text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
z
∂x
∂y
�
�
∂
∂
∂
�
�
∂z
∂x
∂y
�
�
�
�
F1 F2 F3
∂2F2
∂2F3
∂x∂z
∂x∂y
−
=
=
−
∂2F3
∂y∂x
+
∂2F1
∂y∂z
+
∂2F2
∂z∂x
−
∂2F1
∂z∂y
= 0.
This is (2).
Example 18.8. The gravitational field
�
�
F (x, y, z) =
cx
(x2 + y2 + z2)3/2
ˆı+
cy
(x2 + y2 + z2)3/2
jˆ+
cz
(x2 + y2 + z2)3/2
ˆ
k,
is a gr... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
2f = div(grad f ) =
∂2f
∂x
+
∂2f
∂y
+
∂2f
.
∂z
3
A solution of the differential equation
�2f = 0,
is called a harmonic function.
Example 18.11. The function
f (x, y, z) = −
c
,
(x2 + y2 + z2)1/2
is harmonic.
4
MIT OpenCourseWare
http://ocw.mit.edu
18.022 Calculus of Several Variables
Fall 2010
For... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
Figure removed for copyright reasons.
See Fig. 2 in Bockstaller et al. "Size-selective Organization of Enthalpic Compatibilized Nanocrystals in Ternary
Copolymer/Particle Mixtures." J. Amer. Chem. Soc. 125 (2003): 5276-5277.
Figure removed for copyright reasons.
Figure removed for copyright reasons.
See Fig. 2 in Bocks... | https://ocw.mit.edu/courses/3-012-fundamentals-of-materials-science-fall-2005/2812d169e1ca0ba01f2a6fa47f872f3e_lec02t_note.pdf |
Perfect Conductivity Lecture 2
Terry P. Orlando
Dept. of Electrical Engineering
MIT
September 13, 2005
Massachusetts Institute of Technology
6.763 2005 Lecture 2
Outline
1. Persistent Currents
2. Parts of a Physical Theory
3. Circuits and Time Constants
4. Distributive Systems and Time
constants
A.Quasista... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/285aa1ec51ef050027112c396c6eba73_lecture2.pdf |
2. Constitutive Relations
Massachusetts Institute of Technology
6.763 2005 Lecture 2
4
3. Summary Relation
1
jωC
jωL
R
Massachusetts Institute of Technology
6.763 2005 Lecture 2
Simpler Circuits and Time Constants
L
R
LC
RC
Energy stored
in inductor
Resonant transfer of
energy between L and C
Energy s... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/285aa1ec51ef050027112c396c6eba73_lecture2.pdf |
5 Lecture 2
Quasistatic Limit
Length scale
of system
Wavelength of
E&M wave
Speed
of light
Frequency
(angular)
If the dimensions of a structure are much less than the
wavelength of an electromagnetic field interacting with it, the
coupling between the associated electric and magnetic fields is
weak and a qu... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/285aa1ec51ef050027112c396c6eba73_lecture2.pdf |
34, from Orlando, T., and
K. Delin. Foundations of Applied Superconductivity.
Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234.
Magnetic Diffusion Equation
Massachusetts Institute of Technology
6.763 2005 Lecture 2
10 | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/285aa1ec51ef050027112c396c6eba73_lecture2.pdf |
Electronics A
Joel Voldman
Massachusetts Institute of Technology
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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� × = ⇒
0
E
E r
( , )
t
= −∇
V
( , )
r
t
( )
V b V a
−
( )
= −
b
∫
a
E l
d
⋅
( )
V b V a
−
( )
=
V Eg
=
⇒ =
E
V
g
g
Q
=
ε
A
V
g
=
V CV
=
A
ε
g
A
ε
g
C
=
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
dI
dt
V L
=
-
-
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 6E - 6
Source elemen... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
1
Cs
V IZ
=
L
=
ILs
V IZ
=
R
=
IR
-
-
-
-
V
Z
V s
( )
=
I s Z s
( )
( )
I
+
I
+
I
+
+
i
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD M... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
C
V i Z
−
C C
i
= − =
L
i
C
0
i
R
i Z
R
R
=
0
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007,... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
CourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 6E - 13
Nodal analysis
> Element law approach becomes tedious for circuits
with multiple loops
> Nodal analysis is a KCL-based approach
Cite as: Joel Voldman, course materials... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
−
Z
C
i
1
+ + =
v V
=
1
0
i
0
3
v
2
=
0
i
2
0
−
Z
R
v
2
+
−
Z
L
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
determine
2. Replace elements with
complex impedances
3. Assign node voltages &
ground node
4. Write KCL at each node
v1
i1
C
i2
i3
v2
L
V0
+
-
R
LRs
v
2
=
V
0
LRs
+
R Ls
1
Cs
1
+
Cs
2
LRCs
2
+
LRCs
Ls R
+
5. Solve for node voltages
v
2
=
V
0
6. Use node voltages to
find what you care about
V
C
=
v
1
−
v
2
=
V V
−
0... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
Z
1
i
i1
i2
i
Z1
Z2
Z
+
V
-
+
V
-
V i Z
=
1 1
=
i
Z
=
Z Z
1
2
Z
Z
+
1
1
Z
2
1
Z
1
=
Z Z
1
2
Z
Z
+
1
2
=
Z
1
//
Z
2
+
1
Z
2
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
in parallel
Z3 and Z4 NOT in parallel
Z3 and Z4 in parallel
Z1 and Z3 NOT in series
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
R L
Z
Z
+
R
L
+
1
ZZ
C
L
3. Collapse circuit in terms of
series/parallel relations till
circuit is trivial
4. Re-expand to find signal of
=
V
0
=
V
0
interest
Z Z
R
L
+
L
Z Z
R L
(
Z
+
R
RLs
1
)
Z Z Z
C
L
L
1
Ls
RLs
+
(
R Ls
+
1
)
Cs
i
L
=
V
0
RCs
2
+
RLCs
Ls R
+
Cite as: Joel Voldman, course materials for 6.777J / ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
anical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 6E - 26
Superposition
> For circuits with multiple
v1
v2
sources,
• Turn off all independent
sources except one
• Solve circuit
• Repea... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 6E - 28
Conclusions
> There are many ways to analyze equivalent circuits
> Use t... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
18.336 spring 2009
lecture 15
02/13/08
Finite Difference Methods for the One-Way
Wave Equation
�
ut = cux
u(x, 0) = u0(x)
Solution: u(x, t) = u0(x + ct)
Information travels to the left
with velocity c.
