bluuebunny/crossref_metadata_2025 huggingface dataset:
On the solutions of clamped Reissner-Mindlin plates under transverse loads
Authors: Assiff Thomas C. , Yen David H. Y.
Published: December 2016

The governing equations in the Reissner—Mindlin theory may be written in a form such that a small parameter

ϵ
\epsilon

is involved. This parameter

ϵ
\epsilon

depends on a combination of the shear modulus and the plate thickness. The governing equations are singularly perturbed with respect to

ϵ
\epsilon

. However, as

ϵ
→
0

\epsilon \to 0

one does recover the biharmonic equation of the classical plate theory. In a previous work of ours [1] the behavior of solutions for clamped Reissner—Mindlin plates as

ϵ
→
0

\epsilon \to 0

was studied and it was shown there that these solutions tend continuously, in various functional norms, to their corresponding solutions in the classical plate theory. This paper deals with two specific questions concerning the detailed dependence of these solutions on

ϵ
\epsilon

as

ϵ
→
0

\epsilon \to 0

. We shall show the nonexistence of regular asymptotic expansions of the solutions in integral powers of

ϵ
\epsilon

for general clamped Reissner—Mindlin plates. We shall also construct an exact solution for a circular plate which exhibits dependence on fractional powers of

ϵ
\epsilon

. This latter solution shows a boundary layer phenomenon, decaying away from the boundary, often encountered in singular perturbation problems.
Nonlinear acoustic wave propagation in atmosphere
Authors: Hariharan S. I.
Published: December 2016

In this paper we consider a model problem that simulates an atmospheric acoustic wave propagation situation that is nonlinear. The model is derived from the basic Euler equations for the atmospheric flow and from the regular perturbations for the acoustic part. The nonlinear effects are studied by obtaining two successive linear problems in which the second one involves the solution of the first problem. Well-posedness of these problems is discussed and approximations of the radiation boundary conditions that can be used in numerical simulations are presented.
A hyperbolic Stefan problem
Authors: Showalter R. E. , Walkington N. J.
Published: December 2016

A free-boundary problem of Stefan type is presented under constitutive assumptions on flux and energy which contain an effective time delay. This contains the hyperbolic telegraphers equation and, hence, has the feature that propagation speed of disturbances is bounded. With the appropriate physically consistent condition on the interface this is shown to lead to a well-posed weak formulation of the problem.
Lyapunov exponents and subspace evolution
Authors: Rokni M. , Berger B. S.
Published: December 2016

Differential equations are derived which describe the evolution of area tensors and normals associated with the subspaces of an

n
n

-dimensional Euclidean phase space,

E
n

{E_n}

. These provide computational methods for determining the Lyapunov exponents of continuous dynamical systems.
The density function of the solution of a two-point boundary value problem containing small stochastic processes
Authors: Xia Ning Mao
Published: December 2016

This paper concerns a two-point boundary value problem for an

m
m

thorder system of ordinary differential equations containing a vector stochastic process
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi left-parenthesis t comma omega right-parenthesis equals xi 0 left-parenthesis t right-parenthesis plus epsilon xi 1 left-parenthesis t comma omega right-parenthesis plus epsilon squared xi 2 left-parenthesis t comma omega right-parenthesis plus dot dot dot period">

ξ

(

t
,
ω

)

=

ξ
0

(
t
)

+
ε

ξ
1

(

t
,
ω

)

+

ε
2

ξ
2

(

t
,
ω

)

+
⋅
⋅
⋅
.

\xi \left ( {t, \omega } \right ) = {\xi _0}\left ( t \right ) + \varepsilon {\xi _1}\left ( {t, \omega } \right ) + {\varepsilon ^2}{\xi _2}\left ( {t, \omega } \right ) + \cdot \cdot \cdot .

\]
When

ε
\varepsilon

is small, the existence and the asymptotic properties of the solution can be obtained by means of the shooting method, and its density function can be determined by solving a sequence of first-order deterministic partial differential equations.
Stabilization of adiabatic rigid body rotation by dissipation
Authors: Boudourides M. A.
Published: December 2016

We consider the adiabatic tangential flow of an incompressible Newtonian fluid with temperature-dependent viscosity between two coaxial cylinders, one of which is rotating, while the other is kept stress-free. We show that every classical solution approaches, as time goes to infinity, a uniform rigid body rotation at fixed temperature, thus establishing the stabilizing role of dissipation. The proof is based on a priori estimates, obtained with the help of certain identities for solutions of the governing equations.
An unperiodic concentrated sonic pulse
Authors: Synge J. L.
Published: December 2016

The purpose of this note is to define an unperiodic source which gives a highly concentrated pulse of scalar radiation. The source involves four constants, and concentration is obtained by giving small values to dimensionless combinations of those constants. The field is axially symmetric, and there is a focal point which travels along the

x
x

-axis with the speed of propagation; the focal field remains constant for some time, decaying after that. The method is that of the retarded potential, but presented in terms of equivalent Minkowskian geometry of space-time, the speed of propagation playing the part of the speed of light in relativity. The obvious application is to sonic communication in water, but no attempt is made to interpret the source in physical terms.
Membrane caps under hydrostatic pressure
Authors: Dickey R. W.
Published: December 2016

