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Showing papers in "Quarterly of Applied Mathematics in 1972"


Journal ArticleDOI
TL;DR: In this paper, a pair of Gauss-Chebyshev integration formulas for singular integrals are developed and a simple numerical method for solving a system of singular integral equations is described.
Abstract: In this paper a pair of Gauss-Chebyshev integration formulas for singular integrals are developed. Using these formulas a simple numerical method for solving a system of singular integral equations is described. To demonstrate the effectiveness of the method, a numerical example is given. In order to have a basis of comparison, the example problem is solved also by using an alternate method.

1,300 citations


Journal ArticleDOI
TL;DR: A survey of the literature relating to the one-dimensional Burgers equation can be found in this paper, where several distinct solutions of this equation are classified in tabular form and the physically interesting cases are illustrated by means of isochronal graphs.
Abstract: The literature relating to the one-dimensional Burgers equation is surveyed. About thirty-five distinct solutions of this equation are classified in tabular form. The physically interesting cases are illustrated by means of isochronal graphs. Introduction and survey of literature. The quasilinear parabolic equation now known as the \"one-dimensional Burgers equation,\" (du/dt) + u(du/dx) = v(d2u/dx), (1) first appeared in a paper by Bateman [4], who derived two of the essentially steady solutions (1.3 and 1.5 of our Table). It is a special case of some mathematical models of turbulence introduced about thirty years ago by J. M. Burgers [10], [11]. The distinctive feature of (1) is that it is the simplest mathematical formulation of the competition between convection and diffusion. It thus offers a relatively convenient means of studying not only turbulence but also the distortion caused by laminar transport of momentum in an otherwise symmetric disturbance and the decay of dissipation layers formed thereby. Moreover, the transformation u = -(2v/6)(dd/dx) (2) relates u{x, t) and d(x, t) so that if 6 is a solution of the linear diffusion equation (86/dt) = v(d'20/dx2), (3) then u is a solution of the quasilinear Burgers equation (1). Conversely, if u is a solution of (1) then 9 from (2) is a solution of (3), apart from an arbitrary time-dependent multiplicative factor which is irrelevant in (2). In connection with the Burgers equation, transformation (2) appears first in a technical report by Lagerstrom, Cole, and Trilling [38, especially Appendix B], and was published by Cole [21]. At about the same time it was discovered independently by Hopf [30] and also—in the context of the similarity solution u = t~l/2S(z), z = (4vt)~1/2x—by Burgers [14, p. 250]. The similarity form of the Burgers equation—the quasilinear ordinary differential equation for S(z)—is a Riccati equation [51], and can thus be regarded as a basis for motivating transformation (2) inasmuch as (2) is a standard means of linearizing the Riccati equation. More general hydrodynamical applications of this transformation have been discussed by Ames [1, chapter 2], Chu [20], and Shvets and Meleshko [55]. Received February 24, 1971. 196 EDWARD R. BENTON AND GEORGE W. PLATZMAN Another virtue of (1) is that although it appears to lack a pressure gradient term, it is in fact a deductive approximation for propagation of one-dimensional disturbances of moderate amplitude in a uniform diffusive compressible medium, if the variables are interpreted in a particular way. In this \"acoustic\" analogy for an ideal gas, the u, v and x, t of (1) are, respectively, Kt + i)u', I [si'' + v'. + (t lVL ^ x' — a0t', t', where u' is the fluid velocity, v' and v[ are respectively the kinematic shear and bulk viscosities, k' is the thermal diffusivity, 7 the specific-heat ratio (assumed constant), x' is an axis fixed in the undisturbed medium, i! is the actual time, and a0 is the speed of a linear sound wave. According to this analogy, t in (1) is the actual time and x is position in a coordinate frame moving with speed a0 relative to the undisturbed medium; thus du/dt in (1) expresses the relatively slow wave-form distortion caused by convection and diffusion, rather than the comparatively fast local changes associated with ordinary propagation. The restriction to moderate amplitude is necessary because the analogy depends upon approximations valid only for small Mach number of the disturbance. Moreover, the analogy is derivable only when the propagation is unidirectional, as typically for a \"simple\" wave or disturbance that moves into an infinite resting medium. Subject to the same restriction, an alternative statement of the acoustic analogy is to take u, v and x, t in (1) as [43], -1(7 + IK, *[V + V. + (7 iy], (5) a0t' — x', x'/a0. The first term of (1) then expresses spatial variation in the translating frame, variation which is equivalent to the temporal changes of the first analogy and corresponds to the same effects of convection and diffusion. Analogy (4) is convenient for initial-value problems, since when t' = 0 we have (x, t) = (x', 0) in this analogy; whereas (5) is convenient for boundary-value problems, since then (x, t) = (a0t', 0) when x' = 0. With varying degrees of rigor the derivation of the one-dimensional Burgers equation from the fundamental gas-dynamic equations under the restrictions inherent in the acoustic analogy has been