Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 1 Notes: Intro to Course
In this course we will consider a number of different types of wave and wave
motions occurring in the ocean and the atmosphere, at many different time and space
scales. In general, wavelike fluctuations are not exact solu
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 3 Notes: Motion Equation
Equations of Motion
du
dt
= constant
D = constant
p gk (1)
u o (2)
Multiply (1) by u
1
(
p
2u u)t u gw o
z
and
t
In the linearized case, at every level z w
[
1
2u u gz]t(pu) 0
or rate of change (kinetic + potential energ
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 2 Notes: Ray Theory
Suppose the medium is not homogeneous (gravity waves impinging on a beach,
i.e. a varying depth). Then a pure plane wave whose properties are constant in space and
time is not a proper description of the wave field.
However, if
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 6 Notes: Rigorous Planning
So far we have considered N = constant, Now we shall assume that N is a slowly varying
function with respect to the phase of the wave (N represents the medium).
N is a much stronger function of z than of (x,y) so let us
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 5 Notes: BruntVaisala Frequency
Consider a continuously stratified fluid with o(z) the vertical density profile.
z
p'
p
z
Figure 1.
At a point P raise a parcel of water by small amount from its equilibrium position P
adiabatically. The change in
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 4 Notes: Deep Waves
2
Deep water waves ( = gk). Again method of stationary phase
Put (x,t) [C(k)e
i (kxt )
i (kxt )
D(k)e
]dk
superposition of waves going in opposite directions
We need to specify at t = o
(x,o) C(k) D(k)e
ikx
dk
and
t (x,o) iwC(
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 10 Notes: Cartesian Coordinates
Considering now motions with L<R, we can write the equations of motion in
Cartesian coordinate:
1
ut  fv px
1.
o
1
2.
vt fu py
o
3.
0 pz
o
1
4.
g
o
ux+vy+wz = 0
t+woz= 0
In the general case:
Again, the vertical mom
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 8 Notes: Reflection Solution
We consider the reflection from a solid boundary which is at some angle with the
horizontal. Consider a twodimensional solution
i t+ikx+imz
e
aligning x with the horizontal wave vector kH
satisfying
2
wzz R wxx = 0.
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 9 Notes: Radius Equations
Thin layer of fluid D/L<1
linearized
v
z
u
p
= longitude (average) of P, = latitude;
u = eastwest velocity
R = earths radius
v = southnorth velocity
w = radial velocity upward
u
1p
2sinv 2cosw
oR cos
t
v
2sinu
1 p
Introduction to Earth Geology and Geographic Movements
GEOG 105

Fall 2005
Lecture 7 Notes: Motion Equation
The equations of motion, linearized, now are:
(1)
u
fv
v
o = o(z), po = po(z)
ox
t
(2)
1 p
fu
1 p
f = constant fplane
oy
t
(3) w 1 pgt
Basic state:
oz o
(4)
po g
o
u v w 0
x
y
z
z
(5) w do 0
t
z
t of (1) and f x (2)