Wave velocity surface

Semenov (S7) simplified the wavy flow equations by omitting the inertia terms, which is permissible in the case of very thin films. Expressions are obtained for the wavelength, wave velocity, surface shape, stability, etc., with an adjoining gas stream the treatment refers mainly to the case of upward cocurrent flow of the gas and wavy film in a vertical tube. 

Another figure often used to display the same information as in the index ellipsoid is the wave surface, or ray velocity surface. This is obtained by plotting all the points where light rays would arrive at a given time after leaving the origin. The advanced references [5, 23, 65] have details on how to use the ray and the wave velocity surface if required. 

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. 

For the average pressure wave velocity in the pipe, compute the distance at which the amplitude falls to 1/e of its original value, the distance at which it falls at one-half of its original value (half depth) and the attenuation in dB/1,000 ft. Compute also the amplitude at surface. Bottomhole amplitude peak to peak 200 psi frequencies 0.2, 12 and 24 Hz. 

Aluminum nitride with a large piezoelectric coupling factor and a high surface acoustic wave velocity (5650 m/sec) (see Ch. 10). 

Zinc oxide with high surface acoustic wave velocity (see Ch. 11). 

Description of exptl procedures is given on pp 1930-34 of Ref 15a. A schematic, arrangement for delivery of plane shock wave and for measuring shock-wave velocities for shock strengths from 10 to 90 kbar in the specimens and the free-surface velocity of the specimen plate is shown in Fig 2,. p 1930. 

The various shock-producing systems were calibrated by using free-surface velocity measurements of specimen plates and corresponding shock-wave velocities obtd from the known equations of state of the specimen plate materials. Accdg to Footnote 4 on p 1931 of Ref 15a, the free-surface velocity for a plane shock wave is almost twice the particle velocity"... 

Measurements of the transit times of weak shock waves ( 10Q bar) were used to obtain sound wave velocities in larger specimens than listed in Table II. In the arrangement of Fig 3 a cylinder (or slab) of the expl was immersed in a Plexiglas container filled with water. Initiation of the detonator produced a shock wave which arrived nearly plane thru the water at the surface of the expl specimen. The motion of wave was recorded by a smear camera using a shadowgraph technique. Plots of Us up relationships showed that the resulting curves were nearly straight lines and that for particle velocities, up, from 0.3 to 1.2 mm/ftsec, shock wave velocities are ... 

Many materials whose elastic properties are of interest are anisotropic, so the surface wave velocity depends on the direction of propagation. In order to be able to make measurements in one direction at a time, a lens with a cylindrical... 

In the surface of a half space that is isotropic, the Rayleigh wave velocity is the same in all directions. If the surface is imagined to be in a horizontal plane, then the Rayleigh wave is composed of a shear wave component polarized in a vertical plane (SV) and a longitudinal wave component. Shear waves polarized horizontally (SH) can also exist, but they do not couple to the Rayleigh wave at all (nor, in the case of fluid loading, would they couple into waves in the fluid). 

A code has been written to enable the velocities of surface waves in multilayered anisotropic materials, at any orientation and propagation and including piezoelectric effects, to be calculated on a personal computer (Adler et al. 1990). The principle of the calculation is a matrix approach, somewhat along the lines of 10.2 but, because of the additional variables and boundary conditions, and because the wave velocities themselves are being found, it amounts to solving a first-order eight-dimensional vector-matrix equation. A... 

Fig. 11.8. Rayleigh and pseudo-surface wave velocities measured on GaAs(OOl), indicated by +. The solid lines are the calculated curves of Fig. 11.4(b) (without the longitudinal wave curve), plotted on the enlarged vertical scale used for the experimental...

Crean, G. M., Somekh, M. G., Golanski, A., and Oberlin, J. C. (1987). The influence of thin film microstructure on surface acoustic wave velocity. IEEE 1987 Ultrasonics Symposium, pp. 843-7. IEEE, New York. [220]... 

It is seen from (65) that the wave velocity is considerably smaller than the value given by the first approximation, (58). From (63), the ratio of the mean film thickness in wavy flow to the thickness of a smooth film at the same flow rate is given by 4>1/3 or, from above, 0.93. The corresponding value obtained by Portalski (P3) was 0.94. It is thus seen that for wavy flow of the type assumed here, the mean thickness of the wavy film is 6-7% smaller than the corresponding smooth film. It is pointed out by Kapitsa that it does not follow that there may not be some other type of film surface configuration which would lead to a greater reduction in thickness and, therefore, to greater stability of flow. 

The presence of a gas stream appears to increase the size and randomness of the waves on the film surface (C4, F2, F7). Countercurrent gas flow leads to a decrease in the wave velocity, with the opposite effect for cocurrent flow. 

Feind (F2) has indicated that, in determining the effect of a liquid film flow on an adjoining gas stream, the velocity of the gas relative to the film surface is an important parameter. At present there are few measurements of film surface velocities for the various cases of gas/film flow, and the problem remains as to whether the gas velocity should be considered relative to the true surface velocity of the film in the wavy regime, or to the wave velocity, or to some effective surface velocity. 

Portalski (P3), 1960 Extensive study of film flow on vertical plates, with and without gas flow. Liquids included water, aqueous glycerol solutions, methanol. Data on effects of surface tension changes and surfactants, wave and surface velocities, increase in interfacial area by waves, etc. 

Mayer (M7), 1961 Experimental and theoretical study of wavy flow of water in open channel (slopes up to 5°). Data on growth of turbulent spots, local depths, surface velocity, length of entry zone, wave velocities, heights, frequencies, effect of surface-active materials. 

---