3
JOHN W. MILES
PROFESSOR OF ENGINEERING AND GEOPHYSICS
UNIVERSITY OF CALIFORNIA, LOS ANGELES
3.1 Introduction
We define
to be an integral transform of the function f(x), with K(p,x), a prescribed function of p and x, as the kernel of the transform; see Chap. 2. The introduction of such a transform in a particular problem may be advantageous if the determination or manipulation of F(p) is simpler than that of f(x), much as the introduction of log x in place of x is advantageous in certain arithmetical operations. The representation of f(x) by F(p) is, in many cases, merely a way of organizing a solution more efficiently, as in the introduction of logarithms for multiplication, but in some instances it affords solutions to otherwise apparently intractable problems, just as in the introduction of logarithms for manually extracting the 137th root of a given number.
Integral transforms in applied mathematics find their antecedents in the classical methods of Fourier and in the operational methods of Heaviside—antecedents that had rather different receptions by contemporary mathematicians. The classical eleventh edition of the “Encyclopedia Britannica” devotes five pages to Fourier series but does not mention Heaviside’s operational calculus; indeed, no direct entry appears for Heaviside in that edition, although his name is mentioned peripherally. The fourteenth edition does contain a brief biographical entry on Heaviside, but the only reference to his operational calculus is the rather oblique statement that “he made use of unusual methods of his own in solving his problems.”
Fourier’s theorem has constituted one of the cornerstones of mathematical physics from the publication (1822) of his “La Théorie analytique de la chaleur,”18 and its importance was quickly appreciated by mathematicians and physicists alike. For example, as quoted by Campbell and Foster,10 Thomson and Tait remarked:
… Fourier’s Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electrical signals along a telegraph wire, and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.
The concept of an integral transform follows directly from Fourier’s theorem (see Secs. 3.2 to 3.8), but the historical approach, at least to modern applications, was largely through operational methods.
Operational methods in mathematical analysis, having been introduced originally by Leibnitz, are nearly as old as the calculus, but their widespread use in modern technology stems almost entirely from the solitary genius of Oliver Heaviside (1850-1923). To be sure, the bases of Heaviside’s method, as he recognized and stated, lay in the earlier work of Laplace (1779) and Cauchy (1823), but it was Heaviside who recognized and exposed its power not only in circuit analysis but also in partial differential equations. Unlike Fourier, however, Heaviside had no university training and was not a recognized mathematician; indeed, he scorned not only mathematical rigor (“Shall I refuse my dinner because I do not fully understand the process of digestion?”) but, it sometimes appeared, mathematicians (“Even Cambridge mathematicians deserve justice.”). This lack of rapport with mathematicians perhaps helped cause the full importance of his work to be appreciated only gradually, and even some modern mathematicians have been reluctant to give Heaviside his due; thus, Van der Pol and Bremmer9 take Doetsch to task for his description6 of Heaviside as merely “ein englischer Electroingenieur,” using methods that were “mathematisch sehr unzulänglich” and “allerdings mathematisch unzureichend.”
Today, the Laplace transform, which is the case K = e−px, a = 0, and b = ∞ in Eq. (3.1), namely†
may be claimed as a working tool for the solution of ordinary differential equations by every well-trained engineer. We shall consider here its application to partial differential equations, along with the Fourier transform (also called the exponential Fourier transform or the complex Fourier transform),
the Fourier cosine and Fourier sine transforms,
and
the Hankel transform (also called the Bessel or Fourier-Bessel transform),
and, rather briefly, the Mellin transform,
Our definitions are those of Erdélyi, Magnus, Oberhettinger, and Tricomi12 (hereinafter abbreviated as EMOT) except for Eq. (3.6); other definitions of the Fourier transforms, differing from those of Eqs. (3.3) to (3.5) by constant factors, are not uncommon. Notations for the transforms themselves vary widely, with 
a frequent variant for F(p); it also may prove expedient to introduce subscripts such as c, s, and n in the definitions of Eqs. (3.4) to (3.6), respectively; finally, symbols other than p often are used in defining the transforms of Eqs. (3.3) to (3.6), and s often appears in place of p in those of Eqs. (3.2) and (3.7).
The transforms defined by Eqs. (3.2) to (3.7) are the only infinite ones in widespread use at this time, but many others have been studied and tabulated (see EMOT,12 vol. 2), and still more may be introduced in the future. If one’s goal is merely to produce formal solutions, usually accompanied by the phrase, so satisfying to mathematicians but so frustrating to physical workers, “The problem is now reduced to quadrature,” it suffices to know the inversion formula that determines f(x) from F(p), but for wide usefulness in applied mathematics, extensive tabulations of f(x) versus F(p) are essential. Returning to our analogue of the logarithm, we note that formal analysis requires only the knowledge that the inverse of y = loga x is x = ay, but for aid in numerical computation a table of x versus y is indispensable.
In addition to the possibility of defining new transforms through new kernels, there is also the possibility of adopting finite limits in Eq. (3.2), thereby obtaining so-called finite transforms. If, for example, we replace the upper limits in Eqs. (3.4) and (3.5) by π, the inversion formulas are ordinary Fourier cosine and sine series summed over integral values of p. More generally, if the kernel K(p,x) in Eq. (3.1) yields a set of functions orthogonal, with suitable weighting function, over the interval a,b for an infinite discrete set of values p, then the inversion formula defines a Fourier-type series.
