Physics
Capillary Waves
Capillary waves are small, short-wavelength waves that occur at the surface of a liquid. They are caused by the surface tension of the liquid and are typically seen in small bodies of water such as ponds or on the surface of a cup of coffee. These waves are important in understanding the behavior of fluids and the dynamics of liquid surfaces.
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10 Key excerpts on "Capillary Waves"
eBook - PDFOcean Engineering Mechanics
With Applications
- Michael E. McCormick(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
As discussed in Section 2.5, because the air velocity is relatively small, the air flow is laminar. This laminar air flow simply drags the water particles on the free surface in the direction of the flow due to the viscosities of both the air and water. This air flow produces no wave, as illustrated in Figure 3.1a. When the speed of the air increases such that the flow in the boundary layer adjacent to the free surface is turbulent, then the pressure fluctuations on the free surface beneath the turbulent air flow deform the free surface, and small waves are created. These waves are called Capillary Waves, and have a profile similar to that sketched in Figure 3.1b. In that figure, we see that the crest of the wave is broad, while the trough of the wave is narrow. The cause of this rather odd wave profile is the surface tension, which is the dominant force. Capillary Waves travel in the direction of the air flow because of the shear stress on the free surface. For most engineering problems involving water waves, Capillary Waves are of little significance because of their low energy content. As the air speed increases, the energy in the air turbulence increases as does that of the surface shear stress. The air flow can now be referred to as a wind rather than a breeze. The water converts the energies transferred to it by wind turbulence and shear stress into longer waves having sinusoidal profiles, as sketched in Figure 3.1c. These sinusoidal waves are called linear waves because they can be analyzed using linearized equations, as discussed in the next section. One of the linear properties of the waves is that of superposition, that is, linear waves of different heights (H) and lengths ( ) can be combined to form other wave profiles. As discussed in Chapter 5, the most basic analysis of random wave phenomena is that which exploits the property of superposition.
eBook - PDFMechanics of Deformable Bodies
Lectures on Theoretical Physics, Vol. 2
- Arnold Sommerfeld(Author)
- 2016(Publication Date)
- Academic Press(Publisher)
CHAPTER V THEORY OF WAVES Ever since waves were studied, water waves have served the natural scientist as a model for wave theory in general, although they are much more complicated than acoustical or optical waves. As surface waves they are bound to the common surface of two media, while the ordinary acoustic and optical waves are three dimensional waves. There is this fundamental difference between vortices and waves: vortices pull the matter along in their own motion while in a wave the average locomotion of an individual fluid particle vanishes; it is not matter that travels but energy and phase. We shall discuss in this chapter waves with different symmetry char-acteristics, such as plane waves, circular waves, ship waves, and Mach waves, starting out with the simplest type, the plane progressive waves. According to the nature of the restoring force we distinguish gravity waves and Capillary Waves. Gravity waves are the large, conspicuous waves which one usually has in mind in talking of water waves. 23. Plane Gravity Waves in Deep Water We assume the wave as a completely periodic phenomenon and express the time dependence as on p. 98 in the form e~ t0>t ; waves of a more general time dependence can be obtained by superposition of partial waves having different circular frequencies (cf. 26). We further assume that the wave motion is generated out of the state of rest, say, by a gust, a mechanical disturbance or the like (in problem VI, 3 we shall investigate under what circumstances an air current that grazes along a horizontal water surface can produce a wave motion). Since the fluid can be considered as invisela, and since we shall consider in this and in the following article only the potential field of gravity, it follows from the conservation law of 18 that the motion possesses a velocity potential.
eBook - PDF- Reinhard Miller, Libero Liggieri, Reinhard Miller, Libero Liggieri(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
Capillary Waves IN INTERFACIAL RHEOLOGY Boris A. Noskov St. Petersburg State University, Chemical Faculty, Universitetsky pr. 26, 198504 St. Petersburg, Russia Contents A. Introduction 104 B. Theory of Capillary Waves 107 1. Solution of hydrodynamic equations 107 2. Boundary conditions 109 3. Dispersion equation 112 4. Scattering of Capillary Waves 115 C. Experimental studies 120 1. Insoluble monolayers 121 2. Solutions of soluble surfactants 123 2.1. Non-micellar solutions 123 2.2. Micellar solutions 126 2.3. Solutions of polymeric surfactants 128 D. Conclusion 129 E. References 130 Capillary Waves 104 A. INTRODUCTION The idea to apply Capillary Waves to studying surface rheological properties seems to be simple and natural. In the case of bulk liquids the determination of the dynamic compressibility and viscosity of compression (expansion) is mainly based on the application of another kind of mechanical waves – ultrasound waves [1-3]. The corresponding experimental methods are well-established in the physical chemistry [2, 3]. One can assume that surface waves, first of all Capillary Waves, have to play an analogous role in surface science of liquid systems. The influence of surfactants on the surface wave properties is indeed a well-known phenomenon, which reveals the possibility to extract information on the surface rheological properties from the characteristics of surface waves. The realisation of this idea, however, proved to be difficult. It looks surprising but the first experimental studies, where the authors tried to determine the surface rheological properties from measurements of the damping and length of Capillary Waves, appeared only in the sixties of the past century [4-10]. Nowadays Capillary Waves are widely used in surface rheological studies. At the same time the number of systems, where the application of Capillary Waves leads to a detailed description of surface dilational rheology, is limited.
