Can we do more of a broad survey of modern physics rather
than go really in depth with 1 or 2 topics? I would rather touch
on many topics than go into minute details on 1 or 2 topics.
The course material is indeed more broad than deep. We cover
quite a bit, but are prevented from going too deep by the fact
that more math is needed to go further in many places.
What do we do with the packet?
These are handouts on statistics and error analysis that will be
needed for your lab work. If the material is unfamiliar to you, you
should read them. If not, keep the handouts as reference.
Can we have more information about the lab procedures? 5/7 labs?
Everyone does Lab 0, and then you will choose 5 ``regular'' labs out
of a possible 7. Each will take two weeks. More details on lab
logistics will be given during the first lab period, and
information is also available on the course lab web
Do we need to bring our text to class?
Could we have an recitation section (optional)
during the week, led by one or both of the TAs?
As for other upper division classes, you're assumed to by now be proficient enough at studying this kind of material that a formal recitation is not needed. However that is not to say you are on your own! You are very much expected to come get help when needed. The TAs are not required to hold recitations, but they do have two hours of office hours per week; in fact these could turn into ad-hoc recitation-style periods. Plus, we are all available at other times by appointment, in case you don't have an overlap with scheduled office hours. Please do not be shy about coming to ask questions. You can email or phone too.
Let me also suggest getting together in informal study groups. You
are encouraged to work with your peers; it's well known that this is
enormously helpful for learning (although of course, I don't
think I need to remind you that all your
assignments must be your own solutions, in your own words).
I'm nervous as to the level of difficulty of the problem
sets without a recitation. How are the psets compared to 41/42?
I've never taught 41/42 so I'm not sure how to answer this. But
please see above question: you are not on your own, and you're
expected to take the initiative to get help from me and the TAs (and
each other) when you have trouble with a problem set.
How much of this do I need to understand exactly?
The ``WUN2K'' at the end of class gives a good guide to what you need
to know. In general in this course, we will be concerned mostly with
physical ideas rather than formal mathematics. In other words, you
should understand the physical meaning of the equations. For
example, for today's lecture, the key thing to understand is generally
how one gets a wave equation (restoring force plus Newton's second
law), and what the solution means, i.e. how
solutions correspond to wavy behavior in a physical system. The
details of the derivations are less important, although you may be
interested to look them up.
How does the restoring force connect to waves on a string?
When Newton's second law is applied to a problem with a linear, restoring force, you typically get an equation of motion- a differential equation for the displacement as a function of space and time - that's a wave equation. What it means to be a wave equation is that the solution gives some sort of wavy motion. The simplest example we saw is the case of a spring in one dimension: in that case, we got an equation of motion that led to a sinusoid. The next more complicated example is that of waves on a string: for this, plugging the restoring force into Newton's second law yields the wave equation (this expression is known as ``'' wave equation.) See link a couple of questions below for the derivation, if you're interested.
Solutions of the wave equation tend to be are wiggly, wavy things,
e.g. traveling or standing waves, depending on boundary conditions.
A single pulse solution is also possible.
What is a restoring force factor?
This is some factor which is related to the restoring force in
the system. Its specific nature depends on the system.
In our simple Hooke's law example, it was , the
Hooke's constant. For the case of a wave on a string, it's the
tension in the spring.
Can you put the derivation up of the wave equation?
Here's an example from the web, for the vibrating string:
An equation like this can be derived for lots of different situations; I hope you've seen a few of them before. It's a typical form that pops up all the time when you apply Newton's 2nd law () to the case of a material that has Hooke's-Law-like internal forces (an ``elastic medium''). Your freshman physics textbook will definitely have some examples. For instance, see section 14-2 in ``Physics for Scientists and Engineers'' by Fishbane, Gasiorowicz and Thornton.
(Try also googling for ``wave equation vibrating string'' or the like).
In the wave equation why did show up in the denominator
on the right hand side of
If you follow the derivation linked above, you can see where it comes from.
To get a feeling for it: we have . The expression is acceleration of a particular piece of string; if the ratio of to the ``space acceleration'' (rate of rate of change of displacement with respect to space) is large, then the pattern moves quickly.
How did the wave equation
and how does it incorporate a restoring force?
I wouldn't say that the wave equation exactly ``generalizes''
the Hooke's law equation. It's more that it's another example of a
similar, slightly more complex situation, for which displacement
is a function of both and (rather than
the Hooke's law case for which displacement is a function
of time only). The specific restoring force
can be tension in a string (see derivations
referred to above), or pressure in a gas, and comes in
to the constant in the equation. For instance
for a string
, where is tension and
is mass density; for a gas (sound wave solution),
where is pressure and is gas density.
It is important to note that the general wave equation
with constant only describes waves in ideal elastic media.
Real media do not have perfectly linear restoring force function.
That is indeed true. In physics we typically start with idealized
situations, which are often a very good approximation
and yield a lot of insight (although
perhaps not in the ``Consider a spherical cow'' example).
How did you get the velocity of the harmonic wave?
the wave equations appears in
writing this as
have defined to be . So the
of the wave equation is . But
since the wave repeats in time for
, and since the wave repeats in space for
these in, we get
The velocity of the wave is the distance
it travels from peak to peak over the time it takes to go
from peak to peak-in other words . So
the in the wave equation is the velocity of the wave.
What is the distinction between the velocity of the traveling
wave solution and the velocity of any given point on the wave (i.e.
velocity as first derivative of )?
The velocity of the traveling wave is the velocity with which the entire
pattern moves along (see above question). The velocity of a
given point on the wave is a different thing: it's the speed
of that little piece of string (or whatever medium). This
is given by
What does mean?
is any function you can dream up; call it (it has to be well enough behaved, e.g. twice-differentiable). It just has to be a function of in order for it to work as a solution of the wave equation (note that can be negative). For instance, take . solves the general wave equation. Similarly works, and works, and so on.