It is shown that the exact nonlinear theory for a rotationally symmetric membrane cap deformed by hydrostatic pressure is statically determinant. A small strain theory is obtained without any assumptions on the relative magnitudes of the displacements. This small strain theory can be reduced to a single second-order ordinary differential equation for the determination of the radial stress. A linear shallow cap theory is obtained and solved explicitly for the case of the shallow spherical cap.
Some identities and the structure of 𝑁ᵢ in the Stroh formalism of anisotropic elasticity
Authors: Ting T. C. T.
Published: December 2016

The Stroh formalism of anisotropic elasticity leads to a

6
×
6

6 \times 6

real matrix N that can be composed from three

3
×
3

3 \times 3

real matrices

N
i

(

i
=
1
,
2
,
3

)

{N_i} \left ( {i = 1, 2, 3} \right )

. The eigenvalues and eigenvectors of N are all complex. New identities are derived that express certain combinations of the eigenvalues and eigenvectors in terms of the real matrices

N
i

{N_i}

and the three real matrices H, S, L introduced by Barnett and Lothe. It is shown that the elements of

N
1

{N_1}

and

N
3

{N_3}

have simple expressions in terms of the reduced elastic compliances. We prove that

−

N
3

- {N_3}

is positive semidefinite and, with this property, we present a direct proof that L is positive definite.
Metastable equilibrium of fluids with surface tension
Authors: Virga Epifanio G.
Published: December 2016

This paper deals with the effects of the surface tension on the equilibrium of fluids. It shows that, when a fluid with an incompressible fluid inclusion is put in a fluid environment kept at constant pressure, equilibrium configurations may arise whose character of stability is affected by the sign of the perturbations of the environmental pressure. Such configurations we refer to as metastable.
Approximate analytical solution of a Stefan’s problem in a finite domain
Authors: Takagi Shunsuke
Published: December 2016

A Stefan’s problem in a finite domain may be given an approximate analytical solution. An example is shown with constant boundary and initial conditions. The solution is initially that of a semi-infinite domain, transits through infinitely many intermediate stage solutions, and finally becomes stationary. The solution is exact in the initial stage and also at the steady final stage, but approximate at the intermediate stages.
Analytical aspects of heat flow in random two-component laminates
Authors: Hill J. M. , Willis J. R.
Published: December 2016

Analytical aspects are examined for the coupled reaction-diffusion equations arising for the expected conditional temperatures for heat flow in a random two-component laminate with heat flowing in a direction perpendicular to the laminae. The basic equations derived by Clarke [1] involve new terms not previously encountered within the contexts of either diffusion or heat flows. Various results and solution procedures are developed for both the coupled system and the underlying fourth-order equation. Basic source solutions for the coupled system are derived which are extended to general solutions of the coupled system with the new terms. These general solutions involve two arbitrary heat functions and display explicitly the dependence of solutions on the various parameters of the model. However, these general solutions appear not to be as useful in the context of the solution of boundary value problems as the corresponding results for the coupled system without the new terms. A simple illustrative example is provided for the utilization of such general solutions for a specific problem.
Some expansions in series of Bessel functions
Authors: Srivastava H. M. , Shreshtha R. M.
Published: December 2016

A general theorem on generating functions is applied to derive a number of interesting expansions for the generalized hypergeometric

r

F
s

_r{F_s}

function in series of Bessel functions. Several further expansion formulas, relevant to the present discussion, are also considered. Many of these expansions in series of Bessel functions stem from (or are motivated by) their applicability in various seemingly diverse fields of applied sciences and engineering. With this point in view, some examples illustrating possible applications of these results are provided.
Error bounds for a uniform asymptotic expansion of the Legendre function 𝑃_{𝑛}^{-𝑚}(𝑐𝑜𝑠ℎ𝑧)
Authors: Shivakumar P. N. , Wong R.
Published: December 2016

For fixed

m
m

with

m
+

1
2

>
0

m + \frac {1}{2} > 0

, an asymptotic expansion for large

n
n

is derived for the Legendre function

P
n

−
m

(

cosh
⁡
z

)

P_n^{ - m}\left ( {\cosh z} \right )

,which is uniformly valid for

z
z

in the unbounded interval

[

0
,
∞

)

\left [ {0, \infty } \right )

. Our method is based on an integral representation of this function. The coefficients in the expansion satisfy a recurrence relation. Simple computable bounds are also constructed for the error terms associated with the expansion.
Asymptotic analysis of surface waves due to oscillatory wave maker
Authors: Faltas M. S.
Published: December 2016

The initial value problem of surface waves generated by a harmonically oscillating vertical wave-maker immersed in an infinite incompressible fluid of finite constant depth is presented. The resulting motion is investigated using the generalized function method, and an asymptotic analysis for large times and distances is given for the free surface elevation.
On the Lagrange stability of spatially periodic solutions of the Ginzburg-Landau equation
Authors: Yang Yi Song
Published: December 2016