Three Approximations:
⎧
c
⎨
n
n
+1 − Uj
Uj
Δx
U n − Uj
n
jc
−1
Δx
⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎪⎪⎪⎪⎪⎪
U n
⎩ c j+1 − U... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/286be9198204bea2f7e7d77f88e7b061_MIT18_336S09_lec15.pdf |
1 c2uxxΔt
2
2
Leading order error:
1
1
2
2
1 2
2
uttΔt − cuxxΔx = c uxxΔt − cuxxΔx = cuxxΔx(r − 1)
1
2
1
2
= 0 if r = 1
r =
cΔt
Δx
First order if r = 1
Courant number
1
�
Downwind: Analogous: first order
Centered: ut + 1 uttΔt − cux − 1 cuxxxΔx2 + O(Δt2) + O(Δx4)
6
2
Δt
→
2
Δx
First order in time
→
Sec... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/286be9198204bea2f7e7d77f88e7b061_MIT18_336S09_lec15.pdf |
− λ
n
+1 + (1 − λ)Uj
2 Uj
Δt
Uj
U n
j+1 − 2Uj
n
n + Uj
−1
+ θ
(Δx)2
n + λ
�
n
2 Uj
−1 = c
where λ = 2
Δt
(Δx)2
θ
How much diffusion?
Lax-Friedrichs:
Eliminate Uj
n by λ = 1
θ = ⇒
(Δx)2
2Δt
U n+1
j =
1 + r
2
� ��
�
n
+1 +
Uj
1 − r
2
� ��
≥0(for |r|≤1)
�
n
Uj
−1
≥0(for |r|≤1)
Monotone scheme
n
... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/286be9198204bea2f7e7d77f88e7b061_MIT18_336S09_lec15.pdf |
+ O(Δx2)
Accuracy: ut+ 1 uttΔt+ 1 utttΔt2−cux− 1 cuxxxΔx2 Δt c2uxx− Δt c2uxxxxΔx2
1 utttΔt2 − 6
= 6
ut − cux = 0
Δt 2
1
uttΔt −
2
2
Stability: λ = r2
c uxx = 0
− 2
24
G =
r2 + r
2
e ikΔx + (1 − r 2) +
r2 − r
2
e−ik(Δx)
= (1 − r 2) + r 2 cos(kΔx) + ir sin(kΔx)
Worst case: kΔx = π ⇒ G = 1 − 2r2
Stable if |r|... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/286be9198204bea2f7e7d77f88e7b061_MIT18_336S09_lec15.pdf |
1
I-campus
pro ject
School-wide
Program
on
Fluid
Mechanics
Modules
on
Waves
in
fuids
T.
R.
Akylas
&
C.
C.
Mei
CHAPTER
SEVEN
INTERNAL
WAVES
IN
A
STRATIFIED
FLUID
1
Introduction.
The
atmosphere
and
o
c
e
a
n
are
continuously
stratifed
due
to
change
in
temperature,
composition
and
pressure.
These
changes
in
the
ocean
and
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
to
satisfy
the
continuity
equation
1
Dp
@u
@ v
@w
+
+
+
0
(2.1)
p Dt
@x
@ y
@ z
and
the
momentum
equations
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
The
motion
that
takes
place
is
assumed
to
be
isentropic
and
without
change
of
phase,
so
that
e
and
q
are
constant
for
a
material
element.
Therefore
Dp
@ p De
@ p Dq
+
:
(2.6)
0
Dt
@ e Dt
@ q Dt
In
other
words,
p
is
constant
for
a
material
element
because
e
and
q
are,
and
p
depends
only
on
e
and
q .
Such
a
fuid
is
said... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
that
the
wave
motion
results
from
the
perturbation
of
a
state
of
equilibrium,
which
is
the
state
of
rest.