accomplished for viscous, non-conducting ideal gases by Lagerstrom, Cole, and Trilling [38, Appendix B], and by Su and Gardner [59]; for viscous, nonconducting fluids with quadratic dependence of pressure on rate of expansion by Mendousse [43]; for viscous conducting ideal gases by Lighthill [40] and by Soluyan and Khokhlov [58]; and for viscous, conducting fluids of general equation of state by Hayes [29]. The equation also describes finite-amplitude transverse hydromagnetic waves [27], longitudinal elastic waves in an isotropic solid [48], and disturbances on glaciers [39]. An equation related very simply to the Burgers equation arises in a problem of number theory [62], In a remarkable series of papers extending over many years, Burgers (see references) studied statistical and spectral aspects of the equation (and related systems of equations) when initial conditions are given stochastically. Various aspects of the energy spectrum have also been investigated by Reid [49], Ogura [45] and Tatsumi [60]. More recently, the new deductive theories of turbulence have been \"tested\" on the Burgers model SOLUTIONS OF THE ONE-DIMENSIONAL BURGERS EQUATION 197 (see [23], [32], [33], [36], [41], [42], [46], [56], [57]) and various numerical experiments have been made on the Burgers equation (for example, see [5], [26], [31]). Saffman [54] has questioned the basis of the Kolmogorov law by using results derived from the Burgers model. The Burgers equation gives an analytic framework for a second-order theory of finite-amplitude dissipative sound propagation [8], [9], [34], [35], [43], [58], It has been used in discussions of shock structure in a Navier-Stokes fluid principally by Lagerstrom, Cole, and Trilling [38], by Lighthill [40] and by Hayes [29], In the notable work of Lighthill, the conflict between the steepening effect of nonlinear convection and the broadening trend of dissipation is made especially clear; this dual process is the essence of the Burgers equation. One of the most interesting solutions of the Burgers equation, the only known exact time-dependent spectral solution (2.6 in the Table and Figure 9), appears first in a paper by Fay [24] where it was derived in the acoustic framework but without the aid of the Burgers equation and with the role of space and time inverted as in (5) (as pointed out by Rudnick [53], there is a minor error in Fay's Eq. (14): the correct numerical factor is 2, not 8). The Fay series was re-discovered by Cole [21] as an approximate solution of the Burgers equation for a sinusoidal initial condition, and by Benton [6], [7] as an exact solution. The relation of Fay's solution to the corresponding in viscid spectral solution of (1), u(x, t) = —2 ^ (nt)~lJn(nt) sin nx, 71= 1 is thoroughly discussed by Blackstock [9] in connection with the sound field generated by sinusoidal motion of a one-dimensional piston (see also [2], [34], [52], [65]). This inviscid solution is known as the Fubini solution in the acoustics literature because of the work of Fubini-Ghiron [25]. It has been rediscovered by workers in several different fields (see [22], [28], [37], [44], [47], [66]). Description of table. The correspondence between (1) and (3) through (2) makes it easy to construct exact solutions of (1) by starting from solutions of (3). Although the general solution of (3) is known for arbitrary initial conditions, and the transformation (2) is trivial to perform, not all special solutions are physically \"interesting.\" It seems worthwhile, therefore, to call attention to those that are. The purpose of the following Table is to present a list of such solutions, arranged in a somewhat systematic way, as a possible aid in further investigations of the Burgers equation. Some new solutions are included, but we have primarily aimed at collecting and organizing numerous results scattered through a somewhat diffuse literature. Eqs. (1, 2, 3) are invariant to a shift of origin x — x0 —> x, t — t0 —> t; u —> u, d —» d, (6) where x0 and t0 are arbitrary, independent constants. They are also invariant under a change of scale: x/a —> x, t/a—*t; alt —> u, /3d —» 0, (7) where a and (i are arbitrary, independent scale factors. Special cases of (7) that are useful in constructing the Table are — x—> x, t —> t; —u —> u (7a) 198 EDWARD R. BENTON AND GEORGE W. PLATZMAN which reverses the direction of the z-axis, and ix —> x, — t —> t; —iu —» u (7b) which rotates the z-axis 90 degrees in the complex z-plane. The third important invariance transformation, x — Ut •—> x, t —> t) u — U ■—> Uj d exp (U(x — Ut)/2v) —> 6, (8) represents translation of the reference frame at the constant speed U (Galilean invariance). In the Table we use nondimensional variables obtained by making the substitutions x/L —> x, vt/L2 —» t) uL/v —> u. (9) The insertion of v here formally has the effect v —> 1 in (1, 2, 3), whereas the length scale enters as in (7) and thus does not alter the equations. It should be noted that in many cases L is not an external parameter (for example, see Note 1.3). If two solutions are related through one or more invariance transformations, we say they are \"equivalent\" (for example, see Note 1.0 in the Table). If all the transformations in question are real, they do not alter the shape of the function on which they operate; we therefore say that such solutions are \"isomorphic\" (for example, Note 1.0). In the Table we list only real solutions (some of which are equivalent through complex transformations such as (7b)), and in the Not