We study the Lagrange stability (i.e., the pointwise boundedness) of the spatially-periodic solutions of the Ginzburg-Landau equation. There is a parameter range where the solutions blow up (instable) if their initial mean amplitudes are beyond a critical value.
The generalized partial correspondence principle in linear viscoelasticity
Authors: Graham G. A. C. , Golden J. M.
Published: December 2016

A result, referred to as the generalized partial correspondence principle, is proved for noninertial viscoelastic boundary value problems. This states that if

B
u

(
i
)

(
t
)

B_u^{\left ( i \right )}\left ( t \right )

,

B
σ

(
i
)

(
t
)

B_\sigma ^{\left ( i \right )}\left ( t \right )

are the complementary regions of the boundary of a viscoelastic medium with a unique Poisson’s ratio, on which displacements

u
i

(

r
,
t

)

{u_i}\left ( {r, t} \right )

and stresses

σ

i
j

(

r
,
t

)

n
j

{\sigma _{ij}}\left ( {r, t} \right ){n_j}

are specified, respectively, then if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript sigma Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis t prime right-parenthesis subset-of-or-equal-to upper B Subscript sigma Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis t right-parenthesis">

B
σ

(
i
)

(

t
′

)

⊆

B
σ

(
i
)

(
t
)

B_\sigma ^{\left ( i \right )}\left ( {t’} \right ) \subseteq B_\sigma ^{\left ( i \right )}\left ( t \right )

for all

t
′

≤
t

t’ \le t

, the stresses satisfying this mixed boundary value problem at time

t
t

are the stresses for the elastic problem with the same boundary regions and the same specified stresses while known functions take the place of the specified displacements. It is noteworthy that

B
σ

(
i
)

(

t
′

)

B_\sigma ^{\left ( i \right )}\left ( {t’} \right )

,

t
′

≤
t

t’ \le t

, is not required to be monotonic increasing.
On the propagation of transients through thermoviscoelastic media
Authors: McCarthy M. F. , Moodie T. B. , Sawatzky R. P.
Published: December 2016

We examine the propagation of thermal and mechanical transients through general linear thermoviscoelastic media. A linear theory of heat conduction in deformable materials with memory is employed to study the one-dimensional problem of a homogeneous thermoviscoelastic half-space subjected to thermal and mechanical disturbances at its boundary. A ray series approach is used to generate asymptotic wave front expansions for the temperature, strain, and stress response of the medium to the disturbances. General properties of the propagation process are obtained simply and directly. As an example, we specialize the solution to the case of a medium where the conduction of heat obeys Cattaneo’s law and the viscoelastic response is that of a standard linear solid. The propagation of transients through this material is depicted graphically for various values of the thermal and mechanical parameters.
Stabilization of linear systems by time-delay feedback controls. II
Authors: Yong Jiong Min
Published: December 2016

The notion of uniform

r
r

-stabilizability for linear autonomous systems is introduced. Some abstract characterizations and easy-to-check necessary conditions and sufficient conditions are given.
The shape of the strongest column is arbitrarily close to the shape of the weakest column
Authors: Barnes David C.
Published: December 2016

We reconsider the problem of determining the shape of the strongest column having a given length

l
l

and volume

V
V

. Previous results [13,7] have given optimal shapes for which the cross section vanishes at certain points. Although these results are mathematically correct, Theorem 1 below explains what is wrong with these anomalous shapes.
The random eigenvalue problem for a differential equation containing small white noise
Authors: Xia Ning Mao
Published: December 2016

By means of the extended Ito integral and the shooting method, this paper concerns the random problem containing white noise

B

(

t
,
ω

)

B\left ( {t, \omega } \right )

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket minus p left-parenthesis t right-parenthesis u prime left-parenthesis t right-parenthesis right-bracket prime plus left-bracket q left-parenthesis t right-parenthesis plus epsilon upper B left-parenthesis t comma omega right-parenthesis right-bracket u equals lamda u comma u left-parenthesis 0 right-parenthesis equals 0 comma u left-parenthesis 1 right-parenthesis equals 0 period">

[

−
p

(
t
)

u
′

(
t
)

]

′

+

[

q

(
t
)

+
ε
B

(

t
,
ω

)

]

u
=
λ
u
,

u

(
0
)

=
0
,

u

(
1
)

=
0.

{\left [ { - p\left ( t \right )u’\left ( t \right )} \right ]’} + \left [ {q\left ( t \right ) + \varepsilon B\left ( {t, \omega } \right )} \right ] u = \lambda u, \qquad u\left ( 0 \right ) = 0, \qquad u\left ( 1 \right ) = 0.

