The decay of a plane shock wave

1970 ◽  
Vol 43 (4) ◽  
pp. 737-751 ◽  
Author(s):  
H. Ardavan-Rhad
Keyword(s):  
Analytic Solution ◽  
Weak Shock ◽  
Simple Wave ◽  
Plane Shock ◽  
One Dimensional ◽  
Conductive Medium ◽  
Expansion Theory ◽  
Particular Solution ◽  
The One ◽  
First Time

An analytic solution of the non-isentropic equations of gas-dynamics, for the one-dimensional motion of a non-viscous and non-conductive medium, is derived in this paper for the first time. This is a particular solution which contains only one arbitrary function. On the basis of this solution, the interaction of a centred simple wave with a shock of moderate strength is analyzed; and it is shown that, for a weak shock, this analysis is compatible with Friedrichs's theory. Furthermore, in the light of this analysis, it is explained why the empirical methods employed by the shock-expansion theory, including Whitham's rule for determining the shock path, work.

scholarly journals AN ANALYTIC SOLUTION FOR ONE-DIMENSIONAL DISSIPATIONAL STRAIN-GRADIENT PLASTICITY

2009 ◽  
Vol 50 (3) ◽  
pp. 407-420
Author(s):  
ROGER YOUNG
Keyword(s):  
Boundary Condition ◽  
Strain Gradient ◽  
Analytic Solution ◽  
Strain Gradient Plasticity ◽  
Numerical Results ◽  
Gradient Plasticity ◽  
One Dimensional ◽  
Numerical Result ◽  
Formulation Analysis ◽  
The One

AbstractAn analytic solution is developed for the one-dimensional dissipational slip gradient equation first described by Gurtin [“On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids48 (2000) 989–1036] and then investigated numerically by Anand et al. [“A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids53 (2005) 1798–1826]. However we find that the analytic solution is incompatible with the zero-sliprate boundary condition (“clamped boundary condition”) postulated by these authors, and is in fact excluded by the theory. As a consequence the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al. also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur.

Uniformly valid analytical solution to the problem of a decaying shock wave

1987 ◽  
Vol 185 ◽  
pp. 153-170 ◽  
Author(s):  
V. D. Sharma ◽  
Rishi Ram ◽  
P. L. Sachdev
Keyword(s):  
Shock Wave ◽  
Analytical Solution ◽  
Weak Shock ◽  
Simple Wave ◽  
Complete Agreement ◽  
Plane Shock ◽  
Propagation Law ◽  
Arbitrary Strength ◽  
Entire Flow ◽  
Basic Equations

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.

scholarly journals A note on inverse problem for strongly damped wave equation with Gaussian white noise

2018 ◽  
Vol 20 ◽  
pp. 02003
Author(s):  
Chu Duc Khanh ◽  
Nguyen Hoang Luc ◽  
Van Phan ◽  
Nguyen Huy Tuan
Keyword(s):  
White Noise ◽  
A Priori ◽  
Gaussian White Noise ◽  
True Solution ◽  
One Dimensional ◽  
Noise Data ◽  
Ill Posed ◽  
The One ◽  
First Time ◽  
Strongly Damped

In this paper, we study for the first time the inverse initial problem for the one-dimensional strongly damped wave with Gaussian white noise data. Under some a priori assumptions on the true solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Error estimates are given in both the L2– and Hp–norms.

scholarly journals An Example of a Plane Shock of Variable Strength

1963 ◽  
Vol 13 (4) ◽  
pp. 297-302 ◽  
Author(s):  
P. Smith
Keyword(s):  
Density Distribution ◽  
Plane Shock ◽  
One Dimensional ◽  
Variable Strength ◽  
Polytropic Gas ◽  
Particular Solution ◽  
Inhomogeneous Gas

AbstractA particular solution of the equations of one-dimensional anisentropic flow of a polytropic gas is linked by a shock to gas at rest in which the density is non-uniform. The approach is inverse in that the density distribution is derived from the position of the shock and the prescribed flow behind it. The velocity and strength of the shock each vary with time. The result is an example of the propagation of a shock through an inhomogeneous gas.

