text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
Note: as h → ∞. tanh kh → 1, and we obtain deep water dispersion relation deduced in our wind-over
water lecture.
Physical Interpretation
Chapter 19. Water waves
• relative importance of σ and g is prescribed by the Bond number Bo = ρg
σk2 = σ(2π)2
2
ρgλ2 = (2π)2 ℓ
c
λ2
where ℓc =
σ/ρg is the capillary length... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf |
so longer waves travel faster, e.g. drop large stone into a pond.
B. Capillary Waves: Bo ≪ 1, c = ρ tanh kh.
σk
2
J
Deep water kh ≫ 1 ⇒ c =
fastest, e.g. raindrop in a puddle.
√
σkρ so short waves travel
Shallow water kh ≪ 1 ⇒ c =
An interesting note: in lab modeling of shallow water waves
(kh ≪ 1) c ≈
=
J
k... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf |
a dispersive system, the energy of a wave component does not propagate
at c = ω/k (phase speed), but at the group velocity:
Image courtesy of Andrew Davidhazy.
Used with permission.
for kmin =
√
ρg
σ
)
)
1/2
cg =
dω
dk
=
d
dk
(ck)
(19.5)
Deep gravity waves: ω = ck =
√
gk. cg =
∂
∂k
ω =
∂
∂k
√
gk =
... | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf |
OCW: 18.357 Interfacial Phenomena
80
Prof. John W. M. Bush
MIT OpenCourseWare
http://ocw.mit.edu
357 Interfacial Phenomena
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/04759cfc15c79928ca9f72a18b3864dd_MIT18_357F10_lec_all.pdf |
LINEAR ALGEBRA: VECTOR SPACES AND OPERATORS
Contents
1 Vector spaces and dimensionality
2 Linear operators and matrices
3 Eigenvalues and eigenvectors
4
Inner products
5 Orthonormal basis and orthogonal projectors
6 Linear functionals and adjoint operators
7 Hermitian and Unitary operators
1 Vector spaces an... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
have a complex vector space. More generally, the numbers
we use belong to what is called in mathematics a ‘field’ and denoted by the letter F. We will discuss
just two cases, F = R, meaning that the numbers are real, and F = C, meaning that the numbers are
complex.
The definition of a vector space is the same for F b... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
a u
V such that v + u = 0 (additive inverse).
4.
For each v
∈
5.
The element 1
∈
∈
F satisfies 1v = v for all v
V (multiplicative identity).
∈
6.
a(u + v) = au + av and (a + b)v = av + bv for every u, v
V and a, b
∈
∈
F (distributive property).
This definition is very efficient. Several familiar propert... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
speak
of the vectors themselves as real or complex. A vector multiplied by a complex number is not said to
−
−
−
∈
be a complex vector, for example! The vectors in a real vector space are not themselves real, nor are
the vectors in a complex vector space complex. We have the following examples of vector spaces:
1. ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
not form a complex vector
space since multiplication of a hermitian matrix by a complex number ruins the hermiticity.
4. The set
(F) of polynomials p(z). Here the variable z
P
has coefficients a0, a1, . . . an also in F:
F and p(z)
∈
∈
F. Each polynomial p(z)
n
p(z) = a0 + a1z + a2z + . . . + anz .
2
(1.3)
By... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
its possible subspaces. A subspace
of a vector space V
is a subset of V that is also a vector space. To verify that a subset U of V
is a
subspace you must check that U contains the vector 0, and that U is closed under addition and scalar
multiplication.
Sometimes a vector space V can be described clearly in terms... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
v2, . . . , vn) of vectors in V contains,
by definition, a finite number of vectors. The number of vectors in the list is the length of the list.
The span of a list of vectors (v1, v2,
, vn), is the set of all linear
vn) in V , denoted as span(v1, v2,
· · ·
· · ·
combinations of these vectors
a1v1 + a2v2 + . . . an... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
For example 1, consider the list of vectors (e1, e2, . . . eN ) with
0
0
.
.
.
1
1
0
.
.
.
0
0
1
.
.
.
0
e1 =
, e2 =
,
. . .
eN =
.
(1.7)
This list spans the space (the vector displayed is a1e1 + a2e2 + . . . aN eN ). This vector space is finite
dimensional.
... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
Finally, we get to the concept of a basis for a vector space. A basis of V
is a list of vectors
in V that both spans V and it is linearly independent. Mathematicians easily prove that any finite
dimensional vector space has a basis. Moreover, all bases of a finite dimensional vector space have the
same length. The di... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
Moreover they span the space since the most general
hermitian matrix, as shown above, is simply a01 + a1σ1 + a2σ2 + a3σ3. The list is linearly independent
4
as a01 + a1σ1 + a2σ2 + a3σ3 = 0 implies that
a0 + a3 a1
a1 + ia2 a0
�
a3
!