315 citations



Journal ArticleDOI
TL;DR: The results of Hakimi and Yau and others in the realization of a distance matrix are generalized to graphs (digraphs) whose branches (arcs) may have negative weights as mentioned in this paper.
Abstract: The results of Hakimi and Yau and others in the realization of a distance matrix are generalized to graphs (digraphs) whose branches (arcs) may have negative weights. Conditions under which such matrices have a tree, hypertree or directed tree realization are given, uniqueness of these realizations is discussed and algorithms for their construction are indicated. 1. Notation. A number of definitions are given so that results will be presented in a self-contained manner. A graph G = (V, B) consists of a finite non-empty set V = ••• , vn\\ of vertices and a set 5 = {bt ,b2, • ■ ■ ,bm} of unordered pairs of distinct vertices of V. Each such pair bk = e(i\\ , v,) of vertices is a branch of G and is said to be incident at both v{ and y,. A directed graph or digraph G = (F, A) consists of a finite non-empty set V = {f'i , v2 , • • • , vn] of vertices and a set A = {al , a2 , ■ ■ ■ , am] of ordered pairs of distinct vertices of V. Each such pair ak — e(v< , v,) of vertices is an arc of G, is directed from v{ to v, and is incident at both v, and v, . A subgraph of a graph (digraph) G is a graph (digraph) which has all its vertices and branches (arcs) in G. The degree of a vertex Vi in G, denoted deg (y, , G), is the number of branches (arcs) incident at i>, in G. The outdegree of a vertex v{ in digraph G, denoted outdeg(y, , G), is equal to the number of arcs incident at v, in G and directed away from v{. The indegree of Vi, denoted indeg(z\\-, G), is equal to the number of arcs incident at v{ in G and directed towards v{ . A weighted graph (digraph) is a graph (digraph) together with a function which assigns a real number wx to each branch h, (arc a,). All graphs (digraphs) presented here are weighted. An edge-sequence in a graph (digraph) between two vertices v, and vf is an alternating sequence of vertices and branches (arcs) • • • M,beginning and ending with Vi and Vj , in which each branch (arc) is incident at the vertex preceding and the vertex following it. A path from to v,is the set of all branches (arcs) in an edge-sequence between v{ and v, . A directed path in a digraph is a path in which each arc is directed from the vertex preceding it to the vertex following it in the corresponding edge-sequence. A path or directed path is called elementary if all vertices in the edge-sequence are distinct. A path (directed path) is a circuit {cycle) if the first and last vertex in the edgesequence are the same and all others distinct. The length of a path (directed path) is the sum of the weights of the branches (arcs) in it. A connected graph (digraph) is a graph (digraph) in which every pair of vertices is joined by a path. * Received July 13, 1971; revised version received October 9, 1971. This work was supported by the USAF Office of Scientific Research under Grant AFOSR-71-2103. 256 A. N. PATRIXOS AND S. L. HAKIMI A tree (directed, tree) is a connected graph (digraph) containing no circuits and thus any two vertices are joined by a unique elementary path. In directed trees not all vertices are joined by a directed path. In fact, if a directed path exists from t>< to , there is no directed path from i», to v, . In a tree, if each branch is replaced by two oppositely directed arcs, the digraph so constructed is a hypertree. Two such arcs in the hypertree form an elementary -pair, and the sum of the weights of the two arcs is the weight of the elementary pair. In a hypertree there is a unique elementary directed path between any pair of vertices. The distance d(v{ , y,-) of vertex t>,from vertex vt in a graph (digraph) is the length of a shortest (i.e. minimum sum of weights) elementary path (directed path) from v{ to v,-. Clearly d(v{ , vf) = d(Vj , i\\) in graphs while in general d(v> , vf) ^ d{pi , v{) in digraphs. We also have d(v{ ,v{)=0 and d(v{, v,) = if there is no (directed) path from v{ to v,. A distance matrix D(V0 = [djy] of a graph (digraph) G = (F, B), Vi = [vkl , vk%, ■ ■ • , C7,isannXn matrix in which entry rf,-, (i, j = 1,2, , n) is the distance of vertex vkj from vertex vki(vti , vkj £ Fj). If v £ Fi then v is an external vertex, otherwise an internal vertex. Any vertex of degree one in a graph (digraph) is a terminal vertex. By a realization of an n X n matrix I) = [d,,] we mean a graph (digraph) G — (F, B), |F| > n, such that for some Fi Q V, |F,| = n, we shall have D(V,) = D. All graphs (digraphs) have distance matrices, graphs having symmetric ones and digraphs, in general, asymmetric ones. All entries in the distance matrix of a connected graph are finite. Connected digraphs may have infinite entries. A branch (arc) in a graph (digraph) is redundant if its removal results in a graph (digraph) with the same distance matrix. An internal vertex v in a graph (digraph) is redundant if it has deg (v, G) < 3 (indeg (v, G) < 2 or outdeg (v, G) < 2). The nullity of a connected graph (digraph) is equal to \\B\\ — |F| + 1 (|A| — |F| + 1) and thus is zero if the graph (digraph) is a (directed) tree. 2. Introduction. Given a weighted graph (digraph), algorithms are available for computing the distance matrix D of (a subset of) its vertices. Of these, the most efficient is due to Floyd [1], The algorithms fail, in general, if the graph has a branch with a negative weight or if the digraph has a cycle whose length is a negative number. A number of papers have also been published on the realizabilitv of a given n X n matrix D by a graph (digraph). Hakimi and Yau [2] gave necessary and sufficient conditions for an»X» symmetric matrix D with non-negative entries to be the distance matrix of a graph. They defined as 'optimum' that realization which has a minimum total sum of weights and proved that a tree realization, if one exists, is the unique optimum realization. Goldman [3] and Murchland [41 extended some of these results to digraphs. Generalizing the above results, we have proved that any (symmetric) square matrix with zero diagonal elements is the distance matrix of some (graph) digraph. Zaretskii [5, 6] gave necessary and sufficient conditions for the existence of a unique unweighted tree with n terminal vertices whose distance matrix equals a given matrix of order n. Simoes-Pereira [7] gave, without proof, a weaker statement of Theorem 2 presented in this work. Theorem 2 also provides a generalization of Zaretskii's results to the weighted case. Boesch [8], considering strictly non-negative weighted graphs, gave some properties of the distance matrix of a tree and suggested two algorithms for a tree realization. We indicate here that one of these algorithms (the one derived from theorem II of his work) can be successfully used in the general case. Shay [9] introduced the 'hypertree' and gave a necessary condition for its realization. We have completed his THE DISTANCE MATRIX OF A GRAPH 257 work on the hvpertree. Finally, we attacked the case of the distance matrix and its realization as a directed tree. Almost all previous work restricted itself to nou-negative entries in the given matrix D and non-negative weights in its realization. We have placed no such restriction in this work and admitted negative weights. Further, we introduce non-redundant internal vertices if they permit a realization which could not have been achieved otherwise. With these considerations in mind, the objectives of this paper are: 1. to find necessary and sufficient conditions for a given matrix to have a tree, hypertree or directed tree realization, 2. to find whether the above realizations are unique, 3. to indicate algorithms for construction of the tree, hypertree or directed tree if such realizations exist. 3. The distance matrix and its realizability. Given an n X n symmetric matrix D = [tZ,,], necessary and sufficient conditions for the existence of an ?;-vertex graph G with non-negative weights having D as its distance matrix were given by Hakimi and Yau [2], Specifically: 1) da = 0 for all i, 2) da + djk > dik for all i, j and k. The ?i-vertex graph G realizing I) can be constructed as follows: pick n vertices, labeling them Vi ,v2, ■ ■ ■ ,vn, and for every entry diS (i ^ j) of D draw a branch e(v{ , Vj) assigning to it the weight . Since du + dik > dik and dki + d,, > dki for any i, j and k we have du > 0. This implies that D must contain strictly non-negative entries. If no restriction is placed on the type of weights in a realization of D then we can state: Theorem 1: Any n X n symmetric matrix D = [fi,,] with zero diagonal elements is a distance matrix of some graph G. Proof: Consider the n X n matrix D' = [(/',], where d'a = dtj + a,+ a,if i ^ j = 0 if i = j and ak = max (max \\{d,, — (drk + dk.)), \\r, s = 1, 2, n If drk + dk, > dr, for some r, k and s then clearly d'rk + d'k, > d'r, . If drk + dk, < dr, for some r, k and s then 0 < §(c/r. — (dTk + dk,)) < ak , or (dr, + ar + a.) < (drk + ar + ak) + (dk, + ak + a,), or d'r, < d'rk + d'k, . Thus there exists an n-vertex graph G' which has D' as its distance matrix. Let G' be such a graph and v[ , v'2 , • • • , v'n be its vertices. To construct a graph G with distance matrix D add to G' vertex v, and connect it with v'i through a branch e(v,-, v[) of weight —a,(for all i). We have in G: d{v{ , v,) = d(vi , v') + d(v'{ , v'^ + d(v\\ , = — a{ + d'f — a, = du (for all i and j). This proves the theorem. 4. The distance matrix and the tree realization. By virtue of Theorem 1 let us call a symmetric matrix D with zero diagonal elements a 'distance matrix'. We have proved that there exists at least one graph G realizing distance matrix D. If G[D] is the set of all 25S A. N. PATRINOS AND S. L. HAKIMI graphs realizing D we would like to know if there exists a tree I such that I £ G[D\\. The following theorem provides an answer to this question. It was given by SimoesPereira [7] in a slightly weaker form. The interested reader should also consult Zaretskii [5, 6] for the unweighted case. Let us set

