Geometric spreading

Sound attenuation is defined as the decrease in intensity of the sound signal as it propagates from the source to the receiver. This definition does not, however, include the change in intensity due to geometrical spreading of the sound wave (such as, for example, the spherical spreading of the sound in the far field of a finite... 

Kendall, J.-M. Thomson, C. J. 1989. Acommenton the form of the geometrical spreading equations, with numerical examples of seismic ray tracing in inhomogeneous, anisotropic media. Geophysical Journal International, 99, 401-413. 

For interstitial treatments there is an analogous dependence of the spatial distribution of the light on the tissue optical properties. However, geometric spreading of the light causes an additional decrease in the fluence with radial distance, r, from the outer surface of the source. In the case of a point isotropic source (equal in all directions), the fluence distribution is of the form ... 

This equation represents the average power per unit area transmitted by the elastic wave. The resulting power can then be multiphed by the time length of the recorded AE waveform. That result is in turn multiphed by the area of the propagating wave front (assumed spherical) at a distance equal to the source-receiver distance. It should be noted that this analysis accounts for attenuation due to geometric spreading, however it does not account for material attenuation. 

For purely diffusive flows, sharp (fresh versus saline water resistivity) discontinuities always smear in time. The dynamics of such flows are very important in log interpretation. For this class of problems, the speed of the fresh-to-saline water interface slows appreciably once the mudcake establishes itself at the borehole walls, as we have demonstrated in Chapter 17. This is especially true in the case of radial flows, where geometric spreading significantly slows the front. For such problems, the speed of the underlying flow U can be neglected after some time, when diffusion predominates. The problem is shown in Figure 21-2. 

Lineal diffusion and undiffusion examples. For simplicity, consider the fresh-to-saline water invasion problem, where mudcake forms and grows at the inlet entrance. At first, mud filtrate motions are extremely rapid, and fluid movements dominate the convection-diffusion process. However, as mudcake forms, the influx of filtrate decreases rapidly with time, and eventually, diffusion dominates the dynamics. For simplicity, we first study lineal flows where the effects of radial geometric spreading are unimportant. In our examples, because fluid convection is negligible, we consider K 5 C/5x = < ) 5C/5t. For numerical purposes, we fix the left-side (x = 1) concentration at C = 10%, while the right (x = 11) is held at C = 90%. For visual clarity, all concentrations to the left side of X = 6 are initially 10%, while those values to the right are 90%. 

Governing saturation equation. Let us now repeat the lineal flow derivation given earlier but include the effects of radial geometric spreading and nonvanishing capillary pressure. Again, analogous Darcy laws apply, namely. 

The path effect P(fr) which is another component of the process that affects the spectrum of motion at a particular site is represented by functions that account for geometrical spreading and attenuation P(f, r) = Z(/ (r)) where... 

Computing the median (rather than mean) amplitude in each time window provides a measurement of tremor amplitude less biased by events or spikes in the data - but is no longer strictly RSAM data. Corrected for the instrument response and geometrical spreading, and then integrating the data, produces reduced displacement. Since 1996 the Alaska Volcano... 

Geometrical spread of the wavefront (divergence) this effect is zero for planar wavefront. 

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