scholarly journals Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem

2019 ◽  
Vol 19 (3) ◽  
pp. 437-473 ◽  
Author(s):  
Julian López-Gómez ◽  
Pierpaolo Omari
Keyword(s):  
Neumann Problem ◽  
Positive Solutions ◽  
Bounded Variation ◽  
Connected Components ◽  
One Dimensional ◽  
Linearized Problem ◽  
Bounded Variation Functions ◽  
The One ◽  
First Time ◽  
Prime 2

Abstract This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem \begin{dcases}-\Bigg{(}\frac{u^{\prime}}{\sqrt{1+{u^{\prime}}^{2}}}\Bigg{)}^{% \prime}=\lambda a(x)f(u)\quad\text{in }(0,1),\\ u^{\prime}(0)=0,\quad u^{\prime}(1)=0,\end{dcases} where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign, and {f\in C^{1}(\mathbb{R})} is positive in {(0,+\infty)} . The attention is focused on the case {f(0)=0} and {f^{\prime}(0)=1} , where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function a, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.

The Derivation of 1/N Energy-Solutions to the harper-Equation and Related Magnetizations

1998 ◽  
Vol 12 (18) ◽  
pp. 1847-1870 ◽  
Author(s):  
C. Micu ◽  
E. Papp
Keyword(s):  
Energy Spectrum ◽  
Fermi Level ◽  
Quantum Gas ◽  
One Dimensional ◽  
Implicit Form ◽  
Field Independent ◽  
The One ◽  
First Time ◽  
Harper Equation

Proofs are given for the first time that the energy-spectrum of the Harper-equation can be derived in a closed implicit form by using the one-dimensional limit of the 1/N-description. Explicitly solvable cases are discussed in some more detail for Δ=1. Here Δ expresses the Harper-parameter discriminating between metallic (Δ<1) and insulator (Δ>1) phases. Related magnetizations have been established by applying both Dingle- and quantum-gas approaches, now for a fixed value of the Fermi-level. The first description leads to large paramagnetic-like magnetizations oscillating with nearly field-independent amplitudes increasing with the temperature. In the second case one deals with magnetization-oscillations centered around the zero-value, such that the amplitudes decrease both with the field and the temperature.

THE FUNDAMENTAL GROUP OF GALOIS COVER OF THE SURFACE 𝕋 × 𝕋

2008 ◽  
Vol 18 (08) ◽  
pp. 1259-1282 ◽  
Author(s):  
MEIRAV AMRAM ◽  
MINA TEICHER ◽  
UZI VISHNE
Keyword(s):  
Fundamental Group ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Generic Projection ◽  
Galois Cover ◽  
The One ◽  
First Time

This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

scholarly journals Wappapello Dam Spillway Overtopping Event of 2011

2018 ◽  
Vol 46 (2018) ◽  
pp. 69-82
Author(s):  
Gregory H. Nail ◽  
Raymond J. Kopsky
Keyword(s):  
Discharge Coefficient ◽  
Open Channel ◽  
Lake Management ◽  
Hydraulic Modeling ◽  
Flow Modeling ◽  
Intense Rainfall ◽  
One Dimensional ◽  
Modeling Software ◽  
The One ◽  
First Time

Abstract The one-dimensional HEC-RAS multi-purpose open channel flow modeling software was successfully used, with ArcMap and HEC-GeoRAS, to simulate flow over the Wappapello Dam limited-use Ogee spillway (Wappapello, Missouri). Initial computational hydraulic modeling results predicted a lake elevation of 132.9 m (405.0 ft) [NAVD 1988] would be required for the resulting floodwaters overtopping the spillway to reach the nearby Wappapello Lake Management Office. An intense rainfall event during 2011 led to the spillway being overtopped for the first time since 1945. Spillway performance during the 2011 event was analyzed afterwards. Results indicated that the spillway crest was not submerged by backwater. A technique was employed which successfully estimated the design energy head of 7.160 m (23.49 ft) for the spillway. Hydraulic modeling developed after the 2011 event incorporated this estimated design energy head, allowing the spillway discharge coefficient to vary with discharge in the course of an unsteady modeling run. Results indicated that, while the spillway did perform as designed, the performance is limited by the shallow approach depth.

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