72 citations



Journal ArticleDOI
TL;DR: In this paper, the wave motion resulting from a line load moving through an unbounded elastic solid is considered, and exact expressions are derived for the displacement potential functions for all places in the body, for all time, by Laplace transform methods.
Abstract: The wave motion resulting from a line load moving through an unbounded elastic solid is considered. The line load suddenly appears in the body, and then moves in a fixed direction with nonuniform speed. Exact expressions are derived for the displacement potential functions for all places in the body, for all time, by Laplace transform methods. The wavefronts, particularly the Mach waves trailing the moving load, are determined by integrating the bicharacteristic equations. The details of the wavefront shapes are illustrated and discussed for a load accelerating or decelerating in a prescribed way. Finally, a sample first-motion calculation is carried out in which the mean normal stress is determined immediately behind the dilatational wavefront by asymptotic methods. Introduction. The problem of determining the response of an elastic solid to moving loads has been of interest for some time. The motivation for studying problems of this kind has been provided by the fields of structural dynamics, seismology, and related areas. The particular moving-load problems which have been studied may be put into one of three categories, according to the way in which the position of the load depends on time. The three categories are: (i) steady wave motion due to a load moving with constant velocity for all time, (ii) transient wave motion due to a load which begins to act at a certain instant and then moves with constant velocity, and (iii) transient wave motion due to a load which begins to act at a certain instant and then moves in some direction with time-dependent speed. The problem considered here is of the third kind. Representative of the first kind of problem is the steady motion of a line load on the surface of an elastic half-space, which was studied by Sneddon [1] and Cole and Huth [2]. The three-dimensional problem, in which the steady motion of a point force in an unbounded body is studied, was considered by Eason, Fulton and Sneddon [3], The transient problem of a line load which suddenly appears on the surface of an elastic half-space and then moves off with constant velocity, which is typical of type (ii), was considered by Ang [4] and by Pay ton [5]. Both employed integral transform methods and a version of the Cagniard-de Hoop [6] technique. The corresponding three-dimensional problem of a pressure-spot suddenly appearing on the surface of an elastic half-space and then moving with constant velocity was considered by Payton [7]. A number of other references to problems of the first and second kinds may be found in the paper by Gakenheimer [8]. * Received August 5, 1971; revised version received October 9, 1971. The support of the National Science Foundation through grant GK26002X to Brown University is gratefully acknowledged.

27 citations


Journal ArticleDOI
TL;DR: In this article, the Cauchy index of a real rational function can be computed by evaluating the signature of a certain Hankel matrix, and alternative procedures for its computation are presented, one of which offers greater computational simplicity.
Abstract: The Cauchy index of a real rational function can be computed by evaluating the signature of a certain Hankel matrix. Alternative procedures for its computation are presented here, one of which offers greater computational simplicity.

27 citations


Journal ArticleDOI
TL;DR: In this paper, a variational statement for large elastic deformations at finite strains is given, which involves independent variation of the displacement, the (nonsymmetric first Piola-Kirclihoff) stress, and the deformation-gradient fields, and which includes both the boundary and the jump conditions.
Abstract: With a view toward a numerical solution by means of the finite-element method, we give here a variational statement for large elastic deformations at finite strains which involves independent variation of the displacement, the (nonsymmetric first Piola-Kirclihoff) stress, and the deformation-gradient fields, and which includes both the boundary and the jump conditions. Then we present, for small deformations superimposed on the large, three variational statements, each involving three independent fields and each including both the boundary and the jump conditions. These statements are such that the first variation of the corresponding functional yields the field equations which characterize the equilibrium of the finitely-deformed state considered and also the field equations that pertain to the incremental deformations. Several specializations of these results are discussed. By way of illustration, finally, we present a finite-element formulation of the large deformation problem, using three independent fields, where each field is approximated by a piecewise-linear function within each element.

