The infinite medium an arbitrary initial distribution

The method of point sources can be extended to any form of the initial distribution C0(x) in the infinite medium. The amount of substance distributed per unit surface between x and x1 + dx is just C0(x )dx. Summing over all the point sources at x from — oo to +oo gives the concentration distribution 

The exponential term which represents the effect of a point source is sometimes called the influence function or Green function of this diffusion problem. The method of sources and sinks easily produces solutions for an infinite medium or for systems of finite dimension when their boundary is kept at zero concentration. Different boundary conditions require a more elaborate formulation (Carslaw and Jaeger, 1959). 

5 The infinite medium with C0(x) being a periodic function of x A useful result is obtained for the initial distribution C0(x) given by 

Inserting this expression into the diffusion equation leads to 

The shorter the wavelength, the faster its decay. Mineral scale heterogeneities in rocks disappear long before meter-scale or even larger heterogeneities. This concept can be extended to any arbitrary combination of periodic functions in Section 2.6, we have already met the idea that any function bounded over an interval can be expanded as a sum of sine and cosine functions. Shorter wavelengths will decay much faster 

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