�
ia2
−
−
=
0 0
�
0 0
!
�
,
(1.... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
V with the properties:
1. T (u + v) = T u + T v, for all u, v
V .
2. T (au) = aT u, for all a
∈
∈
F and u
V .
∈
We call
(V ) the set of all linear operators that act on V . This can be a very interesting set, as we
L
will see below. Let us consider a few examples of linear operators.
1. Let V denote the spa... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
It could not be any other number because the zero
element (a sequence of all zeroes) should be mapped to itself (by linearity).
5
3. For any V , the zero map 0 such that 0v = 0. This map is linear and maps all elements of V to
the zero element.
4. For any V , the identity map I for which Iv = v for all... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
the space of
linear operators is associative: S(T U ) = (ST )U , for S, T, U , linear operators. Moreover it has an
identity element: the identity map of example 4. Most crucially this multiplication is, in general,
noncommutative. We can check this using the two operators T and S of example 1 acting on the
polynom... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
its
range. Given some linear operator T on V
mapped to the zero element. The null space (or kernel) of T
it is of interest to consider those elements of V that are
(V ) is the subset of vectors in V
∈ L
that are mapped to zero by T :
null T =
v
{
V ; T v = 0
}
.
∈
6
(2.18)
Actually null T is a subspace... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
is injective if and only if null T =
.
0
}
{
Given a linear operator T on V it is also of interest to consider the elements of V of the form T v.
The linear operator may not produce by its action all of the elements of V . We define the range of
T as the image of V under the map T :
Actually range T is a subspace o... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
The inverse is actually unique. Say S and S ′ are inverses
of T . Then we have
∈ L
S = SI = S(T S ′ ) = (ST )S ′ = IS ′ = S ′ .
(2.22)
Note that we required the inverse S to be an inverse acting from the left and acting from the right.
This is useful for infinite dimensional vector spaces. For finite-dimensional vec... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
T that acts on a vector space V .
This matrix will depend on the basis we choose for V . Let us declare that our basis is the list
(v1, v2, . . . vn). It is clear that the full knowledge of the action of T on V
T on the basis vectors, that is on the values (T v1, T v2, . . . , T vn). Since T vj is in V , it can be w... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
association between operator multiplication and matrix multiplication. The left-hand side, where we
have the action of the matrix for T on the j-th basis vector, can be viewed concretely as
T vj
←→
j-th position
(2.28)
T11
T21
.
.
.
T1 j
T2 j
.
.
.
· · ·
· · ·
.
.
.
T1n
T2n
.
.
.
· · ·
· · ·
.
.
.
Tn1
... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
�
0
0
0
.
.
.
1
+
. . .
+ Tn j
8
T1 j v1 + . . . Tn j vn .
(2.29)
←→
which we identify with the right-hand side of (2.25). So (2.25) is reasonable.
Exercise. Verify that the matrix representation... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
32)
which is precisely the right formula for matrix multiplication. In other words, the matrix that repre
sents T S is the product of the matrix that represents T with the matrix that represents S, in that
order.
Changing basis
While matrix representations are very useful for concrete visualization, they are basis... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
the effect of a change of basis on the matrix representation of an operator.
Consider a vector space V and a change of basis from (v1, . . . vn) to (u1, . . . un) defined by the linear
operator A as follows:
A : vk
→
uk, for k = 1, . . . , n .
(2.34)
9
This can also be written as
Avk = uk
(2.35)
Si... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
of the matrix representation of B
in the u basis. Replacing the second relation on the first, and then replacing the first on the second
we get
uk = Ajk Bij ui = Bij Ajk ui
vk = Bjk Aij vi = Aij Bjk vi
Since the u’s and v’s are basis vectors we must have
Bij Ajk = δik and Aij Bjk = δik
which means that the B matri... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
{
}
(
)
(
Comparing with (2.43) we get
)A ui .
ik
}
)
u
Tij (
{
) = A−1T (
v
{
}
)A
}
ij
→
u
T (
{
) = A−1T (
v
{
}
)A .
}
This is the result we wanted to obtain.
(
)
(2.45)
(2.46)
The trace of a matrix Tij is given by Tii, where sum over i is understood. To show that the trace
of T is basis independent we wri... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
a better perspective on these matters, we consider the
eigenvalue/eigenvector problem in more generality.
