Helmholtz wave equation

In free space, an electromagnetic field of frequency w is governed by the homogeneous vector Helmholtz wave equation... 

Employing (1) then gives the homogeneous Helmholtz wave equations of the form... 

This well-known Helmholtz wave equation has two linearly independent solutions which we may take to be the spherical Bessel j (Kr) and Neumann n (Kr) functions, respectively. From the small Kr limits... 

One reason for choosing the tails of muffin-tin orbitals as solutions of the translationally invariant Helmholtz wave equation (5.7) is the extremely simple expansion theorem... 

We again consider an electromagnetic wave propagating parallel to the helical axis of an infinite cholesteric medium (k II qo II z) in the geometry corresponding to Fig. 12.3. Therefore non-zero components of the electric field and Ey depend only on z. The Helmholtz wave equation... 

Let u(r)e be a scalar function representing a time-harmonic (e ) electromagnetic field which satisfies the Helmholtz wave equation... 

In order to take account of retardation, we have to apply electrodynamics rather than electrostatics. All fields must satisfy the Helmholtz wave equation rather than the Laplace equation. 

In order to find the reaction potential of the particle under investigation, we have to solve the Helmholtz wave equation in the presence of the current-charge density given by Eq. (7.55). Within the linear approximation used, we find J(r, t) to exhibit the same frequency as the external potential, yielding... 

The incident electromagnetic radiation travels in a direction characterized by the wavevector k whose magnitude is k =k=2ji/X. The EM radiation is a transverse wave with electric (E) and magnetic (H) fields mutually perpendicular and in-phase, and their cross product points in the direction defined by k. The EM wave is a solution to the Helmholtz wave equation, V f- = 0 where v is the speed of the propagation... 

In this application we consider EM fields in free space consequently both E and H are solenoidal and satisfy Trkalian field relations. Thus, taking the curl of (71), both vector fields satisfy Helmholtz vector wave equations ... 

For their Hertz potential, Rodrigues and Vaz chose the factor < )(f,x) = 4>(x) exp (j fl t). Now, since II satisfies the wave equation, we conclude that the factor (<))x) in turn satisfies the Helmholtz equation ... 

Where l is the length of the capillary, a is the radius of the capillary, Jo, is a Bessel function of the first kind and k = sj / r /poS) = 8, is the viscous skin depth, which is the distance at which the amplitude of the vorti-city (transverse) wave has attenuated by a factor of the natural logarithm e . Inserting (3) into (1) and using Poisson s equation for the charges distribution we can solve for the FDSP Helmholtz-Smoluchowski equation [Reppert et al., 2001],... 

The Helmholtz equation resembles the spatial part of the classical wave equation for matter waves (waves in ocean, sound waves, vibrations of a string, electromagnetic waves in vacuum, etc.) of amplitude F = F(r, f) ... 

Helmholtz solvers the second one is to solve the wave equation of Table 8.1, providing a model for the unsteady reaction rate ujt and solving the corresponding equations either in the time domain or in the frequency domain [270 305 368 258 360 331[. 

Green s funetions for the scalar wave equation and for the corresponding Helmholtz equation... 

The simplest way to solve PDEs is to reduce a PDE with n independent variables to n independent ODEs, each depending on just one variable. This is not always possible, but we will limit our consideration to such cases. In the preceding section, we were able to reduce the wave equation and the heat equation to the Helmholtz equation when the time dependence of T(r, t) was separable. Consider the Helmholtz equation in Cartesian coordinates... 

There are several differential equations that are related to Laplace s equation, e.g. the Poisson equation for the distribution of electric potential in the presence of electric charges, the wave equation for the propagation of a disturbance or the Helmholtz differential equation for the time-invariant distribution of harmonic fields. The latter is of particular relevance for scattering phenomena it has the form ... 

The spherical vector wave functions (SVWF) are the general solution of the vectorial Helmholtz differential equation in spherical coordinates (Xu 1995) ... 

The above equation is a different form of the Helmholtz-Smoluchowski equation. Equation (14) shows that if the average electro-osmotic velocity is determined from the experimentally measured current-time relationship, the zeta potential can be calculated. Introducing the electro-osmotic mobility concept, /tteo = Wave/-Ex, Eq. (14) can be further simplified in the following format ... 

Monochromatic Waves (1.14) A monochromatic e.m. wave Vcj r,t) can be decomposed into the product of a time-independent, complex-valued term Ucj r) and a purely time-dependent complex factor expjojt with unity magnitude. The time-independent term is a solution of the Helmholtz equation. Sets of base functions which are solutions of the Helmholtz equation are plane waves (constant wave vector k and spherical waves whose amplitude varies with the inverse of the distance of their centers. 

Metal-vacuum-metal tunneling 49—50 Method of Harris and Liebsch 110, 123 form of corrugation function 111 leading-Bloch-waves approximation 123 Microphone effect 256 Modified Bardeen approach 65—72 derivation 65 error estimation 69 modified Helmholtz equation 348 Modulus of elasticity in shear 367 deflection 367 Mo(lOO) 101, 118 Na-atom-tip model 157—159 and STM experiments 157 NaCl 322 NbSej 332 NionAu(lll) 331 Nucleation 331... 

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