So
the
distribution
of
density
and
pressure
is
the
hydrostatic
equilibrium
distribution
given
by
When
the
motion
develops,
the
pressure
and
density
changes
to
@ pp
@ z
;
g p
p:
(2.12)
p
p(z ) +
p
,
p
(2.13)
0
p
p(... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
form
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
@ z
@ t
;
;
If
we
substitute
equations
(2.21)
and
(2.22)
into
equation
(2.20),
we
obtain
2
0
2
0
2
1
@
p
@
p
@
w
;
pp
@x
@ y
@
t@
z
2
2
+
+
:
(2.24)
0
We
can
eliminate
p
from
(2.23)
by
using
equation
(2.16)
to
obtain
0
2
2
0
@
w
@
p
@ pp
pp
+
g
w :
(2.25)
2
@ t
@ t@ z
@ z
;
Third,
we
apply
the
operator
+
to
equation
(2... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
2
2
2
2
2
@
@
w
@
w
1
@
@w
@
w
@
w
2
2
2
2
2
2
@ t
@x
@ y
pp @ z
@ z
@x
@ y
+
+
pp
+
N
+
,
(2.27)
0
where
we
defne
2
g
@ pp
N
(z )
,
(2.28)
;
p @
z
which
has
the
units
of
frequency
(rad/sec)
and
is
called
the
Brunt-Viaisialia
frequency
or
buoyancy
frequency.
If
we
assume
that
w
varies
with
z
much
more
rapidly
than
pp... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
of
the
momentum
equations.
The
Boussinesq
approximation
leads
to
equation
(2.30)
for
the
vertical
velocity
w .
3
The
Buoyancy
Frequency
(Brunt-Vaaisaalaa
frequency).
Consider
a
calm
stratifed
fuid
with
a
static
density
distribution
pp(z )
which
decreases
with
height
z .
If
a
fuid
parcel
is
moved
from
the
level
z
upward... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
Brunt
V
iasialia
frequency.
This
elementary
consideration
shows
that
once
a
fuid
is
displaced
from
its
equilibrium
position,
gravity
and
density
gradient
provide
restoring
force
to
enable
oscillations.
4
Internal
Gravity
Waves
in
Unbounded
Stratifed
Fluid.
Consider
the
case
in
which
the
buoyancy
(Brunt-Viaisialia)
freq... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
N
.
The
dispersion
relation
for
internal
waves
is
of
quite
a
diferent
c
haracter
compared
to
that
for
surface
waves.
In
particular,
the
frequency
of
surface
waves
depends
only
on
the
magnitude
k
of
the
wavenumb
e
r
,
whereas
the
frequency
of
internal
waves
is
;!
j
j
independent
of
the
magnitude
of
the
wavenumber
and
de... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
p
0
0
p
cos(kx
+
ly
+
mz
! t
):
(4.41)
2
2
1/2
;
;
(k
+
l
)
From
equation
(2.16)
we
have
the
perturbation
density
p
given
by
0
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
a
fxed
point.
For
instance,
if
the
horizontal
velocity
components
and
perturbation
pressure
of
a
progressive
w
ave
are
measured,
the
horizontal
component
o
f
t
h
e
wavenumb
e
r
vector
can
be
deduced
from
(4.44).
A
sketch
showing
the
properties
of
a
plane
progressive
internal
wave
in
the
vertical
plane
that
contains
the... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
to
the
vertical.
The
restoring
force
p
e
r
unit
displacement
(cos
<dp
/dz )
is
less
than
the
case
where
<
,
so
the
frequency
0
0
of
vibration
is
less.
As
<
tends
to /
2,
the
frequency
of
vibration
tends
to
zero.
The
case
<
/
2
i
s
n
o
t
a
n
i
n
ternal
wave,
but
it
represents
an
important
form
of
motion
that
is
often
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
u
u
i
d
a
s
t
i
d
o
s
t
i
d
L
B
G
B
L
o
u
o
s
i
u
o
n
y
g
a
B
n
t
u
F
l
u
i
d
o
y
a
n
c
y
a
i
n
i
n
g
B
u
o
o
y
a
n
t
F
l
u
i
s
i
n
g
B
u
o
y
y
d
a
a
n
c
y
n
c
y
G
r
o
u
p
V
e
l
o
c
i
t
y
c it y
e l o
e V
s
a
h
P
H
i
g
h
P
r
e
s
s
u
r
e
(
u
,
w
,
P
=
0
)
L
o
w
(
u
,
w
,
P
P
H
i
g
h
r
e
s
s
u
r
e
=
0
)
P
r
e
s
s
u
r
e
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
.
Small
arrows
indicate
the
perturbation
velocities,
which
a
r
e
always
parallel
to
the
lines
of
constant
phase.
Large
heavy
arrows
indicate
the
direction
of
phase
propagation
and
group
velocity.
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
e
r
vector.
When
the
group
velocity
has
an
upward
component,
therefore,
the
phase
velocity
has
a
d
o
wnward
component,
and
vice
versa.
The
group
velocity
vector
is
N
C
sin
<(sin
<
cos
e ,
sin
<
sin
e ,
cos
<):
(4.46)
g
Therefore,
the
magnitude
of
the
group
velocity
is
(
)
sin
<,
and
its
direction
is
at
j k
j
k
;
!
j
j
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
12
m
θ
2 2 2 1/2
(k + l + m )
C g
φ
φ
2
(k
2
+ l
)
1/2
m
l
k
Figure
3:
Wavenumb
e
r
vector
and
group
velocity
vector.
13
wave
packet.