The result of introducing a finite Fourier transform in a given problem is merely to mechanize the classical technique of Fourier series, but it is generally true that the direct, though more tedious, solution of the problem by the classical technique is straightforward, albeit often calling for greater ingenuity. (Compare the use of Lagrange’s equations in mechanics.) This is to be contrasted with the application of infinite transforms, which frequently offer entirely new insight and reduce transcendental to algebraic operations, thereby affording solutions to problems that either have resisted the previous forays of classical techniques or that could have been solved only with an ingenuity of Newtonian or Laplacian magnitude.
We conclude this introduction by contrasting the approaches of the pure mathematician and the pragmatist to transform theory. At one extreme we have Titchmarsh’s statement, in the preface to his treatise,8 that “I have retained, as having a certain picturesqueness, some references to ‘heat,’ ‘radiation,’ and so forth; but the interest is purely analytical, and the reader need not know whether such things exist.” At the other, we have Heaviside’s cavalier statement, “The mathematicians say this series diverges; therefore, it should be useful.” In the following condensed presentation of integral-transform theory, we† shall follow the line set down by Lord Rayleigh:19
In the mathematical investigation I have usually employed such methods as present themselves naturally to a physicist. The pure mathematician will complain, 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 to maintain a uniformly high standard in pure mathematics, the physicist may occasionally do well to rest content with arguments which are fairly satisfactory and conclusive from his point of view. To his mind, exercised in a different order of ideas, the more severe procedure of the pure mathematician may appear not more but less demonstrative. And further, in many cases of difficulty to insist upon the highest standard would mean the exclusion of the subject altogether in view of the space that would be required.
INVERSION FORMULAS AND TRANSFORM PAIRS
3.2 Fourier’s Integral Formulas
We first give a formal derivation of Fourier’s theorem in complex form, following in all essential respects the argument offered by Fourier himself.
Let f(x) be represented by the complex Fourier series
where
and
This representation is evidently periodic with a wavelength λ. Now we allow λ to tend to infinity, noting that the consecutive kn are separated by the increment Δk = 2π/λ; then, by combining Eqs. (3.8) and (3.9), we have
The sum may be replaced by an integral in the limit, and we obtain
By expressing the exponential in terms of its trigonometric components and invoking the even and odd nature of cos k(x − ξ) and sin k(x − ξ), respectively, as functions of k, we obtain
which is Fourier’s integral formula. Fourier’s derivation differed from the above only in starting from the trigonometric form of his series. We emphasize that the order of integration in Eqs. (3.11) and (3.12) must be preserved, because its reversal would lead to meaningless integrals; on the other hand, we assume that f(x) vanishes with sufficient rapidity for large |x| to ensure the existence of the double integrals as written. Actually, in typical, not-too-idealized, physical problems, f vanishes exponentially.
If f(x) is either an even or an odd function (and any function that is not even or odd can be split into a sum of two such functions) and the cosine in Eq. (3.12) is expanded, we obtain Fourier’s cosine formula
or Fourier’s sine formula
Equation (3.11) may be resolved directly into the Fourier pair
and
thereby providing the inversion of Eq. (3.3). The location of the factor (2π)−1 is essentially arbitrary; from the viewpoint of establishing an analogy between Fourier series and Fourier integrals, it would appear preferable to place it in Eq. (3.15a) rather than (3.15b), while from an aesthetic viewpoint a symmetric disposal of the identical factors (2π)−½ would be desirable. Each of these conventions has been adopted by various writers, but the form chosen in Eqs. (3.15a) and (3.15b) has two major advantages: First, it agrees with the notation adopted by Campbell and Foster10 in their very extensive table (see also EMOT,12 vol. 2), and, secondly, it affords a direct transition to the accepted definition of the Laplace-transform pair (see Sec. 3.5). A third advantage, of especial interest in electric-circuit or wave-motion problems, is that, if x, implicitly a space variable in the foregoing discussion, be replaced by the time variable t and k by 2πν, where ν is a frequency, Eqs. (3.15a) and (3.15b) go over to the symmetric pair
and
in which f(t) is represented as a spectral superposition of simple harmonic oscillations having the complex amplitude F(ν). We remark that a similar form for the space-variable pair of Eqs. (3.15a) and (3.15b) results from the substitution k = 2π/λ.
3.4 Fourier Cosine and Sine Transforms
Equations (3.13) and (3.14) yield the transform pairs
and
thereby providing the inversions of Eqs. (3.4) and (3.5), respectively. Again, the 2/π factors may be disposed differently—in particular, symmetrically; the notation adopted here is that of EMOT.12
3.5 Laplace Transform
The path of integration for Eq. (3.15b) may be deformed into a complex k plane in any manner that ensures the convergence of the integrals for both F(k) and f(x). Suppose, to cite the most important particular case, that exp (−cx)f(x) vanishes appropriately at both limits; in particular, f(x) may vanish identically at one limit, usually − ∞. Then the modified transform
exists, where 
denotes the imaginary part of k, and f(x) is given by
Thus, Fourier’s integral formula is extended to functions for which Eq. (3.11) might not be valid. In the most important physical applications, c is positive and the path of integration appears as in Fig. 3.1.
