8.03 at ESG - Notes
Table of Contents
This is not the official 8.03
page.
The official page for Spring 2006 is up and running.
Little if any attempt will be made to keep this page current,
unless some more ESG-types sign up for 8.03.
Special for 8.044, Spring 2006
Some
Supplemental Notes regarding Fourier Transforms, at a slightly higher
mathematical level.
- Fourier Transforms,
including Dirty Tricks and an Unattractive but Highly Useful
Example
- Normalization of Fourier
Transforms
- Supplement to Supplementary Notes
on Fourier Transforms
- A Periodic
Delta-"Function" as a Sum of Complex Exponentials
- Partial Answers to B&B 2.12
- New for 8.044! Extension of Liebnitz's Rule. A
non-rigorous expostion of Liebnitz's
Formula at the 18.02 level might be useful as well.
- More on Problem 4 of Problem Set 3,
mainly calculus, but now with a figure.
- Figures to go with
the March 15 lecture, Discrete or Continuous, the point being that even for
these low numbers, the plots are almost indistinguishable, and that
the slopes of the graphs at the extremes are indeed singular,
consistent with zero temperature.
- For those who are interested
in such things, the ASCII commands that generated these plots and one
more are in S(E)
- Details of Baierlein's Equation (9.15)
We're good. We can do this. These notes quote other notes,
specifically Some Sums.
MAPLE
Resources
These might as well go close to the top.
New!
This is where any new stuff for the Spring 2004 term will be put. Much is
retained from earlier terms, so some dates and references to "what was
done in lecture'' may not be operative.
No Longer so New - Spring 2003
Some dates and references to "what was done in
lecture'' may not be operative.
- A Bit of Math Regarding Critical
Damping
Why is critical damping treated
differently, as in requiring "different'' functions? Really, it's
not different. There are several online presentations. (As the date
indicates, these are from Spring 2001; Prof. Mavalvala
used slightly different notation in the lecture of February 10
2003)
- Explanatory
Notes with Neat Color Graphic, PDF only.
- Explanatory Notes without Neat Color
Graphic, PDF or Xdvi, for those who don't want to waste
toner (or loading time). If you view this file, however, first look
at the
- The Maple worksheet that generated the Neat Color Graphs; the commands are
from Release 3, but will run on other releases (Release 3
gives higher contrast).
- Comments on Problem
Sets - Spring 2003
- Comments on the First Problem
Set
Nothing of major consequence. However, if you'd like to get
started using MAPLE for Problem 6(c), maybe take a look at
- Comments on the Second Problem
Set
The plots are getting more interesting.
- Comments on the Third Problem
Set
We're not doing slow pitch any more. If you're as
interested in Math as you should be, check out some of the following
links:
- Comments on the Fourth
Problem Set. Also, an Alternative Solution to
Problem 4.2(c), XDVI or PDF
- Comments on the Fifth
Problem Set. There are some related materials:
- No Comments on the Sixth
Problem Set: These problems are fairly standard, and
straightforward.
- Comments on the Seventh
Problem Set.
A more advanced consideration of the Larmor
Radiated Power, used in past terms at ESG, and swiping a neat
exposition from J.D. Jackson's Classical
Electrodynamics, might be of interest to those more
mathematically inclined.
(Someone has "borrowed" my copy of Jackson, so I can't give a more
precise reference.)
- No Comments on the Eighth Problem Set, either:
These problems are also fairly standard, and straightforward.
- Comments on the Ninth
Problem Set.
Although not part of the problem set this year, it is possible to
make plots of intensity as a function of angle for an antenna array
using Maple; a very basic worksheet (done for Release 3) may be
downloaded here (the commands
should work in other releases). To further investigate the nature of
the intensity pattern, a worksheet that finds the FWHM and compares to
the high-N prediction (itself a neat math problem) may be downloaded here.
For those who find the related math interesting, there's a far
more mathematical treatment for a related subject at Periodic Delta-"Functions"
- Comments on the Tenth
Problem Set.
Mostly from the lectures and reading. The same MAPLE worksheets
mentioned for the Ninth Problem set will be applicable here as well.
No longer New!
