Page 228 - Physical Principles of Sedimentary Basin Analysis

210 Subsidence

ˆ
Note 7.2 The solution of the temperature equation (7.42) for the transient U(ˆz, ˆ t) is
ˆ
found by separation of variables. The transient temperature is then written as U(ˆz, ˆ t) =
V (ˆ t) W(ˆz), where V (ˆ t) is only a function of ˆ t and the function W(ˆz) is only a function of
ˆ z. When V (ˆ t) W(ˆz) is inserted into the temperature equation (7.42) we get

V W − VW = 0. (7.50)
(A prime denotes differentiation.) Equation (7.50) can be rewritten as

V W 2
= =−λ (constant). (7.51)
V W

The equality V /V = W /W implies that the ratios V /V and W /W are equal to the

same constant, because V /V is only a function of ˆ t, and W /W is only a function of
2
ˆ z. The constant is written as −λ , where it is anticipated that the constant is a negative
2
2

number. Equation (7.51) is two equations, V /V =−λ for V (ˆ t), and W /W =−λ for

2
2
W(ˆz). The solution of equation V =−λ V is V (ˆ t) = b exp(−λ ˆ t) as shown in Exer-

cise 6.5, where coefficient b is the value of V at ˆ t = 0. It is seen by inspection that cos(λˆz)
2
and sin(λˆz) are solutions of the equation W =−λ W. Only the sine function fulfills
the boundary condition (7.44)at ˆz = 0, and can therefore be used. The other boundary
condition requires W(ˆz = 1) = 0, which is only possible for λ = nπ when n is an integer
larger than zero. It is not just one product V (ˆ t)W(ˆz) that solves the temperature equation,
but every product:
V n (ˆ t)W n (ˆz) = b n sin(nπ ˆz) e −(nπ)ˆ t , n ≥ 1 (7.52)
is a solution of the temperature equation. The transient temperature can therefore be written
as a linear combination of all such products as the Fourier series
∞
−(nπ)ˆ t
ˆ
U(ˆz, ˆ t) = b n sin(nπ ˆz) e . (7.53)
n=1
The coefficients b n in the Fourier series are obtained from the initial condition (7.43),
which says that
∞
(β − 1)z, 0 ≤ z < 1/β
b n sin(nπ ˆz) = (7.54)
(1 − z), 1/β ≤ z < 1.
n=1
The unknown coefficients b n are found after both sides of equation (7.54) are multiplied
with sin(mπ) and then integrated over ˆz from 0 to 1. We can then utilize that the functions
W n (ˆz) = sin(nπ) are orthogonal with respect to an inner product defined by integration
from 0 to 1,
1
1/2, n = m
(W n , W m ) = sin(nπ ˆz) sin(mπ ˆz)dˆz = (7.55)
0 0, n = m.