Fig. 3.1 Path of integration for Fourier integral inversion,
Fig. 3.2 Path of integration for Laplace transform inversion,
If we now rotate the path of integration through a right angle (see Fig. 3.2), introduce p = jk, and, at the same time, replace F(k) by F(p), Eqs. (3.19a) and (3.19b) go over to the two-sided Laplace-transform pair
and
Finally, we suppose f(x) to vanish for f(x) < 0, so that the lower limit in Eq. (3.20a) may be replaced by zero. The resulting integral converges if the real part of p exceeds some minimum value, say γ not necessarily nonnegative, such that all singularities of F(p) lie in 
; the inverse transform (3.20b) exists for all c > γ, and we have the Laplace-transform pair
and
3.6 Mellin Transform
Put x = − log y in Eqs. (3.20a) and (3.20b) to obtain
Then the replacement of f(− log y) by f(x) and the dummy variable y by x yields the Mellin-transform pair
and
3.7 Multiple Fourier Transforms
The Fourier-transform pair may be extended to a function of two variables to obtain
and
More generally, if r denotes a vector having the cartesian components x1, x2, …, xn in an n-dimensional space, and k a similar vector in the wave-number space k1, k2, …, kn, we obtain
and
3.8 Hankel Transforms
Further transform pairs may be obtained from Eqs. (3.23a), (3.23b), (3.24a), and (3.24b) by coordinate transformation. In particular, if we introduce polar coordinates according to
x = r cos θy = r sin θk1 = k cos αk2 = k sin α
then Eqs. (3.23a) and (3.23b) go over, with appropriate changes in the functional notation, to
and
We remark that if f(r,θ) is multiplied by exp (−jωt), Eq. (3.25b) represents a packet of plane waves having the amplitude distribution F(k,α), the wave speeds ω/k, and wavefront normals inclined at the angle α to the x axis.
We now assume that
and make use of the integral representation
for Bessel’s function of order n to obtain
where
and
the latter result following from the substitution of Eqs. (3.26a) and (3.26b) in Eq. (3.25b); Eqs. (3.27a) and (3.27b) constitute the Hankel-transform pair of order n.
Another form of the Hankel transform, especially important in that it is used in EMOT,12 vol. 2, is given by
This evidently can be reconciled with Eqs. (3.27a) and (3.27b) by setting gn = r½fn and Gn = k½Fn. As it stands, it reduces, except for the constant factor 
, to the Fourier sine or cosine transform for n = ½ or −½, respectively.
The properties of the foregoing infinite transforms are summarized in Table 3.1.
Table 3.1 Infinite Integral Transforms
† Numbers refer to the list of references at the end of the chapter.
3.9 Introduction
We now proceed to consider some special properties and applications of the Laplace transform. There exist many textbooks and treatises dealing with both the theory and the applications of this transform; and, while there are many specific examples in which other transforms prove more expedient, the Laplace transform is generally the most powerful and flexible in dealing with initial-value problems and is the most extensively tabulated. We first consider a few of the properties that invest it with this power and flexibility and then apply it to three typical problems governed by partial differential equations. Many more examples are given in the textbooks listed at the end of the chapter, especially in Ref. 1.
3.10 Transforms of Derivatives
We require†
where f(n)(t) denotes the nth derivative of f(t) and f(0), f′(0), …, f(n−1)(0) are prescribed as initial conditions as t→ 0+. The required result follows upon repeated integration by parts, the explicit results for the first and second derivatives being
and
We emphasize that the Heaviside operational rule
is not valid except in the special case, always implied in Heaviside’s work, for which 
. A related result, which, however, is generally valid, is
3.11 Heaviside’s Shifting Theorem
Disturbances frequently arise at times other than zero, it being implicit that t = 0 is chosen to correspond to the first event of interest, the effects of all prior events being included in the prescribed initial conditions. It follows directly from the definition of the Laplace transform that
and
where F(p) is the transform of f(t) and f vanishes for negative values of its argument. Many closely related theorems are to be found in EMOT,12 vol. 1, and in other references listed at the end of this chapter.
We add that Eqs. (3.30a) and (3.30b) apply also to the exponential Fourier transform with p = jk.
3.12 Convolution Theorem
It frequently proves expedient to resolve a Laplace transform into a product of two transforms, either because the inversions of the latter transforms are known or because one of them represents an arbitrary function—typically an input to some physical system. For a pair of given inverse transforms f1(t) and f2(t), the convolution theorem states that
To prove Eq. (3.31), we multiply the defining integrals for F1 and F2 to obtain
By introducing the change of variable σ = t − τ and invoking the requirement that f1 must vanish for negative values of its argument, we get
whence the transform of the quantity in brackets is F1F2, the inversion of which yields Eq. (3.31).
We remark that, in typical applications, the right-hand side of Eq. (3.31) represents a superposition of effects of magnitude f2(τ), arising at t = τ, for which f1(t − τ) is the influence function, i.e., the response to a unit impulse. Indeed, it constitutes the extension to impulsive inputs of Duhamel’s superposition theorem (1833) for step inputs. Such a unit impulse is known as the Dirac delta function and has the formal properties
and
By letting
f2(τ) = δ(τ)
in Eq. (3.31), we see that
As defined by Eqs. (3.32a) and (3.32b), the delta function is highly improper (see Chap. 1 for a more sophisticated treatment), but it can be defined as the limit of a proper function and was so introduced by both Cauchy and Poisson in their independent but almost simultaneous (1815) derivations of the Fourier integral theorem.† The function used by Cauchy was
where f(x,y) may be identified as a solution to Laplace’s equation for a doublet source located at (x,y) = (a,0).
