Geology Reference
In-Depth Information
{c(x,y,z)/2} {( / x)
2
+( / y)
2
+( / z)
2
} - ½ {c(x,y,z)}
-1
= 0 (6.6)
and, consistently with Equation 1.181, define the function H satisfying
H = {c(x,y,z)/2}{( / x)
2
+( / y)
2
+( / z)
2
} -1/2 c(x,y,z)
-1
(6.7)
Using Equations 1.210, 1.211 and 1.212, with the parameter from
Chapter 1 set to s, and the dependent variable
set to , the characteristic
equations become
dx/ds = H/ ( / x) = c ( / x) (6.8)
dy/ds = H/ ( / y) = c ( / y) (6.9)
dz/ds = H/ ( / z) = c ( / z) (6.10)
Observe that Equations 1.213 to 1.217 are not required because c(x,y,z) does not
depend on time.
Equations 1.214, 1.215, and 1.216 are used to derive the next group of ray
equations. We will illustrate the steps required in order to implement Equation
1.214. If we take partial derivatives of Equation 6.7 with respect to x, and
simplify the result using Equation 6.5, recognizing that H/
= 0 , w e obtai n
successively
d(
/ x)/ds
= - H/ x
= - { 1 / 2 c
x
{( / x)
2
+ ( / y)
2
+ ( / z)
2
} + 1/2 c
-2
c
x
}
= - { 1 / 2 c
x
{1/c(x,y,z)
2
}+ 1/2 c
-2
c
x
}
(6.11a)
and hence
d(
/ x)/ds = - c
-2
c
x
(6.11b)
Similarly, Equations 1.215 and 1.216 lead to
d( / y)/ds = - c
-2
c
y
(6.12)
d( / z)/ds = - c
-2
c
z
(6.13)
Finally, since H/ (
/ t) = 0, Equation 1.218 requires that
d /ds = (
/ x) H/ (
/ x) + (
/ y) H/ (
/ y)
+ (
/ z)
H/ (
/ z)
= (
/ x) {c(
/ x)} + (
/ y) {c(
/ y)} + (
/ z) {c(
/ z)}
= c { (
/ x)
2
+ (
/ y)
2
+ (
/ z)
2
}
(6.14a)
or, using Equation 6.5,
d /ds = 1/c
(6.14b)
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