Geology Reference
In-Depth Information
where ln is the natural logarithm, and
a
δ
t
=
at.
The integrations also yield constants of integration that can be combined into one, so
Eq. (A.3) becomes
ln(
y
)
=
at
+
c.
Taking the exponential of both sides, this becomes
y
=
exp(
at
+
c
)
=
exp(
at
)exp(
c
)
.
(A.4)
To evaluate the constant, we need to know a value of
y
at a particular time,
t
. Suppose
that, at time
t
=
0,
y
has the value
y
0
. Putting these values into Eq. (A.4) gives
y
0
=
exp(0) exp(
c
)
=
exp(
c
)
.
So Eq. (A.4) becomes
y
=
y
0
exp (
at
)
.
(A.5)
This is the basic equation of exponential growth.
If it should happen that
y
is
decreasing
at a rate proportional to its size, the situation
can be covered by taking
a
to be negative. For example, sometimes
a
takes the form
a
=−
1
/τ.
The logic of solution follows exactly as that just described, and the solution takes the form
y
=
y
0
exp(
−
t/τ
)
.
(A.6)
This is the basic equation of exponential decay, including radioactive decay.
We can also define
a
as 1
/τ
in Eq. (A.5). Then
τ
is a growth (or decay) time constant,
an 'e-folding' time. In other words, with each elapse of time
t
=
τ
,
y
increases by the
factor e
2.178. It is related to a half-life or a doubling time (depending on whether it is
decreasing or increasing). From Eq. (A.5), we can deduce that
y
will be twice
y
0
when
t
=
=
0.693
τ
ln(2)
τ
, where ln denotes a natural logarithm. Thus we can say that the doubling
time is
τ
2
=
=
ln(2)
τ .
Similarly the half-life of decay in Eq. (A.6) is
τ
1
/
2
=
ln(2)
τ
.
A.2 Post-glacial rebound
In Section 4.2, a relationship was derived between the depth,
d
, of a depression caused by
pre-existing ice and the rate of uplift,
v
, of the floor of the depression. Equation (4.8)
expresses the relationship as
μ
=
g
(
ρ
m
−
ρ
w
)
dR/v,
which we can rearrange as
v
=
g
(
ρ
m
−
ρ
w
)
Rd/μ.