One way to understand the action of an operator T
(V ) on a vector space V
is to understand
∈ L
how it acts on subspaces of V , as those are smaller than V and thus possibly simpler to deal with. Let
U denote... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
This equation is so ubiquitous that names have been invented to label the objects involved. The
number λ
F is called an eigenvalue of the linear operator T if there is a nonzero vector u
satisfying this equation. Then it follows that cu, for any c
such that the equation above is satisfied. Suppose we find for some s... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
) we conclude that λ is an eigenvalue also means that (T
surjective. We also note that
λI) is not invertible, and not
−
Set of eigenvectors of T corresponding to λ = null (T
λI) .
−
(3.53)
It should be emphasized that the eigenvalues of T and the invariant subspaces (or eigenvectors as
sociated with fixed eigen... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
are not altered at all by the
rotation.
12
Consider now the case where T is a rotation by ninety degrees on a two-dimensional real vector
space V . Are there one-dimensional subspaces left invariant by T ? No, all vectors are rotated, none
remains pointing in the same direction. Thus there are no eigenva... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
than n linearly independent
vectors, no linear operator on V can have more than n distinct eigenvalues.
We saw that some linear operators in real vector spaces can fail to have eigenvalues. Complex
vector spaces are nicer. In fact, every linear operator on a finite-dimensional complex vector space has
at least one e... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
(λ) in λ of degree N called the characteristic polynomial:
f (λ) = det(T
λ1) = (
−
−
λ)N + bN −1λN −1 + . . . b1λ + b0 ,
(3.56)
where the bi are constants. We are interested in the equation f (λ) = 0, as this determines all possible
eigenvalues. If we are working on real vector spaces, the constants bi are real b... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
because (T
λI) non-invertible means it is not injective, and
therefore there are nonzero vectors that are mapped to zero by this operator.
−
4
Inner products
We have been able to go a long way without introducing extra structure on the vector spaces. We
have considered linear operators, matrix representations, t... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
one may think of
a
|
2 as the dot product of a with a:
|
2 = a
|
a
|
·
2
2
a = a1 + . . . a
n
Based on this the dot product of any two vectors a and b is defined by
b = a1b1 +
. . . + anbn .
a
·
(4.58)
(4.59)
(4.60)
If we try to generalize this dot product we may require as needed properties the following ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
by a
can be proven with these axioms holds true not only for the conventional dot product.
·
The above axioms guarantee that the Schwarz inequality holds:
To prove this consider two (nonzero) vectors a and b and then consider the shortest vector joining the
tip of a to the line defined by the direction of b (see the... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
b)2
b
·
·
0 .
≥
Since b is not the zero vector we then have
b)2
(a
·
(a
a)(b
·
b) .
·
≤
(4.64)
(4.65)
Taking the square root of this relation we obtain the Schwarz inequality (4.62). The inequality becomes
an equality only if a⊥ = 0 or, as discussed above, when a = cb with c a real constant.
For compl... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
could be given
by
∗
w1z1 + . . . + w zn ,
∗
n
(4.68)
and we see that we are not treating the two vectors in an equivalent way. There is the first vector,
in this case w whose components are conjugated and a second vector z whose components are not
conjugated. If the order of vectors is reversed, we get for the in... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
= α
u , v
(
, with α
)
∈
F. Homogeneity in the second entry.
u , v
(
)
=
v , u
(
∗ . Conjugate exchange symmetry.
)
This time the norm
of a vector v
v
|
|
V
∈
is the positive or zero number defined by relation
2 =
|
v
|
v , v
(
)
.
(4.69)
From the axioms above, the only major difference is in number five, ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
+
)
∗ +
)
+
=
=
)
∗
)
v, u2
(
v, u2
(
u2, v
(
)
Homogeneity works differently on the first entry, however,
α u , v
(
)
=
∗
v , α u
)
(
) ∗
= (α
v , u
(
)
= α ∗
u , v
(
)
.
) ∗
)
∗
)
.
(4.70)
(4.71)
Thus we get conjugate homogeneity on the first entry. This is a very important fact. Of course,
for... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
+
|
u
|
2 ,
|
v
|
for u, v
∈
V, orthogonal vectors.
