Sommerfeld radiation conditions

In quite a number of practically important problems of wavefield propagation the initial state of the medium happens to be unknown. This situation occurs, for example, when the direct measurements can be carried out, for technical reasons, only 

We derive first the Sommerfeld radiation conditions for the scalar wave equation. Let us analyze a scalar wavefield, satisfying the following equation, 

As a typical example of this type of a wavefield, we can consider the following elementary time-harmonic spherical waves  

Functions Pi(r, t) and P2 r,t) satisfy equation (13.155) (for P (r,f) = 0) throughout the entire space except at the origin of the coordinates, where they have a singularity of type 1/r. 

The first function. Pi (r, t), corresponds to a time-harmonic wave with its wave-front (defined as the constant phase surface) described by a sphere expanding with time, i.e. Pi(r, t) is nothing else but a divergent spherical wave. The function P2(r, t) characterizes a convergent, i.e. arriving from infinity, spherical wave. 

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These conditions, introduced by Sommerfeld (1912), are called Sommerfeld radiation conditions. We will see below that radiation conditions ensure the obvious physical requirement that the energy of the electromagnetic field travels away from the source domain, i.e. the electromagnetic sources radiate energy outwards from... 

In summary, Sommerfeld radiation conditions for a scalar wavefield can be written as follows ... 

In the conclusion of this section we demonstrate that the Sommerfeld radiation conditions can be extended to the case of an elastic wavefield U (Kupradze, 1933, 1934,... 

Since this field may be represented as a superposition of two types of waves, compressional and shear waves, we are faced with the problem of formulating a certain analytical criterion (similar to the Sommerfeld radiation conditions) that provides for the exclusion from the solution of the elastic field equations of compressional and shear waves that are convergent at infinity. It should also be pointed out that the radiation conditions are not included as some kind of heuristic principle in the initial mathematical formulation of the problem. 

We have demonstrated in a previous chapter that, according to the Sommerfeld radiation conditions (14.4), if the radius R is expanded without limit, the surface integral over Or tends to zero. As a result, we arrive at expression (14.14). 

We assume also that all wavefields - the total, the incident, the scattered, and the transmitted fields - satisfy the Sommerfeld radiation conditions, which are... 

Strictly speaking, to obtain a unique solution in an infinite region, condition (6.10.17) must be supplemented by the Sommerfeld radiation condition [234, 317], so as to eliminate waves coming from infinity. The solution of this problem has the form... 

The most physically plausible open boundary condition is Sommerfeld/Orlanski outgoing wave condition.The Orlanski radiation condition, for example, was used... 

The structure factor F hkl) is the Fourier transform of the unit cell contents sampled at reciprocal lattice points, hkl. The structure factor amplitude (magnitude) F is the ratio of the amplitude of the radiation scattered in a particular direction by the contents of one unit cell to that scattered by a single electron at the origin of the unit cell under the same conditions (see Chapter 3). The first report of the structure factor expression was given by Arnold Sommerfeld at a Solvay Conference. The structure factor F has both a magnitude F(hkl) and a phase rel-... 

The Sommerfeld-Wilson postulate was that non-radiating orbits are specified by the condition that the phase integrals... 

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