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Showing papers in "Quarterly of Applied Mathematics in 1952"



Journal ArticleDOI
TL;DR: In this paper, the safe loads for a Prandtl-Reuss material subject to surface tractions or displacements which increase in ratio are extended to any perfectly plastic material and any history of loading.
Abstract: Earlier results [1,2]2 on safe loads for a Prandtl-Reuss material subject to surface tractions or displacements which increase in ratio are here extended to any perfectly plastic material and any history of loading.

636 citations



Journal ArticleDOI
TL;DR: In this article, a treatment of the Brownian motion in velocity space of a particle with known initial velocity based on Boltzmann's integral equation is given, which employs a suitable scattering kernel, and its solution compared with that of the corresponding Fokker-Planck equation.
Abstract: Abstract. In order to describe Brownian motion rigorously, Boltzmann's integral equation must be used. The Fokker-Planck type of equation is only an approximation to the Boltzmann equation and its domain of validity is worth examining. A treatment of the Brownian motion in velocity space of a particle with known initial velocity based on Boltzmann's integral equation is given. The integral equation, which employs a suitable scattering kernel, is solved and its solution compared with that of the corresponding Fokker-Planck equation. It is seen that when M/m, the mass ratio of the particles involved, is sufficiently high and the dispersion of the velocity distribution sufficiently great, the Fokker-Planck equation is an excellent description. Even when the dispersion is small, the first and second moments of the Fokker-Planck solution are reliable. The higher moments, however, are then in considerable error—an error which becomes negligible as the dispersion increases. 1. In the treatment of Brownian motion, it is customary to assume a Langevin equation and simple dynamical statistics of the individual collisions and then to deduce a Fokker-Planck equation describing the random motion of the heavy particle. The Fokker-Planck equation obtained is a second-order partial differential equation and the absence of higher-order differential terms is inferred directly from the above assumptions. As will be seen, the solution of this Fokker-Planck equation does not provide a completely satisfactory physical description. Consequently, the assumptions underlying the equation cannot be correct [1, 2, 3, 4] and the extent of their approximate validity comes under question. That the solution of the Fokker-Planck equation is not a wholly satisfactory representation of Brownian motion may be seen in the following way. Consider a heavy particle known to have the velocity v0 at t — 0. For all subsequent time, there is a finite probability that the particle will have undergone no collision. It must, therefore, be expected that the probability density w(v, t)** describing the stochastic motion in velocity space wall always have a singular component of the form f(t)d(v — v0), where <5(v — v0) is the Dirac delta-function. If one were to try to describe the motion by the FokkerPlanck equation

164 citations


Journal ArticleDOI
TL;DR: In this paper, an approximate formulation of the problem of a scalar wave propagated over an irregular bounding surface is derived in the form of a one-dimensional Volterra integral equation.
Abstract: In this paper there is derived an approximate formulation of the problem of a scalar wave propagated over an irregular bounding surface. This formulation takes the form of a one-dimensional Volterra integral equation, which can, or so it appears, be very useful in numerical calculations of field intensities. In addition, analytical solutions of this equation are obtained in three special cases: an isotropic radiator (1) at the surface of a plane, homogeneous earth, (2) elevated above a plane earth, and (3) at the surface of a homogeneous sphere. The point to be made about these three special cases is that they all have been solved by the other, more exact methods; but the solutions presented here and obtained from the integral equation formulation are almost identical with these classical solutions.