(4.72)
The Schwarz inequality can be proven by an argument fairly analogous to the one we gave above
for dot products. The result now reads
Schwarz Inequality:
u , v
|(
u
≤ |
.
v
| |
|
)|
(4.73)
The inequality is saturated if and only if one vector is a mu... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
an inner product as we have defined is a Hilbert space if it is finite
dimensional. If the vector space is infinite dimensional, an extra completeness requirement must be
satisfied for the space to be a Hilbert space: all Cauchy sequences of vectors must converge to vectors
in the space. An infinite sequence of vectors v... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
|
+ . . . +
2 .
|
an
|
anen , anen
(
)
)
(5.76)
This result implies the somewhat nontrivial fact that the vectors in any orthonormal list are linearly
2 . This
independent. Indeed if a1e1 + . . . + anen = 0 then its norm is zero and so is
|
implies all ai = 0, thus proving the claim.
2 + . . . +
|
an
|
a1
|
An... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
linearly independent vectors in V one can construct a
list (e1, . . . , en) of orthonormal vectors such that both lists span the same subspace of V .
The Gram-Schmidt algorithm goes as follows. You take e1 to be v1, normalized to have unit norm:
e1 = v1/
. Then take v2 + αe1 and fix the constant α so that this vector... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
define a subspace U ⊥, called the
orthogonal complement of U as the set of all vectors orthogonal to the vectors in U :
U ⊥ =
v
{
V
v, u
= 0, for all u
U
.
(5.81)
|(
∈
)
This is clearly a subspace of V . When U is a subspace, then U and U ⊥ actually give a direct sum
decomposition of the full space:
Theore... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
the comments below (1.5)). Let v
U
vector in the intersection U
∩
and in U ⊥ so it should satisfy
v, v
(
)
= 0. But then v = 0, completing the proof.
∈
∩
∈
and w
Given this decomposition any vector v
U
U ⊥ . One can define a linear operator PU , called the orthogonal projection of V onto U ,
that and that a... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
follows from this
that
PU PU = I PU = PU
P 2
U
= PU .
→
(5.84)
The eigenvalues and eigenvectors of PU are easy to describe. Since all vectors in U are left invariant by
the action of PU , an orthonormal basis of U provides a set of orthonormal eigenvectors of P all with
19
eigenvalue one. If we choose on... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
is an orthonormal basis for V.
(5.85)
Exercise: Use PU ei = ei, for i = 1, . . . n and PU fi = 0, for i = 1, . . . , k, to show that in the above basis
the projector operator is represented by the diagonal matrix:
PU = diag 1, . . . 1 , 0, . . . , 0 ) .
(5.86)
n entries
k entries
(
' -v
We see that, as expected... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
related linear operator T † on V called the adjoint of T . This is a very useful operator and is typically
different from T . When the adjoint T † happens to be equal to T , the operator is said to be Hermitian.
To understand adjoints, we first need to develop the concept of a
linear functional.
A linear functional φ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
fact. We can prove that any linear functional φ(v) admits such representation
φ(v) =
u, v
(
.
)
(6.2)
with some suitable choice of vector u.
Theorem: Let φ be a linear functional on V . There is a unique vector u
V .
for all v
Proof: Consider an orthonormal basis, (e1, . . . , en) and write the vector v as
∈
V ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
see
result for all v, then
′
= 0 for all v. Taking v = u
′
u , v
=
′
u , v
(
u = 0 or u = u, proving uniqueness.1
, which implies
)
u, v
(
′
u
(
−
)
)
−
′
that this shows u
−
We can modify a bit the notation when needed, to write
where the left-hand side makes it clear that this is a functional acting o... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
thus write w = T †u where T † denotes a map (not obviously linear) from V to V . So, we think of
T †u as the vector obtained by acting with some function T † on u. The above equation is written as
u , T v
(
)
=
T † u , v
(
)
,
(6.8)
Our next step is to show that, in fact, T † is a linear operator on V . The oper... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
T †(au) = aT † u .
This concludes the proof that T †, so defined is a linear operator on V .
A couple of important properties are readily proven:
Claim: (ST )† = T †S† . We can show this as follows:
u, ST v
(
)
=
S†u, T v
(
)
=
T †S†u, v
(
.
)
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
Claim: The adjoint of... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
action of T † on a vector. Give the matrix representations of T and T † using the
orthonormal basis e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1). Assume the inner product is the standard
on on C3 .
Solution: We introduce a vector u = (u1, u2, u3) and will use the basic identity
The left-hand side of the identity ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
representation we begin with T . Using basis vectors,
we have from (6.15)
T e1 = T (1, 0, 0) = (0, 1, 3i) = e2 + 3ie3 = T11e1 + T21e2 + T31e3 ,
(6.19)
and deduce that T11 = 0, T21 = 1, T31 = 3i. This can be repeated, and the rule becomes clear quickly:
the coefficients of vi read left to right fit into the i-th colum... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
(
)
ei, Tkj ek
(
ji
)
(6.21)
Relabeling i and j and taking the complex conjugate we find the familiar relation between a matrix
and its adjoint:
(T †)ij = (Tji) ∗ .
