Tóth (1962, 1963) has presented two analytical solutions for the boundary-value problem representing steady-state flow in a vertical, two-dimensional, saturated, homogeneous, isotropic flow field bounded on top by a water table and on the other three sides by impermeable boundaries (the shaded cell of Figure 6.1, reproduced in an coordinate system in Figure A7.1).

He first considered the case where the water-table configuration is a straight line of constant slope. For this case, the region of flow in Figure A7.1 is the region *ABCEA*. Since it is not possible to obtain an analytical solution in a trapezoidal region, Tóth approximated the actual region of flow by the shaded region *ABCDA*. He projected the hydraulic-head values that exist along the actual water table *AE* onto the upper boundary *AD* of the region of solution. The approximation is satisfactory for small .

The equation of flow is Laplace’s equation:

(7.1)

For a region bounded by *x = s* and *z* = *z*_{0}, the boundary conditions are

on and

on (7.2)

on

where *c* = tan .

The analytical solution, obtained by separation of variables, is

(7.3)

This equation satisfies both the equation of flow (A7.1) and the boundary conditions (A7.2). When plotted and contoured, it leads to the equipotential net shown in Figure A7.1. Flow is from the recharge boundary *DF* through the field to the discharge boundary *AF*.

Tóth also considered the case where the water-table configuration is specified as a sine curve superimposed on the line *AE* (to represent hummocky topography). The final boundary condition then becomes

on *AD* (A7.4)

where , , and , *a’* being the amplitude of the sine curve and *b’* being the frequency.

The analytical solution for this case takes the form

(A7.5)

The equipotential net described by this function is similar to that of Figure 6.2(b).