Laplace waves

Capillary Ripples Surface or interfacial waves caused by perturbations of an interface. When the perturbations are caused by mechanical means (e.g., barrier motion), the transverse waves are known as capillary ripples or Laplace waves, and the longitudinal waves are known as Marangoni waves. The characteristics of these waves depend on the surface tension and the surface elasticity. This property forms the basis for the capillary wave method of determining surface or interfacial tension. 

Velarde, M. G., and Chu, X.-L. (1988) The harmonic oscillator approximation to sustained gravity-capillary (Laplace) waves at liquid interfaces. Phys. Lett. A 131 430-432. 

When, however, due consideration is given to the full space dependence, the transverse Kelvin-Laplace wave motion is described by the following set of nonlinear partial differential equations... 

This equation gives for the velocity of sound in air at 0° 280 metres per second instead of 331, as obtained by experiment. The discrepancy was explained by Laplace (1822), who pointed out that in the sound wave the changes of volume are so rapid that the conditions are adiabatic, and not isothermal. Hence e = Q,... 

If the system is initially at rest (all derivatives equal zero) and we start to force it with a sine wave the output x, will go through some transient period as shown in Fig. 12,3 and then settle down to a steady sinusoidal oscillation. In the Laplace domain, the output is by definition... 

At Mo = 0, Eq. (12.5) reduces to the standard wave equation. In the 2D Cartesian coordinate system, the Laplace operator A is given by... 

The harmonic potential in the second term in Eq. (3.88) is necessary to avoid unwanted compression or expansion of the wave function in the x-y plane. However, the driving potential in Eq. (3.88) does not satisfy the Laplace equation. The difficulty is overcome by combining this electromagnetic field and the field that compresses (expands) a wave function in a harmonic potential in the z-direction in a fashion that suppresses unwanted excitation. Below we examine the rotation... 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

Only one other general solution exists. Two methods may be used to solve a partial differential equation such as the diffusion equation, or wave equation separation of variables or Laplace transformation (Carslaw and Jaeger [26] Crank [27]). The Laplace transformation route is often easier, especially if the inversion of the Laplace transform can be found in standard tables [28]. The Laplace transform of a function of time, (t), is defined as... 

This equation can be solved by Laplace transform techniques and Mt expressed as modified spherical Bessel functions [28]. However, because the boundary conditions on M are radically symmetric, only the / = 0 (i.e. S-wave) component is of interest. 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

Recent theoretical studies indicate that thermal fluctuation of a liquid/ liquid interface plays important roles in chemical/physical properties of the surface [34-39], Thermal fluctuation of a liquid surface is characterized by the wavelength of a capillary wave (A). For a macroscopic flat liquid/liquid interface with the total length of the interface of /, capillary waves with various A < / are allowed, while in the case of a droplet, A should be smaller than 2nr (Figure 1) [40], Therefore, surface phenomena should depend on the droplet size. Besides, a pressure (AP) or chemical potential difference (An) between the droplet and surrounding solution phase increases with decreasing r as predicted by the Young-Laplace equation AP = 2y/r, where y is an interfacial tension [33], These discussions indicate clearly that characteristic behavior of chemical/physical processes in droplet/solution systems is elucidated only by direct measurements of individual droplets. 

Using the Laplace expansion in terms of the first row of the determinant and changing over the right side of (15.16) again to the wave function of... 

The relation between the CFP with a detached electrons and the reduced matrix elements of operator q>(lNfiLS generating [see (15.4)] the <7-electron wave function is established in exactly the same way as in the derivation of (15.21). Only now in the appropriate determinants we have to apply the Laplace expansion in terms of a rows. The final expression takes the form... 

Due to the isotropic nature of the liquid, the linearized hydrodynamic equations are easily solved when written in the Fourier (wave number) plane. Thus, the basic equations in fluid mechanics in the wavenumber and Laplace frequency (z) plane are written as... 

Consider first the series junction of N waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of A ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance Rj (s), as depicted in Fig. 10.11 for TV= 4. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., V(s) = C v for velocity waves and F(s) = C / for force waves, where jC denotes the Laplace transform. In the discrete-time case, we use the z transform instead, but otherwise the story is identical. 

To obtain Eqs. (2.137) and (2.140), the Dimensionless Parameter Method (DPM) has been used as described in Appendix A and expressions of the concentration profiles have been obtained [52], In the 1960s, a compact analytical solution for the I-E response was obtained by using the Laplace transform method when the oxidized species was the only present in the electrolytic solution, i.e., for a cathodic wave [53, 54], and non-explicit expressions for the concentration profiles and surface concentrations were obtained. 

What is of further interest here, as a model of the hydrogen atom and its angular momentum, is the vibration of a three-dimensional fluid sphere in a central field. As in 2D the wave equation separates into radial and angular parts, the latter of which determines the angular momentum and is identical with the angular part of Laplace s equation. 

This equation can be solved by separation of variables, provided the potential is either a constant or a pure radial function, which requires that the Lapla-cian operator be specified in spherical polar coordinates. This transformation and solution of Laplace s equation, V2 / = 0, are well-known mathematical procedures, closely followed in solution of the wave equation. The details will not be repeated here, but serious students of quantum theory should familiarize themselves with the procedures [15]. 

Both the case where the Laplace transform of K(t) of Eq. (24) diverge (superdiffusion) or vanish (subdiffusion) must be treated with caution. These conditions will be the main subject under study in this review. The existence of environment fluctuations makes it possible for us to interpret the electron transport as resulting from random jumps, without involving the notion of wave-function collapse, but this is limited to the case of Poisson statistics. Anderson... 

A wave may be viewed as a unit of the response of the system to applied input or disturbances. These responses could be in terms of physical deflections, pressure, velocity, vorticity, temperature etc., those physical properties relevant to the dynamics, showing up in general, as function of space and time. Any arbitrary function of space and time can be written in terms of Fourier-Laplace transform as given by,... 

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... 

Figure 2.16 The strip of convergence for the Foiu"ier-Laplace integral in the wave number plane...

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