(6.22)
If we did not, in the equation above the use of
The adjoint matrix is the transpose and complex conjugate matrix only if we use an orthonormal... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
surprising and important
that for complex vector spaces the result is very strong: any such operator T necessarily vanishes.
This is a theorem:
Theorem: Let T be a linear operator in a complex vector space V :
If
v , T v
(
)
= 0 for all v
∈
V, then T = 0.
(7.24)
Proof: Any proof must be such that it fails to work... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
of getting just
.
v , T u
)
(
Here is where complex numbers help, we can get the same two terms but with opposite signs by trying,
u , T v
(
we also got
)
u + iv, T (u + iv)
(
) − (
u
iv, T (u
iv)
)
= 2i
u, T v
(
−
2i
) −
v, T u
(
)
.
−
(7.26)
It follows from the last two relations that
The condition
=... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
mitian:
24
T = T † if and only if
v, T v
(
) ∈
R for all v .
To prove this first go from left to right. If T =
T †
v, T v
(
)
=
T † v, v
(
)
=
T v, v
(
)
=
v, T v
(
∗ ,
)
(7.28)
(7.29)
showing that
v, T v
(
)
is real. To go from right to left first note that the reality condition means ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
The eigenvalues of Hermitian operators are real.
Theorem 2: Different eigenvalues of a Hermitian operator correspond to orthogonal eigenfunctions.
Proof 1: Let v be a nonzero eigenvector of the Hermitian operator T with eigenvalue λ: T v = λv.
Taking the inner product with v we have that
v, T v
(
)
=
v, λv
(
)
= ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
and second, using hermiticity of T ,
v2, T v1
(
)
=
T v2, v1
(
)
=
λ2v2, v1
(
)
= λ2
v2, v1
(
)
.
From these two evaluations we conclude that
(λ1
λ2)
v1, v2
(
)
−
= 0
(7.35)
(7.36)
and the assumption λ1
=
λ2, leads to
v1, v2
(
)
=
0, showing the orthogonality of the eigenvectors.
Let us now consider ... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
. Indeed
|
λIu
|
|
=
λu
|
|
=
u
λ
|
||
=
|
u
|
|
for all u. Moreover, the operator is clearly surjective.
For another useful characterization of unitary operators we begin by squaring (7.37)
U u, U u
(
)
=
u, u
(
)
By the definition of adjoint
u, U †U u
(
=
)
u, u
(
) → (
u , (U †U
I)u
)
−
= 0
for all... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
We readily see that the f ’s are also a basis, because they are linearly
independent: Acting on a1f1 + . . . + anfn = 0 with U † we find a1e1 + . . . + anen = 0, and thus ai = 0.
We now see that the new basis is also orthonormal:
fi , fj
(
)
=
U ei , U ej
)
(
=
ei , ej
)
(
= δij .
The matrix elements of U i... | https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/04b0570b349e84d74129eef504498472_MIT8_05F13_Chap_03.pdf |
Lectures 13 & 14
Packet Multiple Access: The Aloha protocol
Eytan Modiano
Massachusetts Institute of Technology
Eytan Modiano
Slide 1
Multiple Access
• Shared Transmission Medium
– a receiver can hear multiple transmitters
– a transmitter can be heard by multiple receivers
•
the major problem with multi-access... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/04ca100a8b247ecf18d328b752f1b929_Lectures13_14.pdf |
Slotted Aloha Assumptions
• Poisson external arrivals
• No capture
– Packets involved in a collision are lost
– Capture models are also possible
• Immediate feedback
–
Idle (0) , Success (1), Collision (e)
• If a new packet arrives during a slot, transmit in next slot
• If a transmission has a collision, node b... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/04ca100a8b247ecf18d328b752f1b929_Lectures13_14.pdf |
) = 1 - P(idle) - P(success)
Eytan Modiano
Slide 9
Throughput of Slotted Aloha
• The throughput is the fraction of slots that contain a successful
transmission = P(success) = g(n)e-g(n)
– When system is stable throughput must also equal the external
arrival rate (λ)
-1e
g(n)e-g(n)
Departure rate
1
g(n)
– Wh... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/04ca100a8b247ecf18d328b752f1b929_Lectures13_14.pdf |
– Given the backlog estimate, choose qr to keep g(n) = 1
Assume all arrivals are immediately backlogged
g(n) = nqr , P(success) = nqr (1-qr)n-1
To maximize P(success) choose qr = min{1,1/n}
– When the estimate of n is perfect:
idles occur with probability 1/e,
successes with 1/e, and
collisions with 1-2/e.
