Drag of a Plate Planing on the Shallow Water with Formation of Waves

The plane problem of the plate planing at a constant velocity on the surface of a heavy, ideal, incompressible, finite-depth fluid is considered. The approximate, depth-independent expression for the force acting on the plate is derived from the linear distribution of the fluid velocity along the plate and the height of the flow stagnation point, without regard for jet formation near the leading edge. In this approximate formulation the plate drag depends on its velocity and the trailing edge immersion and does not depend on the planing angle. Experiments and numerical calculations in the exact formulation are performed in the near-critical flow regimes. It is shown that the wave patterns in the experiments and numerical calculations coincide, the formula for the drag being in agreement with the numerical experiments. An approximate criterion of the formation of waves going away from the plate in the forward direction is proposed.     

Spectral Stability of Small Shock Waves

This paper is the first in a series of articles that study the eigenvalue problem for small-amplitude shock waves in systems of conservation laws with viscosity or relaxation. The papers show in various contexts that, in the zero-amplitude limit, appropriately scaled versions of the associated “Evans bundles” converge to suspensions of Evans bundles for fixed shock waves in related reduced systems with lower-dimensional state space. In this article the new geometric framework establishing this connection is introduced in the simplest context, that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension.     

Viscous Profiles for Shock Waves in Isothermal Magnetohydrodynamics

We prove existence of viscous profiles for slow and fast shock waves in isothermal magnetohydrodynamics with a general pressure law both using the Conley index and connection matrices.     

On Maximally Dissipative Shock Waves in Nonlinear Elasticity

Shock waves in nonlinearly elastic solids are, in general, dissipative. We study the following question: among all plane shock waves that can propagate with a given speed in a given one-dimensional nonlinearly elastic bar, which one—if any—maximizes the rate of dissipation? We find that the answer to this question depends strongly on the qualitative nature of the stress-strain relation characteristic of the given material. When maximally dissipative shocks do occur, they propagate according to a definite kinetic relation, which we characterize and illustrate with examples.     

On the Structure of MHD Shock Waves in Diffusive-Dispersive Media

We consider the equations of compressible magnetohydrodynamics including the diffusive effects of fluid viscosity, thermal heat conductivity, electrical resistivity, and, in particular, the dispersive influence of the Hall term. The equations describe the dynamics of a plasma as it arises in astrophysical and technical applications. For a model problem we prove an existence result for shock profiles for all values of the Hall parameter. The same question for the complete six-dimensional system seems to be only solvable for small values of the Hall parameter, that is, viewing the Hall effect as a perturbation. To obtain substantial information on the complete system for all ranges of the dissipation parameters we present a careful numerical study. In particular the study confirms a well-known conjecture on the existence and bifurcation of orbits and illustrates the breaking of symmetry due to the Hall term.     

General Relativistic Shock Waves that Extend the Oppenheimer-Snyder Model

In earlier work we constructed a class of spherically symmetric, fluid dynamical shock waves that satisfy the Einstein equations of general relativity. These shock waves extend the celebrated Oppenheimer-Snyder result to the case of non-zero pressure. Our shock waves are determined by a system of ordinary differential equations that describe the matching of a Friedmann-Robertson-Walker metric (a cosmological model for the expanding universe) to an Oppenheimer-Tolman metric (a model for the interior of a star) across a shock interface. In this paper we derive an alternate version of these ordinary differential equations, which are used to demonstrate that our theory generates a large class of physically meaningful (Lax-admissible) outgoing shock waves that model blast waves in a general relativistic setting. We also obtain formulas for the shock speed and other important quantities that evolve according to the equations. The resulting formulas are important for the numerical simulation of these solutions. (Accepted January 19, 1996)     

