Shock wave - SpeedyLook encyclopedia

A problematic definition

A shock according to the everyday usage, it is a transition brutal, not-progressive, without intermediary. One can define this idea in a way mathematical, formally rigorous, with the concept of Continuité. By definition in a continuous space there are always intermediate points. A shock, for the mathematician, it is thus a discontinuous transition.

The existence of the shocks seems incompatible with a principle of Leibniz: nature does not make jumps. Current observations seem to contradict this principle. The space transition to the interface between a liquid and its vapor east seems it rather brutal. Or there is liquid or there is vapor, and one passes brutally from the one to the other. The truth is not so simple. One can model the interface liquid-steamer by a continuous transition where the concentration of the molecules (or their probability of presence) passes continuously from its value in the liquid to that in the vapor. More generally, for very discontinuous model, one can find a model continuous which is very similar for him (as similar as it is wanted). The distinction between continuity and discontinuity thus does not have much direction from the point of view of experimental physics. One can choose the models which one wants. That depends only on what one wants to do. But from the mathematical point of view, the principle of Leibniz is rather right: all that exists can be described with functions.

A Onde with the physical direction it is a Champ. The agitated surface (without splash nor surge) of a lake in is a rather intuitive example. The height of water can vary in space (the surface of the lake) and in time. The movement of surface is described mathematically by a function H of three variables X, there and T.H (X, there, T) is the height of water at the point (X, there) at the moment T.

a shock wave, it is a field where there is a discontinuous space transition moving.

In the case of the surface of the lake, a shock wave would be a water wall which moves (a kind of tidal wave).

A simple theory of the formation of the shock waves

A mathematical discovery of Siméon Denis Poisson on one of the the simplest partial derivative equations led to the development of a mathematical theory of the shock waves. The partial derivative equations are the principal equations which make it possible to study the dynamics of the waves, i.e. the laws of the movement of the waves.

In the case studied by Poisson, one can describe the movement in an intuitive way: each point of the wave seems to move at a characteristic speed: if one follows the wave at this speed on the basis of this point, the state of the wave (the height of water,…) do not change. The characteristic concept of Speed spreads with more complicated cases (spaces with two and three dimensions) for which the explanation above is not more valid. In the case of the its, characteristic speed is the speed of sound (the sound is a wave of pressure in gases, the liquids and the solids.) In a general way, characteristic speed is the propagation velocity of the small disturbances.

In the case studied by Poisson, one can simply envisage the evolution of the form of a Vague. That is to say a wave moving in a direction and let us suppose that characteristic speed varies with the height of vagueness. If the speed at the top is higher than at the base, the top catches up with the base, the front face of vagueness becomes increasingly abrupt. If speed at the top is on the contrary smaller than at the base, it is the back face of the vagueness which becomes moreover el more abrupt. In both cases, one of the faces of vagueness becomes vertical at the end of a finished time. All occurs as if all the parts of vagueness concentrated in the same point. There is a kind of implosion of vagueness on itself. One can also think of a Compression. This is why such shock waves are called compressive.

Such shocks are known as compressive. Speed characteristic with the back of the shock is larger than the speed of the shock, which is itself larger than characteristic speed in front of the shock. These two inequalities are the conditions of Peter Lax. They are always checked for the acoustic shock waves . For a long time one believed that they were always checked, for several reasons.

One did not know a counterexample.
They are a consequence of the compressive character of the formation of a shock wave
an intuitive reason: if speed characteristic with the back of the shock were smaller than the speed of the shock, of the small disturbances could be detached and remain behind, the shock would lose all its energy thus. The same would apply if characteristic speed in front of the shock were larger than the speed of the shock.

One defines a compressive shock as a shock wave which obeys in the conditions of Lax.

There exist shock waves, which one can study in experiments, and which do not obey in the conditions of Lax. They are called under-compressive shocks.