The
phase
velocity
is
given
by
the
equation
(4.45),
where
the
wavenumb
e
r
vector
k
makes
an
angle
<
with
the
horizontal
direction
(see
fgure
1,
but... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
-dimensional
Fourier
transform
pair
considered
is
1
1
u^(k
, m
)
dx
dz
exp(
ikx
imz )u(x,
z )
(4.49)
;1
;1
f
;
;
g
Z
Z
and
1
1
1
u(x,
z )
dk
dm
exp(
ikx
imz )
u^(k
, m
)
:
(4.50)
2
4
;1
;1
f
;
;
g
Z
Z
The
Fourier
transform
of
the
equation
(2.30)
is
given
by
the
equation
2
2
2
@
w^
N
k
+
w^
,
(4.51)
0
2
2
2
@ t
k
+
m... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
^ :
(4.54)
0
^
2
p
N
0
p
(k ,
m,
t)
iA(k
, m
)
ex
p
(
i! t) +
iB (k
, m
)
exp(i! t)
,
(4.55)
g
!
(k
, m
)
f;
g
where
the
constants
A
and
B
are
determined
from
the
Fourier
transform
of
the
initial
conditions
for
p
,
given
by
the
equations
0
0
p
(x,
z ,
0)
f
(x,
z ),
(4.56)
0
@
p
@ t
(x,
z ,
0) 0,
(4.57)
which
implies
t... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
2
2
~
2
2
2
;
;
1
1 (k
k
)
1 (m
m~ )
1 (k
+
k
)
1 (m
+
m~ )
2
2
2
2
~
~
^
f
(k
, m
)
exp
+
exp
:
;
;
2
2
2
2
2<T
2
<
2
T
2
<
2
T
;
;
;
;
(
�
!
�
!)
(4.62)
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
p
a
c
ket
has
the
same
modulation
along
the
x
and
z
directions
(<
T
),
the
two
parts
of
the
initial
wave
packet
travel
towards
the
middle
of
the
second
and
fourth
quadrants,
as
we
see
in
the
animation.
For
the
wave
packet
in
the
second
(fourth)
quadrant
the
group
velocity
vector
p
o
i
n
ts
away
from
the
origin
towards
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
wave
packets
for
early
times
is
more
intense
than
what
was
observed
in
the
previous
example,
as
we
can
see
in
the
animation.
To
see
it,
click
here.
The
third
animation
has
as
initial
condition
a
Gaussian
wave
packet
with
<
/2,
1
T
/20
and
k
m~
.
This
initial
wave
packet
has
a
shape
of
an
elongated
1
ellipse
2
~
�
in
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
16
4.2
Saint
Andrew's
Cross.
Here
we
discuss
the
wave
pattern
for
internal
waves
produced
by
a
localized
source
on
a
sinusoidal
oscillation,
like
an
oscillating
cylinder
for
example,
in
a
fuid
with
constant
density
gradient
(the
buoyancy
frequency
is
constant).
For
sinusoidal
internal
waves,
the
wave
energy
fux
I
p u
(... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
gravity
surface
waves.
Instead,
the
crests
and
other
surfaces
of
constant
phase
stretch
radially
outward
from
the
source
because
wave
energy
travels
with
the
group
velocity,
which
is
parallel
to
surfaces
of
constant
phase.
For
a
source
of
defnite
frequency
!
N
(less
than
the
buoyancy
frequency),
those
surfaces
are
all
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
is
orthogonal
to
the
group
velocity,
a
n
d
that
17
C
+
C
cos
e ,
sin
e ,
0
:
(4.66)
g
N
;
! !
;
f
g
k
;
!
j
j
Then,
given
the
direction
of
the
group
velocity,
the
orthogonality
of
the
phase
and
group
velocity
plus
the
condition
(4.66),
the
direction
of
the
phase
velocity
i
s
specifed.
If
the
group
velocity
has
a
positi... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
e
h
a
vior.
In
this
section
we
study
free
wave
propagation
in
a
continuously
stratifed
fuid
in
the
presence
of
boundaries.
Attention
is
restricted
to
the
case
in
which
the
bottom
is
fat.
The
equilibrium
state
that
is
being
perturbed
is
the
one
at
rest,
so
density,
and
hence
buoyancy
frequency,
is
a
function
only
of
the... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
18
C+C
g
C g
C
C
C g
C + C
g
Figure
4:
Phase
and
group
velocities.
C+C
g
C g
C
C
C g
C+C
g
19
Figure
5:
St
Andrew's
Cross
in
a
stratifed
fuid.
In
the
top
fgure
!
/N
0:9
and
in
the
left
bottom
fgure
!
/N
:7.
0
© sources unknown. All rights reserved. This content is excluded from our Creative
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
tain
a
free-surface
0
boundary
condition
for
w^(z ).
We
apply
the
operator
to
the
equation
(2.24),
and
then
@
t
@
we
substitute
equation
(5.70)
into
the
resulting
equation.
As
a
result,
we
obtain
the
equation
3
2
2
@
w
@
w
@
w
2
2
2
@ t @ z
@x
@ y
g
+
at
z
(5.71)
0
Now,
if
we
substitute
equation
(5.67)
into
the
equati... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
efect
of
confning
the
wave
energy
to
a
region
of
fnite
extent,
so
the
ocean
can
b
e
considered
as
a
that
causes
the
energy
to
waveguide
propagate
horizontally.