Summary of the 24th International Symposium on Shock Waves

The 24th International Symposium on Shock Waves (ISSW24) was held at the Beijing Friendship Hotel from July 11 to July 16, 2004, in Beijing, China, after a one-year delay due to the SARS outbreak in Beijing shortly before the Symposiums originally scheduled date in 2003. The event achieved success due to the continuous support and kind understanding from all the delegates and the International Advisory Committee. During the last three years, I have communicated constantly with so many people who encouraged me by providing their suggestions and advice whenever I was in need, from which I feel a sense of community: the community being full of friendship and understanding. It is very heart-warming to have such an experience and I am very happy to have served as chairman of the Symposium for such a community. On behalf of the Local Organizing Committee I would like to thank all of you for the contributions and help that you have given us, without which we would not have had the Symposium.After the announcement of the ISSW24 was sent out, the response from the international shock wave research community was very encouraging. A total of 460 abstracts were submitted to the ISSW24. Each of the abstracts was evaluated by three members of the Scientific Review Committee and the decision on acceptance was made based on the reviewers reports. 195 oral papers, including nine plenary lectures, were accepted to be presented in three parallel sessions, and 135 poster papers in three dedicated poster sessions. Topics discussed in these papers cover all aspects of shock wave research. Among the topics, supersonic and hypersonic aerodynamics; shock wave reflection, diffraction and focusing; and detonation phenomena and pulse detonation engines were the most popular. Such topics not only include interesting fundamental shock wave physics, but also have important application backgrounds. The plenary lectures of the ISSW24 were selected based on the recommendations from all the IAC members, and review the state of the art of the recent shock wave research. It was also found that a good number of papers were the result of international research collaboration. These facts have demonstrated that the ISSW24 is really international and scientific, and that shock wave research becomes an important research field of continuously increasing interest.The final programme consisted of eight plenary lectures, 123 oral papers and 75 poster papers. The total number is much smaller than that originally accepted due to the change of date, and even though this was chosen carefully, it still overlapped with other conferences, for example, the 24th International Symposium on Rarefied Gas Dynamics from July 10-14, 2004. There were 233 participants registered at ISSW24 from 20 countries and regions: Australia 9, Canada 11, China 58, France 8, Germany 14, India 21, Iran 1, Israel 7, Japan 57, Korea 2, Morocco 1, Netherlands 2, Norway 2, Russia 8, Singapore 3, South Africa 4, Chinese Taiwan 2, Thailand 1, UK 7, and USA 15.The Proceedings is a valuable resource because it brings the recent information of shock wave research together in one place, acts as an introduction to many researchers and students, and serves as a tool for promotion of the ISSW. This is the reason why the Local Organizing Committee works hard to manage to publish it. Two hundred and six papers in total are published in the final proceedings and are organized in such a way that all the papers relating to similar topics are grouped together. Both oral and poster papers are considered to be equal without making any distinction between them. The Proceedings are more comprehensive than the CD-ROM that all the participants received during the Symposium, which only contains the full papers that were received before June 15, 2004, and was intended as a tool for the prompt exchange of research information.The ISSW24 was delayed one year because of the SARS outbreak in Beijing last year. In keeping with the pattern set at the ISSW over the last 40 years, the International Advisory Committee of the ISSW24 has decided that the ISSW25 will be held in the year 2005. The organizer of the ISSW25 has been chosen by voting of all the IAC members at the end of the year 2002 to gain more time for the next organizer. Prof. K. P. J. Reddy from Indian Institute of Science has been selected to be the Chairman of the ISSW25. I sincerely express my congratulations to him and wish ISSW25 a success.We were greatly assisted in planning the ISSW24 by continuous support from various sponsors: the Chinese Academy of Sciences, the Nature Sciences Foundation of China, the Chinese Society of Shock Waves, the Chinese Society of Theoretical and Applied Mechanics and the Chinese Society of Aerodynamics. On behalf of the Local Organizing Committee, I would like thank all members of the International Advisory Committee for their guidance and suggestions, and the Scientific Review Committee for their careful and efficient evaluation of the abstracts. My greatest thanks have to be to the delegates for providing high quality papers and actively participating in the Symposium. For editorial assistance, I wish to thank Dr. C. Wang for re-editing all the papers according to the required style of the ISSW24, and for compiling these proceedings. I must also thank Miss Q. Pu and X. Wang, my secretaries, for taking care of so many details in organizing the Symposium. Many thanks also go to the staff and students of the Key Laboratory of High Temperature Gas Dynamics, the Institute of Mechanics, Chinese Academy of Sciences for helping and supporting me over the past three years. Finally, I want to express my appreciation to Dr. Chris Caron from Springer-Verlag for his kind cooperation in publishing these proceedings.Received: 30 November 2004, Accepted: 30 November 2004, Published online: 24 February 2005[/PUBLISHED]Zonglin Jiang: Chairman of the ISS24Correspondence to: Zonglin Jiang     