– When... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/04ca100a8b247ecf18d328b752f1b929_Lectures13_14.pdf |
otted) Aloha
• New arrivals are transmitted immediately (no slots)
– No need for synchronization
– No need for fixed length packets
• A backlogged packet is retried after an exponentially distributed
random delay with some mean 1/x
• The total arrival process is a time varying Poisson process of rate
g(n) = λ + nx ... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/04ca100a8b247ecf18d328b752f1b929_Lectures13_14.pdf |
assume only two users involved in collision
Practical algorithm must allow for collisions involving unknown number
of users
Eytan Modiano
Slide 17
Tree algorithms
• After a collision, all new arrivals and all backlogged packets not
in the collision wait
• Each colliding packet randomly joins either one of two gro... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/04ca100a8b247ecf18d328b752f1b929_Lectures13_14.pdf |
Lecture Notes on Geometrical Optics (02/18/14)
2.71/2.710 Introduction to Optics –Nick Fang
Outline:
A. Optical Invariant
B. Composite Lenses
C. Ray Vector and Ray Matrix
D. Location of Principal Planes for an Optical System
E. Aperture Stops, Pupils and Windows
A. Optical Invariant
-What happens to an arbitra... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
points. For two points that are separated far apart,
there is a limiting angle to transmit their information across the imaging system.
B. Composite Lenses
To elaborate the effect of lens in combinations, let’s consider first two lenses
separated by a distance d. We may apply the thin lens equation and cascade the ... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
as a condenser for illumination.
2
f1f2f1f2d
Lecture Notes on Geometrical Optics (02/18/14)
2.71/2.710 Introduction to Optics –Nick Fang
Practice Example: Huygens eyepiece
A Huygens eyepiece is designed with two plano-convex lenses separated by the
average of the two focal length. Ideally, s... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
𝑜𝑢𝑡
(
𝜃𝑜𝑢𝑡
These matrices are often (uncreatively) called ABCD Matrices.
Since the displacements and angles are assumed to be small, we can think in terms
of partial derivatives.
𝐴 𝐵
𝐶 𝐷
𝑖𝑛
𝜃𝑖𝑛
) = [
] (
)
𝐴 = (
) : spatial magnification;
Therefore, we can connect the Matrix components with the f... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
�O1O3O2 𝑜𝑢𝑡𝜃𝑜𝑢𝑡 𝑖𝑛𝜃𝑖𝑛 𝑜𝑢𝑡𝜃𝑜𝑢𝑡= 2 1 𝑖𝑛𝜃𝑖𝑛
Lecture Notes on Geometrical Optics (02/18/14)
2.71/2.710 Introduction to Optics –Nick Fang
© Pearson Prentice Hall. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
�
5
n1n2xin12insRzouts
Lecture Notes on Geometrical Optics (02/18/14)
2.71/2.710 Introduction to Optics –Nick Fang
Example 2: matrix of a ray propagating in a medium (changes x but not )
Example 3: refraction matrix through a thin lens (combined refraction)
Example 4: Imaging matrix thr... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
Location of Principal Planes for an Optical System
A ray matrix of the optical system (composite lenses and other elements) can give
us a complete description of the rays passing through the overall optical train. In
this session, we show that the focusing properties of the composite lens, such as
the principal pla... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
is located at a distance from the output plane given by:
𝐵𝐹𝐿 − 𝐸𝐹𝐿 = −(𝐴 − 1)/𝐶.
Likewise, we can find FFL and the first principal plane by the matrix components.
