Wave solution

In contrast to the planar wave solution, the penetration depth in case of a coil can be described —besides the dependence on co and a— as a function of the depth z and the coil diameter R ... 

Health and Safety. The dermal toxicology of alkaline solutions of thioglycolic acid has been reviewed extensively (63—65). The reagent has been found harmless to normal skin when used under conditions adopted for cold waving. Some irritation is observed on abraded skin but this appears to be associated with the alkaline component of the waving solution (65). Hand protection is recommended for the professional hairdressers who routinely handle these products. 

Using the isotherm to calculate loadings in equilibrium with the feed gives rii = 3.87 mol/kg and ri2 = 1.94 mol/kg. An attempt to find a simple wave solution for this problem fails because of the favorable isotherms (see the next example for the general solution method). To obtain the two shocks, Eq. (16-136) is written... 

Equation (9-82) admits of plane wave solutions of the form... 

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

Figure 1. Underlying structure of the "pulsating wave" solution two saddle-type standing waves S i and S2 that are n shifts of each other are...

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

The Uij are the eigenfunctions of the energy and momentum operators, with eigenvalues of hcu and hk respectively. Substitution of the plane-wave solutions and the 4 x 4 a. and j3 matrices into (17) give a set of secular equations [63]... 

Expressions for the electric and magnetic fields can likewise be obtained. These plane-wave solutions are then expanded in terms of spherical harmonics... 

This ensemble of spherical waves forms a complete set. The plane-wave solution of a particle of momentum hk and energy E is therefore represented by... 

Stationary, traveling wave solutions are expected to exist in a reference frame attached to the combustion front. In such a frame, the time derivatives in the set of equations disappear. Instead, convective terms appear for transport of the solid fuel, containing the unknown front velocity, us. The solutions of the transformed set of equations exist as spatial profiles for the temperature, porosity and mass fraction of oxygen for a given gas velocity. In addition, the front velocity (which can be regarded as an eigenvalue of the set of equations) is a result from the calculation. The front velocity and the gas velocity can be used to calculate the solid mass flux and gas mass flux into the reaction zone, i.e., msu = ps(l — e)us and... 

Uses Removing carbon dioxide and hydrogen sulfide from natural gas in emulsifiers, hair waving solutions, polishes softening agent for hides agricultural sprays pharmaceuticals, chemical intermediates corrosion inhibitor rubber accelerator nonionic detergents wool treatment. 

Uses. In the formulations of permanent wave solutions and depilatories in pharmaceutical manufacture as a stabilizer in vinyl plastics... 

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

Fig. 1.25. The s-wave-tip model. The tip was modeled as a spherical potential well of radius R. The distance of nearest approach is d. The center of curvature of tip is To, at a distance (R + d) from the sample surface. Only the 5-wave solution of the spherical-potential-well problem is taken as the tip wavefunction. In the interpretation of the images of the reconstructions on Au(llO), the parameters used are R = 9 A, d = 6 A. The center of curvature of the tip is 15 A from the Au surface. (After Tersoff and Hamann, 1983.)...

The electron is not allowed outside the box and to ensure this we put the potential to infinity outside the box. Since the electron cannot have infinite energy, the wave function must be zero outside the box and since it cannot be discontinuous, it must be zero at the boundaries of the box. If we take the sine wave solution, then this is zero at =0. To be zero at x=a as well, there must be a whole number of half waves in the box. Sine functions have a value of zero at angles of nn radians where n is an integer and so... 

Let us look for plane-wave solutions to the Maxwell equations (2.12)- (2.15). What does this statement mean We know that the electromagnetic field (E, H) cannot be arbitrarily specified. Only certain electromagnetic fields, those that satisfy the Maxwell equations, are physically realizable. Therefore, because of their simple form, we should like to know under what conditions plane electromagnetic waves... 

Consider a plane wave propagating in a nonabsorbing medium with refractive index N2 = n2, which is incident on a medium with refractive index A, = w, + iky (Fig. 2.4). The amplitude of the incident electric field is E(, and we assume that there are transmitted and reflected waves with amplitudes E, and Er, respectively. Therefore, plane-wave solutions to the Maxwell equations at... 

This can be seen directly by substituting into it the plane wave solution,... 

The first equation is scalar, and has a wave solution with velocity Vi = -J c /p). This is the longitudinal wave of eqn (6.7). It is sometimes called an irrotational wave, because V x u = 0 and there is no rotation of the medium. The second equation is vector, and has two degenerate orthogonal solutions with velocity v = s/(cu/p)- These are the transverse or shear waves of eqn (6.6) the degenerate solutions correspond to perpendicular polarization. They are sometimes called divergence-free waves, because V u = 0 and there is no dilation of the medium. Waves in fluids may be considered as a special case with C44 = 0, so that the transverse solutions vanish, and C = B, the adiabatic bulk modulus. 

One may obtain traveling wave solutions with other kinds of boundary conditions. This is, for example, the case when the reaction medium can be visualized as a closed curve in a two-dimensional space, or a closed surface in three-dimensional space (periodic boundary conditions).2... 

Fig. 18. Rotating wavelike solution at various times during the period of oscillation, B = 5.8. Curves of equal concentration for X. The other parameters are as in Fig. 17. The stabilization of the rotating wave solution occurs at B>5.4.

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