25 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived an expression for the mean square error in stresses obtained from the solution of a boundary-value problem in the classical theory of plates with respect to the solution in the theory of elasticity.
Abstract: Introduction. In a recent paper [1] we derived an expression for the mean square error in stresses obtained from the solution of a boundary-value problem in the classical theory of plates with respect to the solution of a corresponding problem in the theory of elasticity. Further, the relative mean square error was bounded by a quantity proportional to the plate thickness. The derivation in [1] employs the hypersphere theorems of Prager and Synge [2], [3] in elasticity which lead to the equations of plate theory via energy minimization techniques. Subsequently, Simmonds [4] bounded the relative mean square error by a quantity proportional to the plate thickness squared for isotropic plates. The derivation of this improved bound is based on a direct application of the main hypersphere theorem [2] with a specially constructed kinematically admissible stress field of a more elaborate form than in [1], In the present note we derive an expression for the mean square error and a bound on the relative error for Reissner's theory of plates [5] which includes the effect of shear deformation. The derivation is similar to that of [1], although we employ a kinematically admissible stress field of nearly the same form as in [4]. The relative error is again bounded by a quantity proportional to plate thickness squared. This lack of essential improvement over the error bound for classical theory is not surprising in view of the well-known fact that Reissner's theory offers improvement over classical theory only near the edge of the plate. Of particular interest is the value obtained here for the numerical constant in the constitutive equation for the shear stress resultant, namely f. This value is in agreement with Reissner's original derivation [5] and most subsequent derivations,1 e.g., the direct derivation of Green [7]. In the present derivation the value f follows from minimization of both the potential energy and the complementary energy whose sum is the mean square error in stress [1], [3], Therefore, f can be regarded as the best value for the shear constant within the context of mean square error minimization for static problems. Also, we find that the constitutive equation for stress couple need not include a term proportional to the normal surface load, as in [5] and [7], Finally, the error bound of Simmonds [4] for classical theory of isotropic plates is extended to anisotropic plates with midsurface elastic symmetry. For brevity we do not repeat Sees. 1 and 2 of [I] which deal, respectively, with function space concepts in elasticity and the statement of a class of boundary value

21 citations



Journal ArticleDOI
TL;DR: In this article, a number of optimization problems are posed and solved for supersonic aircraft flight subject to the condition that a shock wave appears only incipiently in the sonic boom signal at a given point.
Abstract: A number of optimization problems are posed and solved for supersonic aircraft flight subject to the condition that a shock wave appears only incipiently in the sonic boom signal at a given point. The principal result is one giving the maximum effective gross weight of an aircraft of given effective length under given flight conditions. The calculus of variations with inequality constraints is used, with the novel features of a non-local isoperimetric relation and of only an upper bound on a control variable.

Book ChapterDOI
TL;DR: In this paper, Truesdell and Toupin proposed the principle of equipresence, which states that a variable present as an independent variable in one constitutive equation should be so present in all.
Abstract: In 1960, Truesdell and Toupin [1] proposed that in formulating constitutive equations for phenomenological theories in which there is more than one constitutive equation, the following principle, which they named the Principle of Equipresence, must be satisfied: A variable present as an independent variable in one constitutive equation should be so present in all. In explaining the application of this principle, they state “Let it not be thought that this principle would invalidate the classical separate theories in the cases for which they are intended, or that no separation of effects remains possible. Quite the reverse: The various principles of invariance, stated above, when brought to bear upon a general constitutive equation have the effect of restricting the manner in which a particular variable, such as the spin tensor or the temperature gradient, may occur. The classical separations may always be expected, in one form or another, for small changes—not as assumptions, but as proven consequences of invariance requirements. The principle of equipresence states, in effect, that no restrictions beyond those of invariance are to be imposed in constitutive equations”.

Journal ArticleDOI
TL;DR: In this paper, the authors develop formal asymptotic series to represent the inflated membrane and find the series are power series in the parameter p~2 and that the lowest-order term verifies the formula in [1].
Abstract: characterizing the meridian curve of a closed membrane of revolution at its undeformed state. By a closed membrane we mean that the curve C, together with the axis of symmetry, r — 0, encloses a simply connected domain. Suppose the membrane, composed of a Mooney material [2], is inflated by a properly nondimensionalized pressure p. Isaacson [1] has shown that as p —> the shape of the inflated membrane becomes spherical. We develop formal asymptotic series to represent the inflated membrane and find the series are power series in the parameter p~2 and that the lowest-order term verifies the formula in [1]. The inflated membrane approaches a spherical surface of radius R„ asymptotically as p —> °°, with




Journal ArticleDOI
TL;DR: In this article, a modified version of the geometric optics asymptotic expansion, involving a Bessel function, is given for the fundamental point source solution, which is proven to be uniformly valid in the large, until a caustic is reached.
Abstract: The physical problem of steady-state heat conduction in a thin shell is described by the \"reduced wave equation\" in which the differential operator is the (generally noneuclidean) Laplacian for the surface. A similar equation gives the approximation for steady-state waves in a prestressed curved membrane. A modification of the \"geometric optics\" asymptotic expansion, involving a Bessel function, is given for the fundamental point source solution. This is proven to be uniformly valid in the large, until a \"caustic\" is reached. Various features of the solution for a surface, which do not occur for the plane, are discussed.