A
useful
piece
of
imaginary
is
to
picture
internal
waves
propagating
obliquely
through
the
ocean,
refections
at
the
upper
and
lower
boundaries
ensuing
no
loss
o... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
by
the
equation
w^ (z )
sin[m(z
+
H )]
with
m
(k
+
l
),
(5.76)
;
2
!
2
2
2
2
2
(N
!
)
which
already
satisfes
the
bottom
boundary
condition.
If
we
substitute
equation
(5.76)
into
the
free-surface
boundary
condition
(5.72),
we
obtain
the
dispersion
relation
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
,
so
equation
(5.72)
reduces
to
w^ (z )
0 at
z
:
(5.78)
0
This
b
o
u
n
d
a
r
y
condition
gives
a
dispersion
relation
of
the
form
sin(mH ) 0
(5.79)
or
2
(k
+
l
)N
H
2
2
2
2
!
, n
,
2,
3, : : :
,
(5.80)
1
2 2
2
2
2
n
+ (
k
+
l
)H
which
is
close
to
the
result
given
by
the
dispersion
relation
given
by
the
free-surface
bo... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
modes.
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
in the range 0
! N
,
i
.
e.,
there
:
is
a
continuous
spectrum
of
solutions.
Superposition
of
such
solutions
can
b
e
used
to
solve
initial-value
problems,
and
have
the
from
of
Fourier
integrals.
When
N
varies
with
z ,
there
is
another
type
of
solution
possible,
namely,
one
that
satisfes
the
condition
at
the
ground
yet
d... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
umbers
m
and
m
.
For
this
case,
the
spectrum
is
continuous
1
2
and
!
can
assume
any
value
b
e
t
ween
0
and
N
.
This
is
not
true
for
the
case
when
2
N
! N
,
when
the
frequency
!
can
assume
only
a
fnite
set
of
values
in
the
range
2
1
N
! N
.
In
this
case,
the
solution
of
equation
(5.73)
for
the
frst
layer
is
given
by
2... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
2
2
2
2
2
(!
N
)
At
t
h
e
intersection
z H
b
e
t
ween
the
two
l
a
yers,
the
perturbation
pressure
p
and
the
0
vertical
velocity
w
should
be
continuous.
Alternatively,
this
condition
can
be
expressed
in
terms
of
the
ratio
0
p
Z
,
(5.85)
p
w
0
which
must
b
e
the
same
on
both
sides
of
the
boundary.
It
is
convenient
to
ref... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
modes.
The
relative
amplitude
of
the
diferent
modes
depends
on
the
initial
state.
6
Energetics
of
Internal
Waves.
The
energy
equation
for
internal
waves,
under
the
assumption
of
small
perturbations,
incompressible
and
inviscid
fuid
and
irrotational
fow,
can
b
e
obtained
by
multiplying
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
gp
@ (p
u)
@ (p v )
@ (p
w)
2
2
2
0
0
0
0
pp u
+
v
+
w
+
+
+
+
:
(6.87)
0
@ t
2
2
pNp
@x
@ y
@ z
2
The
term
pp(u
+
v
+
w
)/2
stands
for
the
perturbation
kinetic
energy
density.
The
;
)
2
2
2
term
(p
u,
p
v
, p
w)
stands
for
the
perturbation
energy
density
fux
and
the
term
0
0
0
2
0
2
1
g
�
2
2 ��N
stands
for
the
per... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
the
energy
due
to
the
perturbation,
which
is
equal
to
1
1
2
2
p
g
+
g (p
p
)h
dxdy ,
(6.89)
2
1
2
2
2
;
Z Z
�
�
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
.91)
becomes
2 0
2
1
g p
2
pNp
2
Z Z Z
dxdydz :
(6.93)
The
connection
with
(6.87)
is
now
clear.
For
periodic
waves
in
a
medium
with
uniform
properties,
the
integral
over
each
wavelength
is
the
same,
and
so
the
mean
over
a
large
volume
becomes
equal
to
the
mean
over
one
wavelength
in
the
limit
as
the
volume
tends
to
inf... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
the
average
over
one
wavelength,
so
it
is
convenient
to
consider
the
spatial
mean
for
the
fuxes
as
well.
Thus
the
energy
fux
density
vector
F
is
defned
by
the
equation
;
!
0
27
0
0
0
0
0
F
;!
(u
,
v
,
w
)
>,
(6.95)
p
where
(u
,
v
,
w
)
is
the
perturbation
velocity
v
ector.
0
0
0
7
Mountain
Waves.
Internal
waves
in
the... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
velocity
over
single
ridge
in
a
fnite
atmosphere
or
o
c
e
a
n
(waveguide
case).
We
assume
that
the
atmosphere
or
ocean
density
stratifcation
is
such
that
the
buoyancy
frequency
is
constant.
7.1
Governing
Equation.
We
consider
an
air
or
water
moving
at
a
constant
velocity
U
(z )
over
a
periodic
or
localized
topography
g... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
write
the
horizontal
velocity
i
n
t
h
e
form
1
D
p
p
D
t
(7.100)
0
u(x,
z ) U
(z )
+
u
(x,
z ),
(7.101)
0
We
substitute
equation
(7.101)
in
the
equations
(7.97)
to
(7.100).
We
assume
the
velocities
u
and
w
as
small
quantities,
so
we
can
linearize
the
resulting
equations.
0
The
linearized
form
of
the
continuity
equatio... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
hydrostatic
equilibrium
distribution
given
by
equation
(2.12).
