Geology Reference
In-Depth Information
2.6.2 The ray equations.
On comparing Equations 2.144 to 2.146 with Equations 2.141 to 2.143, we
obtain the coupled set of nonlinear ordinary differential equations
dk x /dt = -
r x
(2.147)
dk y /dt = -
r y
(2.148)
dk z /dt = -
r z
(2.149)
dx/dt =
r kx
(2.150)
dy/dt =
r ky
(2.151)
dz/dt = r kz (2.152)
Equations 2.147 to 2.152 are solved by numerical integration using standard
computational schemes for ordinary differential equations. Only when the right
sides of Equations 2.147 to 2.149 identically vanish (that is, no explicit
dependences on x, y, and z) do the wavenumbers k x , k y , and k z remain
unchanged along rays. Then, since the right-side arguments in Equations 2.150
to 2.152 depend at most on k x , k y and k z , and are thus constant, the rays
therefore remain straight.
2.6.3 Frequency variation.
We now determine how frequency varies along rays in inhomogeneous
media. Since the function r (k x ,k y ,k z ,x,y,z,t) depends on seven independent
variables, namely, k x , k y , k z , x, y, z, and t, its total differential d
r is known
from calculus as
d
r =
r kx dk x +
r ky dk y +
r kz dk z
+
r x dx +
r y dy +
r z dz +
r t dt (2.153)
Division by dt gives
d r /dt = r kx dk x /dt + r ky dk y /dt + r kz dk z /dt
+ r x dx/dt + r y dy/dt + r z dz/dt + r t (2.154)
The terms in Equation 2.154 can be regrouped in the form
d r /dt = ( r kx dk x /dt + r x dx/dt) + ( r ky dk y /dt + r y dy/dt)
+ ( r kz dk z /dt + r z dz/dt) + r t (2.155)
Equation 2.155 can be simplified using Equations 2.147 to 2.152, so that
d
r /dt
= (-
r kx
r x +
r x
r kx ) + (-
r ky
r y +
r y
r ky )
+ ( -
r kz
r z +
r z
r kz ) +
r t
(2.156)
=
r t
(2.157)
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