Geoscience Reference
In-Depth Information
Fig. A1
Example of an impulse function of magnitude
I
, uniform intensity
I
/
Δ
t
, and duration
Δ
t
.
I
Δ
t
Δ
t
t
0
t
Fig. A2
Illustration of the limit with a rectangular
impulse (i.e. with uniform intensity), making
Δ
t
gradually approach zero to obtain a
(Dirac) delta function.
1
t
Δ
Δ
t
t
0
t
The unit impulse or the (Dirac) delta function can be obtained from (A3), in the limit,
as
t
is made to approach zero, as illustrated in Figure A2. Thus one has in this case
⎧
⎨
<
t
0
−
/
0
for
t
t
2
δ
(
t
−
t
0
)
=
lim
t
→
0
1
/
t
for
t
0
−
t
/
2
<
t
<
t
0
+
t
/
2
(A4)
⎩
0
for
t
>
t
0
+
t
/
2
A similar procedure can be applied for other excitation functions as well; this is illustrated
in Figure A3 for a triangular function.
In general, following these preliminaries, the delta function is often expressed a
0
for
t
=
t
0
δ
(
t
−
t
0
)
=
(A5)
∞
for
t
=
t
0
+∞
δ
(
t
−
t
0
)
dt
=
1
(A6)
−∞
