Plane wave solutions

We can show by direct substitution that the scalar wave equation (2.20) has solutions of the form (Problem 2.1) 

In terms of the wavelength A we have k = 2tt/A. At any instant t, the amplitude of the wave is constant over the surface 

This is the equation of a plane normal to the vector k and the solution (2.24) therefore represents a plane wavefront travelling with velocity v in the direction k (Problem 2.2) However, the electric and magnetic fields are vector quantities and we therefore assume that the plane-wave solutions of equations (2.19) are of the form  

This notation is very useful since all operations which are linear in the fields can be performed directly on the complex expressions of equations (2.27) and (2.28) and the real part is taken at the end of the calculation. Unfortunately expressions involving products of the fields cannot be treated in this simple way, as we explain in section 2.5 below. 

This means that both E and H are perpendicular to the direction of propagation k, and these solutions to Maxwell s equations therefore describe transverse waves. Important relations connecting the magnitudes and directions of E and H are obtained by the application of equation (2.17), giving 

From the eigenvectors of h(p) we obtain the solutions of the Dirac equation that correspond to the plane waves of the Schrodinger equation. The four linearly independent functions 

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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... 

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... 

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 relation (803) interpreted as one between eigenvalues is compatible with the plane-wave solutions... 

Consider a non-plane-wave solution of Maxwell s equations, whose direction of propagation varies with respect to the z axis. In general it holds that... 

Up to now, we have examined how the Beltrami vector field relation surfaces in many electromagnetic contexts, featuring predominantly plane-wave solutions (PWSs) to the free-space Maxwell equations in conjunction with biisotropic media (Lakhtakia-Bohren), in homogeneous isotropic vacua (Hillion/Quinnez), or in the magnetostatic context exemplified by FFMFs associated with plasmas (Bostick, etc.). 

Using plane-wave solutions we get the following dispersion relation in a covariant form... 

Non-plane-wave solutions of the Klein-Gordon equation using unconventional basic functions and coupling ansatz ... 

PROBLEM 2.7.6. For the situation of Problem 2.7.4, find the plane-wave solution. 

Looking for plane wave solutions amounts to testing nontrivial solutions of the type... 

The equations of motion are constructed from this model and plane-wave solutions are sought. The frequencies of vibrational modes with a given q can be determined by the diagonalisation of F(q)X(q) ... 

There is a well known expansion theorem to expand the plane wave solution (which are the solutions to the free particle equation) in terms of Bessel functions as well [39] ... 

Subsequently, (2.117) is discretized by the operators of (2.116) and Hz is replaced with plane-wave solutions H e k x+k< y where knum = 7(uimx + 7""" y is the numerical wavenumber. These steps lead to... 

For L — 0 case eq.(34) will be equivalent to the eigenequation for a free noninteracting electron gas. The projection weights will become Dirac delta functions P ( r) = 5(r-ra). Inserting the exact plane wave solutions into eqs.(19) and (21) leads to [56]... 

In Section V.B dispersion relations were given for the squeezing and bending modes, assuming a plane wave solution for the amplitude t) of both modes ... 

We propose the following plane wave solution (propagating in the x direction) for the Electric Field ... 

The equations of motion, Eqs. (36) and (37) in Section 5, contain many coefficients (a , bi, Ci, etc.) that are defined mathematically in that section. Similarly, in Section 6, when a particular solution is written, the coefficients (B, c, C, D ) are defined explicitly in terms of physical quantities. In Section 7, where the plane wave solution for the porosity diffusion wave is given, all the symbols and coefficients are defined immediately after they are written down, or have been previously defined. 

We note that these conditions are satisfied for the plane wave solutions which are considered below. 

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