When
the
motion
devel-
ops,
the
pressure
and
density
are
given,
respectively,
b
y
equations
(2.13)
and
(2.14),
and
0
0
p
and
p
are
the
pressure
and
density
perturbations
of
the
\background
state".
Now
the
momentum
equation
assume
the
form
29
0
0
@
u
@
u
@
U... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
continuity
equation
(7.102)
to
obtain
2
0
2
@
u
@
w
+
:
(7.109)
0
@ t@x
@ t@ z
Second,
we
take
the
x
derivative
of
the
equation
(7.106)
to
obtain
2
0
2
0
2
0
@
u
@
u
@w @U
@
p
pp
+
U
+
(7.110)
@x@ t
@x
@x @ z
@x
;
2
2
�
�
0
Third,
we
eliminate
the
u
variable
from
the
equation
(7.110)
above.
To
do
so,
we
use
equation
(... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
@
@
30
2
2
2
2 0
2
0
0
0
@ w
@ w
@ w
@ p
@ p
@ p
@ p
2
pp
+ 2
U
+
U
U
g
+
U
,
(7.112)
2
2
@ t
@x@ t
@x
@ t@ z
@x@ z
@ t
@x
;
;
;
�
�
�
�
and
with
equation
(7.105)
we
can
eliminate
p
from
equation
(7.112).
The
result
is
the
0
equation
2
2
2
2
0
2 0
@ w
@ w
@ w
@ p
@ p
@ pp
2
pp
+ 2
U
+
U
U
gw
,
(7.113)
2
2
@ t
@x@ t
@x
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
the
vertical
velocity
w .
We
can
simplify
the
equation
above.
We
can
write
it
in
the
form
@
@
@ w
1
@
@w
@
@
1
@
@U
@w
@ w
2
2
2
2
+
U
+
pp
+
U
pp
+
N
(z )
,
0
@ t
@x
@x
pp @ z
@ z
@ t
@x
pp @ z
@ z @x
@x
;
2
2
�
�
�
�
(7.115)
where
N (z )
is
the
buoyancy
frequency
defned
according
to
equation
(2.28).
If
we
assume
t... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
t
@x
@x
@ z
@ t
@x
@ z
@ z @x
@x
;
2
2
2
�
�
�
�
(7.118)
We
can
simplify
this
equation
further
by
assuming
that
the
velocity
U
is
constant.
In
this
case
we
end
up
with
an
equation
of
the
form
@
@
@
w
@
w
@
w
2
2
2
2
2
+
U
+
+
N
(z )
,
(7.119)
0
@ t
@x
@x
@ z
@x
2
2
2
�
�
Next,
we
discuss
boundary
conditions
for
the... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
) U
on
z
(7.121)
0
@
(
@
x
For
an
infnite
atmosphere
we
need
a
radiation
condition,
which
ensures
that
the
energy
fux
is
away
from
the
ground.
In
other
words,
energy
is
radiated
away
from
the
ground
by
t
h
e
i
n
ternal
waves
generated
by
the
topography.
For
the
case
of
a
fnite
atmosphere,
we
need
a
boundary
condition
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
z ,
t) A
U k
cos(kx
)
at
z
(7.123)
0
0
In
this
example,
we
assume
a
constant
b
u
o
yancy
frequency
N
for
the
entire
atmosphere.
Under
such
condition,
we
can
assume
a
solution
for
the
steady
state
regime
of
the
form
w(x,
z ,
t) A
cos(kx
+
mz ),
(7.124)
where
k
is
the
topography
wavenumber,
since
the
solution
(7.124)
ha... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
33
7.2.1
Case
.
> k
U
N
There
are
two
solutions
for
equation
(7.126)
(plus
or
minus
sign
of
the
square
root
of
the
right
hand
side
of
equation
(7.126)),
and
to
decide
which
o
n
e
w
e
should
use
to
represent
energy
being
radiated
away
from
the
ground
we
are
going
to
consider
the
energy
fux
in
the
vertical
direction.
Acc... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
away
from
the
ground,
we
chose
m
as
given
by
the
equation
2
N
2
m
k
:
(7.130)
+
2
U
;
r
Since
energy
is
b
e
i
n
g
radiated
away
from
the
ground,
there
is
drag
exerted
by
the
topography
due
to
the
generation
of
the
internal
waves.
The
magnitude
of
the
drag
force
p
e
r
unit
area
is
equal
to
the
rate
T
p
e
r
unit
area
at... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
k
equal
/
to
the
horizontal
distance
traveled
by
a
fuid
c
particle
in
one
buoyancy
period.
From
equation
(7.126)
we
have
that
Thus,
the
angle
<
b
e
t
ween
wave
crests
and
the
vertical
changes
according
to
the
equation
0
N
U
2
2
2
k
+
m
k
)
:
(
c
The
angle
<
given
by
the
equation
above
is
illustrated
in
the
fgure
7.
0... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
drag
force
exerted
by
the
topography.
To
see
this,
we
consider
the
expression
for
the
pressure
in
this
case,
which
is
given
by
the
equation
0
A
p
ip1p U
exp(
1 z
+
ikx)
(7.136)
if;
;
g
k
This
is
out
of
phase
with
the
vertical
velocity,
i.e.,
is
zero
when
w
is
a
maximum
or
a
minimum,
and
is
a
maximum
or
a
minimum
when
w... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
speed
U
),
we
will
see
the
localized
topography
moving
with
speed
U
,
and
we
will
see
waves
with
;
phase
speed
U
matching
the
speed
on
the
topography
(steady
disturbance
downstream
;
to
the
topography
in
the
fxed
reference
frame).
