## Solution of the linearised moisture flow equation

A general method to solve analytically the one-dimensional linearized Richards Equation (8.31) is discussed next.

Since (Sfi/St) is negative during the drying process, from Equation (8.27) it is clear that an inflection point can be present along the soil moisture profile only if D(ft) has a minimum. This behaviour of the diffusivity function may be related to vapour diffusivity, which increases as the soil dries (Hillel 1980b).

In this example, a soil water content function is assumed, which may or may not present the inflection point depending on the parameter a:

The second derivative of (8.28) with respect to z shows that ft(z) profile does not present the inflection point if a < 1. If a > 1, the profile has an inflection point moving downward: z/ = ((a — \)/a)lla ■ V4 ■ D0 ■ t. Replacing (8.28) in Equation (8.19), we obtain:

If a < 1, X(u) is an increasing function of u; X(u) has a minimum if 1 < a < 2. X(u) is a decreasing function of u if a > 2.

where & is the normalised soil water content (8.12); D is the hydraulic diffusivity and V is the derivative of the hydraulic conductivity with respect to water content. Here, D and V are taken as constants. The gravity term is taken into account. In some cases, the constant V may assume a different meaning. For example, Equation (8.31) may describe the diffusion from a fixed source in a moving homogeneous medium with velocity V.

The limited utility of the linearized Richards equation is well known, but it may be justified by its simplicity or it may be reasonable in some specific situations (Warrick 1975; Basha 1999; Chen etal. 2001). In the following, the solution of (8.31) is obtained as the sum of two classes of solutions derived for complementary boundary conditions. Therefore, any initial condition and boundary condition can be used to solve Equation (8.31).

The first class of solutions results from choosing the following boundary and initial conditions.

& = &i(z),z > 0,t = 0; & = 0,z = 0,t> 0

&i is the vertical profile of the soil water content at the origin of the time integration.

In order to solve the linearized Richards Equation (8.31) with the (8.32) conditions, the method of separation of variables is used. Following the known procedure to solve the heat diffusion equation (similar to Equation (8.31) with V = 0; see, for example, Carslaw and Jaeger (1986)), the first class of solutions is obtained:

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