Usual concept of shock wave

Any movement imposed in an unspecified way on a gas can be interpreted by considering a succession of small disturbances which are propagated with the speed of sound. If their intensity is sufficient, they impress our ears.

Under certain conditions, they can be confined in a zone outside which no sound is audible. This phenomenon, called shock wave , meets in many problems of dynamics of gases, particularly in supersonic aerodynamics. Nearby phenomena are also observed in branches very different from physics.

A mobile creates shock waves when its speed becomes higher than the speed of sound. It is allowed to say that a shock occurs when the mobile meets gas particles which were not prevented of its arrival.

History

In 1808, Poisson found a solution with discontinuity of the equations of Euler which satisfied the conservation of the mass and the momentum. Bernhard Riemann, in its thesis of 1860, could not say if it were of a realistic solution or a simple mathematical curiosity.

It is Ernst Mach which elegantly solved the problem by publishing in 1876 a photograph of shock wave produced by a bullet and while posing that the relevant parameter is the speed ratio of the mobile to the speed of sound (the Mach number).

William John Macquorn Rankine (1870) and Pierre-Henri Hugoniot (1887) established independently the equations of the shock wave right based on the conservation of the mass, the momentum and energy.

They did not give any indication on the direction of the shock and the problem was solved by Ludwig Prandtl in 1908. The appearance of a shock wave is related to an increase in the Entropie during the compression which makes pass from subsonic to the supersonic one. The reverse transformation of supersonic with subsonic is done through an isentropic phenomenon called range of relaxation.

Description

The cone of Mach describes in Mach number is a simplified but relevant image of a real shock wave. As long as a mobile infinitely small moves at a speed lower than the speed of sound, the disturbances which it creates move away from him in all the directions. When it exceeds Mach 1, those line up in a cone having the mobile for top. Thus a discontinuity is introduced, which one can describe as shock wave, between the interior of the disturbed cone and outside. It is nevertheless about an infinitesimal shock wave: outside and the interior have behaviors far from different.

A true shock wave appears with a mobile of finished size. One can consider that the cone of Mach previously associated with a point breaks up into lines of Mach. In the case of a considerable mobile of size, each point has its own system of lines of Mach. These various systems combine to give the wave or the shock waves which, superimposing the effects of the various points on the body, now have a finished intensity. For indications on these more complex phenomena which occur in the vicinity of a wing of plane, to see Supersonique and Transsonique.

The shock wave is thus the place of brutal modifications of the component normal speed to the shock, pressure and temperature. In addition, a shock wave is a physical being which cannot obviously have a null thickness. In this one particularly brutal phenomena at the origin of the “sonic” bang observed at any supersonic speed occur. This thickness is nevertheless enough low to be neglected in the concrete applications, which makes it possible to compare the shock wave to a mathematical surface.

The phenomenon of shock wave is universal in nature. Put aside the shock produced in the air by the passage of a supersonic aircraft, our planet itself is surrounded of a shock wave to the interface of the Solar wind and the terrestrial Magnétosphère. In a more general way the shocks are very present in the astrophysical mediums. Thus connection between the solar wind and the local interstellar medium is marked by the shock heliospheric that the probes Voyager I and Pioneer 10 have just crossed. Explosion of Supernova to the starts gamma a whole range of object could draw part of the emission which they produce because of kinetic dissipation of energy by shock waves.

Close phenomena

The preceding considerations were largely developed for objects being driven in the free air (planes, machines, bullets,…). They also apply with some modifications to other gas phenomena (unidimensional shock in the air of a tube pushed by a piston, explosion,…).

The Effet Tcherenkov relates to the light waves. When a nuclear particle comparable to a point moves more quickly than the light in a transparent medium, it occurs a phenomenon completely similar to a cone of Mach.

A boat also produces a wake which locks up all the disturbances that it created previously. The phenomenon differs from the precedents in the sense that the angle which delimits the disturbed zone is 39° independently speed.

See too

Effect Tcherenkov
Mach number

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