Here
we
consider
the
second
reference
frame,
since
in
this
reference
frame
we
can
easily
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
related
according
to
the
equation
0
x
x
+
U
t:
(7.139)
The
boundary
condition
on
the
ground,
given
by
the
equation
(7.121),
for
this
ref-
erence
frame
assumes
the
form
0
0
@ (
w(x , z
, t
) U
(x
+
U t
)
at
z
:
(7.140)
0
@x
To
solve
the
boundary
value
problem
given
by
equations
(7.138)
and
(7.140)
with
the
appropriate
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
37
w^ (k
, z , t
) ikU
(
(k)
exp(ikU t)
(7.144)
^
Next,
we
consider
a
time
dependence
of
the
form
w^ (k
, z , t
) w^
(k
, z
)
exp(
i! t)
(7.145)
0
;
If
we
substitute
equation
(7.145)
into
the
governing
equation
(7.143)
we
end
up
with
the
governing
equation
for
w^
,
which
follows:
0
2
0
@
w^
2
2
2
2
2
0
!
+ (
N
k
!
k
)
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
ersion
relation,
which
follows
kN
!
(7.149)
2
2
p
m
+
k
From
the
dispersion
relation
above
we
can
obtain
the
group
velocity
v
ector.
The
group
velocity
is
the
speed
with
which
energy
is
propagated
by
the
internal
waves,
it
is
the
gradient
o
f
t
h
e
wave
frequency
with
respect
to
the
wavenumber,
then
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
need
m
to
b
e
negative
and
for
k
0
we
need
m
to
b
e
positive.
As
a
result,
we
chose
the
appropriate
branch
of
m
given
by
equation
(7.148)
as
follows:
For
k
0
w
e
have
that
•
2
m(k , N
/U
)
,
(7.152)
;
;
j
j
:
2
1/2
U
N
2
N
U
k
if
k
8
2
N
N
1/2
h
i
i
k
if
k
>
;
)
U
U
{
:
;
;
j
j
[
;
)J
For
k 0
we
have
that
•
m(k , N
/U... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
39
2
m
C
U
,
(7.155)
gx
;
k
;
!
2
j
j
which
implies
a
negative
v
alue
for
the
horizontal
component
of
the
group
velocity.
From
this
equation
we
also
realize
that
the
horizontal
component
of
the
group
velocity
is
in
the
same
direction
as
the
horizontal
component
of
the
phase
speed,
but
smaller
magnitude.
This
implies
th... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
apply
the
inverse
Fourier
transform
to
the
result-
ing
equation,
we
obtain
0
0
U
N
^
2
w(x
,
z
,
t
)
ik
( (k)
s
i
n
(
k
z
+
k
U
t
+
k
x
)dk
N/U
2
;
8
U
;
0
s
Z
{
1
:
2
N
(7.158)
+
ik
( (k)
e
x
p
(
k
z )
sin(+k
U
t
+
k
x
)dk
^
2
0
N/U
s
;
;
U
Z
9
}
In
terms
of
the
fxed
reference
frame,
the
vertical
velocity
w
is
given
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
( (x) (fxed reference
fram
e)
is
given by the equation
A
0
( (x)
,
(7.160)
1 +
(
x/b)
2
and
its
Fourier
transform
is
given
by
the
equation
^
( (k)
b
exp(
A
kb
):
(7.161)
0
;j
j
For
this
particular
example,
the
vertical
velocity
is
given
by
w(x,
z )
U A
b
k
exp(
b
k
)
sin
(
k
z
+
kx
)dk
0
N/U
2
N
2
;
;
j
j
;
U
8
0
s
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
1 (x,
z ) A
b
exp(
b
k
)
sin
(
k
z
+
kx
)dk
0
8
;
j
j
;
U
0
s
N/U
2
N
2
Z
{
1
:
2
2
N
+
exp(
b
k
k
z )
sin(+kx
)dk
N/U
s
;
j
j ;
;
U
9
Z
}
41
(7.164)
By
inspecting
equation
(7.164),
if
the
buoyancy
frequency
N
is
zero,
the
stream
function
1
has
a
simple
expression
given
by
the
equation
1 (x,
z )
,
(7.165)
1 +
[
x/(b
+... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
velocity
w .
For
this
governing
equation
we
h
a
ve
the
boundary
condition
at
the
ground,
given
by
equation
(7.121),
and
we
also
consider
a
boundary
condition
at
z
h,
which
is
the
top
of
the
fnite
atmosphere.
This
boundary
condition
is
given
by
the
equation
w(x,
z )
0 at
z
h:
(7.166)
In
this
case
no
energy
is
radiated
i... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
x axis
Figure
8:
Stream
lines
for
the
case
of
zero
value
for
the
buoyancy
frequency
N
.
A
1
0
and
b
.
4
43
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
s
i
x
a
z
-10
0
10
20
x axis
Figure
9:
Stream
lines
for
the
case
of
non-zero
value
for
the
buoyancy
freque... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
(7.166).
This
implies
that
we
should
have
!
kU
.
