Saunders, 1966).Perhaps the most extensive table of Laplace transforms.



INDEX

(See also Index of Notations, pp. 87-88; Glossary, p. 89; and Bibliography, pp. 90-93.)

Aerodynamic problems, 36-38Airfoil, oscillating, 36-38Asymptotic approximations, 14, 29, 30,

40, 50-53, 62, 63, 66Asymptotic limit, 39 (Exercise 2.9)

Bar, 30-33,41,70, 71Beam, 55Bessel functions, 40,57-66,72-75,77,78Bessel's equation, 59Boundary-value problems, procedure for

solution of, 14Branch cuts, 28, 35, 40Branch points, 27-30, 35, 40

CAUCHY, A., 5, 18, 48H

Cauchy-Poisson problem, 48-53,61-65Cauchy's residue theorem, 8, 26Circuit analysis, 2, 3, 25, 39Complex Fourier transforms; see

Fourier transformsConduction; see Heat conductionContour integrals, 25-29Convolution theorem, 17, 18, 38, 39,

45, 46, 84(T2.2.9), 85, 89Cooling of bar, 73-74Cosine transform; see Fourier trans-

forms

Damped oscillator, 39Delta function, 18, 55, 83(T2.1.8)Derivatives, transforms of, 14, 15, 43,

44, 68, 69, 85, 86Differential equations, solution of, 14,

24,25Diffusion, of heat; see Heat conduction

viscous, 74, 75, 78Diffusion equation; see Heat conductionDiffusivity, 34Dini series, 73Discontinuous functions, 18, 19Duhamel's superposition theorem, 17

Eigenvalues, 67, 86Electric circuits, 2, 3, 25, 39Error function, 35, 87

Faltung theorem; see Convolutiontheorem

Finite transforms, 3, 4, 67-78, 86Flux of heat, 34FOURIER, J., 4 , 6

Fourier series, 4, 6, 67Fourier transform, 2, 7-10, 43-56, 85;

see also Finite Fourier transformsFourier-cosine transform, 2, 85Fourier-sine transform, 2, 85

95



96 / INDEX

Fourier transform (Continued)in two dimensions, 53

Galilean transformation, 37Generalized functions, 18Gravity waves; see Water waves

Hankel transform(s), 2, 57-66, 85finite, 72-75, 86

Harmonic time dependence, 37, 59Heat conduction, 34

in a bar, 73, 74in an infinite solid, 55in a semi-infinite solid, 34-36, 40, 47,

48,66in a slab, 40, 41, 71, 72, 77

HEAVISIDE, O., 1, 4, 5, 15, 26Heaviside's expansion theorem, 26Hooke's law, 30

Impulse functions, 17, 18, 55Infinite transforms, 1-66, 85Influence function, 17Initial conditions, 14, 15, 18, 19, 37Integral equations, \4nIntegrals, transforms of, 15Integral transform, 1Inversion integral (inverse Laplace trans-

form), 24-30; see also Transform pairs

Jordan's lemma, 26, 89

Kernel of transform, 1, 45

Laplace transform, 1, 10, 11, 13-42, 49,63,85

operational theorems, 84tables, 83in two dimensions, 54two-sided, 11

Laplace's equation, 48, 54, 62, 77Laplacian operator, 57Legendre transform, 86LEIBNIZ, G., 5

L'Hospital's rule, 89Linearization, 37, 48

Loudspeaker, 59-61

Mellin transform, 12, 42, 85, 89Meromorphic transform, 25, 26, 89Modal expansion, 76

Operational theorems, 24for Fourier transforms, 45, 46for Laplace transforms, 17-24, 84

Operators, transform, 7, 11, 57, 85, 86Orthogonal functions, 67, 68, 78Oscillator, 15-17, 18, 20, 24, 39

Parseval's theorem, 45Partial-differential equations, solution

of, 14Partial-fraction expansions, 13, 14, 16,

23, 79-82Periodic functions, 20-24, 84 (T2.2.8)Poles, 8, 25-27

double poles, 27, 39,81,82Potential, velocity, 36, 48Potential theory, 54, 77Power spectrum, 45Procedure for transform solution, 14Products of transforms; see Convolution

theorem

Radiation, of heat, 40, 69, 71of sound, 59-61,65, 89

RAYLEIGH, LORD, 5 - 6 , 59«, 61

Residues, 8, 26, 89; see also PolesResonance, 24

Saddle-point approximation, 52Scaling theorems, 84(T2.2.1, T2.2.7)Separation of variables, 68, 75Shifting theorems, 18-20, 45, 84(T2.2.5,

T2.2.6)Signum function, 51, 54Sine transform; see Fourier transformsSound; see Wave propagationSpecific heat, 34Stationary-phase approximation, 50-53,

62, 63, 65, 66Step function, 19



INDEX / 97

String, 41, 54, 55, 76Sturm-Liouville problems, 68Supersonic flow, 36-38Superposition; see Convolution

theoremSymmetry considerations, 43, 44

Tables, 24, 83-86Temperature; see Heat conductionThermal conductivity, 34THOMSON, W. (LORD KELVIN), 4

Transform pairs, 7-12, 24, 83-86Traveling waves, 33Triangular wave, 20-24, 33

Vibrating string, 41, 54, 55, 76Vibration problems, 15-17, 25; see also

Oscillator; Wave propagation

Water waves, 48-53, 55, 56, 61-66Watson's lemma, 29, 30Wave equation, 31, 36, 46, 47, 60Wave-number spectrum, 50Wave propagation, 46, 47

in bar, 30-33,41,70, 71on beam, 55sound waves, 36, 42, 59-61, 65on string, 41, 54, 55, 76surface waves, 48-53, 55, 56, 61-66



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