\]
When

ε
\varepsilon

is small, the existence and asymptotic expansions for the solutions can be obtained, and the normal properties for the first correction terms can be proved. Formulas for evaluation are derived and one example of the Schrödinger equation is given to illustrate the whole procedure.
Spherical waves in odd-dimensional space
Authors: Kingston J. G.
Published: December 2016

The general solution is given of the

(

2
N
+
1

)

\left ( {2N + 1} \right )

-dimensional wave equation with spherical symmetry,

u

t
t

−

u

x
x

−

2
N

x

u
x

=
0

{u_{tt}} - {u_{xx}} - \frac {{2N}}{x}{u_x} = 0

, in terms of two arbitrary functions and their first

N
N

derivatives. Simple transformations then yield the general solutions to the Euler-Poisson-Darboux equation,

u

x
y

+

N

(

x
+
y

)

(

u
x

+

u
y

)

=
0

{u_{xy}} + \frac {N}{{\left ( {x + y} \right )}}\left ( {{u_x} + {u_y}} \right ) = 0

, for integer

N
N

, and also the one-dimensional wave equation,

u

t
t

−

c
2

u

x
x

=
0

{u_{tt}} - {c^2}{u_{xx}} = 0

, for certain variable wave speeds

c

(
x
)

c\left ( x \right )

.
On a new extension of Liapunov’s direct method to discrete equations
Authors: Carvalho L. A. V. , Ferreira Raimundo R.
Published: December 2016

In this paper, a new procedure is given for applying Liapunov’s direct method to autonomous discrete equations. This procedure is based on an idea that is closely related to Razumikhin’s principle and it includes Liapunov’s direct method as a special case. Examples are given.
Identifiability of an unknown coefficient in a nonlinear diffusion equation
Authors: Choulli M.
Published: December 2016

We consider the problem of identifying an unknown coefficient in a one-dimensional diffusion equation from overspecified data measured in the interior of the domain. We obtain an identifiability result which is used to derive a uniqueness result for the inverse problem.
Patterns of ship waves
Authors: Yih Chia-Shun , Zhu Songping
Published: December 2016

Patterns of water waves created by a moving disturbance representing a moving body, floating or submerged, can be found by applying (1) the principle of stationary phase, (2) the principle that the phase lines are normal to the wave-number vector, and (3) the perception that the local phase velocity of the waves must be equal to the component of the velocity of the disturbance normal to the phase line. The three equations thus obtained are solved, and formulas for the phase lines are derived, which depend explicitly on the dispersion equation, and on that equation only. These formulas are applied to deep-water surface waves, surface waves in water of finite depth, internal waves, and capillary waves in thin sheets to obtain the wave patterns sufficiently far from the moving disturbance.
A numerical method for semilinear singular parabolic quenching problems
Authors: Chan C. Y. , Chen C. S.
Published: December 2016

For the problem given by

u

x
x

+
b

u
x

/

x
−

u
t

=
−

(

1
−
u

)

−
1

{u_{xx}} + b{u_x}/x - {u_t} = - {\left ( {1 - u} \right )^{ - 1}}

for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than x greater-than a comma 0 greater-than t greater-than upper T Subscript a Baseline less-than-or-equal-to normal infinity comma u left-parenthesis x comma 0 right-parenthesis equals 0 equals u left-parenthesis 0 comma t right-parenthesis equals u left-parenthesis a comma t right-parenthesis">

0
>
x
>
a
,

0
>
t
>

T
a

≤
∞
,
u

(

x
,
0

)

=
0
=
u

(

0
,
t

)

=
u

(

a
,
t

)

0 > x > a, \\ 0 > t > {T_a} \le \infty , u\left ( {x, 0} \right ) = 0 = u\left ( {0, t} \right ) = u\left ( {a, t} \right )

, where

b
b

is a constant less than one, a lower bound of

u
u

is used to estimate the critical length

a
a

beyond which quenching occurs, and an upper bound for the time when quenching happens. An upper bound of

u
u

, given by the minimal solution of its steady state, is constructed by using a modified Picard method with the construction of the appropriate Green’s function. To determine the critical length numerically, it is shown that for a given length

a
a

, all iterates attain their maximum values at the same

x
x

-coordinate; the largest interval for existence of the minimal solution corresponds to the critical length for the parabolic problem. As illustrations of the numerical method, the critical lengths corresponding to four given values of

b
b

are computed.
A place for philosophy? The rise of modeling in statistical science
Authors: Diaconis Persi
Published: December 2016

Large statistical models seem to have reached epidemic proportions. I will document the harm they are currently causing and contrast it with some success stories of smaller modeling efforts. A case study involving a model for flipping coins shows how “adding a few bells and whistles” can lead to nonsense. Finally, an effort to list the “real uses” of models is offered as a start to making sense of current modeling efforts.
Quasi-transonic flow past delta wings
Authors: Schleiniger Gilberto
Published: December 2016

Linearized supersonic theory for flows past delta wings must be corrected for the cases of wing sweepback angles close to the Mach angle. Results of calculations based on a local transonic expansion valid near the leading edge are presented.
Asymptotic expansions in nonlinear rotordynamics
Authors: Day William B.
Published: December 2016

This paper is an examination of special nonlinearities of the Jeffcott equations in rotordynamics. The immediate application of this analysis is directed toward understanding the excessive vibrations recorded in the LOX pump of the SSME during hot-firing ground testing.
Asymptotic solutions for unsteady flow in cascades
Authors: Bendiksen Oddvar O.
Published: December 2016

