Here's an odd bit of business that landed on my desk recently:
Suppose you have an RLC circuit, or a spectral line, and
you want to know how much thermal noise there is. You do
the experiment, and do the analysis, and they don't agree!
They're off by a lot. You check everything.
To make a long story short, there are *two* notions of
bandwidth in this problem. Neither one is wrong; they're
just different.
-- There is the FWHM of the resonance curve, which is fine
as far as it goes, and is a key parameter in the equation
of motion.
-- However, that doesn't tell you the whole story. Think
about the shape of the response function, i.e. the lineshape,
sometimes called a Lorentzian. It has a *ton* of weight
in the wings. The system has a nontrivial response at
frequencies far from resonance. In particular, the
effective bandwidth "B" that shows up in Nyquist's formula
for the Johnson noise (4 kT B) is *not* the FWHM of the
resonance. It's bigger by a factor of π/2.
This shows up in RLC circuits, in mechanical oscillators,
in spectroscopic lines, etc. etc. etc.
The resonant lineshape stands in dramatic contrast to a
Gaussian. The latter has very little weight in the wings,
and its area is nearly equal to the height times FWHM, to
within a few percent.
I wrote up some notes on the subject. https://www.av8n.com/physics/rlc.htm
This is not meant to be an introductory tutorial, but almost
the opposite. If you already know the basic idea, this is
a compendium of details and ramifications that might be of
use on occasion.
Philosophical and pedagogical remark: It is a recipe for
disaster when there are two concepts masquerading behind
the same name. This happens all the time in the classroom
and in real life. Sometimes I think I spend half my time
trying to identify and untangle messes of this kind.
You might think that «the» bandwidth of an oscillator would
be a unique, well-defined thing ... but no such luck.
Also: Data visualization is a Big Deal. This is often
discussed in terms of experimental data, but the idea
applies to theory as well. It really helps to visualize
what the theory is telling you. For example, it is easy
to see that the skinny triangle in this figure: https://www.av8n.com/physics/rlc.htm#fig-rlc6lin
is a good match to the FWHM, but seriously underestimates
the area under the resonance curve.
Along the same lines: I get a lot of mileage out of symbolic
algebra programs like Macsyma (or the open-source generic
equivalent, maxima).
A lot of people complain that they can never get the program
to do what they want. Well, it helps if you don't ask the
program to do too much. In particular, it will *not* solve
the whole problem, at least not for the kind of problems I
care about. That is:
*) If you are at a certain point in the calculation, it
will not tell you what comes next. However, if you
know how to do the calculation, you can write down the
next equation and the computer will *check* it.
*) If you think expression X is a more-convenient version
of expression Y, do not expect the computer to cough up
X. It knows nothing about convenience. However if you
type in X and Y, the machine can easily tell you whether
or not X-Y is zero.
This is a big win, because there are a lot of people (including
me!) who have a tendency to make «small» mistakes in the algebra.
Of course there is no such thing as small; the first mistake
sends you off into an alternate universe and it's hard to get
home from there.
Usually I don't just type things at the program; instead I
edit up a script and feed that to the program. That way
there is a record showing exactly what I did. This further
facilitates checking the work. I can check it, and I can
give it to colleagues to check. It also provides a head
start the next time I need to attack a similar problem.