Report on the 21st International Symposium on Shock Waves

Shock wave symposium Received 7 September / Accepted 7 September 1998     

Gasdynamic Analogies in Problems of the Oblique Interaction of MHD Shock Waves

Gasdynamic analogies are constructed for the oblique interaction of MHD shock waves (counter colliding or overtaking). These analogies fairly adequately describe the complex dependences of the gas dynamic parameters of the medium on the magnetic field strength and inclination. The complete gas dynamic analogy in which the MHD interaction is simulated by the interaction of two gas dynamic shock waves with Mach numbers calculated on the basis of the fast magnetosonic speeds adequately describe the state of the medium for weak and moderate magnetic fields. The hybrid model, in which the state behind the interacting shock wave is calculated from the MHD relations on discontinuities and the gas dynamic analogy is then used, gives satisfactory results in a stronger field.     

Particle Trajectories in Linear Water Waves

We prove that in linear periodic gravity water waves there are no closed orbits for the water particles in the fluid. Each particle experiences per period a backward-forward motion that leads overall to a forward drift. This paper was written while both authors participated in the program “Wave Motion” at the Mittag-Leffler Institute, Stockholm, in the Fall of 2005.     

Spectral Stability of Small-Amplitude Viscous Shock Waves in Several Space Dimensions

A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues correspond to zeros of an essentially analytic function E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ,ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished that relate the small strength e{\epsilon} of the shock wave to appropriate ranges of values of the spectral parameters (ρλ, ρ ω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality.     

Surface Water Waves and Tsunamis

Because of the enormous earthquake in Sumatra on December 26, 2004, and the devastating tsunami which followed, I have chosen the focus of my mini-course lectures at this year’s PASI to be on two topics which involve the dynamics of surface water waves. These topics are of interest to mathematicians interested in wave propagation, and particularly to Chilean scientists, I believe, because of Chile’s presence on the tectonically active Pacific Rim. My first lecture will describe the equations of fluid dynamics for the free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this, we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. In fact, the signal given by ocean waves generated by the Sumatra earthquake was felt globally; within 48 h distinguishable tsunami waves were measured by wave gages in Antarctica, Chile, Rio di Janeiro, the west coast of Mexico, the east coast of the United States, and at Halifax, Nova Scotia. To describe ocean waves, we will formulate the full nonlinear fluid dynamical equations as a Hamiltonian system [19], and we will introduce the Greens function and the Dirichlet-Neumann operator for the fluid domain along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations, we will derive the known Boussinesq-like systems and the KdV and KP equations, which govern the asymptotic behavior of tsunami waves over an idealized flat bottom. When the bottom is no longer assumed to be perfectly flat, a related theory [6, 13] gives a family of model equations taking this into account. My second lecture will describe a series of recent results in PDE, numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE), the Euler equations for a free surface do not admit clean (‘elastic’) interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical simulations which quantify the degree to which solitary waves lose energy during interactions [3, 4]. It is striking that this degree of ‘inelasticity’ is remarkably small. I will describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon [14, 18].     

Problems on Nonstationary Interaction of Structural Elements with Shock Waves

Experimental investigations into the interaction of rigid and elastic structure elements with a shock wave in water and in air are reviewed. Investigation techniques are described and the nonstationary behavior of elements is analyzed     

On the Motion of Shock Waves at a Constant Speed in Multimodulus Elastic Media

For piecewise linear models of multimodulus elastic media, exact analytical solutions of one-dimensional dynamic deformation problemswith plane or sphericalwave surfaces are presented. Compression-extension regimes with the appearance of a centered Riemann wave and extensioncompression with formation of a shock wave are considered. As the main solution method for the compression phase, we use the method of inverse determination of the boundary condition from known information on the nature ofmotion of the shock wave. Essential qualitative differences of the solutions obtained from the corresponding classical results for linearly elastic media are especially important in the study of the dynamics of porous and cohesive granular media.