Rayleigh integral

It can be demonstrated that the boundary values of the wavefield and its normal derivative on surface S are not independent (Zhdanov, 1988). Indeed, if we substitute any two continuous functions, / (r, t) and g (r, t), instead of the wavefield U (r, t) and its normal derivative dU (r, t) /dnm expression (15.188), we will still generate some wavefield in the lower half-space. This wavefield can be continuously extrapolated back to the surface of observation S. However, this field and its normal derivative may not be equal to the original functions / (r, t) and g (r, t) on S. In order for formula (15.188) to reproduce the correct wavefield values in the lower half-space and on the surface S, it is necessary and sufficient that integral (15.188) be identically equal to zero at the points of the upper half-space, from which we have at once 

The solution of the integral equations represented by the relationships (15.192), (15.193), and (15.194) with respect to functions du r,t)/dn or du r,uj)/dn proves to be a difficult problem in the case of an arbitrary surface S. However, if the surface 5 is a horizontal plane, these integral formulae can be simplified using a simple geometrical method. 

Then the following equalities hold true for any point r located on S  

Writing equality (15.192) for horizontal surface S z = 0), and replacing the Green s function and its normal derivative at the point r by the Green s function at the symmetrical point r, according to equations (15.195) and (15.196), we obtain the following identity  

Substituting relation (15.197) into (15.188), we arrive at an important modification of the Kirchhoff integral known as the Rayleigh integral formula (Schneider, 1978 Berkhout, 1980)  

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The field radiated into the coupling medium by such a distribution of sources may be obtained by means of the well-known Rayleigh integral. The field at the considered point r is computed by a simple integral over the whole radiating surface of the contributions of each elementary source acting as a hemispherical point source. 

The Champ-Sons model is based upon this approximation. It results into a modified Rayleigh integral where specific terms appear. The resulting formula for the refracted field (e.g., displacement vector-field), is given by... 

The last modification in the Rayleigh integral concerns the delay of propagation between the two points which is simply the time taken by the energy to propagate along the path of stationary phase. It is denoted T and given by,... 

Thus, at points in the lower half-space V z > 0), the Rayleigh integral describes an upgoing wavefield which assumes the given values U(r, t) on the observation plane S, i.e. it may serve as an analytical tool for utilization of the main element of migration - backward extrapolation (continuation) of the observed scattered wave-field. 

In a similar way we can obtain the Rayleigh integral formula in the frequency domain ... 

Another method of practical realization of the Kirchhoff type reverse-time migration is based on the Rayleigh integral formula in the frequency domain (15.200). Applying an inverse Fourier transform to both sides of the Rayleigh formula, we obtain for f = 0... 

Thus we have demonstrated that the spectral extrapolation based on Stolt s method (formulae (15.222) and (15.223)) is equivalent to the reverse-time wave equation migration using the Rayleigh integral formulae (15.203) and (15.205). 

This is known as the Stefan-Boltzmaim law of radiation. If in this calculation of total energy U one uses the classical equipartition result = k T, one encounters the integral f da 03 which is infinite. This divergence, which is the Rayleigh-Jeans result, was one of the historical results which collectively led to the inevitability of a quantum hypothesis. This divergence is also the cause of the infinite emissivity prediction for a black body according to classical mechanics. 

As the attenuation of the incident beam per unit path through the solution, the turbidity is larger than the Rayleigh ratio by the factor Ibrr/S, since T is obtained by integrating Rg over a spherical surface. Thus, if Eq. (10.54) is written in terms of Rg rather than r, the proportionality constant H must also be decreased by l6n/3, in which case the constant is represented by the symbol K ... 

This equation was first given by Lord Rayleigh and is called the Rayleigh equation. Integration between the initial number of moles n o in the still with composition over any time yields the following ... 

By using vapor-liquid equilibrium data the above integral can be evaluated numerically. A graphical method is also possible, where a plot of l/(y - xj versus Xr is prepared and the area under the curve over the limits between the initial and fmal mole fraction is determined. However, for special cases the integration can be done analytically. If pressure is constant, the temperature change in the still is small, and the vapor-liquid equilibrium values (K-values, defined as K=y/x for each component) are independent from composition, integration of the Rayleigh equation yields ... 

Figure 8-38. Graphical integration of Rayleigh or similar equation by Simpson s Rule, for Example 8-14.

In more modem terms, the "Rayleigh criterion" states that positive energy is transferred to the acoushc wave if the pressure fluctuation and heat release fluctuation are in phase. This criterion is usually written in an integral form ... 

Figure 14.8 Integration of the Rayleigh Equation for constant reflux ratio. 

This is the fundamental distillation equation, often referred to as the Rayleigh law when in its integrated form (Rayleigh, 1896). As far as Dt is considered to be a function of F, this equation applies to the change of any species concentration in the course of phase separation. Liquid-vapor or solid-solid fractionations are liable to the same formulation. 

U nlike Rayleigh s original example of a collapsing empty cavity, this bubble will reduce to a minimum size, on compression, after which it will expand to Rj and subsequently it will oscillate between the two extremes R and Rf in. Obviously at the two extremes of radii, motion of the bubble wall is zero - i. e. R = 0. To determine these radii it is necessary to integrate Eq. A.25. With Z = (R /R), the integration yields ... 

The previous section dealt with a horizontal scale change in one domain resulting in both horizontal and vertical changes in the other domain. Rayleigh s theorem, on the other hand, makes a statement about the area under the squared modulus. This integral, in fact, has the same value in both domains ... 

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