Journal ArticleDOI
TL;DR: This article made some observations on the teaching of applied mathematics, both to nonmathematicians and to mathematicians, as they grew out of my experiences in teaching the subject, both in this country and in Switzerland.
Abstract: In my talk today I shall make some observations on the teaching of applied mathematics, both to non-mathematicians and to mathematicians, as they grew out of my experiences in teaching the subject, both in this country and in Switzerland. Hardly anything of what I shall say is especially new or startling, and I am sure that many experienced teachers will agree with most of my points. But to pass on even ordinary fruits of experience may be of some benefit to a younger generation. Let me begin with some statistics. In the academic year 1970-71, the number of first year students at the ETH (Eidgendssische Technische Hochschule at Zurich) who took some kind of instruction in mathematics was 1308 [7]. The distribution of these students among the various fields was as follows:

Journal ArticleDOI
TL;DR: In this paper, a review of the work carried out at the Weizmann Institute during the past 25 years in the fields of terrestrial spectroscopy, the dynamo theory of the earth's magnetic field, the tides in the world oceans, theoretical seismograma, hydrodynamynamynamics, atomic Spectroscopy and the Boltzmann integral equation is given.
Abstract: A review is given of the work carried out at the Weizmann Institute during the past 25 years in the fields of terrestrial spectroscopy, the dynamo theory of the earth's magnetic field, the tides in the world oceans, theoretical seismograma, hydrodynamic stability, atomic spectroscopy, and the Boltzmann integral equation. Some open problems in the solution of the Schrodinger wave equation are formulated.


Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of a two-mode discontinuity on a dielectric slab, where one half of the structure is allowed to be slightly thick while the other half is perfectly conducting.
Abstract: The phenomenological theory of multi-mode surface wave propagation is applied to a plane structure having a multi-mode discontinuity in impedance. The resulting boundary-value problem is reduced to the solution of a Wiener-Hopf equation whose factorization is given in terms of the factorization that occurred in the one-mode case. Despite the complexity of the solution, the magnitudes of the surface wave excitation coefficients are elementary functions, as is the cylindrical power flow. On computing the power flow through the impedance surface, a definition of subsurface power flow \"inside\" the structure is suggested. The form can be taken so that the concept of modal power flow separability is maintained wherever the associated exterior field is primarily that of surface waves. It is further observed that without consideration of this term, power flow coupling occurs. Analogous results appear in the exact case of a dielectric slab having multiple simultaneously propagating surface wave modes. Lastly, conservation of power is verified by actual evaluation of the closed contour integral used to define the various components of power flow (surface wave, cylindrical, junction, and boundary). In fact, the following physical interpretation can be made for the magnitudes of the respective power flow distributions: the incident surface wave power (including the associated subsurface power) is equal to the excited surface wave power (including the associated subsurface power) plus the cylindrically radiating power. Introduction. Most exact problems of electromagnetic wave propagation are intractable. As a result, numerous approximate methods have been devised to discuss various aspects of the phenomena. In certain problems involving surface waves, it is known that this feature may be investigated by replacing the details of the structure with an impedance boundary condition. The class of problems to which we direct ourselves is characterized by discontinuous plane structures (infinite in extent) having the capability of supporting several surface wave modes. The orientation and geometry are such that the electromagnetic field produced is determined by solving a two-dimensional problem and all field components are derivable from a single unknown scalar wave function u(x, y). The plane structure will be replaced by the boundary conditions put forth by Karp * Received September 18, 1971. The research reported in this paper was supported partially by St. John's University and partially by the office of Naval Research under Contract No. N00014-67A0467-0075. Reproduction in whole or part permitted for any use of the U. S. Government. 300 R. C. MORGAN AND S. N. KARP and Karal [1], [2], These conditions have the form of generalizing the classical impedance condition to products of this type. Furthermore, they are derivable by approximating the reflection coefficient due to a plane incident wave (which may be known theoretically or experimentally). Previously, these conditions were applied to continuous structures in the papers [2], [3] and [4]. A discontinuity in the boundary condition for a two-mode problem on a right-angled wedge was treated in [5]. This paper will be directed toward solving the resulting mathematical problem of a two-mode discontinuity on a plane structure. A similar single-mode discontinuity problem was discussed by Kay [6]. It is interesting that despite the complexity of the solution, the physically important magnitude of the surface wave excitation coefficients is easily obtained (Kay observed this in the single-mode case). We take as our incident field an appropriate surface wave, and for simplicity, one half of the structure is allowed to be slightly thick (see [4]) while the other half is perfectly conducting. The various components of the power flow above the structure are computable by choosing an appropriate contour in physical space. In particular, on computing the power flow through the impedance surface, we are led to a definition of power flow inside and down the structure. The form is such that the concept of modal power flow separability is maintained wherever the associated exterior field is primarily that of surface waves. It is well known that, in the exactly solvable case of an infinite dielectric slab having simultaneously several propagating surface wave modes, surface wave modal power flow separability results if the power flow across a plane perpendicular to the slab is computed by including the associated power flow inside. Otherwise, there is in general a coupling of power when it is computed only above the slab. This phenomenon is observed also in our impedance model. For the case of the single real impedance, a consideration of this type does not occur because power is not transferred across the impedance surface and the model is one for which the power flowing inside the structure is negligible. In the case of the dielectric slab mentioned above, this corresponds to a slab of vanishing thickness. However, for a slab of finite thickness, the power flow inside may be significant. Thus, by our model, we provide an approximation that accounts for such power flow while the structure is also discontinuous. The exact problem of such a discontinuous structure has not been solved. However, we point out that the phenomenological theory is broader in context than the problem of the dielectric slab. Thus, we conjecture that our results herein are also. As a partial verification of our computations, conservation of power is checked for the contour used to define the various component field power distributions (i.e., surface wave, cylindrical and boundary). It is shown that the surface wave power (including that amount associated with flow inside the boundary) exactly cancels out the cylindrically radiated flow. The remaining boundary terms give zero power flow. As a result, we can state the following physical theorem for the magnitudes of the power flow distributions: the incident surface wave power flow (including \"inside\" term) is equal to the excited surface wave power flow (including \"inside\" term) plus the cylindrically radiated power flow. The boundary-value problem formulation is modeled after that used previously in proving uniqueness for a multi-mode right-angled wedge problem [11]. In addition, the junction condition assumed arises naturally upon reducing the problem to the solution of a Wiener-Hopf equation. Here the factorization is given in terms of the factorization that occurred in the one-mode case. SURFACE WAVE INCIDENCE ON A PLANE STRUCTURE 301 1. B.V.P. formulation and solution. The problem (see Fig. 1) we shall solve is posed by the following conditions: (i) (A + k)u{x, y) = 0, all x, y > 0 (ii) (d/dy + Xi)(d/dy + X2)w = 0, x < 0, y = 0, where Xi , X2 are taken as real positive constants. (iii) u = 0, x > 0, y = 0. (iv) u and its derivatives satisfy d'u (a) X) Z dx' ' dy' < M for r > R0 , where M is independent of r and 8 and R0 is some positive constant, (b) d2u/dy2 is integrable at the origin. (v) 11 ^incident *~i~ Wexcjted ~i~ Wradiftted winc. = A exp I — \\^y + i(k2 + X?)1/2a:], x < 0, y > 0, = 0 , x > 0, y > 0, and 2 w«=ited = X) Cm exp [-X„2/ i{k2 + X2)1/2z], x < 0, y > 0, m -1 = 0 , x > 0, y > 0. Here A represents the given incident surface wave amplitude and the C„ are constants that must be determined. (vi) wauled = u — wiD0. — uCTc,ted obeys the radiation condition lim \\/r (dWrad.M\" — ikurlli) = 0 r—*co uniformly in 6, 0 < 6 < tr. Following a method employed by Kane and Karp [7] and Kane [8], the solution is reduced to solving a Wiener-Hopf problem. The critical factorization required is that of the function x, , x.) f[ [l + ^ i->y,] (l.l)



