The method of ripples

Two dynamic methods have been developed for measurement of the surface tensions, the method of ripples first employed by Rayleigh and the vibrating jet method developed by Bohr. 

The method of ripples. Waves on the surface of a liquid cause periodic local extensions and contractions of the surface. The theory of these was first fully investigated by Thomson (Lord Kelvin),2 who gave the formula, for considerable depths of liquid,... 

Rayleigh L (1890) On the tension of water surfaces, clean and contaminated, investigated by the method of ripples. Phil Mag 30 386-400... 

The dynamic methods depend on the fact that certain vibrations of a liquid cause periodic extensions and contractions of its surface, which are resisted or assisted by the surface tension. Surface tension therefore forms an important part, or the whole, of the restoring force which is concerned in these vibrations, and may be calculated from observations of their periodicity. Dynamic methods include determination of the wave-length of ripples, of the oscillations of jets issuing from non-circular orifices, and of the oscillations of hanging drops. Dynamic methods may measure a different quantity from the static methods, in the case of solutions, as the surface is constantly being renewed in some of these methods, and may not be old enough for adsorption to have reached equilibrium. In the formation of ripples there is so little interchange of material between the surface and interior, and so little renewal of the surface, that the surface tension measured is the static tension ( 12. ... 

Within the computational scheme described in the course of this work, the available information about the atomic substructure (core+valence) can be taken into account explicitly. In the simplest possible calculation, a fragment of atomic cores is used, and a MaxEnt distribution for valence electrons is computed by modulation of a uniform prior prejudice. As we have shown in the noise-free calculations on l-alanine described in Section 3.1.1, the method will yield a better representation of bonding and non-bonding valence charge concentration regions, but bias will still be present because of Fourier truncation ripples and aliasing errors ... 

In the ripple method a series of ripples is caused to travel over the surface of the liquid, the ripples being formed by means of an an electrically driven tuning fork dipping into the liquid. If viewed by means of intermittent illumination conveniently arranged by periodic interception of the light by interposition of a screen attached to one limb of the fork, apparently stationary waves may be observed and the mean wave length readily determined. 

As we shall have occasion to note in dealing with solutions, the composition of the surface phase is very different from that of the bulk liquid. When a liquid interface is newly formed the system is unstable until the surface phase has acquired its correct excess or deficit of solute by diffusion from or into the bulk of the solution. This process of diffusion is by no means instantaneous and, as has been observed in discussing the drop weight method, several minutes may elapse before equilibrium is established. In the ripple method the surfece is not renewed instantaneously but may be regarded as undergoing a series of expansions and contractions, thus we should anticipate that the value of the surface tension of a solution determined by this method would lie between those determined by the static and an ideal dynamic method respectively. 

In addition to the methods discussed here and in Section 6.2, there are a few other methods for measuring surface tension that are classified as dynamic methods as they involve the flow of the liquids involved (e.g., methods based on the dimensions of an oscillating liquid jet or of the ripples on a liquid film). As one might expect, the dynamic methods have their advantages as well as disadvantages. For example, the oscillating jet technique is ill-suited for air-liquid interfaces, but has been found quite useful in the case of surfactant solutions. A discussion of these methods, however, will require advanced fluid dynamics concepts that are beyond our scope here. As our primary objective in this chapter is simply to provide a basic introduction to surface tension and contact angle phenomena, we shall not consider dynamic methods here. Brief discussions of these methods and a comparison of the data obtained from different techniques are available elsewhere (e.g., see Adamson 1990 and references therein). 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

It is interesting to know that the distribution of the ethylene oxide segments is rather independent of the salt concentration, but the point of the pistachio ripples was to investigate how the presence of the polymer affects the distribution of the counterions. To elucidate this in more detail, we employed the modeling technique used to fit the raspberry ripples [4], We recall that this essentially crystallographic technique is a method for producing a single-particle distribution function, p(z),... 

These ideas of Godunov and van Leer were later generalized via the concept of total variation diminishing (TVD) schemes, introduced by Harten [69], whereby the variation of the numerical solution is controlled in a non-linear way, such as to forbid the appearance of any new extremum. Such methods give higher order accuracy without dispersive ripples. 

There exist different methods for impedance measurement. Normally, a sine-wave current is injected and the voltage response is measured. In some special applications, it is possible to use an existing current ripple instead of the sine-wave current source. For example, the a.c. ripple from an alternator that is a part of the system can, perhaps, be used [11]. 

The capillary wave method is based on the generation of harmonic waves on the surface of a bulk volume of liquid [28], The wavelength of the ripples formed, A, is a function of the surface tension, which can be evaluated from expressions given by Kelvin ... 

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