97 citations





Journal ArticleDOI
TL;DR: In this paper, it was shown that the error value cannot be bounded by any function of x 0, • • •, xn, y 0 = f(x0), • ••, y n = f (xn) = f n, and /(J) − p(|) = k.
Abstract: Scarborough, 2nd edition, pp. 99-103, the mistake arising from identification of two ^-values in the interval which may be distinct.) Consider the given data / (J) — j for j = 0, 1, • • • , n, for which the polynomial approximation is p(x) = x. Can the error be bounded by any function of x0 , • • • , xn , y0 = f(x0), • • • , y„ = f (xn) ? That it cannot is clear from the function f(x) = x + k sin ttx. We have /(J) = f + k, p(§) = §, and /(J) — p(|) = k. Since k is independent of x,and /(x,), this error value cannot be bounded by any function of these without strong hypotheses on the function /

31 citations


Journal ArticleDOI
TL;DR: In this article, a torsion solution based on the Michell-Foppl theory is presented for an infinite circular cylinder having a symmetrically-located spherical cavity.
Abstract: This paper presents a torsion solution, based on the Michell-Foppl theory, .for an infinite circular cylinder having a symmetrically-located spherical cavity. Numerical values are also given to show, in particular, the effect of the cavity on the maximum shear stress in the cylinder. The general theory. Let (r, 6, z) be the cylindrical coordinates of a point. For convenience, r and z will be considered as dimensionless quantities each measured by a unit of certain typical length a. The solution of the torsion problem for a solid of revolution, requires a function \p which satisfies .the following differential equation.1 + a) dr r dr dz in which the axis of revolution is taken as the z axis. The two non-vanishing stress components are expressed by _ M d± Te' r2 dr' T" r2 dz' [ } where fi is the modulus of rigidity. The condition that the surface is free from traction takes the form ip = const. (3) on the entire surface. The constant for \p at the internal surface will henceforth be taken as zero as no generality is lost in doing so. The terminal couple acting on the cylinder is then equal to T = 2irfj.a\pe , (4) where \pe is the value of ip at the external surface. The above theory is due to Michell and was rediscovered by Foppl. Solutions in cylindrical and spherical coordinates. If we put \p = r2 = Fir) . (7) cos kz *Received April 11, 1951. 'See A. E. H. Love, Mathematical theory of elasticity, 4th edition. Dover Publications, New York, 1944, pp. 325-326. 150 CHIH-BING LING [Vol. X, No. 2 It then appears that the function F satisfies Bessel's equation ? + ® This equation is satisfied by I2(kr) and K2(kr) which are modified Bessel functions of order two, of the first and second kinds, respectively. The above solution holds for any value of k. A more general solution of \p is therefore expressed by the following four integrals: I2{kr) sin kz >P = r2 I J(k) X dk, (9) 0 K2(kr) cos kz where in each integral / is an arbitrary function of k. Now we proceed to find the solution in the spherical coordinates (p, 0, 6) which are connected with the cylindrical coordinates by z = p cos , r = p sin , 6 = 0. (10) Consequently, the differential operators are transformed to d , d sin d — = COS —, az dp p d d d , COS — + —• dr dp p d (11) The differential equation for * becomes d2* 2d* 1 d2* cot <£d* 4* a 2 ~T „ "T" 2 „ ,2 "r 2 a , 2 -2 , — (J (1^) op p ap p a