eBook - PDFMechanics of Deformable Bodies
Lectures on Theoretical Physics
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
CHAPTER V T H E O R Y O F W A V E S Ever since waves were studied, water waves have served the natural scientist as a model for wave theory in general, although they are much more complicated than acoustical or optical waves. As surface waves they are bound to the common surface of two media, while the ordinary acoustic and optical waves are three dimensional waves. There is this fundamental difference between vortices and waves: vortices pull the matter along in their own motion while in a wave the average locomotion of an individual fluid particle vanishes; it is not matter that travels but energy and phase. W e shall discuss in this chapter waves with different symmetry char-acteristics, such as plane waves, circular waves, ship waves, and Mach waves, starting out with the simplest type, the plane progressive waves. According to the nature of the restoring force we distinguish gravity waves and Capillary Waves. Gravity waves are the large, conspicuous waves which one usually has in mind in talking of water waves. 23. Plane Gravity Waves in Deep Water We assume the wave as a completely periodic phenomenon and express the time dependence as on p. 98 in the form e~ t03t ; waves of a more general time dependence can be obtained by superposition of partial waves having different circular frequencies (cf. 26). W e further assume that the wave motion is generated out of the state of rest, say, by a gust, a mechanical disturbance or the like (in problem VI, 3 we shall investigate under what circumstances an air current that grazes along a horizontal water surface can produce a wave motion). Since the fluid can be considered as inviscid, and since we shall consider in this and in the following article only the potential field of gravity, it follows from the conservation law of 18 that the motion possesses a velocity potential.
eBook - PDF- Ira M. Cohen, Pijush K. Kundu(Authors)
- 2004(Publication Date)
- Academic Press(Publisher)
73 cm . (7.68) Only small waves (say, λ < 7 cm for an air–water interface), called ripples , are there-fore affected by surface tension. Wavelengths < 4 mm are dominated by surface ten-sion and are rather unaffected by gravity. From equation (7.66), the phase speed of these pure Capillary Waves is c = 2 πσ ρλ , (7.69) where we have again assumed tanh ( 2 πH/λ) 1. The smallest of these, traveling at a relatively large speed, can be found leading the waves generated by dropping a stone into a pond. 8. Standing Waves So far, we have been studying propagating waves. Nonpropagating waves can be gen-erated by superposing two waves of the same amplitude and wavelength, but moving 8. Standing Waves 223 in opposite directions. The resulting surface displacement is η = a cos (kx − ωt) + a cos (kx + ωt) = 2 a cos kx cos ωt. It follows that η = 0 for kx = ± π/ 2 , ± 3 π/ 2 . . . . Points of zero surface displacement are called nodes . The free surface therefore does not propagate, but simply oscillates up and down with frequency ω , keeping the nodal points fixed. Such waves are called standing waves . The corresponding streamfunction, using equation (7.50), is both for the cos (kx − ωt) and cos (kx + ωt) components, and for the sum. This gives ψ = aω k sinh k(z + H) sinh kH [cos (kx − ωt) − cos (kx + ωt) ] = 2 aω k sinh k(z + H) sinh kH sin kx sin ωt. (7.70) The instantaneous streamline pattern shown in Figure 7.14 should be compared with the streamline pattern for a propagating wave (Figure 7.7). A limited body of water such as a lake forms standing waves by reflection from the walls. A standing oscillation in a lake is called a seiche (pronounced “saysh”), in which only certain wavelengths and frequencies ω (eigenvalues) are allowed by the system. Let L be the length of the lake, and assume that the waves are invariant along y . The possible wavelengths are found by setting u = 0 at the two walls.