3.13 Inversion Procedures
The most direct manner of inversion is through a suitable table of transform pairs, but it frequently happens that the required entry cannot be found in any existing table. Aside from such extensions of the tables as may be achieved through the shifting, convolution, and other theorems, the most powerful methods of inversion are provided by complex-variable theory.
We consider first the case where F(p) is a meromorphic function of the complex variable p—that is, its only singularities in the entire finite p plane are poles—that exhibits the asymptotic behavior
F(p) = 0(|p|−b) |p| → ∞ b > 0
Here, the relationship w = O(z) in the neighborhood of z = a means 
; the most frequently occurring values of a, usually clear from the context, are 0 and ∞. If F(p) satisfies the foregoing condition, the line integral of Fig. 3.2 may be closed at infinity as in Fig. 3.3; it then follows from Cauchy’s residue theorem that f is given by 2πj times the sum of the residues at the poles of F(p)ept, all of which, by hypothesis, must lie in 
. In particular, if
where G has no singularities and H has N simple zeros at p1, p2, …, pN, N finite or infinite, we have
a result due essentially to Heaviside.
If the transform F(p) contains branch-point singularities, the contour of Fig. 3.3 must be deformed around the appropriate branch cuts, as illustrated in Fig. 3.4 for the important special case of a branch point at p = 0. More generally, branch points of Laplace transforms are likely to be on the imaginary axis, but only occasionally elsewhere. The contributions of the poles, if any, may be evaluated as before, in particular from Eq. (3.34), but it also is necessary to include the integrals over both sides of the branch cut. A specific example will be considered in Sec. 3.15, but the ultimate form of the integrals over the cut of Fig. 3.4 will be
Fig. 3.3 Contour of integration for meromorphic Laplace transform.
Fig. 3.4 Contour of integration for Laplace transform having branch point at the origin.
where ϕ is derived from F(p). It is possible that this integral may be evaluated from a table of Laplace transforms with I appearing as the transform of ϕ, but whether or not this is true, it often suffices to obtain an asymptotic approximation for large t. Actually, such approximations sometimes prove satisfactory for surprisingly small values of t.
The asymptotic evaluation (see Chap. 5) of I is especially simple if ϕ(x) is an analytic function of x, regular in some circle (|x| < R), where it has the representation
If positive real numbers C and a exist such that |ϕ| < Ceax, x → ∞, then Watson’s lemma states that I has the asymptotic expansion
If μ is a positive integer, the same result may be obtained by repeatedly integrating Eq. (3.35a) by parts. More generally, the expansion of ϕ(x) about x = 0 may contain logarithmic terms, and a formal asymptotic expansion for I may be obtained by integrating term by term.
Other methods of evaluating branch-cut integrals, of which I is only one—albeit the most important—form, are discussed in the monographs of Erdélyi16 and Copson.15 We also remark that approximations valid for small t may be obtained by an inverse application of Watson’s lemma to the Laplace transform F(p); that is, if
then
Finally, we note that the numerical evaluation of integrals such as that in Eq. (3.35a) may be entirely practical by virtue of the exponential convergence, although it may be necessary first to separate out the singularity at the origin.
3.14 A Problem in Wave Motion
A uniform bar of unit cross section is at rest and unstressed for t < 0, with one end fixed at x = 0 (see Fig. 3.5). At t = 0, a force of magnitude P is applied to the free end, where x = l, in the direction of the positive x axis. We require the subsequent motion.
Fig. 3.5 A uniform bar to which a load P is applied abruptly at time t = 0.
Let y(x,t) denote the displacement from equilibrium of any section initially at x; it is known that y satisfies the wave equation
where c, E, and ρ denote bar velocity, Young’s modulus, and density, respectively. The initial conditions are
y = yt = 0at t = 0 and 0 < x < l
while the boundary conditions are
y = 0at x = 0andEyx = Pat x = l t > 0
Transforming Eq. (3.37) subject to the initial conditions yields
while transforming the boundary conditions yields
The solution that satisfies these boundary conditions is
as may be verified by direct substitution.
We may invert Eq. (3.38) via Eq. (3.34), with
We note that H = 0 at p = 0 and at p = ±(2n + 1)(πjc/2l), n = 0, 1, 2, …. Substitution in Eq. (3.34) yields
The first term in Eq. (3.39)—viz., Px/E—represents the ultimate static deflection of the bar, while the remaining terms represent standing waves that gradually would die out if friction were admitted. In the absence of friction, however, the displacement continues to oscillate about the static displacement, as shown for the loaded end (x = l) in Fig. 3.5.
An alternative solution to the problem may be obtained by expressing the hyperbolic functions in terms of exponentials and using the expansion
Now the inverse transform of p−2 is t, whence the shifting theorem, Eq. (3.30b), yields
where, by definition, the square brackets vanish identically if their contents are negative.
Equation (3.40) 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 solution in a finite number of terms (since only a finite number of the square brackets are positive at any finite time), thereby rendering numerical computation simpler for small ct/l, but also because it provides additional insight into the physical problem. It is, indeed, one of the great virtues of the Laplace-transform solution that it comprises both the standing- and traveling-wave representations.
Fig. 3.6 Displacement of the loaded end of the bar shown in Fig. 3.5.
At x = l, where the load acts, the displacement given by Eq. (3.40) reduces to
corresponding to the triangular wave shown in Fig. 3.6, with a mean value equal to the static displacement Pl/E.