), which are Legendre's associated functions of degree n and order two, of the first and second kinds, respectively. It is noted that the same Legendre's associated equation is obtained if we assume instead * = (15) P Since the preceding solution is valid for any integral value of n including zero, a more general solution of ip is expressed by the following four series : P2(cos 4>) p" X , (16) "=0 ,Q2( cost) P"n_1 where A„ are parametric coefficients. Note that P\ vanishes when n = 0 or 1. 1952] TORSION OF A CYLINDER HAVING A SPHERICAL CAVITY 151 Method of solution. Now, consider an infinite circular cylinder of radius a (i.e., r = 1) having a symmetrically-located spherical cavity of radius \a (i.e., p = X), which is under torsion by terminal couples about the axis of the cylinder. For convenience, the centre of the cavity will be placed at the origin. The method of solution is to assume \p as being composed of two parts as follows: i , (17) where \J/0 is the solution of a corresponding cylinder without a cavity, while 4>x is an auxiliary solution which vanishes on the surface of the cylinder. The latter is added to yp0 so that the remaining boundary condition on the surface of the cavity is satisfied by adjusting the parametric coefficients involved in the function. The function \f/0 is io = bar*, (18) where r is a constant. Since \pt vanishes on the surface of the cylinder, it follows that the terminal couple is equal to |x/ira4. The auxiliary function 4*i may be constructed by combining linearly a set of functions each of which satisfies the given differential equation and vanishes on the surface of the cylinder. It appears that the set of functions can be derived from the following particular solution for \p, namely: 2 f* co \p* = ^3Pl(cos) + r2 / f(k)I2(kr) cos kzdk. (19) p J 0 It is obvious that the function so constructed satisfies the given differential equation. Note that this function is even in z or cos 4> and, besides, it has a singularity at the origin. According to the definition by Hobson,2 Pi (cos 4>) is equal to 3 sin2 . Hence \p* vanishes when r = 1 provided that [ f(k)I2(k) cos kz dk = 3 (20) Jo (1 -(z) A Fourier transform gives {(h, _2_3_ f" cos kz dz _ 2k2K2(k) R) rrI2(k)J0 (1 + zy>* Th{k) 121> The last result is a particular case of the integral considered by Poisson and Malmsten.The function \p* is thus fully determined. It is obvious that differentiation with respect to z gives functions with the desired properties on the surface of the cylinder, but odd derivatives must be excluded since they are not even in z and cannot enter into the required solution. The function \p* itself may also be included. The set of functions is therefore d2\l/* dV* **< (22) 2See E. W. Hobson, Theory of spherical and ellipsoidal harmonics, Cambridge University Press, 1931, p. 94. Also cf. E. T. Whittaker and G. N. Watson, Modern analysis, Cambridge University Press, 1927, p. 325. The definition given in the latter is slightly different though it is also attributed to Hobson. 'See G. N. Watson, Theory of Bessel functions, 2nd edition, Cambridge University Press, 1944, p. 185. 152 CHIH-BING LING [Vol. X, No. 2 Since the use of any constant multiplier does not affect the desired properties, we may write the set of functions as follows: /* 1 . ^2s (2s-2)! dz*«-* ^ the initial function being \p* = ip*. To express the functions in terms of the spherical coordinates, the following relations are useful.4

16 citations


Journal ArticleDOI
TL;DR: In this article, the authors estimate the effect of gravity on the pressure exerted by a liquid on a missile moving through it with wetted area trailed by a cavity (Fig. 2).
Abstract: 6. Correction for hydrostatic force. Since gravity has been neglected above, it is interesting to have a rough estimate of the effect of gravity on the pressure exerted by a liquid on a missile moving through it with wetted area W' trailed by a cavity (Fig. 2). We suppose the liquid incompressible, and bounded by W', container walls W\", and a free surface S. The additional instantaneous acceleration b due to a vertical gravity field with intensity g satisfies b = gVB, where V2B = 0, B = y (depth coordinate) on S, and dB/dn = 0 on W + W\" = W; the associated hydrostatic pressure is pg(y — B). For given boundary configurations S and W, the resulting \"hydrostatic acceleration potential\" gB can be most easily estimated using an electrolytic tank, and the results interpreted in terms of the dimension! ess cavity buoyancy coefficient


Journal ArticleDOI
TL;DR: In this paper, the Hurwitz criterion of stability and the reduction of stable operator polynomials in p to such of a lower degree are discussed. But the reduction is restricted to polynomial functions.
Abstract: where z(t) may be considered as an arbitrary disturbance function. For instance, let z{t) = 1 for t < 0. At t = 0, z(t) may step down to z(t) = 0 for t > 0. The response y(t) then is a solution of (2), and the integral Y measures, how fast the systems lines up with the stepping of z. The knowledge of F makes it possible to choose the coefficients a, of (1) under given conditions in order to minimize F**. Two examples of such a minimization will be given in Sec. 4. The development of this formula will also yield a new approach to the well known Hurwitz criterion of stability and to reductions of \"stable\" operator polynomials in p to such of a lower degree, including the reduction of Schur [1], 1. Auxiliary theorems and algorithms of reduction. Notation. Let J be the imaginary axis of the complex plane, J' the set of all points wi, of J with w > 0, J\" the set of all points m of J with co < 0, and Re x the real, Im x the imaginary part of x. Definitions. Let f(x) = b0xm + b^x'\"'1 + • • • + bm a polynomial with real or complex coefficients. We call m the proper degree of f(x), if b0 0. We define now 1) F{x) = b0xm + bm as the \"simplification\" of f(x), if f(x) has the proper degree m,


