Analytical Groundwater Mechanics by Otto D. L. Strack

Author:Otto D. L. Strack [Strack, Otto D. L.]
Language: eng
Format: azw3, pdf
Publisher: Cambridge University Press
Published: 2017-08-31T04:00:00+00:00

7.5.1Verification

We demonstrate that (7.92) satisfies the partial differential equation (7.88) as follows. We obtain for ∂Φ/∂x

(7.94)

and for ∂2Φ/∂x2

(7.95)

We combine terms, simplify,

(7.96)

and use (7.93):

(7.97)

We obtain the derivative with respect to time as

(7.98)

We see that indeed equals ∂2Φ/∂x2. We verify that the boundary condition at x = 0 is satisfied by substituting zero for x in (7.92). We also observe that the potential approaches the average value for large values of x; the exponential function approaches zero.

This solution is restricted to positive values of x. If the aquifer occupies the domain −∞ < x ≤ 0, the solution must be modified to

(7.99)

This solution is obtained simply by replacing x by −x. That the solution satisfies the partial differential equation follows from the general solution given in Appendix A, equation (A.14).

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