text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
))
Let
g(θ1, θ2) = θ1x1 + θ2x2 −
1
2
From
d
dθj
g(θ1, θ2) = 0, we have that
(θ1
2 + θ1θ2 + θ2
2).
x1 − θ1 − θ2 = 0,
1
2
x2 − θ2 − θ1 = 0,
1
2
2
from which we have
θ1 = x1 − x2,
θ2 = x2 − x1
4
3
2
3
4
3
2
3
Then
So we need to find
2
I(x1, x2) = (x1 + x2 − x1x2).
2
2
3
2
(x1 + x 2 − x1x2)
2
inf
x1,x2... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
3
3
0, then µ
0 and further x1
x2
=
=
=
=
min
2
3
2
x1 +
2
3
(5 − 2x1)2 − x1(5 − 2x1)
which gives x1 =
10
11
, x2 =
35
11
and I(x1, x2) = 5.37. Thus
lim sup
n
1
n
log P(
Sn
n
∈ F ) ≤ −5.37
Applying the lower bound part of the Cram´er’s Theorem we obtain
lim inf
n
1
n
log P(
Sn
n
∈ F )
≥ lim... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
exists a broad set of con
ditions under which the large deviations bounds hold. Thus consider a general
sequence of random variable Yn ∈ Rd which stands for (1/n)Sn in the i.i.d.
case. Let φn(θ) = 1 log E[exp(n(θ, Yn))]. Note that for the i.i.d. case
n
φn(θ) =
=
1
n
1
n
log E[exp(n(θ, n−1Sn))]
log M n(θ)
=... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
0, we
have
1P( Yn ≥ a) = P(exp(θYn) ≥ exp(θna))
n
≤ exp(−n(θa − φn(θ)))
So we can get an upper bound
sup(θa − φn(a))
θ≥0
In the i.i.d. case we used the fact that supθ≥0(θa − M (θ)) = supθ(θa − M (θ))
when a > µ = E[X]. But now we are dealing with the multidimensional case
where such an identity does not make ... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
-
Frobenious eigenvalue, which satisfies the following properties.
5
1. ρ > 0 is real.
2. For every e-value λ of B, |λ| ≤ ρ, where |λ| is the norm of (possibly
complex) λ.
3. The left and right e-vectors of B denoted by µ and ν corresponding to
ρ, are unique up to a constant multiple and have strictly positive co... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
ρnνi)
The second identity is established similarly.
Now, given a Markov chain Xn, a function f : Σ → Rd and vector θ ∈ Rd ,
consider a modified matrix Pθ = (e(θ,f (j))Pi,j , 1 ≤ i, j ≤ N ). Then Pθ is an
irreducible non-negative matrix, since P is such a matrix. Let ρ(Pθ) denote its
Perron-Frobenious eigenvalue.
T... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
n
1≤j≤N
where P n(i, j) denotes the i, j-th entry of the matrix Pθ
applying Corollary 1, we obtain
θ
n . Letting φj = 1 and
lim φn(θ) = log ρ(Pθ).
n
Thus the G¨artner-Ellis can be applied provided the differentiability of log ρ(Pθ)
with respect to θ can be established. The Perron-Frobenious theory in fact can
... | https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/27fd98c78d35c40fd821548867809ed5_MIT15_070JF13_Lec5.pdf |
3.032 Mechanical Behavior of Materials
Fall 2007
Using U(r):
Measure parameters for U(r) in physical model to predict stresses that are high enough for
elastic instabilities to occur (e.g., nucleation of defects in crystals).
Images removed due to copyright restrictions.
Please see: Fig. 1c in Gouldstone, Andrew, et a... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/280e4e7a60634a54a4a94650acce94f7_lec13.pdf |
ofibers: Do physical and mechanical properties differ from bulk polymers?
Courtesy Gregory Rutledge. Used with permission.
Images removed due to copyright restrictions.
Please see Fig. 1 and 4 in Curgul, Sezen, et al. "Molecular Dynamics
Simulation of Size-Dependent Structural and Thermal Properties of
Polymer Nanof... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/280e4e7a60634a54a4a94650acce94f7_lec13.pdf |
18. Div grad curl and all that
Theorem 18.1. Let A ⊂ Rn be open and let f : A −→ R be a differ
entiable function.