Journal ArticleDOI
TL;DR: In this paper, a central difference method based on a formula closely associated with Simpson's rule was proposed for numerical integration of ordinary differential equations of the first order with one-point boundary conditions.
Abstract: The difference methods for the numerical integration of ordinary differential equations of the first order are discussed by using operator calculus and symbolic expansions. A new straightforward central difference method is developed, which is based on a formula closely associated with Simpson's rule. The main features of the method are that, for each step of integration, the largest unknown term is determined by an algebraic equation and that the remaining difference correction is extremely small. The method can directly be applied even to systems of the first order with onepoint boundary conditions. A numerical example is given.


Journal ArticleDOI
TL;DR: In this article, it was shown that the failure of the momentum principle in the case of a bar containing a hinge moving along its length is due to the wrong choice of partial and converted derivatives.
Abstract: for example an induction motor. The magnetic force producing the torque moves with the synchronous angular velocity of the alternating current, whereas the rotor speed falls below this by the slip, so that the magnetic force is moving relative to the rotor on which it is producing a torque. The rate of work done on the rotor is the torque multiplied by the rotor speed, and not the torque multiplied by the synchronous speed. In many cases, as in this case, the difference in these two quantities may be lost as mechanical work; in this case it appears as eddy current loss. The energy balance at the point of application is a function of the detailed method of application of the force. A problem, in which a similar difficulty in the use of partial and converted derivatives arose, appeared in the discussion of the motion of a bar containing a hinge moving along its length.2 In this case the wrong choice involved an apparent paradox: the failure of the momentum principle.

Journal ArticleDOI
TL;DR: In this paper, the authors considered a case of combined radial and axial heat flow in the unsteady state in finite cylinders composed of two coaxial parts of different materials.
Abstract: Introduction. Although several problems of heat flow in composite cylinders have been studied, all the cases considered treat the heat flow in the radial direction only [1, 2, 3]. The case of combined radial and axial heat flow in composite cylinders presents an interesting boundary value problem which has also considerable significance in the theory of vibrations and propagation of electromagnetic waves [4, 5, 6]. In this paper, we consider a case of combined radial and axial heat flow in the unsteady state in finite cylinders composed of two coaxial parts of different materials. The temperature distribution in the cylinder at any instant under the assumed boundary and initial conditions has been obtained by the use of the Laplace transformation. The procedure is illustrated by a numerical calculation in a particular case. The Problem. Composite cylinder made of two different materials, the inner cylinder 0 < r < a and the outer cylinder a < r 0 (see Fig. 1).