Asymptotic methods are presented for obtaining approximate solutions to unsteady flows in oscillating cascades. The formulation is in the framework of linearized potential flow, and the problems considered include low-frequency and low-solidity expansions for subsonic and supersonic cascades. For the supersonic cascade, simple formulas are obtained for the unsteady lift and moment, valid to first order in a frequency parameter. It is shown that terms of successive orders can be obtained by solving a sequence of quasi-static problems, with the effective upwash at each step modified by the lower-order solutions. The approach is in the spirit of matched asymptotic expansions, and different expansions based on different limit processes are sought for the subresonant and superresonant regions. For cascades in subsonic axial flow, the acoustic resonance phenomenon leads to a nonuniformity with respect to the interblade phase angle. The location of the singularity can be moved by suitably redefining the limit process, permitting uniformly valid expansions to be obtained separately for the subresonant and superresonant regions. Numerical comparisons with the full unsteady solution indicate that the present approximations are remarkably accurate in the range of reduced frequencies of interest in aeroelastic analyses.
A theory of electromagnetic induction in layered media: Linear motor modeling
Authors: Rae I. C.
Published: December 2016

A general theory of electromagnetic induction is developed based on the superposition of electric current filaments. The theory has applications in both geophysical and engineering problems where current sources are used to induce electromagnetic fields in plane layered media. As an example, three-dimensional integral formulae are derived for the source and induced fields of a linear induction motor configuration.
Stokes flow in a rectangular well: Natural convection and boundary-layer function
Authors: de Socio L. M. , Gaffuri G. , Misici L.
Published: December 2016

Free convection in a rectangular well of infinite depth at low Rayleigh number is considered. A pair of opposite vertical walls are at different temperatures, whereas the other walls are adiabatic. The three-dimensional Stokes flow regime is analyzed and the boundary-layer function for the adiabatic walls is determined.
An adaptive algorithm to accelerate multi-parameter Monte Carlo computations
Authors: Hitzl Donald L. , Zele Frank
Published: December 2016

Estimator expansions in terms of orthonormal Hermite polynomials show particular promise for variance reduction in Monte Carlo computer simulations. This paper briefly describes some new adaptive refinements to the basic variance reduction procedure so that the acceleration towards convergence of multi-parameter Monte Carlo computations is further improved. Simulation results using the adaptive algorithm are presented for a five-parameter model problem.
On effective moduli of an elastic body containing periodically distributed voids
Authors: Nemat-Nasser S. , Taya M.
Published: December 2016

For an elastic body containing periodically distributed voids, several effective techniques are presented which can be used to obtain the effective elastic moduli with any desired degree of accuracy. The results include the effects of void geometry as well as void interactions. For a body containing spherical voids, numerical results are presented and compared with those obtained by other methods.
Finite-amplitude surface waves in electrohydrodynamics
Authors: Kant Rama , Jindia R. K. , Malik S. K.
Published: December 2016

The stability of weakly nonlinear waves on the surface of a fluid layer in the presence of an applied electric field is investigated by using the derivative expansion method. A nonlinear Schrödinger equation for the complex amplitude of quasi-monochromatic traveling wave is derived. The wave train of constant amplitude is unstable against modulation. The equation governing the amplitude modulation of the standing wave is also obtained which yields the nonlinear cut-off wave number.
The non-uniform motion of a thin smooth rigid wedge into an elastic half-plane
Authors: Brock L. M.
Published: December 2016

A two-dimensional problem of non-uniform motion by a smooth, thin, symmetric rigid wedge into the surface of an elastic half-plane is analyzed. Both the wedge starting on the surface and from an embedded position are considered. Based on the integral transforms for a more general imposed displacement, the half-plane surface and wedge path behavior yields explicit information on the various component waves appearing in the solution. Logarithmic singularities are found at the wedge apex for subsonic speeds.
On oscillatory necking in polymers
Authors: Matkowsky B. J. , Sivashinsky G. I.
Published: December 2016

The phenomenon of oscillatory necking in the stretching of polyethylene films is described. We propose an extension of a model of Barenblatt for isothermic necking in polymers, and we show that oscillatory necking may arise, for example, in polymers for which the effect of diffusion of stresses exceeds that of diffusion of oriented material. In such polymers, at sufficiently low temperatures, uniform necking is no longer stable and self-oscillations are observed.
A radiation-conditioned Gaussian induced MGD conefield
Authors: Chee-seng Lim
Published: December 2016