If �r : I −→ A is a flow line for �f : A −→ Rn, then the function
f ◦ �r : I −→ R is increasing.
Proof. By the chain rule,
d(f �r)
◦
dt
(t) = �f (�r(t)) · �r�(t)
= �r�(t) �r�(t) ≥ 0.
·
�
Corollary... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
∂f
∂y
jˆ+
ˆı +
on vector fields:
Definition 18.5. Let A ⊂ R3 be an open subset and let F� : A −→ R3
be a vector field.
The divergence of F� is the scalar function,
which is defined by the rule
div F� : A −→ R,
div F� (x, y, z) = � · F� (x, y, z) =
∂f
∂x
+
∂f
∂y
+
∂f
.
∂z
1
The curl of F� is the vector... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
and it is called incompressible if
the divergence is zero, div F� = 0.
: −→
�
3R
Proposition 18.7. Let f be a scalar field and F� a vector field.
(1) If f is C2, then curl(grad f ) = �0. Every conservative vector field
is rotation free.
(2) If F� is C2, then div(curl F� ) = 0. The curl of a vector field is
incompres... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
�
�
�
�
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 gradient vector field, so that the gravitat... | https://ocw.mit.edu/courses/18-022-calculus-of-several-variables-fall-2010/2811ddbe75dcb771e8ff4b7d4e62dcac_MIT18_022F10_l_18.pdf |
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 information about citing these materials or our Terms ... | 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 |
R
Current conservation: i=i
C + iL
iL =iR
Energy Conservation v = vC = vR + vL
Massachusetts Institute of Technology
6.763 2005 Lecture 2
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... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/285aa1ec51ef050027112c396c6eba73_lecture2.pdf |
Gauss Law
Gauss’ Magnetic Law
Conservations laws
Charge conservation
Also Poynting’s
Massachusetts Institute of Technology
6.763 2005 Lecture 2
7
Distributied systems con’t
2. Constitutive Relations
Local in space,
linear time invariant
Ohm’s Law
3. Summary relations
Complex: Search first for first order i... | https://ocw.mit.edu/courses/6-763-applied-superconductivity-fall-2005/285aa1ec51ef050027112c396c6eba73_lecture2.pdf |
QuasiStatics
Solve first
Image removed for copyright reasons.
Please see: Figure 2.9, page 34, from Orlando, T., and
K. Delin. Foundations of Applied Superconductivity.
Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234.
Solve for E once B is found
Boundary conditions:
Massachusetts Institute of Technology
6.... | 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].
JV: 2.372J/... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
devices
> The electrical capacitor
• What is the relation
between Q and V?
A
I
+
-
V
ε
E a
d
⋅
=
Q
(cid:118)
∫
closed
surface
ε
EA Q
=
∇ × = ⇒
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 ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 6E - 5
Elements and element laws
> Do this with all three basic
elements
> Resistor
> Capacitor
> Inductor
I
+
R
V
V RI=
-
I
+
I
+
C
V
dV
dt
I C
=
L
V
dI
dt
V L
=
-
-
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabricatio... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
7J / 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 - 8
Complex impedances
> For LTI systems, use
complex impedances
instead
• Impl... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
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 - 10
Let’s analyze a circuit
1. Figure out what are
you trying to determine
2. Replace eleme... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
i Z
R
L
−
i Z
R
R
=
0
i
R
=
V
0
Z
L
=
+
Z
R
Z
C
+
V
0
Ls R
+
+
1
Cs
i
R
=
2
LCs
Cs
+
RCs
V
0
+
1
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... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
. Figure out what are you
trying to determine
+
-VC
2. Replace elements with
complex impedances
V0
+
-
3. Assign node voltages &
ground node
4. Write KCL at each node
5. Solve for node voltages
6. Use node voltages to
find what you care about
v1
V0
+
-
C
C
R
R
L
L
v2
Cite as: Joel Voldman, course materials for 6.77... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
777J Spring 2007, Lecture 6E - 16
Nodal analysis
1. Figure out what are you
trying to determine
v1
i2
i3
i1
C
R