This is from the Fall
2002 term, and it will stay around for a while, but not
long. Much is retained from
previous terms, so some dates and references to "what was done in
lecture'' may not be operative.
- Comments on the first problem
set.
- Further Comments
relating the lecture of September 11, 2002 to the first
problem set.
- MAPLE commands to make plots of
frequency response curves; these commands should work for any
release, and may be copied and pasted into your MAPLE worksheet.
- A Related Problem from
18.03.
(Problem 1 of Part II, on the second page; references
are to the Edwards & Penney Differential Equations text.)
- Comments on the Second Problem set,
due September 20. (Updated Spetmeber 18)
- Comments on the Third Problem Set.
For those who care and have lots of free time, there's also
- Comments on the Fourth Problem Set.
These comments refer to notes below on Adiabatic
Bulk Moduli. For even more, see Thermodynamics
from the 18.023-ESG material.
- Comments on the Fifth Problem Set are being
assembled. For now, you might want to see Correction to French 6-13.
- No Comments on the Sixth Problem Set, but there
are some Comments on Notation for
Circular Polarization, related to the lecture of October 30,
2002.
- No Comments on the Seventh Problem Set, either.
Comments on the related lecture were sent to those who were there.
- Comments on the Eighth Problem
Set. Wow, is it mid-November already?
- Comments on the Tenth Problem
Set.
Some related notes are at Stuff from Chapter
4, but beware; the angle psi in those notes is not the same used
in Chapter 8.
Today's (December 2, 2002) lecture ended with a claim that the
diffraction pattern from a circular aperture can be represented in
terms of a Bessel Function. For those who would like to see a
variation of the derivation, see the last few pages of Airy
Transformations with an Illustrative Example,.
Related Notes
- Driving at Resonance, using an example from
B&B.
- Animations of the Waves shown on
Februrary 28, 2001
These animations, done in MAPLE, use the "Heaviside
Function,'' H(u), where
H(u) = 0 for x > 0 and
H(u) = 1 for x < 0.
- An Online
animation, with explanation and instructions for making your own
animations, relating to Energy
Densities in a Standing Wave
- From math (18.02), The Fundamental
Theorems of Vector Calculus, with garish figures.
From Previous Terms
I may want to make frequent reference to these.
These are notes and comments from previous terms or other classes,
sometimes cited above, and often extended as needed.
These notes were developed
for a version of
8.03, Vibrations and Waves, as taught at the Experimental Study
Group (ESG) at MIT. This version was for those students
more mathematically inclined, and makes more use of differential
equations and linear algebra.
Specific references are made to the
primary texts, Electromagnetic Vibrations, Waves, and Radiation
by George Bekefi and Alan H. Barrett (cited as B&B
in these notes) and Vibrations and Waves by
A. P. French. Cross-references are not exhaustive. The
order presented here is roughly the order in which the material
appears in B&B.
All of the notes have been converted to Portable Document Form
(PDF), which requires Adobe Acrobat Viewer.
- Adiabatic Bulk
Moduli
A derivation of Equation 1.40 in B&B,
Equation 3-25 in French, from basic principles of thermodynamics.
French gives a derivation for a monatomic gas in Pages 176-178.
- More on Cyclotron
Fields
A more general consideration of driven
oscillations, using B&B Pages 84-90 as motivation.
- Coupled Linear
Oscillators
Mainly for those who wish to see how the
results of linear algebra can be used to our advantage. These notes
rederive some of the preliminary results pertinent to normal modes,
and introduces some basic terminology from linear algebra that is
often used by physicists without apology. Coupled Linear
Oscillators may be consulted independently of
Normality of Modes of Discrete Coupled Oscillators, below.
- Normality of Modes
of Discrete Coupled Oscillators
The properties of the
oscillations of discrete coupled oscillators are summarized in French
Page 141, Eqs. (5-25), (5-26) and (5-27). However, although
these modes are correctly identified as normal modes, the normality is
never shown explicitly. (The result is quoted without derivation on
Page 195.) Using some tricks that are well-known to physicists,
these notes demonstrate the normality explicitly.
- Normal Modes of Coupled
Pendulums
Formulas corresponding to Equations (5-25)
and (5-26) in French are found for coupled oscillators that are not
constrained at the extremes.