Explicit radiation-conditioned solutions are derived for magnetically-aligned MGD flow past a Gaussian source. If the latter’s principal dimension is small, then corresponding to a supersonic-superAlfvénic (or restricted subsonic-subAlfvénic) flow, a strong perturbation field is encountered downstream from a downstream (or upstream from an upstream) tangent cone to the “fast” sheet (or one of the two “slow” cusped sheets) of the Friedrichs wavefront, while a weak estimable field appears upstream (or downstream) from this cone. The strong field is represented by an expansion with accessible asymptotic development. For a finite principal source dimension, other asymptotic modes can be extracted from a separate uniformly valid formulation. In particular, the asymptotic behavior near the dividing tangent cone is nonsingular and suggests the possible existence there of a conical sheath within which the field is probably most concentrated, and across which its transition from strong to weak occurs gradually.
On deviations from linear wave motion in inhomogeneous stars
Authors: Melvin P. J.
Published: December 2016

The hydrodynamic equations for the large-amplitude, adiabatic pulsations of a spherically symmetric, inhomogeneous star are solved by a method of approximation in which the form of the fluid velocity is specified a priori. The assumed velocity is a nonlinear function of the radius and contains two arbitrary functions of time. These two functions are determined by a pair of second-order, quasi-linear, ordinary differential equations, and an analytic, periodic solution to these equations is constructed. This solution corresponds to large amplitude, anharmonic, nonlinear pulsations of a star in which the fluid velocity is a travelling wave. A specific inhomogeneous star is studied to demonstrate the feasibility of numerically solving the pair of differential equations and of constructing the periodic solution.
Natural stress rate
Authors: Durban David , Baruch Menahem
Published: December 2016

A three-dimensional definition of the natural stress rate is suggested. The behavior of a hypoelastic material of grade zero based on the natural stress rate is analyzed for the case of simple extension. The results of the theoretical analysis agree with existing experimental data for urethane rubber. An interesting relation between the natural stress rate and stress rates suggested by Truesdell and Hill is given. The paper is written in full tensorial notation which is especially suitable when rates of tensor are considered.
Interaction of two vortices in a flued with lower stagnation pressure
Authors: Ting Lu
Published: December 2016

The flow field induced by two point vortices of strength

Γ
\Gamma

at a distance

2
h

2h

apart in a quiescent field with lower stagnation pressure is investigated. The flow field around the vortices is bounded by a free streamline with constant velocity

U
U

determined by the difference in the stagnation pressures. For the parameter

Λ
=
2
π
h
U

/

Γ
>
1

\Lambda = 2\pi hU/\Gamma > 1

, the flow field around each vortex is isolated from the other and is bounded by a circular free streamline of radius

Γ

/

(

2
π
U

)

\Gamma /\left ( {2\pi U} \right )

. When

Λ
=
1

\Lambda = 1

, the two circular free streamlines just touch each other. When

Λ
>
1

\Lambda > 1

the two flows merge and they are enclosed inside a single convex free streamline. Analytical solutions are presented for the merged flow field and the free streamline for the entire range of the parameter

Λ
\Lambda

.
Viscous effects on perturbed spherical flows
Authors: Prosperetti Andrea
Published: December 2016

The problem of two viscous, incompressible fluids separated by a nearly spherical free surface is considered in general terms as an initial-value problem to first order in the perturbation of the spherical symmetry. As an example of the applications of the theory, the free oscillations of a viscous liquid drop and of a bubble in a viscous liquid are studied in some detail. It is shown that the oscillations are initially describable in terms of an irrotational approximation, and that the normal-mode results are recovered as

t
→
∞

t \to \infty

. In between these asymptotic regimes, however, the motion is significantly different from either approximation.
Application of Khas’minskii’s limit theorem to the buckling problem of a column with random initial deflections
Authors: Videc B. P. , Sanders J. Lyell
Published: December 2016

An approximate asymptotic expression is obtained for the buckling load of an imperfect column resting on a nonlinear elastic foundation. The result holds for a large range of imperfection shapes, which are assumed to be stationary random functions of position. The asymptotic analysis is based on application of Khas’minskii’s limit theorem to equations for the slowly varying part of the deflection of the column. Previous results obtained for Gaussian imperfection shapes are shown to be valid also for the larger class of random imperfections considered here.
Transient disturbances in a half-space during the first stage of frictionless indentation of a smooth rigid die of arbitrary shape
Authors: Robinson A. R. , Thompson J. C.
Published: December 2016

A solution is obtained by the method of self-similar potentials for the time-spatial distribution of the contact stress between a homogeneous, isotropic, linearly elastic half-space and a smooth rigid die having an arbitrary indenting velocity and shape. The solution holds as long as the outward speed of the contact zone does not fall below the speed of the dilatational wave in the elastic medium. A proof is given that the instantaneous value of the force required to indent the die during this stage of contact is directly proportional to the product of the area of contact and the velocity of indentation at that instant.
The transverse vibration of a rotating beam with tip mass: The method of integral equations
Authors: Jones Louise H.
Published: December 2016

An integral equation method is used to obtain improvable lower bounds for the second eigenvalue of the second-order “reduced” problem obtained from the problem described in the title by singular perturbation methods. These lower bounds are compared with results obtained directly by invariant embedding. The computational aspects of the integral equation method are stressed. The method is shown to be quite general and can be applied to a variety of boundary-value problems including those in which the eigenvalue parameter appears in the boundary conditions as well as in the differential operator.
On the breaking of water waves on a sloping beach of arbitrary shape
Authors: Gurtin Morton E.
Published: December 2016