v2
L
V0
+
-
2. Replace elements with
complex impedances
3. Assign node voltages &
ground node
4. Write KCL at each node
5. Solve for node voltages
6. Use node voltages to
find what you care about
v
2
⎛
⎜... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
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
0
2
LRCs
2
+
LRCs
Ls R
+
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/), Massachuset... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
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 of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Sprin... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
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 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 |
are you
trying to determine
2. Replace elements with
complex impedances
=
V
0
Z Z
R L
Z
Z
+
R
Z Z
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
+... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
one
• Solve circuit
• Repeat for all sources, then add
responses
> Turning off a voltage source gives
a short circuit
V0
+
-
short
> Turning off a current source gives
I0
open
an open circuit
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/2869c3087b8604d32f8e855919c1bdbb_07lecture06e.pdf |
, then add
v
2
=
I
0
Z Z
R C
+
Z
Z
+
V
0
C
Z
R
+
Z
C
Z
R
responses
v1
+
-
V0
v2
C
R
I0
Find v
2
v
2
=
v
2
=
+
V R
0
R
1
I R
0
Cs
1
+
R
Cs
I R V RCs
0
+
RCs
0
+
1
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http:... | 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 |
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
→
Second order in space
Stability:
Upwind:
G − 1
Δt... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/286be9198204bea2f7e7d77f88e7b061_MIT18_336S09_lec15.pdf |
Uj
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
n
+1 − Uj
Uj
−1
2Δx
r =
cΔt
Δx
Accuracy: ... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/286be9198204bea2f7e7d77f88e7b061_MIT18_336S09_lec15.pdf |
uxx− 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| ≤ 1
Image by MIT OpenCourseWare.
4
MIT OpenCourseWare
http://... | 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 |
denoted
as
u, v
and
w .
The
fuid
particle
has
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 |
q .
Then
for
fxed
e
and
q
(or
s),
p
is
:
independent
of
pressure
p
p(e , q
):
(2.5)
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
mater... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
p
p
,
(2.9)
@ t
@x
;
@ v
@ p
p
,
(2.10)
@ t
@ y
;
@w
@ p
p
gp:
(2.11)
@ t
@ z
; ;
Next,
we
consider
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
d... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
that
the
density
perturbation
at
a
point
is
generated
by
a
v
ertical
advection
of
the
background
density
distribution.
The
continuity
equation
(2.7)
for
incompressible
fuid
stays
the
same,
but
the
momentum
equations
(2.9)
to
(2.11)
assume
the
form
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
,
respectively,
of
the
equations
(2.17)
to
(2.19),
and
we
obtain
2
2
0
@
u
@
p
pp
,
(2.21)
@x@ t
@x
;
2
2
2
0
@
v
@
p
pp
,
(2.22)
@ y@ t
@ y
;
2
2
2
0
0
@
w
@
p
@ p
pp
g
:
(2.23)
2
@ t
@ t@ 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
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
Next,
we
use
equation
(2.24)
to
eliminate
p
from
equation
(2.26),
which
gives
the
0
following
partial
diferential
equation
for
w :
5
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... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
z
@x
@ y
+
+
+
N
+
:
(2.30)
0
The
assumption
above
is
equivalent
to
the
Boussinesq
approximation,
which
applies
when
the
motion
has
vertical
scale
small
compared
with
the
scale
of
the
background
density.
It
consists
in
taking
the
density
to
b
e
constant
in
computing
rates
of
change
of
momentum
from
accelerations,
but
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
6
g
[
pp(z
+
( )
pp(z )]
g
( ,
(3.31)
d pp
;
�
dz
and
it
is
negative.
Applying
Newton's
law
t
o
the
fuid
parcel
of
unit
volume,
we
have
2
@
(
d pp
pp
g
(
(3.32)
2
@ t
dz
or
where
2
@
(
2
@
t
2
+
N
(
,
(3.33)
0
2
g
d pp
N
(z )
,
(3.34)
;
pp
dz
which
is
called
the
buoyancy
frequency
or
the
Brunt
V
iasialia
frequency.
T... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
is
the
frequency.
In
order
for
(4.35)
to
satisfy
the
governing
equation
(2.30)
for
the
vertical
perturbation
velocity,
!
and
k
must
be
related
by
the
dispersion
;!
relation
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
37)
;!
j
j
l
k
cos(<)
sin
(e)
(4.38)
;!
j
j
m
k
sin(<)
(4.39)
;!
j
j
The
coordinate
system
in
the
wavenumb
e
r
space
is
given
in
the
fgure
1.
The
dispersion
relation
given
by
equation
(4.36)
reduces
to
2
!
N
cos(<):
(4.40)
Now
w
e
can
write
expressions
for
the
quantities
p
, p
, u
and
v .
From
equation
(2.20)
we
0
0
ca... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
8
m
2 2 2 1/2
(k + l + m )
φ
2
(k
2
+ l
)
1/2
θ
m
l
k
Figure
1:
Coordinate
system
in
the
wavenumb
e
r
space.
9
0
2
N
p
p
w
sin(kx
+
ly
+
mz
! t
):
(4.42)
0
0
;
!g
;
The
horizontal
velocity
components
can
b
e
found
from
equations
(2.17)
and
(2.18),
which
give
(u,
v )
(k
, l
)(k
+
l
)... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
along
wave
crests,
and
there
is
no
pressure
gradient
in
this
direction.
The
restoring
force
on
a
particle
is
therefore
due
solely
to
the
component
g
cos
<
of
gravity
in
the
direction
of
motion.
The
restoring
force
is
also
proportional
to
the
component
of
density
change
in
this
direction,
which
is
cos
<
per
unit
displac... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
an
important
form
of
motion
that
is
often
observed.
For
instance,
it
is
quite
common
on
airplane
journeys
to
see
thick
layers
of
cloud
that
are
remarkably
fat
and
extensive.
Each
cloud
layer
is
moving
in
its
own
horizontal
plane,
but
diferent
layers
are
moving
relative
t
o
e
a
c
h
other.
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
X - axis
Figure
2:
The
instantaneous
distribution
of
velocity,
pressure,
and
buoyancy
perturba-
tions
in
an
i... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
11
4.1
Dispersion
Efects.
In
practice,
internal
gravity
waves
never
have
the
form
of
the
exact
plane
wave
given
by
equation
(4.35),
so
it
is
necessary
to
consider
superposition
of
such
waves.
As
a
consequence,
dispersion
efects
become
evident,
since
waves
with
diferent
frequencies
have
diferent
phase
and
group
velociti... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
;
N
;
!
an
angle
<
to
the
vertical,
as
illustrated
in
the
fgure
3.
To
illustrate
the
efects
of
dispersion,
we
consider
the
case
of
two
dimensional
mo-
tions.
We
consi... | 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 |
ed
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
which
has
solution
of
the
form
w^(k ,
m,
t)... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
0
^
2
@ p
p
N
0
@
t
g
w
^ :
(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... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
by
the
equations
^
f
(x,
z )
exp
x
<
z
T
cos(k
x
+
mz~ ),
(4.61)
1
1
1
2
2
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 |
2
~
�
parts
as
time
increases.
This
two
parts
propagate
in
opposite
directions
from
each
other.
Since
the
x
and
z
components
of
the
main
wavenumb
e
r
are
equal
and
positive
and
the
wave
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
mid... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
initial
wave
packet
looks
almost
without
variation
in
the
x
direction.
The
wave
packet
splits
in
two
parts
as
time
increases.
These
two
parts
propagate
in
opposite
directions
from
each
other,
in
a
way
similar
to
the
previous
example.
The
interference
efect
b
e
t
ween
these
two
wave
packets
for
early
times
is
more
inten... | 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 |
k
+
l
f
;
g
which
is
parallel
to
the
group
velocity,
according
to
equation
(4.46).
Therefore,
for
internal
waves
the
energy
propagates
in
the
direction
of
the
group
velocity,
which
is
parallel
to
the
surfaces
of
constant
phase.