eBook - PDF- P.H. LeBlond, L.A. Mysak(Authors)
- 1981(Publication Date)
- Elsevier Science(Publisher)
Using the exact form for ps given in (11.2) and the kinematic boundary condition (12.38d), together with a modified form for (12.38b) which includes the influence of surface tension, but not that of gravity, formulate the problem of capillary wave motion in a form analogous to (12.36)-(12.38). 9. Choosing the wavelength X for a vertical as well as for a horizontal scale, a period T = (p/uki) for a time scale and an appropriate scale for the potential 9 (to be dis- covered), reduce the capillary wave problem to a form similar to (12.41)-(12.43) in terms of the single small parameter e = a/X. Solve to the first nonlinear correction r#~(l). 10. Show that plane wave solutions of & + V& = 0, where V is a complex quantity, contain a part which grows exponentially in time. 11. Using (12.76), calculate the waveheight h, measured from trough (x' = 0) to crest (x' = n/6) of the progressive wave of maximum amplitude; show that h / h = 0.1374. Also calculate, again from (12.76), the mean level Z and show that the height of the wave crest above the mean level satisfies (h -Z)/h = 0.0926. 1 08 13. AVERAGE PROPERTIES OF HIGH-FREQUENCY GRAVITY AND Capillary Waves Waves consist of oscillations about some equilibrium state and are described by the time and space dependence of these variations from equilibrium. They are also charac- terized by average properties, such as frequency, wavenumber, speed, energy density and flux, which generally vary over space and time scales much larger than those describing the fluctuations about equilibrium. Some of these average properties have already been discussed in general terms in Section 6. We shall now describe the more significant average properties of gravity-Capillary Waves discussed in the previous sections.
eBook - PDF- J. F. Danielli, M. D. Rosenberg, D. A. Cadenhead, J. F. Danielli, M. D. Rosenberg, D. A. Cadenhead(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
Both damping coefficient and wave number of these waves depend sensitively on surface elastic modulus from quite low to (substantially) arbitrarily high values (as pre-viously mentioned, the transverse or Capillary Waves are sensitive to this property only in the neighborhood of the damping maximum). The longitudinal waves can also be investigated experimentally over many decades in frequency, so they are potentially of great interest in the study of monolayer relaxation phenomena. The longitudinal waves are contained in hydrodynamic theories of the Levich type; Lucassen was the first to recognize that these theories contained a second root of physical importance and to generate the motion corresponding to this root. B. The problem The capillary wave problem is illustrated in Fig. 1 with representative parameter values (vertical and horizontal axes should be imagined differently scaled, so that the actual ratio of amplitude to wavelength is 10 3 or less). A probe moving perpendicular to the surface with fre-quency ν (angular frequency ω = 2πν) generates a wave (here shown 4 ROBERT S. HANSEN AND JAMIL AHMAD Fia. 1. Interfacial wave propagating to the right, with characteristics to be measured and explained indicated. Ripple characteristics: (1) wavelength λ = 2π/κ; (2) damping coefficient α, ζ = a COS(KX — wt)e~ ax for 200-Hz waves, λ ~ 0.2 cm, α ~ 0.5 cm 1 , κ ~ 30 c m -1 . propagating to the right) which has a characteristic wavelength λ (or wave number κ = 2π /λ) and which damps exponentially with distance so that there is a logarithmic decrement α (the damping coefficient) with unit increase in distance χ from the probe. The theoretical problem has to do first with understanding how such a simple motion can arise, and second with understanding how the wavelength and damping coefficient are related to the frequency, to properties of the fluids meeting at the interface, and to properties of the interface.
eBook - PDF- Pijush K. Kundu, Ira M. Cohen(Authors)
- 2001(Publication Date)
- Academic Press(Publisher)
Waves are the means by which information is transmitted between two points in space and time, without movement of the medium across the two points. The energy and phase of some disturbance travel during a wave motion, but motion of the matter is generally small. Waves are generated due to the existence of some kind of restoring force that tends to bring the system back to its undisturbed state, and of some kind of inertia that causes the system to overshoot after the system has returned to the undisturbed state. One type of wave motion is generated when the restoring forces are due to the compressibility or elasticity of the material medium, which can be a solid, liquid, or gas. The resulting wave motion, in which the particles move to and fro in the direction of wave propagation, is called a compression wave, elastic wave, or pressure wave. The small-amplitude variety of these is called a sound wave. Another common wave motion, and the one we are most familiar with from everyday experience, is the one that occurs at the free surface of a liquid, with gravity playing the role of the restoring force. These are called surface gravity waves. Gravity waves, however, can also exist at the interface between two fluids of different density, in which case they are called internal gravity waves. The particle motion in gravity waves can have components both along and perpendicular to the direction of propagation, as we shall see. In this chapter, we shall examine some basic features of wave motion and illustrate them with gravity waves because these are the easiest to comprehend physically. The wave frequency will be assumed much larger than the Coriolis frequency, in which case the wave motion is unaffected by the earth's rotation. Waves affected by planetary rotation will be considered in Chapter 14. Wave motion due to compressibility effects will be considered in Chapter 16.