3.15 A Problem in Heat Conduction
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. We require a solution to the diffusion equation
where κ is the diffusivity, subject to the initial condition
v = 0at t = 0 and x > 0
and the boundary conditions
v = v0at x = 0andv < ∞as x → ∞
By taking the Laplace transform of these equations, we obtain
to which the solution is
This may be inverted directly from standard tables, but we shall use it to illustrate the general procedure.
The right-hand side of Eq. (3.42) has a square-root branch point at p = 0; accordingly, we choose a branch cut and close the path of integration as in Figs. 3.4 and 3.7. The function V has no singularities inside the closed contour, whence the contour integral of V exp (pt) around the closed path is zero; further, since the arcs at infinity contribute null values, the inversion integral for V is equal to the integral over the branch cut plus that over the small circle around the origin taken in a counterclockwise direction, viz.,
Fig. 3.7 Contour ABCD of integration for
In the neighborhood of the origin, V tends to infinity like p−1v0, and the contribution of the path BC as its radius tends to zero is simply v0 [as may be proved by setting p = 
exp (jθ) in the integrand, letting → 0, and integrating between −π and π]. We emphasize that V does not have a Laurent-series representation in the neighborhood of p = 0, and the contribution of the path BC must be treated more or less ad hoc. If F(p) is of the form p−1F1(p), where F1(p) has a branch point at p = 0 but is finite there, the contribution to f(t) of the path BC as its radius tends to zero is simply F1(0), but if the infinity is of higher order, the limiting forms of the integrals over AB, BC, and CD all may be improper, in which case the limit as → 0 may be taken only after the contributions of AB and CD have been combined with that of BC.
To evaluate the contributions of AB and CD, we set
p = re−jπandp = rejπ
respectively, to obtain
This last integral may be transformed to the error-function integral (see Ref. 1, page 94), and the end result is
Having this representation, we could obtain numerical results directly from tabulations of the error function, while approximations useful for small or large values of (κt)−½x may be obtained from the appropriate series representations. It is only in the simplest problems, however, that the branch-cut integrals lead to tabulated functions, and it generally is necessary to revert directly to series approximations; more generally, asymptotic approximations to these integrals may be deduced from Watson’s lemma. Thus, by comparing the integral on the right-hand side of Eq. (3.43a) to I in Eq. (3.35), with r substituted for x therein, we have
whence
μ = −½
and
We remark that this same result could have been obtained directly by expanding the exponential in Eq. (3.42) in ascending powers of p and inverting term by term, but then we would have had to assume that the inverse transform of a positive integral power of p vanishes identically for t > 0.
3.16 A Problem in Supersonic Flow
The unsteady flow past a two-dimensional supersonic airfoil is governed, in linear approximation, by the partial differential equation
where ϕ denotes the perturbation velocity potential, x and y cartesian coordinates in a reference frame moving with the airfoil, c the velocity of sound, and U the flight speed directed along the negative x axis, with U > c. We shall solve Eq. (3.44) subject to the boundary condition
ϕy = w(x)ejwtat y = 0andx > 0−∞ < t < ∞
corresponding to a prescribed transverse velocity of complex amplitude w(x) for an oscillating airfoil having leading edge at x = 0. The airfoil may be assumed to terminate at the trailing edge x = l; but the flow in x < l cannot be influenced by conditions in x > l, by virtue of the supersonic flight speed.
We first remark that ϕ may be assumed to exhibit the time dependence exp (jωt); taking the Laplace transform of Eq. (3.44) with respect to x then yields
while the transform of the boundary condition is
Φy = W(p)ejωt
The required solution, subject to the condition that ϕ must remain bounded as |y| → ∞, is
Φ = −λ−1W(p)ejωt−λyy > 0
where
with
That branch of λ for which 
is implied, while if y < 0, the sign of λ must be changed.
We may invert Φ with the aid of the convolution theorem. Considering first the factor λ−1 exp (−λy), we apply the shifting theorem to obtain
which then may be inverted from tables (see, for example, Ref. 10, No. 866) to yield, by means of the convolution theorem, the end result
where
K(x,y) = ej(ωt−νMx)J0{ν[x2 − (M2 − 1)y2]}
FOURIER TRANSFORMS
3.17 Introduction
We have seen that the Laplace transformation is especially suited to initial-value problems,17 in that the transform of the nth derivative incorporates the initial values of the first n − 1 derivatives. The Fourier transformation, on the other hand, appears to best advantage in boundary-value problems associated with semi-infinite or infinite domains, with the appropriate selection depending on the boundary conditions or symmetry considerations. We also recall that the convergence restrictions for Fourier transforms generally are more stringent than for Laplace transforms.
3.18 Transforms of Derivatives
With the three types of Fourier transforms defined as in Eqs. (3.15) to (3.18), integration by parts yields
Analogous results may be established for the cosine and sine transforms of higher derivatives of even order, but the cosine (sine) transform of an odd derivative involves the sine (cosine) transform of the original function. Thus, as implied directly by their trigonometric kernels, these transforms are intrinsically suited to differential equations having only even derivatives with respect to the variable in question. Moreover, the cosine (sine) transform of such a differential equation incorporates only the values of the odd (even) derivatives at x = 0; the values of other derivatives at x = 0 could be incorporated as constants to be determined, but the most satisfactory applications are those in which the unincorporated boundary conditions are null conditions at x = ∞.
The exponential transform, on the other hand, may be applied to all derivatives, but it incorporates no boundary values and therefore arises naturally only for infinite domains; to be sure, it may be applied to semi-infinite domains, but then it becomes essentially a Laplace transform.