Greenspan [1] considered water waves of finite amplitude on a beach of constant slope. He proved that: (

(

G
1

)

\left ( {{G_1}} \right )

) A wave of elevation with nonzero slope at the front propagating shoreward into quiescent water always breaks before the shore.†(

(

G
2

)

\left ( {{G_2}} \right )

) Under the same conditions a wave of depression never breaks.
Application of hill functions to circular plate problems
Authors: Kao Robert
Published: December 2016

A type of finite element—hill functions—is applied to solve circular plate problems in conjunction with the method of Lagrange multipliers which is used to treat various constraint conditions. Results obtained compare very nicely with the exact solutions.
Stochastic models of the scattering of sound by bubbles in the upper ocean
Authors: Wang Peter C. C. , Medwin Herman
Published: December 2016

Stochastic models are developed to relate the statistics of sound speed fluctuations and bubble density variations as a function of sound frequency in the upper ocean. These predictions from the stochastic model have been compared with ocean experimental data of sound speed modulation in the frequency range 15 to 150 kHz, and show satisfactory agreement. Future experiments and further modification of this model are discussed.
Infinitely stretched Mooney surfaces of revolution are uniformly stressed catenoids
Authors: Wu Chien-Heng
Published: December 2016

Axially and radially stretched Mooney surfaces of revolution are found to tend to catenoids as the stretching tends to infinity. Moreover, two catenoids are found to exist for any given set of stretching parameters. A formal two-term asymptotic solution is obtained explicitly and the stretching of a cylindrical surface is given as an example.
Applications of invariant transformations in one-dimensional non-steady gasdynamics
Authors: Castell S. P. , Rogers C.
Published: December 2016

Certain reciprocal and adjoint transformations available for one-dimensional non-steady gasdynamic flow are applied to an existing solution to construct new exact solutions of the governing equations. The particle trajectories and the pressure-density relations on these trajectories are calculated. An application of the adjoint transformation in studying the flow between a piston and a non-uniform shock wave is indicated.
Motion of a compressible plate driven by shock or detonation
Authors: Rivard W. C. , Engelke Ray
Published: December 2016

The method of characteristics is used to obtain solutions for the time-dependent motion of a compressible plate driven by a one-dimensional shock or detonation followed by a constant state. The plate can either move in contact with or separate from the driver, depending on the values of the initial parameters. The motion of the surfaces is described by a system of linear, first-order, ordinary differential equations in one independent variable. The interior waves and interactions are determined from the surface motions. The equation of state of the plate has the form described by Murnaghan, specialized to produce straight characteristics in the

x
,
t

x,t

plane. The equation of state for the driver is arbitrary. Closed-form solutions are obtained for the surfaces during their first reverberations for particular driver equations of state. Numerical examples are given which closely represent real systems. The results can be used as standards for finite-difference calculations.
A note on the stability of an immiscible liquid layer in a porous medium
Authors: Raghavan R.
Published: December 2016

The Saffman—Taylor instability of a liquid layer is studied by the standard first-order perturbation method. The viscosity and density of the layer and the bounding fluids are assumed to be constant. Interfacial tension is taken into account. An expression for the rate of growth of perturbations at the interface is determined. Results obtained by previous analyses are shown to be particular cases of this solution. The reduction in the rate of growth of the perturbations due to the presence of the liquid layer is demonstrated.
Asymptotic analysis of the buckling of externally pressurized cylinders with random imperfections
Authors: Amazigo John C.
Published: December 2016

The buckling of finite circular cylindrical shells with random stress-free initial displacements which are subjected to lateral or hydrostatic pressure is studied using a perturbation scheme developed in an earlier paper [1]. A simple approximate asymptotic expression is obtained for the buckling load for small magnitudes of the imperfection. This result is compared with earlier results obtained for localized imperfections and imperfections in the shape of the linear buckling mode.
The transient temperature field in a composite semi-space resulting from an incident heat flux
Authors: Grimado P. B.
Published: December 2016

The two-dimensional, transient temperature field in a composite semi-space resulting from an incident heat input is constructed using operational techniques. Examples of the temperature field are presented with general conclusions that allow a qualitative assessment of the temperature distribution given the heat input and thermal properties of the constituent materials.
On the secular equation for anisotropic wave motions
Authors: Hayes M.
Published: December 2016

In a previous paper [7] it was shown that for waves propagating in certain linear anisotropic mechanical systems there exist universal connections between the phase speeds of harmonic plane waves. Here the results are shown to hold in a wider context.
Analytical structure of a generalized direct-interaction approximation
Authors: Lee Jon
Published: December 2016

As a mathematically tractable example, we have investigated the stochastic dynamic problem of an irreversible second-order chemical reaction. A generalized direct-interaction approximation has been devised to close off the hierarchy of moment equations at the arbitrary moment level, and then the results of such a closure technique have been compared term-by-term with the exact moment solutions. This shows qualitatively how the expansion terms summed up in the direct-interaction approximation are different from the classes of expansion terms present in the exact moment solutions. A quantitative comparison of the covariances indicates that the direct-interaction equations which are closed at the triple moment level represent a meaningful statistical approximation of the lowest order for the second-order reactive problem at hand.
On the boundary-value problem of a spheroid
Authors: Fikioris John G.
Published: December 2016

