Homogeneous wave equation

If Eq. (16) is substituted into (27), then an identity follows, which suggests that Eq. (27) is not an independent condition. Of course, when Jd = 0 the conventional homogeneous wave equations obtain... 

Let wavefield F(r, t) satisfy the homogeneous wave equation in that domain ... 

The resonator with the highest symmetry is the spherical resonator. A treatment of oscillations in a spherical cavity was already given by Rayleigh in 1894 A recent discussion may be found in Ref. In the simple loss-free model, sound is described by the homogeneous wave equation ... 

Now we consider the wave equation for the propagation of sound in a cylinder of radius Tq and length 1. For cylindrical coordinates (r, 9, z) we obtain the homogeneous wave equation ... 

The starting point in all these models is the calculation of the time-dependent pressure field using the following three-dimensional homogeneous wave equation (Junger and Feit, 1986) ... 

Normal mode A free, or resonant, oscillation of the atmosphere the solution to the homogeneous wave equation with homogeneous (unforced) boundary conditions. 

Combine Maxwell s equations in vacuum with Eqs. 1.39 and 1.42 to generate the homogeneous wave equation for the vector potential in the Coulomb gauge,... 

The practical problems associated with SHG may be appreciated by considering a classical theory for wave propagation in the medium. By combining the Maxwell equations (1.37c) and (1.37d), we obtain the homogeneous wave equation [1]... 

The electromagnetic fields (x,r) and H(x, t) associated with scattering from a microsphere satisfy Maxwell s equations. For a homogeneous, isotropic linear material the time-harmonic electrical held E and the magnetic held H satisfy vector wave equations, which in SI units are (Bohren and Huffman, 1983)... 

We recall that our wave equation represents a long wave approximation to the behavior of a structured media (atomic lattice, periodically layered composite, bar of finite thickness), and does not contain information about the processes at small scales which are effectively homogenized out. When the model at the microlevel is nonlinear, one expects essential interaction between different scales which in turn complicates any universal homogenization procedure. In this case, the macro model is often formulated on the basis of some phenomenological constitutive hypotheses nonlinear elasticity with nonconvex energy is a theory of this type. 

The loss of observable THG in the far field with tight focusing of the beam in homogenous normal dispersion media can be described with the paraxial wave equation [Equation (4.2)] assuming slow spatial variation of electric field amplitudes along the beam propagation direction (z direction). The solution of the paraxial wave equation for the amplitude of third harmonic (A3 J can be written as follows (Boyd 1992) ... 

The wave vector of a homogeneous wave may be written k = (k + zk")e, where k and k" are nonnegative and is a real unit vector in the direction of propagation. Equation (2.45) requires that... 

We showed in Chapter 3 that a physically realizable time-harmonic electromagnetic field (E, H) in a linear, isotropic, homogeneous medium must satisfy the wave equation... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

A second question concerns the existence of longitudinal components of the magnetic field. Maxwell s equations in free space are (completely ) equivalent to two homogeneous uncoupled wave equations for the vector fields E and B. The uncoupled wave equations admit longitudinal components for both fields E and B. However, longitudinal components are prohibited in the conventional interpretation of Maxwell s equations. 

The propagation of light in a nonlinear medium is governed by the wave equation, which was derived from Maxwell s equations for an arbitrary homogeneous dielectric medium,... 

The Hooke s law problem described in Section 2.5 can be revisited. Consider longitudinal waves in a homogeneous line described by x, the position of a particular point on that line, and u, the longitudinal displacement of that point from its equilibrium position. It can be shown (see Problem 5.7.1) that the displacement u obeys a one-dimensional mechanical wave equation ... 

Since a and 3 are represented by 4 x 4 matrices, the wave function / must also be a four-component function and the Dirac wave equation (3.9) is actually equivalent to four simultaneous first-order partial differential equations which are linear and homogeneous in the four components of P. According to the Pauli spin theory, introduced in the previous chapter, the spin of the electron requires the wave function to have only two components. We shall see in the next section that the wave equation (3.9) actually has two solutions corresponding to states of positive energy, and two corresponding to states of negative energy. The two solutions in each case correspond to the spin components. 

For longitudinal plane waves propagating in the x direction through a homogeneous medium of mass density p, the wave equation is... 

Homogeneous dissipation by chemical or molecular relaxation processes in the gas may be addressed on the basis of a formulation like that given in Section 4.3.4. Wave equations arise in which a relaxation time i... 

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

Let us consider in conclusion of this section the case where Cp = Cg = c and U satisfies a homogeneous vector wave equation ... 

Assuming that all external forces are located outside the homogeneous domain V c = const), we arrive at the Kirchhoff integral formula for the vector wave equation... 

We have found in Chapter 13, that the compressional and shear waves in a homogeneous domain satisfy the vector wave equations (13.44) and (13.46). Using the vector identity... 

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