- Some Sums
We can use
Fourier Series techniques to evaluate infinite sums that arise
frequently. Knowing how to do this will allow you to amaze your
friends and confound your enemies.
- Related to the above, but with (gulp!)
Complex Variables is Using a
Sommerfeld-Watson
Transformation. The example presented is Laplace's
Equation in a square. Other interested parties are welcome to
ivestigate higher dimensions avd less humble boundaries.
- Normal Modes for Continuous
Systems
That's a pretty broad topic, and the notes
discuss one case where, even though the frequencies must be found from
solution of a transcendental equation, orthogonality of normal modes
may be demonstrated explicitly. A suggested link to the related
online Differential Equations topic is 18.03-ESG
Notes,which deals with orthoganility of Bessel Functions.
- Fourier
Transforms including More Dirty
Tricks and an Unattractive but
Highly Useful Example
That title comes from reading
too much Dickens. Fourier Transforms, as opposed to Fourier Series,
are barely introduced at the end of B&B Chapter 2, and the
text necessarily leaves out a lot. These notes cannot be complete,
and introduce some of the terminology and uses of Fourier Transforms.
There's a lot of math, which makes it fun, of course.
- Normalization of Fourier
Transforms
A supplement to the previous notes, which
were kind of long anyway.
- Supplement to
Supplementary Notes on Fourier Transforms
Fourier
transforms are not my life, just a large part of it. These notes use
Fourier transforms to derive d'Alembert's solution to the Wavy
Quation.
- A Periodic
Delta-"Function'' as a Sum of Complex Exponentials,
Xdvi or PDF. These
notes, right now a work-in-progress, consider the possibility of the
reprsentation
The
cited figures are first and second.
A minor-league MAPLE worksheet may
be downloaded here, or the ASCII commands for
the simple animation may be copied and pasted from here (but these use a very large number of points,
400, which might be changed as desired).
- Partial Answers to
B&B 2.12 with even
More Dirty Tricks
Just what it says: The alleged
"dirty tricks'' are those developed in the other notes on Fourier
Transforms. These notes investigate the relations between the widths
of the functions and the transforms.
- Larmor Radiated
Power
The fact that an accelerated charge radiates
energy begs the question of what force or other agency does work on
the charge. If it's the force that causes the acceleration, then
conservation of energy needs to be used carefully, with some perhaps
surprising results.
A MAPLE animation related to the two figures may be generated using
the commands in the ASCII file Larmor
Animation; these commands, for any of the Releases 3 through 8,
and may be downloaded here. (The commands,
written for Release 3, will work on later releases, although if
you download the worksheet, you'll be told that you're using older
commands. It doesn't matter for this worksheet.)
Further documentation for Maple is available at
- An After-the-Fact
Dicsussion of Part of B&B Problem 4.5
and
a Glimpse into the Future
In
considering radiation from antennas or other arrays of oscillators,
different geometries will necessitate different mathematical
approximations and terminology. These notes give a general expression
which may be simplified as circumstances dictate.
- Math Tricks needed for the Planck
Formula
As B&B is not a text on Quantum Mechanics, much
is left out of the derivation of the Planck Formula for the energy
density of a blackbody radiator. These notes fill in some of the
gaps. Originally written in conjunction with specific assignments,
there are references to other class material that may be spurious.
Other sets of these notes are cited.
- Blackbody
Radiation
B&B get almost to, but not quite to, the
Stefan-Boltzmann constant, 
where a is the combination of physical
constants that is in 
at the end of Page 312. These notes fill in
the details.
- Electromagnetic Fields in
Matter
B&B assume a good deal of familiarity with
basic 8.02,
but not everyone takes the same version of 8.02. The needed results
given on Pages 419 and 420 are presented without proof or
apology. These notes fill in the steps.
- Plasma
Oscillations
The dispersion relation for plasma waves
may be found in many ways. These notes show how two ways,
specifically (a) treating the electron motion as a conduction
current and (b) using an imaginary conductivity give the same
dispersion relation obtained by treating the electron displacement as
a time-varying polarization.
Comments and suggestions,
especially corrections of errors of all degrees, are
always welcome.
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