Phase speed

Wilson PS, Roy RA, Carey WM (2005) Phase speed and attenuation in bubbly liquids inferred from impedance measurements near the individual bubble resonance frequency. J Acoust Soc Am 117 1895-1910... 

Acceleration of rotation speed is, roughly speaking, wave pattern (phase) speed (20-100km/s), which is enough to enforce the equatorial rotation velocity to reach its break-up velocity. 

Fig.4.38. Separation of carboxylic acids by ion-pair chromatography. Stationary phase N,N-dimethyl-protriptyline, 0.036 M, pH 9.0 (30% on cellulose). Mobile phase cyclohexane-chloroform-1-pentanol (15 4 1). Mobile phase speed 2 mm/sec. Column 300 X 2.7 mm I.D. Peaks B benzilic acid (0.7 nmole) P = phenylbutyric acid (1.1 nmoles) S = salicylic acid (1.4 nmoles).

Another chromatographic parameter is the retention ratio Rr. It is the relative average speed v of an analyte through a chromatographic system compared to the average mobile phase speed or velocity u. 

Presence of the imaginary part with negative sign implies temporal instability for all wave lengths. Also, to be noted that since the group velocity and phase speed in y-direction is identically zero, therefore the Kelvin-Helmholtz instability for pure shear always will lead to two-dimensional instability. 

Then the phase speed of the disturbance field is given by, Cph = tojdr. Additionally, if one defines dj = a + and = tan fiifai), then... 

One can readily testify that for the TS mode the group velocity is positive, showing the TS mode to propagate downstream. For this new mode, the corresponding variation of phase speed and group velocity with are calculated numerically and shown in Fig. 2.26. This testify that the new mode... 

Figure 2.26 Variation of phase speed (top) and group velocity (bottom) of the modes shown in Fig. 2.25, with wq for the Blasius boundary layer at Re = 1000...

Figure 2.29 (a) Phase speed of first five modes as functions of (jSr), (b) Streamwise component of group velocity of first five modes as functions of (/ r) (c) spanwise component of group velocity of first five modes as functions of (/ r)... 

For convected vortical disturbance field in the freestream, the growth process is also qualitatively different as compared to the growth of disturbances that are created at the wall or inside the shear layer- as reported in Schubauer Skramstad (1947). For monochromatic wall excitation, the real frequency of the disturbance field is held fixed and the phase speed adjusts itself continually to the local stability property of the shear layer. Contrarily, for the disturbance held generated by convecting vortices, it is the phase speed or group velocity that is an invariant of the input distur-... 

Mode h real h- tmag Phase speed Group velocity... 

While the wave length, phase speed and group velocity is similar for the first modes in Tables 6.1 and 6.2, the spatial growth rate has increased significantly due to added instability via buoyancy effect. The second mode of these two tables are also similarly related, while the third mode of Table 6.1 has disappeared for the case of mixed convection. Disappearance of modes have been identified in Sengupta et al. (1997) as related to waves attaining phase speed equal to the free stream speed. The third mode of Table 6.2 can be related to the thermal mode (fourth) of Table 6.1. We note that the thermal mode propagate at lower speeds compared to hydrodynamic modes. 

In non-dispersive systems, such as acoustic waves in a fluid or simple tension waves on a string, the wave speed does not vary with frequency. Thus the energy speed Cg is the same as the phase speed c, so that... 

The uncertainties are approximately 0.2% in density at temperatures up to 320 K, 0.5% in density at higher temperatures, 2% in heat capacity above 250 K, 4% in heat capacity at lower temperatures, 0.1% in the vapor phase speed of sound, 3% in the liquid-phase speed of sound, and 0.4% in vapor pressure at temperatures above 200 K. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4-6%... 

The uncertainties in the equation are 0.05% in the saturated-liquid density between 280 and 335 K and 6.2% in density in the liquid phase below 430 K and 10 MPa. The uncertainty increases to 0.3% up to 100 MPa and 0.5% up to 800 MPa. In the vapor phase and at supercritical state points, the uncertainty in density is 1%, whereas in the liquid phase between 430 K and the critical point it is 0.5% in density. Other uncertainties are 0.2% in vapor pressure between 300 and 430 K, 0.5% in vapor pressure at higher temperatures, 2% in heat capacities below 550 K, 5% at higher temperatures, and 1% in the liquid-phase speed of sound below 430 K. The estimated uncertainty in viscosity is 1.0% along the saturated-liquid line, 5% elsewhere. Uncertainty in thermal conductivity is 3%, except in the supercritical region and dilute gas which have an uncertainty of 5%. 

Typical uncertainties in density are 0.02% in the liquid phase, 0.05% in the vapor phase and at supercritical temperatures, and 0.1% in the critical region, except very near the critical point, where the uncertainty in pressure is 0.1%. For vapor pressures, the uncertainty is 0.02% above 180 K, 0.05% above 1 Pa (115 K), and dropping to 0.001 mPa at the triple point. The uncertainty in heat capacity (isobaric, isochoric, and saturated) is 0.5% at temperatures above 125 K and 2% at temperatures below 125 K for the liquid, and is 0.5% for all vapor states. The uncertainty in the liquid-phase speed of sound is 0.5%, and that for the vapor phase is 0.05%. The uncertainties are higher for all properties very near the critical point except pressure (saturated vapor/liquid and single pliase). The uncertainty in viscosity varies from 0.4% in the dilute gas between room temperature and 600 K, to about 2.5% from 100 to 475 K up to about 30 MPa, and to about 4% outside this range. Uncertainty in thermal conductivity is 3%, except in the critical region and dilute gas which have an uncertainty of 5%. 

The uncertainties in the equation of state are 0.1% in density (except near the critical point), 0.25% in vapor-pressure, 1% in heat capacities, 0.2% in the vapor-phase speed of sound, and 3% in the liquid speed of sound. The liquid speed of sound uncertainty is an estimate and cannot be verified without experimental information. The uncertainties above 290 K in vapor pressure may be as high as 0.5%. 

The uncertainties in the equation are 0.5% in density for liquid and vapor states and 1% in density or pressure for supercritical states. For vapor pressure, the uncertainty is 0.3%, that for vapor-phase speed of sounds is 0.2%, and the uncertainty for heat capacities is 5%. 

Fennel, W., Seifert, T, Kayser, B., 1991. Rossby radii and phase speeds in the Baltic Sea. Continental Shelf Research, 11, 23-36. 

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [205]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. 

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