The cosine or sine transform also may be advantageously applied to an infinite domain if f(x) is an even or odd function of x, respectively, in which case f′(0) or f(0) vanishes in consequence of symmetry.
3.19 Application to a Semi-infinite Domain
It is evident from the foregoing remarks that the cosine and sine transforms are less flexible than the Laplace transform when applied to a semi-infinite domain; nevertheless, where applicable they may offer distinct advantages. We shall illustrate this last assertion for the heat-conduction problem of Eqs. (3.41) et seq., although it should be emphasized that we are comparing the application of the Laplace transform relative to t with the Fourier transform relative to x; the Laplace transform is not well suited to x, nor is the Fourier transform to t.
By taking the sine transform of Eq. (3.41) and incorporating the boundary condition at x = 0 in accordance with Eq. (3.45c), we obtain
to which the solution is
This solution could be obtained via a Laplace transform with respect to t, but would be a rather trivial application thereof. Substituting in the inversion formula (3.18b) and using the known result
yields
in agreement with the results of Eq. (3.43b).
3.20 Initial-value Problem for One-dimensional Wave Equation
A classical problem in wave motion requires the solution to the wave equation
for the initial values
ϕ = f(x)andϕt = g(x)for − ∞ < x < ∞
We may imagine f(x) and g(x) to be the initial displacement and velocity of an infinitely long string.
By taking the Fourier transform with respect to x and using the result (3.45a), we obtain
Φtt + (kc)2Φ = 0
and
Φ = F(k)andΦt = G(k)at t = 0
the solution to which is
By applying the shifting theorem (3.30b) with p = jk and a = ±jkc and noting that division by jk transforms to integration with respect to x [cf. Eq. (3.29)], we get
HANKEL TRANSFORMS
3.21 Introduction
The Hankel transformation arises naturally in connection with the wave equation in cylindrical polar coordinates, viz.,
or, under the assumption that ϕ varies as either cos nθ or sin nθ,
Taking the nth-order Hankel transform of Eq. (3.51b), we require
as follows after twice integrating by parts and invoking Bessel’s differential equation
3.22 Problem of an Oscillating Piston
We shall illustrate the Hankel transformation by applying it to the problem of an oscillating piston of radius a in an infinite baffle at x = 0, as shown in Fig. 3.8; this frequently is used as a simple model of a loudspeaker. On introducing the velocity potential ϕ, we require a solution to the wave equation (3.51a) subject to the boundary condition
where c denotes the velocity of sound and ω the angular frequency of oscillation. In accordance with the usual convention, A is the complex displacement, assumed small, of the piston from its equilibrium position, and the actual displacement is given by the real part of A exp (jωt); similarly, only the real part of ϕ is to be retained in the end result.
Fig. 3.8 An oscillating circular piston in an infinite baffle.
We first observe that, by virtue of radial symmetry, ϕ must be independent of θ and therefore ϕ satisfies Eq. (3.51b) with n = 0; accordingly, we introduce a Hankel transform of order zero. We also remark that ϕ must be proportional to exp (jωt) in order to satisfy Eq. (3.53), so that ϕtt = − ω2ϕ. Transforming Eqs. (3.51b) and (3.53) then yields
and
to which a solution is
where λ must be a positive real number for k > ω/c and a positive imaginary number (λ = iμ, μ > 0) for k < ω/c, the latter condition following from the fact that the disturbance must satisfy the radiation conditions of propagating outward from x = 0. Inverting Φ via Eq. (3.27b) then yields
The pressure on the piston is given by
while the total force thereon is expressed as
After some manipulation, this last integral may be reduced to the complex sum of a Bessel function and a Struve function, in agreement with the result obtained originally, in an entirely different fashion, by Lord Rayleigh.19
FINITE FOURIER TRANSFORMS
3.23 Introduction
The transforms considered thus far have been applicable to semi-infinite or infinite domains and have had a common antecedent in Fourier’s integral formula (3.12). It is natural to inquire whether transforms can be defined by Eq. (3.1) and their inverses derived from the theory of Fourier series. The essential result of this theory (Chap. 2) is that if ψ1(x), ψ2(x), … constitute a complete orthogonal set of functions for the interval a < x < b and the weighting function w(x), corresponding to a discrete set of eigenvalues p1, p2, … —that is, if
where δnm is the Kronecker delta, defined by
δnm = 0, m ≠ n; δnm = 1, m = n
and N(pn) denotes the integral of [ψn(x)]2w(x)—then it may be shown that
and
corresponding to K(p,x) = ψ(x,p)w(x) in Eq. (3.1). Equations (3.56a) and (3.56b) evidently constitute a generalized finite-Fourier-transform pair.
The choice of the orthogonal functions ψ(x,p) depends both on the differential equation and on the boundary conditions to be satisfied by f(x), just as with the infinite transforms. We shall consider here only finite sine and cosine transforms, appropriate to differential equations containing only even derivatives with respect to x and implying a weighting function equal to unity. The only other finite transform that has been applied extensively is that of Hankel (see especially Ref. 7), which is appropriate to the radial derivatives in ∇2f, viz., frr + r−1fr, and implies a weighting function w = r corresponding to that in the element of area r dr dθ in polar coordinates. The technique is, however, clearly applicable to all functions and boundary conditions of the Sturm-Liouville type and serves to mechanize much of the time-consuming detail associated with the determination of the unknown coefficients in the classical procedure that begins with separation of variables.
