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In ecology, rarefaction is a technique to assess species richness from the results of sampling. Rarefaction allows the calculation of species richness for a given number of individual samples, based on the construction of so-called rarefaction curves. This curve is a plot of the number of species as a function of the number of samples.
This is called a rarefaction, and indicates the solution typically exists only in a weak, i.e. integral equation, sense. The direction of the characteristic lines indicates the flow of values through the solution, as the example above demonstrates.
The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations.
Rarefaction is the reduction of an item's density, the opposite of compression. [1] Like compression, which can travel in waves ( sound waves , for instance), rarefaction waves also exist in nature. A common rarefaction wave is the area of low relative pressure following a shock wave (see picture).
In supersonic and hypersonic flows rarefaction is characterized by Tsien's parameter, which is equivalent to the product of Knudsen number and Mach number (KnM) or M /Re, where Re is the Reynolds number. [4] [5] In these rarefied flows, the Navier-Stokes equations can be inaccurate. The DSMC method has been extended to model continuum flows (Kn ...
Derivation from linearized MHD equations [1] [2] [3]. In an ideal electrically conducting fluid with a homogeneous magnetic field B, the closed set of MHD equations consisting of the equation of motion, continuity equation, equation of state, and ideal induction equation (see Magnetohydrodynamics § Equations) linearized about a stationary equilibrium where the pressure p and density ρ are ...
NOTE: The equations provided below are only correct when rarefaction takes place on left side of domain and shock happens on right side of domain. The different states of the solution are separated by the time evolution of the three characteristics of the system, which is due to the finite speed of information propagation. Two of them are equal ...
The Gurney equations give a result that assumes the shell or sheet of material remains intact throughout a large portion of the explosive-gas expansion such that work can performed upon it. For some configurations and materials this is true; explosive welding, for example, uses a thin sheet of explosive to evenly accelerate flat plates of metal ...