We
assume
a
general
solution
for
w^ of
the
form
0
;
0
w^
B
sinh(m(h
z )),
(7.168)
;
which
already
satisfes
the
Fourier
transform
of
equation
(7.166).
If
we
substitute
equa-
tion
(7.167)
into
the
ground
boundary
condition
given
by
equation
(7.144),
we
obtain
ikU
(
(k)
^
B... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
the
integration
contour
of
the
inverse
Fourier
transform
of
w^(k
, z , t
).
By
substituting
equation
(7.169)
into
equation
(7.168),
we
obtain
for
w^ (k
, z , t
)
the
expression
45
w^ (k
, z , t
) ikU
(
(k)
exp(ikU t),
(7.172)
;
sinh(m(z
h))
^
sinh(mh)
0
and
its
inverse
Fourier
transform
gives
the
expression
for
w(x
, z... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
2
k
l
with
l
,
2, : : :
(7.176)
1
±
;
U
h
(
)
( r
These
are
frst
order
poles
which
are
usually
pure
imaginary
numb
e
r
s
,
but
we
can
have
real
poles
for
non-zero
buoyancy
frequency
values
if
>
for
some
values
of
l
(smaller
U
h
N
l�
values
of
l).
The
real
p
o
l
e
s
are
associated
with
waves
generated
by
the
presence
o... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
half
part
of
the
complex
k
plane
for
(U t
+
x
)
>
0,
or
a
semi-circle
0
of
infnite
radius
lying
in
the
lower
half
part
of
the
complex
k
plane
for
(U t
+
x
)
0.
0
We
consider
a
deformation
also
of
the
real
axis
to
take
into
account
the
real
poles
that
we
may
have.
We
have
to
decide
if
we
deform
the
real
axis
to
pass
abo... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
the
case
(U t
+
x
)
>
0.
Then,
we
deform
the
real
axis
0
to
pass
below
the
real
poles
that
we
m
i
g
h
t
h
a
ve.
The
integration
contours
are
illustrated
in
fgure
10.
The
contribution
of
the
integral
along
the
part
of
the
contour
that
lies
at
the
infnite
in
the
complex
k
plane
is
zero.
Therefore,
the
only
contribution
c... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
case
(U t
+ x
)
>
0,
(B)
- Deformed
0
integration
contour
for
the
case
(U t
+
x
)
0.
0
48
L
0
(ik(U t
+x
))
i
sinh(m(k , N
/U
)(h
z ))
^
0
w(x , z
, t
)
Res
ikU
(
(k)
e
;
2
2
;
2
sinh(m(k , N
/U
)h)
j
k= (N/U
)
;(j�
/h
)
p
j=1
X
L
�
�
i
sinh(m(k , N
/U
)(h
z ))
^
(... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
t
)
iRes
ikU
(
(k)
e
;
2
2
sinh(m(k , N
/U
)h)
j
k=;i
(j�
/h
)
;(N/U
)
p
j=L+1
X
�
�
(7.181)
Since,
the
minimum
value
of
l
is
one,
the
critical
speed
for
a
given
value
of
the
buoyancy
frequency
N
is
U
.
For
current
values
U >
,
there
is
no
wave
�
�
N
h
N h
disturbance
downstream
of
the
localized
topography.
To
illust... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
Case
x >
0.
We
assume
that
the
frst
L
poles
are
real,
and
the
poles
for
l >
L
are
•
pure
imaginary.
We
have
that
49
L
i
sinh(m(k , N
/U
)(h
z ))
^
1 (x,
z )
Res
(
(k)
exp(ikx)
;
2
2
;
2
sinh(m(k , N
/U
)h)
j
k= (N/U )
;(j�
/h)
p
j=1
X
L
�
�
i
sinh(m(k , N
/U
)(h
z ))
^
+
Res
(
(k)
exp(ikx)
;
2
2
;
2
sinh(m(k , N
/U
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
(j�
/h)
;(N/U )
p
j=L+1
X
�
�
(7.184)
The
equations
(7.183)
and
(7.184)
can
b
e
written
in
a
simple
way
in
terms
of
the
quantities
e
, 1
and
a
defned
in
the
appendix
A.
j
j
j
Case
x >
0,
where
the
frst
L
poles
are
assumed
real
numbers,
and
the
other
poles
•
are
in
the
upper
part
of
the
complex
k
plane.
L
1
1 (x,
z )
/... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
1
exp(a
x),
(7.187)
j
j
j
;
j=L+1
X
We
chose
for
( (x)
the
same
topography
we
considered
in
the
previous
section.
The
stream
lines
for
this
fow
are
illustrated
in
fgure
11,
12
and
13.
For
stream
speeds
that
approach
the
critical
values
from
below,
the
group
velocity
l�
N
h
C
for
the
l-th
lee
wave
approaches
the
stream
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
j
ia
i
(j /h
)
(N
/U
)
(A.189)
j
2
2
±
±
;
p
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
i
x
a
z
15
10
5
-10
-5
0
5
10
15
20
25
x axis
30
35
40
45
50
Figure
13:
Stream
functions
for
the
\Witch
of
Agnasi"
topography
in
a
fnite
atmosphere
with
U
/N
1
and
A
/2.
We
expect
to
see
a
superposition
of
nine
lee
wave
for
this
1
0
value
of
N
/U
.
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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