This
fact
means
that
internal
waves
generated
by
a
localized
source
could
never
have
the
fam... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
of
the
group
velocity
and
by
the
fact
that
the
phase
velocity
N
cos
<
C
cos
<
cos
e ,
cos
<
sin
e ,
sin
<
(4.65)
;
!
f
g
k
;
!
j
j
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
isotropy
has
been
verifed
in
dramatic
experiments
by
Mowbray
and
Stevenson.
By
oscillating
a
long
cylinder
at
various
frequencies
vertically
in
a
stratifed
fuid,
equal
phase
lines
are
only
found
along
four
beams
forming
\St.
Andrew's
Cross",
see
fgure
5
for
!
/N
:7
and
!
/N
:9.
It
can
be
verifed
that
the
0
0
angles
a... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
state
as
the
state
of
rest,
fuid
properties
are
constant
on
horizontal
surfaces
and,
furthermore,
the
boundaries
are
horizontal.
Solutions
of
the
perturbation
equation
(2.27)
can
b
e
found
in
the
form
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
67)
;
The
equation
for
w^ (z )
can
be
found
by
substitution
of
equation
(5.67)
into
the
governing
equation
(2.27).
We
obtain
1
@
@w^
(N
!
)
2
2
2
2
pp @ z
@ z
!
;
2
pp
+
(k
+
l
)
w^(z )
0
(5.68)
The
boundary
conditions
for
this
equation
are
the
bottom
condition
of
no
fux
across
it,
given
by
the
equation
a
n
d
a
t
t
h
e... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
2
2
+ (k
+
l
)
w^(z ) 0 at
z
:
(5.72)
0
@ z
!
2
To
simplify
the
governing
equation
for
w^(z ),
we
make
the
Boussinesq
approximation,
such
that
equation
(5.68)
simplifes
to
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
of
equation
(5.73)
under
the
boundary
condi-
tions
(5.72)
and
(5.69).
We
frst
consider
the
case
where
!
> N
.
For
this
case
the
2
2
general
solution
has
the
form
sinh[m(z
+
H )]
(!
N
)
2
2
2
2
2
w^ (z )
with
m
(k
+
l
),
(5.74)
sinh(mH )
!
;
2
which
already
satisfes
the
bottom
boundary
condition.
The
free-surface
bound... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
22
2
2
N
!
gm
tanh(mH ):
(5.77)
;
For
a
given
value
of
the
frequency
! ,
this
dispersion
relation
gives
a
countable
set
of
values
for
the
modulus
of
the
horizontal
component
(
k
+
l
) of the
wavenumber, or for
2
2
a
given
value
of
the
modulus
of
the
horizontal
component
of
the
wavenumber,
we
have
a
c
o
u
n
table
set
of... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
!
, 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
boundary
condition
(5.77).
The
value
of
m
for
the
case
with
a
free-surface
is
slightly
larger
than
the
case
with
the
rigid
lid
approximation.
If
the
ocean
is
perturbed
with
a
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
23
5.2
Free
Waves
in
a
semi-infnite
region.
The
atmosphere
does
not
have
a
defnite
upper
boundary
as
does
the
ocean,
so
solutions
of
equation
(5.73)
will
now
b
e
considered
for
the
case
of
a
semi-infnite
domain
z >
0.
In
this
case
there
are
two
types
of
solutions,
the
frst
b
e
i
n
g
typifed
by
the
case
N
constant.
The
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
decays
as
z
.
These
are
waveguide
modes,
! 1
and
there
are,
in
general,
only
a
fnite
numb
e
r
possible.
A
simple
example
is
p... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
1
the
equation
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
refer
t... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
discrete
waveguide
modes
and
the
continuous
spectrum
of
sinusoidal
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,... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
7),
(2.18)
and
(2.19),
respectively,
b
y
u,
v
and
w ,
b
y
m
ultiplying
equation
(2.16)
by
g
p pN
/ p
,
and
then
adding
the
result
and
by
taking
into
account
the
equation
2
0
2
of
continuity
(2.7)
and
the
defnition
of
the
buoyancy
frequency.
