Scalar wave equation

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

This wave equation is tire basis of all wave optics and defines tire fimdamental stmcture of electromagnetic tlieory witli tire scalar function U representing any of tire components of tire vector functions E and H. (Note tliat equation (C2.15.5) can be easily derived by taking tire curl of equation (C2.15.1) and equation (C2.15.2) and substituting relations (C2.15.3) and (C2.15.4) into tire results.)... 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

Maxwell s equations can be combined (61) to describe the propagation of light ia free space, yielding the following scalar wave equation ... 

These equations (14) and (15) determine the scalar and vector potentials in terms of p and J. When p and J are zero, these equations become wave equations with wave velocity c = y/l/pe. That is, A and are solutions of decoupled equations, where they are related by the wave operator... 

Invariance of the fields with respect to changes in potential is known as gauge invariance. It is used to simplify Maxwell s equations in regions where there is no free charge. In this case ip itself is a solution of the wave equation, so that it can be adjusted to cancel and eliminate the scalar potential. This means that in (13) V A= 0 and, as before... 

A wave is described by a wave function y(f, /), either scalar (as pressure p) or vector (as u or v) at position r and time t. The wave function is the solution of a wave equation that describes the response of the medium to an external stress (see below). 

Propagation of non-stationary light beam in a nonlinear medium with material dispersion is described by the scalar wave equation for the linearly-polarized y-component of electrical field E x,z,t) ... 

The solution of the vector wave equation can be written in terms of the generating function ij/, which is a solution of the scalar wave equation... 

Thus, M and N have all of the properties required of the electromagnetic field. Furthermore, ij/ satisfies the scalar wave equation in spherical coordinates. 

Therefore, M and N have all the required properties of an electromagnetic field they satisfy the vector wave equation, they are divergence-free, the curl of M is proportional to N, and the curl of N is proportional to M. Thus, the problem of finding solutions to the field equations reduces to the comparatively simpler problem of finding solutions to the scalar wave equation. We shall call the scalar function ip a generating function for the vector harmonics M and N the vector c is sometimes called the guiding or pilot vector. 

The scalar wave equation in spherical polar coordinates is... 

We have now done enough work to construct generating functions that satisfy the scalar wave equation in spherical polar coordinates ... 

The Maxwell equations become wave equations for ( and ,. In the absence of externally applied currents, conductivity, and externally inserted charges, with scalar electric and magnetic susceptibilities s and // that are constant in each region, we have10... 

Equation (2) is recognized as the four equations of electromagnetism modified by a wave-like scalar field. Equation (1) represents the 10 Einstein equations of general relativity, equated to energy and momentum derived from the fifth dimension. In short, KK theory is a unified account of gravity, electromagnetism and a scalar field. Kaluza s case, 744 = — 2 = — 1, together with the identification 1... 

To understand better the connection between the geometrical optics approach and wave equation solutions, we will discuss in this section the basic equations describing high frequency scalar wavefield propagation. Following Bleistein (1984) and Bleistein et al. (2001), we represent the solution of the scalar wave equation (13.56) outside of the source in the form of the Debye series... 

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

Green s functions appear as the solutions of seismic field equations (acoustic wave equation or equations of dynamic elasticity theory) in cases where the right-hand side of those equations represents the point pulse source. These solutions are often referred to as fundamental solutions. For example, in the case of the scalar wave equation (13.54), the density of the distribution of point pulse forces is given as a product,... 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

As in the scalar case (equation (13.66)), we can conclude that, using Green s tensor G of the vector wave equation, one can find the solution to this equation with an arbitrary right-hand side F (r, t), as the convolution of the Green s tensor G with the function F (r, t), i.e.,... 

Using considerations similar to the one discussed above for a scalar wave equation, one can demonstrate that the Green s tensor for the vector wave equation is... 

Let us consider an acoustic medium. The propagation of acoustic waves can be described by the scalar wave equation... 

The spin of elementary particles is not described by the square-root Klein-Gordon equation. The solutions of the square-root Klein-Gordon equation are scalar wave functions. Real electrons have spin and should be described by a matrix-wave equation. 

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