′0
(
0
) = [
𝐴 𝐵
𝐶 𝐷
] (
− ′𝑓
−𝜃′𝑓
)
8
dFFLx0Input Plane Output Plane 1stPPx’fEFL-’f-’f
Lecture Notes on Geometrical Opt... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
��
Therefore
𝐴 𝐵
[
𝐶 𝐷
] =
𝐴 𝐵
[
𝐶 𝐷
] = (
1
𝑜𝑢𝑡
1
𝜃𝑜𝑢𝑡
2
𝑜𝑢𝑡
2 ) (
𝜃𝑜𝑢𝑡
1
𝑖𝑛
1
𝜃𝑖𝑛
−1
2
𝑖𝑛
2 )
𝜃𝑖𝑛
1
2 − 𝑖𝑛
1 )
2 𝜃𝑖𝑛
( 𝑖𝑛
1 𝜃𝑖𝑛
(
1 𝜃𝑖𝑛
𝑜𝑢𝑡
1 𝜃𝑖𝑛
𝜃𝑜𝑢𝑡
2 − 𝑜𝑢𝑡
2 − 𝜃𝑜𝑢𝑡
1
2 𝜃𝑖𝑛
1
2 𝜃𝑖𝑛
2 𝑖𝑛
𝑜𝑢𝑡
2 𝑖𝑛
𝜃𝑜𝑢𝑡
1 − 𝑜𝑢𝑡
1 − 𝜃𝑜𝑢𝑡
2
1 �... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
10
ff
Lecture Notes on Geometrical Optics (02/18/14)
2.71/2.710 Introduction to Optics –Nick Fang
E. Aperture Stops, Pupils and Windows
o The Aperture Stops and Numerical Aperture
o Numerical Aperture(NA):
-
- also defines the resolution (or resolving power) of the optical syste... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
Fang
Practice Example: Single lens camera:
© Pearson Prentice Hall. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
-
-
-
Please determine the position and size of the image.
Please determine the entrance and exit pupils.
Plea... | https://ocw.mit.edu/courses/2-71-optics-spring-2014/04d4bff4ff4a9ae5e2c8ba7500905557_MIT2_71S14_lec4_notes.pdf |
Day 3
Hashing, Collections,
and Comparators
Wed. January 25th 2006
Scott Ostler
Hashing
Yesterday we overrode .equals()
Today we override .hashCode()
Goal: understand why we need to, and how
to do it
What is a Hash?
An integer that “stands in” for an object
Quick way to check for inequality, construct
... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
ance");
Name jack2 = new Name("Jack", "Nance");
System.out.println(kyle.equals(jack));
System.out.println(jack.equals(jack2));
System.out.println(kyle.hashCode());
System.out.println(jack.hashCode());
System.out.println(jack2.hashCode());
⇒ false, true, 6718604, 7122755, 14718739
Objects are equal, hashCodes ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
is no longer equal to what it was
Two equal objects must have an equal
hashCode
It is good if two unequal objects have distinct
hashes
Ex: Jack Nance will be different from Nance Jack
Before We Switch Topics
Any questions about hashCode, please ask!
It will be an important point later today
It will cau... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
Why Use Generics?
List untyped = new ArrayList();
List<String> typed = new ArrayList<String>();
Object obj = untyped.get(0);
String sillyString = (String) obj;
String smartString = typed.get(0);
Retrieving objects
Given Collection<Foo> coll
Iterator:
Iterator<Foo> it = coll.iterator();
while (it.hasNext) {
Foo ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
Set Overview
No set size, no set order
No duplicate objects allowed!
Set<Name> names = new HashSet<Name>();
names.add(new Name(“Jack”, “Nance”));
names.add(new Name(“Jack”, “Nance”));
System.out.println(names.size()); => 1
Set Contract
A set element cannot be changed in a way
that affects its equality
Th... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
ns.containsKey(“scotty.mit.edu”));
System.out.println(dns.containsValue(“18.227.0.87”));
dns.remove(“scotty.mit.edu”);
System.out.println(dns.containsValue(“18.227.0.87”));
// => “18.227.0.87”, true, true, false
Other Useful Methods
keySet() - returns a Set of all the keys
values() - returns a Collection of ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
-1234”);
Now changes to dennis don’t mess up map
But the keys themselves can still be changed
For (Name name : map.keySet()) {
name.first = “u r wrecked”; // uh oh
}
Make Immutable Keys
public class Name {
public final String first;
public final String last;
public Name(String first, String last) {
this.firs... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
�� > 0 if a > b
Comparison Example
Integer one = 1;
System.out.println(one.compareTo(3));
System.out.println(one.compareTo(-50));
String frank = “Frank”;
System.out.println(frank.compareTo(“Booth”));
System.out.println(frank.compareTo(“Hopper”));
// => -1 , 1, 4, -2
Sorting a List Alphabetically
List<String> n... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
we can define Comparator classes
A Comparator takes in two objects, and
determines which is bigger
For type Foo, a Comparator<Foo> has:
int compare(Foo o1, Foo o2);
A First-Name-First Comparator
public class FirstNameFirst implements
Comparator<Name> {
public int compare(Name n1, Name n2) {
int ret = n1.fi... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
Documentation:
http://java.sun.com/j2se/1.5.0/docs/api