We
obtain
@
1
1
gp
@ (p
u)
@ (p v )
@ (p
w)
2
2
2
0
0
0
0
pp u
+
v
+
w
+
+
+
+
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
H
]
dxdy
2
2
1
2
1
1
2
2
2
2
Z Z Z
Z Z
�
�
2
2
;
;
;
;
1
1
2
2
2
2
p
g [
(H
h)
] +
p
g [(H
h)
H
]
dxdy
(6.88)
2
2
1
2
2
2
;
;
;
;
Z Z
�
�
If
we
skip
the
constant
terms
in
the
equation
above,
we
end
up
only
with
the
potential
energy
associated
with
the
energy
due
to
the
perturbation,
which
is
equal
to
1
1
2
2
p
g
+
g ... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
pNp
h
dxdydz ,
(6.91)
;
2
@ z
2
Z Z Z
Z Z Z
where
h
is
the
displacement
of
a
fuid
element
from
its
equilibrium
position.
Since
the
density of a fuid element at its
p
ertu
rb
ed
level
z
+
h
is
equal
to
the
density
pp(z )
at
its
equilibrium
position,
the
perturbation
density
p
is
given
by
0
0
@ pp
p
pp(z )
pp(z
+
h)
h
,
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
0
2
E
pp
(u
+
v
+
w
)
>
+
(6.94)
2
2
pN
p
2
When
integrated
over
a
large
volume,
equation
(6.87)
shows
that
the
rate
of
change
of
energy
over
that
volume
is
equal
to
the
fux
of
energy
across
the
sides.
since
this
fux
is
also
periodic,
the
average
over
a
large
plane
area
is
approximately
the
same
as
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
ariety
o
f
m
e
c
hanisms.
Often
the
source
region
is
approximately
horizontal,
so
the
vertical
velocity
component
can
efectively
b
e
specifed
on
some
horizontal
surface,
and
the
motion
away
from
the
source
can
b
e
calculated
from
the
equations
of
motion.
We
frst
consider
the
case
in
which
air
or
water
is
moving
with
un... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
and
the
momentum
equations
28
@
u
@
u
@
u
@
p
p
+
u
+
w
,
(7.98)
@
t
@
x
@
z
@
x
;
�
�
@
w
@
w
@
w
@
p
p
+
u
+
w
pg :
(7.99)
@
t
@
x
@
z
@
z
;
;
�
�
The
fuid
density
has
to
satisfy
the
equation
Now
we
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
substitut... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
equation,
its
linearized
form
is
0
@u
@u
@U
@ p
p
+
U
+
w
,
(7.103)
@ t
@x
@ z
@x
;
�
�
@w
@w
@ p
p
+
U
pg :
(7.104)
@ t
@x
@ z
;
;
�
�
For
the
density
equation
(7.100)
we
obtain
1
@ p
@ p
@ p
@ p
0
+
U
+
u
+
w
(7.105)
0
p
@ t
@x
@x
@ z
�
�
We
consider
the
wave
motion
as
a
result
from
the
perturbation
of
the
state
of
e... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
um
equation
assume
the
form
29
0
0
@
u
@
u
@
U
@
p
pp
+
U
+
w
,
(7.106)
@
t
@
x
@
z
@
x
;
�
�
@
w
@
w
@
p
0
0
pp
+
U
p
g
,
(7.107)
@
t
@
x
@
z
;
;
�
�
and
the
density
equation
assumes
the
form
1
@
p
@
p
@
p
0
0
0
+
U
+
w
:
(7.108)
0
pp
@
t
@
x
@
z
�
�
As
we
did
in
section
2,
we
would
like
to
reduce
the
system
of
equati... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
(7.109)
and
the
x
derivative
of
the
continuity
equation
(7.102).
After
the
u
0
variable
is
eliminated,
equation
(7.110)
assumes
the
form
2
2
2
0
@
w
@
w
@w @U
@
p
pp
U
+
(7.111)
;
;
;
@ z@ t... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
@ t@ z
@x@ z
@ z
;
;
;
�
�
Next,
we
apply
the
operator
to
equation
(7.113).