3.24 Finite Cosine and Sine Transforms
The simplest finite cosine and sine transforms, introduced originally (1935) by Doetsch, are those for which a = 0, b = π, and p = n in Eqs. (3.56a) and (3.56b), which then reduce to
and
The corresponding transforms of the second derivatives are given by
and
whence these transforms are expedient for problems where f′(x) or f(x), respectively, is prescribed at the end points of the interval. The generalization to an interval of length other than π merely requires a scale transformation.
More general forms of these transforms, corresponding to Fourier’s own generalization of his series, are given by
where either upper or lower alternatives must be taken together. These results are applicable to problems in heat conduction in which radiation takes place at x = l and to problems involving lumped parameters at the boundaries of electrical or mechanical systems, all of which prescribe f′(l) + hf(l).
We remark that h usually is nonnegative in physical problems, by virtue of which Eq. (3.59c) has only real roots, which occur in pairs of equal magnitude and opposite sign, with only the positive values being included in the summation of Eq. (3.59b). But if h is negative, Eq. (3.59c) has a pair of conjugate imaginary roots, one of which must appear in Eq. (3.59b) and accordingly must introduce a hyperbolic function therein; we note, however, that f(x) depends only on k2 and therefore remains real. If h = 0, then the value k = 0 appears as a nontrivial root of Eq. (3.59c) for the cosine transform.
These results all have their counterparts for Hankel transforms. See Ref. 5 or Ref. 7 for h > 0. Hankel transforms for h < 0 do not appear to have been considered in the literature, but the properties of the corresponding orthogonal expansions, the Dini series, are well known; see pages 596ff. of Ref. 20.
3.25 A Problem in Wave Motion
We now consider, by way of illustration, the application of the finite sine transform of Eq. (3.59) to the problem of Sec. 3.14. The prescribed boundary conditions for this problem are on f(0) and f′(1), whence Eq. (3.60) indicates a sine transform with h = 0 and kl an odd multiple of π/2, viz.,
where
Transforming the differential equation (3.37) then yields
to which a solution satisfying the initial conditions Y = Yt = 0 is given by
Substituting this result in Eq. (3.59b) with Y = F, h = 0, and
The terms in the series that are not time-dependent may be identified as the Fourier series representation of Px/E, whence Eq. (3.62) may be reduced to Eq. (3.39). A comparison of the detailed solutions, only outlines of which have been given here, indicates the superiority of the finite-transform method in obtaining the modal expansion.
3.26 Conclusion
The superiority of the finite-transform method, in obtaining solutions as expansions of natural modes, over either the classical procedure of separation of variables or the Laplace transformation tends to increase with the complexity of the problem and is even more evident for the Hankel transform. In particular, the finite-transform method always provides the modal expansion of the static solution, for example, Px/E in the above problem, although this may not always be an advantage. We emphasize, nevertheless, that the Laplace transform is both more flexible and more powerful. Not only does it incorporate alternative interpretations, such as the traveling-wave expansion of Eq. (3.40), but it also places less stringent conditions on the boundary conditions that may be accommodated. To be sure, the expansions provided by Laplace transforms for non-Sturm-Liouville problems might be used to develop finite transforms of a more general type, but such an ad hoc procedure scarcely appears worthwhile.
The properties of several finite transforms are summarized in Table 3.2.
EXERCISES
1. A uniform bar of unit cross section is at rest and unstressed for t < 0 and has free ends at both x = 0 and x = l. When t = 0, a force P is applied at x = l. Use both the Laplace-transform and finite-Fourier-transform methods to show that the subsequent displacement of any section initially at x is given by
where c2 = E/ρ, m = ρl, and ρ and E denote density and Young’s modulus. (Hint: The center of mass of any body, not necessarily rigid, moves as if all external forces were applied there; this component of the motion may be separated out in the finite-transform approach.) Obtain also an expression for the motion in terms of traveling waves.
Table 3.2 Finite Integral Transforms
† Numbers refer to the list of references at the end of the chapter.
2. Let v0 be a function of t in the problem of Sec. 3.15. Use the convolution theorem to obtain the solution
3. Obtain the asymptotic solution to the problem of Sec. 3.15 by expanding Eq. (3.42) in ascending powers of p and inverting term by term.
4. Use the Fourier sine transform to obtain a solution to Laplace’s equation
ϕxx + ϕyy = 0in x > 0 and 0 < y < b
subject to the boundary conditions
Answer:
5. Steady, axially symmetric heat conduction is governed by the equation
Heat is supplied over a circular area on the surface of a semi-infinite solid, x > 0, yielding the boundary condition
Kvx = −Qfor 0 ≤ r < a and x = 0
Show that the steady temperature distribution is given by
6. A slab of infinite area extends from x = 0 to x = 1 and is initially at zero temperature. The temperature of the face at x = 0 is raised to v = v0 when t = 0, while radiation takes place at the other face in accordance with
vx + hv = 0 at x = l
Show that the subsequent temperature is given by
where the k are determined by
k cot kl = −h
Annotated Bibliography on Integral Transforms
a. Texts
1. Carslaw, H. S., and J. C. Jaeger, “Operational Methods in Applied Mathematics” Oxford University Press, New York, 1953. Deals only with the Laplace transform but contains extensive collection of worked and unworked problems involving both ordinary and partial differential equations.