Journal ArticleDOI
TL;DR: In this paper, a conformal mapping of the flow region onto the interior of a semicircular region is restricted to flows with at least one stagnation point, and the model of a single vortex with circulation such that the force on the vortex is upward is useful in the description of some of the features of a flow past a submerged lifting body; this flow is adequately described by Gurevich's solution.
Abstract: Recently Gurevich [1] found an exact solution for the problem posed by the plane flow past a vortex of an inviscid, incompressible fluid which is bounded above by a free surface and below by a horizontal wall. This solution, which is found by using a conformal mapping of the flow region onto the interior of a semicircular region, is restricted to flows with at least one stagnation point. The model of a single vortex with circulation such that the force on the vortex is upward is useful in the description of some of the features of the flow past a submerged lifting body; this flow is adequately described by Gurevich's solution. For the case in which the force on the vortex is downward, Gurevich's solution has been extended to include flows without stagnation points but with bifurcation points on the free surface which correspond to singular points of the mapping. This extension describes the flow of a vortex lowered into the fluid from above.

Book ChapterDOI
TL;DR: In this article, it was shown that the complex slowness of a sinusoidal wave propagated in an isotropic nonlinear viscoelastic material subject to a static pure homogeneous deformation is independent of the orientation of the direction of propagation in the principal plane.
Abstract: A small-amplitude plane sinusoidal wave is propagated in an isotropic nonlinear viscoelastic material Subjected to a Static pure homogeneous deformation. The wave is polarized along one of the principal directions for the pure homogeneous deformation. The normals to the planes of constant phase and amplitude for the wave are perpendicular to each other and lie in the principal plane normal to the direction of polarization. It is found that the complex slowness for such a wave is independent of the orientation of the direction of propagation in the principal plane. The three complex slownesses corresponding to a particular class of waves of this type polarized along the three principal directions satisfy a relation which is independent of the detailed form of the constitutive equation.



Journal ArticleDOI
TL;DR: In this paper, it was shown that Moseley's method applies in greater generality to any linear partial differential equation, and, indeed, to linear operator equation, where L is a linear operator and w is any solution to the equation.
Abstract: In [1] Moseley derived some nonseparable solutions of the Helmholtz equation, and in [2] he demonstrated how these solutions could be generated by applying a certain differential operator to a separable solution. In the present paper we point out that Moseley's method applies in greater generality to any linear partial differential equation, and, indeed, to any linear operator equation. Suppose L is a linear operator and that w is any solution to the equation