Then,
we
elimina... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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
that
w varies
with
z much
more
rapidly
than
pp(z ),
then
we
can
write
1
@
@w
@
w
2
pp
(7.116)
pp @ ... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
@ 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
equatio... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
,
a
n
d
w
e
obtain
the
linear
boundary
0
condition
w(x,
z ,
t) 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... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
)
0
where
k
is
the
wavenumb
e
r
of
the
topography.
The
boundary
condition
(7.121)
at
the
ground
for
this
case
assumes
the
form
w(x,
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
stea... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
to
the
equation
above,
if > k
,
them
m
is
real
and
we
obtain
waves
which
U
N
propagates
through
the
atmosphere.
If
k ,
them
m
is
imaginary
and
we
have
a
U
solution
which
decays
exponentially
away
from
the
ground.
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
velocity
given
by
equation
(7.124),
we
have
0
m
p
pU A
p
cos(kx
+
mz )
(7.128)
k
Now,
we
compute
the
average
vertical
energy
fux,
and
we
obtain
the
expression
0
2
1
m
F
pU A
p
,
(7.129)
z
2
k
and
since
we
need
a
positive
vertical
average
energy
fux
for
the
energy
to
b
e
radiated
away
from
the
ground,
we
chose
m
as
give... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
For
the
vertical
wavenumb
e
r
m
equal
to
zero,
we
h
a
ve
the
horizontal
cut-of
wavenumber
k
,
given
by
the
equation
c
34
N
k
:
(7.132)
c
U
This
wavenumb
e
r
divides
the
two
types
of
solutions
(N
/U
> k
and
N/U
k),
and
corresponds
to
a
wavelength
2
k
equal
/
to
the
horizontal
distance
traveled
by
a
fuid
c
particle
in
o... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
7.134)
if
;
g
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
a
horizontal
constant
speed
U
passing
over
a
localized
topography
z
( (x).
We
assume
that
the
buoyancy
frequency
N
is
constant.
For
a
reference
frame
fxed
to
the
ground,
the
governing
equation
(7.119)
assumes
the
form
2
2
2
2
@
@
w
@
w
@
w
2
2
U
+
+
N
:
(7.137)
0
2
2
2
2
@x
@x
@ z
@x
In
this
reference
frame
we
will
ob... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
case.
The
governing
equation
in
the
moving
reference
frame
is
given
by
equation
(2.30)
without
the
y
component.
In
other
words,
36
2
2
2
2
@
@
w
@
w
@
w
2
2
0
2
2
0
2
@ t
@ (x
)
@ z
@ (x
)
+
+
N
,
(7.138)
0
where
x
is
the
horizontal
axis
in
the
moving
reference
frame.
The
horizontal
axis
in
the
0
fxed
and
moving
frame
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
^
0
0
f
(k)
f
(x)
ex
p
(
ikx
)dx
(7.141)
;1
;
Z
and
+1
1
^
0
f
(x)
f
(k)
exp(ikx
)dk :
(7.142)
2
;1
Z
We
apply
the
Fourier
transform
to
the
governing
equation
(7.138)
and
to
the
boundary
condition
(7.140).
The
governing
equation
(7.138)
in
the
wavenumb
e
r
domain
assumes
the
form
2
2
@
@
^w
2
2
2
2
2
@ t
@ z
;
;
k
+
... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
(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
)
w^
(7.146)
0
2
@ z
;
Solutions
of
the
equation
(7.146)
are
in
the
form
0
w^
(k
, z
) A
exp(
im(k)z ),
(7.147)
;
where
m(k)
can
b
e
obtained
in
terms
of
k
and
!
by
substituting
equation
(7.147)
into
equation
(7.... | https://ocw.mit.edu/courses/2-062j-wave-propagation-spring-2017/2870676df2a0100adddd62ce6d852589_MIT2_062J_S17_Chap7.pdf |
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 |
:
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
)
,
(7.153)
;
j
j :
2
1/2
U
N
2
N
U
k
if
k
where
we
used
in
equation
(7.148)
the
fact
that
8
2
N
N
1/2
h
i
i
k
if
k
>
;
)
U
U
{
:
;
;
j
j
[
;
)J
!... | 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 |
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