Water waves frequency

The molecular collective behavior of surfactant molecules has been analyzed using the time courses of capillary wave frequency after injection of surfactant aqueous solution onto the liquid-liquid interface [5,8]. Typical power spectra for capillary waves excited at the water-nitrobenzene interface are shown in Fig. 3 (a) without CTAB (cetyltrimethy-lammonium bromide) molecules, and (b) 10 s after the injection of CTAB solution to the water phase [5]. The peak appearing around 10-13 kHz represents the beat frequency, i.e., the capillary wave frequency. The peak of the capillary wave frequency shifts from 12.5 to 10.0kHz on the injection of CTAB solution. This is due to the decrease in interfacial tension caused by the increased number density of surfactant molecules at the interface. Time courses of capillary wave frequency after the injection of different CTAB concentrations into the aqueous phase are reproduced in Fig. 4. An anomalous temporary decrease in capillary wave frequency is observed when the CTAB solution beyond the CMC (critical micelle concentration) was injected. The capillary wave frequency decreases rapidly on injection, and after attaining its minimum value, it increases... 

Recently, the newly developed time-resolved quasielastic laser scattering (QELS) has been applied to follow the changes in the surface tension of the nonpolarized water nitrobenzene interface upon the injection of cetyltrimethylammonium bromide [34] and sodium dodecyl sulfate [35] around or beyond their critical micelle concentrations. As a matter of fact, the method is based on the determination of the frequency of the thermally excited capillary waves at liquid-liquid interfaces. Since the capillary wave frequency is a function of the surface tension, and the change in the surface tension reflects the ion surface concentration, the QELS method allows us to observe the dynamic changes of the ITIES, such as the formation of monolayers of various surfactants [34]. 

Taylor et al. (T6, T7) have reported on wave velocities in upward cocurrent gas/film flow. It was found that the wave velocity increased rapidly with increasing gas flow rate but varied little with liquid flow rate. It was found, furthermore, that the individual wave velocities were not uniformly distributed around the mean value under given flow conditions, but that certain preferred velocities appeared to exist. The reasons for such behavior are not clear at present. More recently, Nedderman and Shearer (Nla) have carried out similar studies over a wider range of gas and liquid flow rates. Although the results were similar in many respects, it seems that the wave frequencies, of the large disturbance waves in particular, are dependent on the geometry of the apparatus. These results showed that, at constant water flow rate, the wave velocity for upward cocurrent flow varied with the square root of the air velocity relative to the waves. 

This is not really very complicated and it applies equally well to water waves or electromagnetic radiation. What is almost needlessly complicated is the variety of units commonly used to express k and v for electromagnetic radiation. One problem is tradition, the other is the desire to avoid very large or very small numbers. Thus, as Figure 9-7 shows, we may be interested in electromagnetic wavelengths that differ by as much as a factor of 1016. Because the velocity of electromagnetic radiation in a vacuum is constant at 3 X 108 meters sec-1, the frequencies will differ by the same factor. 

Fig. 4 General solution for the dispersion equation on water at 25 °C. The damping coefficient a vs. the real capillary wave frequency o> , for isopleths of constant dynamic dilation elasticity ed (solid radial curves), and dilational viscosity k (dashed circular curves). The plot was generated for a reference subphase at k = 32431 m 1, ad = 71.97 mN m-1, /i = 0mNsm 1, p = 997.0kgm 3, jj = 0.894mPas and g = 9.80ms 2. The limits correspond to I = Pure Liquid Limit, II = Maximum Velocity Limit for a Purely Elastic Surface Film, III = Maximum Damping Coefficient for the same, IV = Minimum Velocity Limit, V = Surface Film with an Infinite Lateral Modulus and VI = Maximum Damping Coefficient for a Perfectly Viscous Surface Film...

Thus, the dispersion relation for Eqn. (1.4.3), is the statement of governing equation in the spectral plane and tells us that the scale of space variation and the scale of time variation are not independent and they are related. For many other problems, the dispersion relation will be consequence of boundary conditions, as is often derived for water waves developing for an equilibrium solution given by the Laplace s equation. Equation (1.4.5) implies that each frequency component will travel in space with the... 

To elucidate the dynamic molecular behaviour in the phase transfer catalytic reaction, we investigated the time courses of the capillary wave frequencies after the injection of the TEAB, TPRAB and TBAB solutions into the water phase. The time just prior to... 

FIGURE 3.5. Time course of capillary wave frequency after an injection of a quaternary ammonium bromide solution (1 mM) at 283K. (a)TEAB, TPRAB and (c) TBAB. The concentration of CsHsONa in the water phase and DPPC in nitrobenzene phase was 70 mM and 1 mM, respectively. 

Fig. 5. Typical power spectrum of the optically mixed light intensity. O/W, ripplon (capillary wave) frequency at nitrobenzene-water interface W/A, ripplon frequency at water-air interface. (Reprinted from [76] with permission. Copyright The Japan Society of Analytical Chemistry).

Consider first transverse waves with zero dllational modulus K°, low viscosity (rj pco/k ) and m k. These conditions are valid for water at not too high wave frequencies. In this case there is no strong damping, p is small (see 3.6.53j) and so is the vorticity of the flow, I B I I A I. Then, from 13.6.61],... 

The measured relative damping coefficients, i.e., the damping coefficient at a given surfactant concentration, normalized by the damping coefficient y0 for a clean water surface is shown in Figure 5 at different wave frequencies for oleic acid, oleyl alcohol and Emkarox films. 

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