The surface charge density of a charged spheroid is obtained in exact, closed form using a Green’s function expansion in spherical coordinates. The possibility is thus established of solving boundary-value problems analytically using coordinates that do not correspond to boundary shapes. The present approach, used previously in numerical solutions of related problems, requires the potential function to be constant in the interior of the conductor. Its advantages from the theoretical standpoint and its further possibilities are discussed.
Unsymmetric wrinkling of circular plates
Authors: Cheo L. S. , Reiss Edward L.
Published: December 2016

The branching of unsymmetric equilibrium states from axisymmetric equilibrium states for clamped circular plates subjected to a uniform edge thrust and a uniform lateral pressure is analyzed in this paper. The branching process is called wrinkling and the loads at which branching occurs are called wrinkling loads. The nonlinear von Kármán plate theory is employed. The wrinkling loads are determined by solving numerically the eigenvalue problem obtained by linearizing about a symmetric equilibrium state. The post-wrinkling behavior is studied by a perturbation expansion in the neighborhood of the wrinkling loads.
Approximations for large deflection of a cantilever beam
Authors: Bisshopp K. E.
Published: December 2016

Rational approximations for the end slope and deflection of a cantilever beam are derived from the exact solution in terms of elliptic integrals [1] by asymptotic methods.
Flattening of membranes of revolution by large stretching—Asymptotic solution with boundary layer
Authors: Perng Daniel Y. P. , Wu Chien-Heng
Published: December 2016

The problem is solved explicitly by the method of matching asymptotics. The stretching of a flat membrane with an inclusion is studied as an example. Asymptotic stress and strain concentration ratios are derived. It is shown that the stress concentration ratio tends to unity as the stretching tends to infinity. This is justified by the exact numerical solution.
A class of waves in a deformed viscoelastic solid
Authors: Hayes M. A. , Rivlin R. S.
Published: December 2016

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.
A note on the potential vortex in a wall jet
Authors: Shaw Susann J. N.
Published: December 2016

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.
Optimum configurations for bangless sonic booms
Authors: Hayes Wallace D. , Weiskopf Jr.
Published: December 2016

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.
Data analysis, computation and mathematics
Authors: Tukey John W.
Published: December 2016

"Data analysis” instead of “statistics” is a name that allows us to use probability where it is needed and avoid it when we should. Data analysis has to analyze real data. Most real data calls for data investigation, while almost all statistical theory is concerned with data processing. This can be borne, in part because large segments of data investigation are, by themselves, data processing. Summarizing a batch of 20 numbers is a convenient paradigm for more complex aims in data analysis. A particular summary, highly competitive among those known and known about in August 1971, is a hybrid between two moderately complex summaries. Data investigation comes in three stages: exploratory data analysis (no probability), rough confirmatory data analysis (sign test procedures and the like), mustering and borrowing strength (the best of modern robust techniques, and an art of knowing when to stop). Exploratory data analysis can be improved by being made more resistant, either with medians or with fancier summaries. Rough confirmatory data analysis can be improved by facing up to the issues surrounding the choice of what is to be confirmed or disaffirmed. Borrowing strength is imbedded in our classical procedures, though we often forget this. Mustering strength calls for the best in robust summaries we can supply. The sampling behavior of such a summary as the hybrid mentioned above is not going to be learned through the mathematics of certainty, at least as we know it today, especially if we are realistic about the diversity of non-Gaussian situations that are studied. The mathematics of simulation, inevitably involving the mathematically sound “swindles” of Monte Carlo, will be our trust and reliance. I illustrate results for a few summaries, including the hybrid mentioned above. Bayesian techniques are still a problem to the author, mainly because there seems to be no agreement on what their essence is. From my own point of view, some statements of their essence are wholly acceptable and others are equally unacceptable. The use of exogeneous information in analyzing a given body of data is a very different thing (a) depending on sample size and (b) depending on just how the exogeneous information is used. It would be a very fine thing if the questions that practical data analysis has to have answered could be answered by the mathematics of certainty. For my own part, I see no escape, for the next decade or so at least, from a dependence on the mathematics of simulation, in which we should heed von Neumann’s aphorism as much as we can.
The synthesis of dynamical systems
Authors: Brockett R. W.
Published: December 2016

A significant part of contemporary applied mathematics is concerned directly with communication, control and computation. In these fields many of the central problems involve the synthesis of algorithms, or dynamical systems, as opposed to the analysis of dynamical systems which predominates in mathematical physics. Arithmetic and numerical algorithms, finite-state machines and electrical filters are examples of the types of dynamical systems which are frequently needed to operate on data, in continuous or discrete form, and to produce data on a compatible time scale. In this paper we discuss the scope and success of some of the synthesis procedures currently available to treat these problems.