No one can keep Java in their head
Everytime you code, have this page open | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/050457e9d4f48a612421484aa1ee573c_lecture3.pdf |
6.776
High Speed Communication Circuits
Lecture 3
Wave Guides and Transmission Lines
Massachusetts Institute of
Technology
February 8, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H. Perrott
Maxwell’s Equations in Free Space
Take Curl of (1):
E
×−∇=×∇×∇
⎛
⎜
⎝
µ
H
∂
t
∂
⎞
−=⎟
⎠
µ
∂
t
∂
(
×∇
H
)
From (2)
µ
∂
t
∂... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/05126e43ce52f3dc8ba9809b0692eadb_lec3.pdf |
∂
z
∂
H
2
+
µε
2
∂
t
∂
H
2
0=
(11)
(12)
(11) and (12) can be satisfied by any function in the form
(
zf
±
)vt
where
=v
1
µε
H.-S. Lee & M.H. Perrott
MIT OCW
Calculating Propagation Speed
(cid:131) The function f is a function of time AND position
(cid:131) Velocity calculation
vt
=
constant
±=
v
z... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/05126e43ce52f3dc8ba9809b0692eadb_lec3.pdf |
.-S. Lee & M.H. Perrott
MIT OCW
Now Put All the Pieces Together
(cid:131) Solve Maxwell’s Equation (1)
H.-S. Lee & M.H. Perrott
MIT OCW
Now Put All the Pieces Together
(cid:131) Solve Maxwell’s Equations (1) and (2)
H.-S. Lee & M.H. Perrott
MIT OCW
Freespace Values
(cid:131) Constants
(cid:131) Impedance
(cid:131) P... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/05126e43ce52f3dc8ba9809b0692eadb_lec3.pdf |
y
b
a
H.-S. Lee & M.H. Perrott
MIT OCW
Voltage and E-Field
(cid:131) Approximate E-field to be uniform and restricted to lie
between the plates
J
x
y
z
b
Hy
Ex
a
J
b
x
EV
a
y
H.-S. Lee & M.H. Perrott
MIT OCW
Back to Maxwell’s Equations
(cid:131) From previous analysis
(cid:131) These can be equivalently written as
(... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/05126e43ce52f3dc8ba9809b0692eadb_lec3.pdf |
in
C
C
C
(cid:131) Calculate input impedance
H.-S. Lee & M.H. Perrott
MIT OCW
LC Network Analogy of Transmission Line (TEM)
(cid:131) LC network analogy
L
L
L
L
Zin
C
C
C
(cid:131) Calculate input impedance
H.-S. Lee & M.H. Perrott
MIT OCW
How are Lumped LC and Transmission Lines Different?
(cid:131) In transmission ... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/05126e43ce52f3dc8ba9809b0692eadb_lec3.pdf |
18.354J Nonlinear Dynamics II: Continuum Systems Lecture 2
Spring 2015
2 Dimensional analysis
Before moving on to more ‘sophisticated things’, let us think a little about dimensional
analysis and scaling. On the one hand these are trivial, and on the other they give a simple
method for getting answers to problems tha... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/051766b6177c971b4cd0f005ddfdfe17_MIT18_354JS15_Ch2.pdf |
mass m, hanging in a gravitational field of strength g. What is the period of the pendulum?
time involving these numbers. The only
We need a way to construct a quantity with units of
(cid:112)
possible way to do this is with the combination
L/g. Therefore, we know immediately
that
τ = c(cid:112)L/g.
This result might se... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/051766b6177c971b4cd0f005ddfdfe17_MIT18_354JS15_Ch2.pdf |
star
It is known that the sun, and many other stars undergo some mode of oscillation. The
question we might ask is how does the frequency of oscillation ω depend on the properties
of that star? The first step is to identify the physically relevant variables. These are the
density ρ, the radius R and the gravitational co... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/051766b6177c971b4cd0f005ddfdfe17_MIT18_354JS15_Ch2.pdf |
−1 giving us a characteristic
frequency of 3Hz for a raindrop.
One final question we might ask ourselves before moving on is how big does the droplet
have to be for gravity to have an effect? We reason that the crossover will occur when the
two models give the same frequency of oscillation. Thus, when
we find that
(cid:11... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/051766b6177c971b4cd0f005ddfdfe17_MIT18_354JS15_Ch2.pdf |
capillary waves are equal when
(cid:112)k ∼ ρg/γ.
(34)
This gives a wavelength of 1cm for water waves.
11
MIT OpenCourseWare
http://ocw.mit.edu
18.354J / 1.062J / 12.207J Nonlinear Dynamics II: Continuum Systems
Spring 2015
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/ter... | https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/051766b6177c971b4cd0f005ddfdfe17_MIT18_354JS15_Ch2.pdf |
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