2. Churchill, R. V., “Operational Mathematics,” 2d ed., McGraw-Hill Book Company, Inc., New York, 1958. Elementary text dealing with the Laplace transform, finite Fourier transforms, and, briefly, complex-variable theory. Reasonable mathematical rigor is maintained, but the problems are more elementary than in Ref. 1.
3. McLachlan, N. W., “Complex Variable Theory and Transform Calculus,” Cambridge University Press, New York, 1953. Excellent, albeit specialized, presentation of complex-variable theory required for handling inversion integrals. Uses Heaviside notation for Laplace transforms.
4. Thomson, W. T., “Laplace Transformation,” Prentice-Hall, Inc., Englewood Cliffs, N.J., 1950. Similar in scope to Ref. 2; less rigorous mathematics but more elaborate engineering applications.
5. Tranter, C. J., “Integral Transforms in Mathematical Physics,” Methuen & Co., Ltd., London, 1956. A brief (133-page) but not elementary coverage of all the commonly used transforms with more sophisticated physical examples than Refs. 1–4, above; an excellent supplement to any of Refs. 1–4.
b. Treatises
The following references do not contain exercises for the student but are otherwise more extensive and more advanced than Refs. 1–4, above.
6. Doetsch, G., “Theorie und Anwendung der Laplace-transformation,” Dover Publications, New York, 1943. Originally published by Springer-Verlag, Berlin, Vienna, 1937. Standard mathematical treatise on the Laplace transform; extensive bibliography of pre-1937 works.
7. Sneddon, I. N., “Fourier Transforms,” McGraw-Hill Book Company, Inc., New York, 1951. Extensive applications of various transforms to physical problems at research-paper level.
8. Titchmarsh, E. C., “Introduction to the Theory of Fourier Integrals,” Oxford University Press, New York, 1948. Standard treatise on Fourier, including Fourier-Bessel or Hankel, integrals and transforms; largely complementary to Ref. 6; extensive bibliography of pre-1948 works.
9. Van der Pol, B., and H. Bremmer, “Operational Calculus Based on the Two-sided Laplace Integral,” Cambridge University Press, New York, 1950. A modern, rigorous presentation of Heaviside’s operational calculus as operational calculus. Advanced and stimulating applications in such diverse fields as electric circuits and number theory.
c. Tables
10. Campbell, G. A., and R. M. Foster, “Fourier Integrals for Practical Application,” John Wiley & Sons, Inc., New York, 1948. The most extensive table of exponential Fourier integrals; many entries are effectively Laplace-transform pairs and are presented as such.
11. Erdélyi, A., and J. Cossar, “Dictionary of Laplace Transforms,” Department of Scientific Research and Experiment, Admiralty Computing Service, London, 1944. Most of the material from these tables has been included in Ref. 12.
12. [EMOT] Erdélyi, A., (ed.), with W. Magnus, F. Oberhettinger, and F. Tricomi, “Tables of Integral Transforms,” 2 vols., McGraw-Hill Book Company, Inc., 1954. The most comprehensive tables of integral transforms presently available. Volume 1 contains Fourier-exponential, -cosine, and -sine, Laplace, and Mellin transforms; vol. 2 contains Hankel transforms, along with many transforms not introduced in the foregoing treatment.
13. Magnus, W., and F. Oberhettinger, “Formulas and Theorems for the Special Functions of Mathematical Physics,” Chelsea Publishing Co., New York, 1949. This indispensable (for the applied mathematician) compendium contains short but well-selected tables of both Fourier- and Laplace-transform pairs. Ideal for graduate students.
14. McLachlan, N. W., and P. Humbert, “Formulaire pour le calcul symbolique,” Gauthier-Villars, Paris, 1950. Extensive table of Heaviside (p-multiplied Laplace) transforms.
d. Asymptotic Expansions
15. Copson, E. T., “The Asymptotic Expansion of a Function Defined by a Definite Integral or Contour Integral,” Department of Scientific Research and Experiment, Admiralty Computing Service, London, 1946.
16. Erdélyi, A., “Asymptotic Expansions,” Dover Publications, New York, 1956. The treatments of integrals given in Ref. 15 and in Chap. 2 of Ref. 16 are approximately similar in scope and give a coordinated treatment of material that otherwise can be found only in widely scattered sources.
Other References
17. Barnes, John L., Functional Transformations for Engineering Design, chap. 14 in “Modern Mathematics for the Engineer,” First Series, edited by E. F. Beckenbach, McGraw-Hill Book Company, Inc., New York, 1956.
18. Fourier, Joseph, “La Théorie analytique de la chaleur,” Paris, 1822, translated by A. Freeman, Cambridge, 1878. Reprinted by Dover Publications, New York, 1955.
19. Rayleigh, Lord, “The Theory of Sound,” London, 1894. Reprinted by Dover Publications, New York, 1945.
20. Watson, G. N., “Bessel Functions,” The Macmillan Company, New York, 1948.
† In Heaviside’s form of operational calculus, the right-hand side of Eq. (3.2) appears multiplied by p, but the Laplace form gradually has gained the ascendancy.
† The reader should note that the writer of this chapter is an engineer, sometimes defined as “one who assumes everything but the responsibility.”
† We now use t as the independent variable in recognition of the fact that usually that variable is time; nevertheless, initial-value problems may be encountered in which t is a space variable—e.g., those in linearized, supersonic flow.
† Adumbration might be more accurate than derivation, in that the equivalent of Fourier’s integral theorem appeared as one step in obtaining a general solution to the wave equation. Its use in the present context is, of course, due to Fourier himself.