Journal ArticleDOI
TL;DR: In this article, it is proved that the integrals of these equations include a class of solutions of the boundary layer type; that is to say, solutions which asymptotically converge (with vanishing viscosity) towards flow patterns entirely different from those which are obtained when from the start viscoship is neglected.
Abstract: Foreword. The usual way to deal with the so-called shoch phenomenon in compressible fluids is the following. On the one hand there is the fact that in many cases of observable flow there exist narrow zones across which pressure, density and velocity undergo rapid changes. On the other hand, it is well-known that the differential equations of inviscid perfect fluids fail to supply solutions satisfying certain boundary conditions that can be realized physically. One therefore makes the assumption that these differential equations are valid in regions of the (x, y, z, t) space which are separated from each other by discontinuity surfaces whose shape is a priori unknown. From physically plausible hypotheses one then derives necessary conditions for the values assumed by the physical variables on either side of the discontinuity surfaces. Such conditions were first given by Riemann, and later modified by Rankine and Hugoniot. It is finally assumed—-and confirmed at least in special cases—that the differential equations combined with these transition conditions are sufficient to determine both the discontinuity surfaces themselves and the continuous flows in the regions between them (see [4], pp. 116-118, 134138). A different approach, as suggested by R. von Mises to the author, is followed in the present paper. The sole basis is formed by the system of partial differential equations (Navier-Stokes equations) which govern the motion of a viscous, heat-conducting, compressible fluid. No additional assumptions of any kind are introduced. It is proved that the integrals of these equations include a class of solutions of the boundary layer type; that is to say, solutions which asymptotically converge (with vanishing viscosity) towards flow patterns entirely different from those which are obtained when from the start viscosity is neglected. For a small viscosity coefficient n, these flows have rapid changes of the physical variables across certain narrow regions, the widths of which converge to zero as /j, —> 0. In the limit the values of the variables on the two sides of the transition are subject to equations which are identical with the Rankine-Hugoniot conditions. These conditions, obtained here without any hypotheses, are thus proved to be not only necessary but also sufficient for the existence of shocks.



Journal ArticleDOI
TL;DR: In this article, the finite amplitude wave problem is considered for a channel of finite depth and it will be shown that Parts I and II are special cases of this problem, and the basic approach will not be explained in detail here since it is given in Part I, but the problem will be solved independently of the earlier parts.
Abstract: Introduction. Part I of this series** is concerned with the theory of finite amplitude gravity waves in a channel of infinite depth, Part II deals with the problem of the solitary wave. In the present paper, the finite amplitude wave problem is considered for a channel of finite depth and it will be shown that Parts I and II are special cases of this problem. The basic approach will not be explained in detail here since it is given in Part I, but the problem will be solved independently of the earlier parts. 1. Statement of problem. We deal with an incompressible fluid in a channel with a horizontal base, the fluid having been set in motion in such a way that the flow is two-dimensional and irrotational. The fluid motion is periodic with wave length X in 0




Journal ArticleDOI
TL;DR: In this article, the authors derived the characteristic function of the sum of the squares of the deviations of the displacement of the particles from their given initial positions, also the corresponding characteristic function for the continuous string.
Abstract: where the qk represent charges, Lik inductances, Rik resistances, Gik reciprocals of capacitance, the Ejk are random e.m.f's**, and primes denote differentiations with respect to time. The theory of such systems of equations, quite carefully examined during the war years in applications to noise in electrical networks, turns out also to be applicable to the various mechanical systems—in particular, it is applicable to the system of a vibrating string with fixed end points. A general theory for the system of equations (1) has been developed by Uhlenbeck and Wang [2], Some of the results which we shall obtain in this article have been derived without making use of the general theory. We shall make comparison with these results and in addition we shall derive several more results. The results of Uhlenbeck and certain of his co-authors [1, 5], have been derived directly from the differential equation of motion for the string. In order to apply the general theory to the vibrating string, it is necessary to discretize the string. We therefore assume that the stringf is made up of n + 2 particles, (2 fixed, n vibrating) of equal mass m harmonically bound together by means of massless springs. Furthermore let us assume that this system of particles has random forces acting on it and that as a result the system vibrates, the vibration taking place in a plane. As a last assumption we suppose that the vibration takes place in a viscous medium so that each of the particles undergoes a damped vibration. When we have obtained our results for the discretized system, we can derive the results for the case of a continuous string by a limiting procedure, namely that of letting n the number of vibrating particles go to infinity while the total mass and length of the system remains constant. In this article we shall derive the following: 1) the characteristic function of the sum of the squares of the deviations of the displacement of the particles from their given initial positions—also the corresponding characteristic function for the continuous string,