Attenuation equations

Important parameters, characterizing fibre properties, are the attenuation a and numerical aperture NA. The attenuation (Equation 1) represents optical losses caused by absorption or scattering of the light guided through the unit fibre length. It is given by ... 

To minimize signal loss due to spin-spin relaxation, it is often preferable with larger solute molecules to use a stimulated-echo-based sequence (see Fig. 9C) in lieu of the Hahn spin-echo sequence, for which the attenuation equation is given by... 

An analytical approach to determine the current and potential distribution as a function of distance x along a coated pipeline of infinite length are given by the attenuation equations [95] ... 

Estimation of PGA for all earthquake sources considered using appropriate attenuation equation... 

Equation (30a) shows that the relevant peak undergoes a simultaneous migration and diffusion process but is free from kinetic attenuation. Equation (30b) shows a kineti-cally fast-attenuated mode which yields a negligible contribution to the electrophoretic pattern. 

Now, the overpotentials are approximated sis t = j. - E, rfo = (f>o Eeorr o due to Eeorr 0 sincc the pipeline is designed against corrosion. Thus, these assumptions lead to the cathodic protection attenuation equation known as Uhlig s equation for an infinite pipeline which is compared with the disk model equation in mathematical and graphical form (Figure 8.13)... 

The current and potential distribution on structures with cathodic protection will be described for the example of underground pipelines belonging to the group of linear objects. The calculations are based on the relation where potential and current are a function of the distance x along the steel pipelines covered with a coating, and are given by the so-called attenuation equations... 

Solutions of Eqs. (8-38) and (8-39) take the form of well known attenuation equations (Sec. 8.2.2). Current changes, as well as voltage changes in the case of a linear object take an exponential form. The character of changes of current and potential as a function of distance depend mainly on the attenuation constant a given by Eq. (8-10). The derived relationships are of fundamental importance in the design of cathodic protection of linear objects. 

Under these idealized conditions the following attenuation equations apply... 

As sub-THz radiation penetrates a lossy dielectric (e">0), energy is dissipated and, therefore, the power of the radiation is attenuated. Equations relating the absorbed dose rate and the penetration depth with e and e" can be derived from Maxwell s equations details (not given here) for this are available from the relevant literature [3,6],... 

Lin PS, Lee CT (2008) Ground-motion attenuation relationships for subduction-zone earthquakes in northeastern Taiwan. Bull Seismol Soc Am 98(l) 220-240. doi 10.1785/0120060002 Tsai YB, Wen KL, Chen KB, Kuo JY (2000) Summary of development of the Taiwan earthquake catalog and strong ground motion attenuation equations. Report for the National Project on the Disaster Prevention, 79 pp (in Chinese)... 

The development of attenuation equations for specific seismic regions and corresponding derivation of seismic hazard maps in terms of spectral ordinates, as opposed to macroseismic intensity or PGA, has given impetuous to the development of analytical methods. These methods tend to feature more detailed and transparent... 

Total cross-section determinations of sample materials are made by measuring the transmission of the beam through the material. The transmission is defined as the ratio of the intensities of the beam with and without the sample material inserted in the path of the neutrons. After corrections for background (epi-cadmium neutrons) are made on the counting rates, the total cross section for thermal neutrons can be calculated from the basic attenuation equation. This expression states that the fractional loss of the beam intensity per unit path length is constant. 

Attenuation of radiation as it passes through the sample leads to a transmittance of less than 1. As described, equation 10.1 does not distinguish between the different ways in which the attenuation of radiation occurs. Besides absorption by the analyte, several additional phenomena contribute to the net attenuation of radiation, including reflection and absorption by the sample container, absorption by components of the sample matrix other than the analyte, and the scattering of radiation. To compensate for this loss of the electromagnetic radiation s power, we use a method blank (Figure 10.20b). The radiation s power exiting from the method blank is taken to be Pq. 

Reconstmction of the object wave is achieved by illumination of the developed hologram with the reference wave as shown in Figure 3a. The diffracted wave amphtude from the hologram is given by equation 3, where the first term represents the attenuated reference wave after passage through the hologram. 

The transducers on most ultrasound imaging systems operate at a frequency between 1 and 20 MH2. The attenuation, of ultrasound by tissues is both frequency and tissue dependent. The attenuation coefficient, a, of a tissue is defined by equation 5 ... 

Ahrens, T.J., and O Keefe, J.D. (1977), Equation of State and Impact-Induced Shock-Wave Attenuation on the Moon, in Impact and Explosion Cratering (edited by Roddy D.J. et al.), Pergamon Press, New York, pp. 639-656. 

The attenuation factor for a receptor at the center of a finite plane uniform source of radiation is given by equation 8.3-7, where b is the radius and z is the height of the receptor above the plane. If there is... 

This discussion of geometric effects ignored the attenuation of radiation by material through which the radiation must travel to reach the receptor. The number of particles, dN, penetrating material, equals the number of particles incident N times a small penetration distance, dx, divided by the mean free path length of the type of particle in the type of material (equation 8.3-8). Integrating gives the transmission coefficient for the radiation (equation 8.3-9). 

Combining Tr + Tr for a point source gives equation 8.3-10. Radiation is attenuated by the distance from the source, the shielding material for the type of radiation and the thickness of shield that it must penetrate. 

These complex processes are treated as an attenuation process with a build-up factor, B(E,r), in equation 8.3-9 to give equation 8.3-11. 

These coefficients must be multiplied by the density of air and tissue, respectively. Figure 15.7.2-1 depicts a radiation fallout field. Let C be the curie activity/m. The radiation into a unit area receptor at z = 1 m above the ground. The area is emitting C r dr d0 gammas/s. These are attenuated in the air as exp(-p, R) and geometrically as l/(4 7t R). The radiation received by the receptor is given by equation 15.7.2-1 which becomes 15.7.2-2 by a change of... 

The proportion of ionized and unionized forms of a chemical compound can be readily calculated according to the above equation. It can be easily seen that pK is also a pH value at which 50% of the compound exists in ionized form. The ionization of weak acids increases as the pH increases, whereas the ionization of weak bases increases when the pH decreases. As the proportion of an ionized chemical increases, the diffusion of the chemical through the biological membranes is greatly impaired, and this attenuates toxicokinetic processes. For example, the common drug acetosalicylic acid (aspirin), a weak acid, is readily absorbed from the stomach because most of its dose is in an unionized form at the acidic pH of the stomach. 

This equation gives higher transmissivity values than those calculated with methods described earlier. Presumably, Lihou and Maund s transmissivity is to be used for conditions of low relative humidity, in which dust particles (haze) are the main cause of attenuation. A conservative approach is to assume = 1. 

An examination of the attenuation hy an area ratio equation will indicate that for any practical installation, the term... 

These equations should give a check as to the degree of attenuation that may be expected from a given filter system. 

The attenuation of the pressure waves increases with depth and with the mud pressure wave velocity. More attenuation is observed with oil-base muds, which are mostly used in deep or very deep holes, and can be calculated with the mud and pipe characteristics [108] according to the equations... 

The above equations are useful for calculating distance-attenuation effects. 

Complex structures such as tank farms, tank bottoms and marine installations have complicated attenuation patterns and it is not feasible to use complex equations to determine the effect. This must be determined by a current drain test or by practical experience. For pipelines the attenuation depends on the linear resistance of the pipe, coating resistance, and to some extent the resistivity of the soil. Connections to foreign structures and the method of termination of the line also affect the attenuation (see Section 10.1). 

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