Wave velocity

Equation 34 has the form of the kinematic wave equation and represents a transition traveling with the wave velocity given by... 

Eor a linear system f (c) = if, so the wave velocity becomes independent of concentration and, in the absence of dispersive effects such as mass transfer resistance or axial mixing, a concentration perturbation propagates without changing its shape. The propagation velocity is inversely dependent on the adsorption equiUbrium constant. 

Adsorption Chromatography. The principle of gas-sohd or Hquid-sohd chromatography may be easily understood from equation 35. In a linear multicomponent system (several sorbates at low concentration in an inert carrier) the wave velocity for each component depends on its adsorption equihbrium constant. Thus, if a pulse of the mixed sorbate is injected at the column inlet, the different species separate into bands which travel through the column at their characteristic velocities, and at the oudet of the column a sequence of peaks corresponding to the different species is detected. 

Both ultrasonic and radiographic techniques have shown appHcations which ate useful in determining residual stresses (27,28,33,34). Ultrasonic techniques use the acoustoelastic effect where the ultrasonic wave velocity changes with stress. The x-ray diffraction (xrd) method uses Bragg s law of diffraction of crystallographic planes to experimentally determine the strain in a material. The result is used to calculate the stress. As of this writing, whereas xrd equipment has been developed to where the technique may be conveniently appHed in the field, convenient ultrasonic stress measurement equipment has not. This latter technique has shown an abiHty to differentiate between stress reHeved and nonstress reHeved welds in laboratory experiments. 

The basic wave velocity is a function of the bulk modulus and density of the fluid, where K = bulk modulus, Pa. 

Fig. 7. Relations between elastic constants and ultrasonic wave velocities, (a) Young s modulus (b) shear modulus (c) Poisson s ratio and (d) bulk...

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

For a given fixed flow rate Q = Vbh, and channel width profile b(x), Eq. (6-56) may be integrated to determine the liquid depth profile h(x). The dimensionless Fronde number is Fr = VVg/j. When Fr = 1, the flow is critical, when Fr < 1, the flow is subcritical, and when Fr > 1, the flow is supercritical. Surface disturbances move at a wave velocity c = V they cannot propagate upstream in supercritical flows. The specific energy Ejp is nearly constant. 

AV = change in liquid velocity a = pressure wave velocity... 

The numerator gives the wave velocity for perfec tly rigid pipe, and the denominator corrects for waU elasticity. This formula is for thin-walled pipes for thick-walled pipes, the factor D/b is replaced by... 

Successive reflections of the pressure wave between the pipe inlet and the closed valve result in alternating pressure increases and decreases, which are gradually attenuated by fluid friction and imperfect elasticity of the pipe. Periods of reduced pressure occur while the reflected pressure wave is travehng from inlet to valve. Degassing of the liquid may occur, as may vaporization if the pressure drops below the vapor pressure of the liquid. Gas and vapor bubbles decrease the wave velocity. Vaporization may lead to what is often called liquid column separation subsequent collapse of the vapor pocket can result in pipe rupture. 

Steady wave A propagating transition region that connects two uniform states of a material. The wave velocities of all parts of the disturbance are the same, so the profile does not change with time, and the assumptions that go into the jump conditions are valid. 

Prompt instrumentation is usually intended to measure quantities while uniaxial strain conditions still prevail, i.e., before the arrival of any lateral edge effects. The quantities of interest are nearly always the shock velocity or stress wave velocity, the material (particle) velocity behind the shock or throughout the wave, and the pressure behind the shock or throughout the wave. Knowledge of any two of these quantities allows one to calculate the pressure-volume-energy path followed by the specimen material during the experimental event, i.e., it provides basic information about the material s equation of state (EOS). Time-resolved temperature measurements can further define the equation-of-state characteristics. 

Thus, by using two VISARs, and by monitoring two beams at 6, both the longitudinal velocity and the shear-wave velocity can be determined simultaneously by solving the above two equations. With a lens delay leg VISAR (Amery, 1976), a precision in determining F(t) to 2% can be achieved. The longitudinal and transverse particle velocity profiles obtained in a study of aluminum are indicated in Fig. 3.12. 

Figure 4.23. Elastic wave velocities as a function of pressure along the Hugoniot of iron. The solid curve is the calculated bulk sound velocity. (From Brown and McQueen (1982).)...

McQueen, R.G., Hopson, J.W., and Fritz, J.N. (1982), Optical Technique for Determining Rarefaction Wave Velocities at Very High Pressures, Rev. Sci. Instrum. 53, 245-250. 

J.N. Johnson, Wave Velocities in Shock-Compressed Cubic and Hexagonal Single Crystals Above the Elastic Limit, J. Phys. Chem. Solids 35, 609-616 (1974). 

If we accept the assumption that the elastic wave can be treated to good aproximation as a mathematical discontinuity, then the stress decay at the elastic wave front is given by (A. 15) and (A. 16) in terms of the material-dependent and amplitude-dependent wave speeds c, (the isentropic longitudinal elastic sound speed), U (the finite-amplitude elastic shock velocity), and Cfi [(A.9)]. In general, all three wave velocities are different. We know, for example, that... 

The parameters for the model were originally evaluated for oil shale, a material for which substantial fracture stress and fragment size data depending on strain rate were available (see Fig. 8.11). In the case of a less well-characterized brittle material, the parameters may be inferred from the shear-wave velocity and a dynamic fracture or spall stress at a known strain rate. In particular, is approximately one-third the shear-wave velocity, m has been shown to be about 6 for various brittle materials (Grady and Lipkin, 1980), and k can then be determined from a known dynamic fracture stress using an analytic solution of (8.65), (8.66) and (8.68) in one dimension for constant strain rate. 

Fig. 2.21. Melting in solid density materials occurs at very high pressures. The release wave velocities measured as a function of pressure in tantalum show a shift from elastic values to bulk values at pressures approaching 300 GPa. Such a behavior is indicative of a melt (after Brown and Shaner [84B02]).

A detonation shock wave is an abrupt gas dynamic discontinuity across which properties such as gas pressure, density, temperature, and local flow velocities change discontinnonsly. Shockwaves are always characterized by the observation that the wave travels with a velocity that is faster than the local speed of sound in the undisturbed mixtnre ahead of the wave front. The ratio of the wave velocity to the speed of sound is called the Mach number. 

The attenuation of the pressure waves increases with depth and with the mud pressure wave velocity. More attenuation is observed with oil-base muds, which are mostly used in deep or very deep holes, and can be calculated with the mud and pipe characteristics [108] according to the equations... 

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