eBook - PDF- Pijush K. Kundu, Ira M. Cohen(Authors)
- 2010(Publication Date)
- Academic Press(Publisher)
73 cm . (7.68) Only small waves (say, λ < 7 cm for an air–water interface), called ripples , are there-fore affected by surface tension. Wavelengths < 4 mm are dominated by surface tension 8. Standing Waves 237 and are rather unaffected by gravity. From equation (7.66), the phase speed of these pure Capillary Waves is c = 2 πσ ρλ , (7.69) where we have again assumed tanh ( 2 πH/λ) 1. The smallest of these, traveling at a relatively large speed, can be found leading the waves generated by dropping a stone into a pond. 8. Standing Waves So far, we have been studying propagating waves. Nonpropagating waves can be gen-erated by superposing two waves of the same amplitude and wavelength, but moving in opposite directions. The resulting surface displacement is η = a cos (kx − ωt) + a cos (kx + ωt) = 2 a cos kx cos ωt. It follows that η = 0 for kx = ± π/ 2 , ± 3 π/ 2 . . . . Points of zero surface displacement are called nodes . The free surface therefore does not propagate, but simply oscillates up and down with frequency ω , keeping the nodal points fixed. Such waves are called standing waves . The corresponding streamfunction, using equation (7.50), is both for the cos (kx − ωt) and cos (kx + ωt) components, and for the sum. This gives ψ = aω k sinh k(z + H) sinh kH [ cos (kx − ωt) − cos (kx + ωt) ] = 2 aω k sinh k(z + H) sinh kH sin kx sin ωt. (7.70) The instantaneous streamline pattern shown in Figure 7.14 should be compared with the streamline pattern for a propagating wave (Figure 7.7). A limited body of water such as a lake forms standing waves by reflection from the walls. A standing oscillation in a lake is called a seiche (pronounced “saysh”), Figure 7.14 Instantaneous streamline pattern in a standing surface gravity wave. If this is mode n = 0, then two successive vertical streamlines are a distance L apart. If this is mode n = 1, then the first and third vertical streamlines are a distance L apart.
eBook - PDFAir and Water
The Biology and Physics of Life's Media
- Mark Denny(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
These waves can be used in a form of echolocation. as fast as the animal itself! This can be of biological importance. For example, as whirligig beetles swim, they produce Capillary Waves that move off ahead (fig. 13.17). These leading waves may serve to warn prey of the beetle's approach, but they may also be used by the beetle itself. Because Capillary Waves move faster than the animal, the beetle can use its own waves in a form of echolocation. Waves reflected by objects in front of the beetle return to the beetle and can presumably be sensed (Tucker 1969). The jerky, intermittent pace with which these beetles swim may serve to send out discreet pulses of Capillary Waves in much the same way that bats and dolphins send out pulses of sound. Capillary Waves may also serve to signal the presence of stationary objects in a moving stream. For instance, a fishing line or reed held stationary through the surface of a stream produces upstream of itself a series of Capillary Waves. Aside from providing the knowledgable observer with visual evidence that the group velocity of Capillary Waves is indeed faster than the wave celerity, the presence of these waves may serve as a warning to whirligig beetles and water striders that there is an obstruction ahead. 13.13 Group Velocity—A General Formula If gravity waves in deep water have a group velocity half that of the wave celerity and Capillary Waves a c g half again larger than c, what happens with waves that are influenced by both gravity and surface tension? As one might expect, the group velocity of these intermediate waves is itself intermediate. Lamb (1945) has shown that C O =C 1 A 2 -X l (13.68) where A m ^ n (eq. 13.30) is the wavelength at minimum wave celerity (i.e., 17 mm). A brief inspection of this equation shows that when A A min , c g approaches c/2 as it should. When A = A min , c g = c. Eq. 13.68 is graphed in figure 13.18.
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