(Note that any old function won't work for specific physical
situations, for which boundary conditions determine what
solutions are allowed!)
How did you find the general solution
Here the wave equation's general solution was sort of pulled out of a hat for the lecture, because the solution takes a fair amount of time and would sidetrack the story. Here is some more info for those interested:
The wave DE is asking you: ``I'm a function of and ... differentiate me twice with respect to time, divide by and you get my 2nd derivative with respect to space... what function am I?'' And you have to come up with such a function that also respects the boundary conditions.
Here is d'Alembert's general solution to the wave equation,
for the more math-inclined:
Lots of solutions (an infinite number, of course)
of the general form, , are possible and
different ones are
appropriate for different boundary conditions and situations.
To get the specific solutions we covered, traveling and standing
waves, you don't really ``solve'' the equation the way you
would some algebraic equation to solve for ;
the answers are figured out from knowledge of what functions
have specific properties. These traveling and standing wave
functions have specific properties that apply for specific physical
work for all ?
You can see it for yourself by plugging into the wave equation.
, where .
Then, employing the Chain Rule of derivatives,
We also have,
Then the of the equation is
and the is
no matter what is.
You mentioned that
is the most general
solution to the wave equation. However what about ? Isn't
this an independent solution, assuming is constat?
, is a constant which can be positive or negative.
So is indeed a function of this form (just redefining
as the negative of itself). represents a wave
traveling in the direction, while represents a wave
traveling in the direction.
Where do the solutions of the differential equations come from?
I didn't derive them: I selected them. Any function of the form
will be a solution to the general wave equation.
I wrote down two types
of common solutions which have this form, the traveling wave
and the standing wave. These two types of solutions have different
properties appropriate to describe different situations.
Given a 2nd order, linear, 2 variable partial differential equation,
what must the boundary conditions look like in order to get quantized
I do not know the formal math answer to this off the top of my head
(the question probably needs to be defined carefully). Standing wave
solutions to wave equations with finite boundary conditions will
typically have quantized solutions; we'll see lots of examples.
What was the difference between the time and space waves and which
part affected the phase for each? Why did it change?
I'm assuming you're asking about the traveling wave. For the harmonic wave (traveling wave), , first consider a point in space, . You are looking at what happens to that point in space (say, a point on a string) as a function of time. The function describing displacement as a function of time is a sinusoid, , where here is a constant. A constant inside the sinusoid acts as a phase: it tells you how much the sinusoidal pattern is shifted along the axis. In other words it tells you when the wiggle starts.
Now consider a snapshot in time, . You are now
looking at the space pattern at a particular time, like a frame
in a movie. Now the function describing the space pattern is
. In this case, acts
as a constant phase: it tells you how much the wave is shifted in space.
At a different time, , the phase is different:
now it's . The wave is shifted
along. In other words, the wave pattern changes in time: it moves.
It's a traveling wave!
For the standing waves, how did you derive everywhere?
For the case of the case of the standing wave on a string fixed at both ends,
we must have at for all values of time . Since
, the sine part
has to be zero for . So we must have
For the standing waves, where did the come from?
The string is fixed at and , so its displacement is
zero there. So
must be zero for all values of at and . Since
in general the cosine term isn't zero, we must have the sine term
equal to zero for and . Take :
so we must have to satisfy that boundary condition. Now take
: plugging that in, we find we must have to satisfy
the boundary condition. This means that .
Can you explain how standing waves are generated from
A standing wave is a superposition of traveling waves in opposite directions
(see also below).
I don't think I said that today; we'll get to that in a bit when
we talk about interference next class.
What's the difference between traveling and standing waves?
A traveling wave moves in time along . In contrast, a standing wave does not travel along, but rather wiggles up and down ``in place''. To see how the standing wave equation, with space and time separated out, matches the physical situation: think of a snapshot in time, say : as a function of you see a sinusoid. But unlike for the harmonic wave case, a different snapshot in () does not correspond to a different phase ( vs ) and a shift of the wave along the axis. Rather, the displacement at each is multiplied by the new time sinusoid value. If you imagine watching each spot in time, you will see it thrashing up and down (sinusoidally). This is exactly what a standing wave does.
Examples of standing waves everyday life are a vibrating string, or a
column of gas (pressure waves creating sound). A standing wave can be
created from two traveling harmonic waves moving in opposite
directions (you can show this yourself using a few trig identities).
This can happen when one wave is reflected from a boundary and
interferes with (is added to) itself after reflection: see
http://physics.usask.ca/hirose/ep225/animation/standing1/anim-stwave1.htm. We'll discuss superposition and interference next class.
Can you explain the space/time separation of the standing wave?
Basically, when the time and space parts are separated out,
rather than intermingled in the same sinusoid argument,
the phase of the space wiggle does not change with time. So the standing
wave doesn't march along the axis. See above question also.
How does the standing wave equation have the form
You can manipulate the standing wave equation with trig identities to get it in this form. The easiest way to see it is perhaps as follows: a standing wave is actually a superposition of two traveling waves with the same amplitude, traveling in opposite directions:
Using the trig identity
you can show that
This can be shown more generally with phases, too, using a bit more algebra.
In the standing wave equation,
, how do you know
to separate space and time wiggle to get a standing wave as opposed
to a traveling wave? Where did this solution come from?
I'm not sure this kind of solution is exactly ``derived'': it's
written down as a particular function we happen to know about
that has the
desired properties (sometimes called ``solution by inspection'').
It's sort of like having a job to do and already
having the right tool for it, so you don't have to go out and