Supervisions: Miscellaneous Information and Links
This page contains optional extra work (mostly aimed at IA Nat. Sci. maths), interesting links, and assorted other things.
- For extra graph-plotting practice try questions Y1-6 from the graph-plotting questions. In order to check your answers as you go along, or to help if you get stuck on the overall shape of a function, it can be useful to try plotting graphs on a graphical calculator or a computer. The best program for the task is Pyxplot, but it only works on computers with sufficiently Unix-like operating systems (I think currently just Linux and possibly Mac OS X). There are several web-based graphing programs such as Wolfram Alpha which should work with any operating system and browser.
- There is a sheet of extension questions for the IA Nat. Sci. maths course. I'm happy to mark any attempts made at these.
- If you'd like to enthuse the general public about science, and/or play with science toys, then you might be interested in CHaOS. The Cambridge Science Centre also wants volunteers for similarly exciting events. (And if you're interested in demonstrations of scientific ideas using everyday things, then the website of the Naked Scientists is also worth a look, especially the kitchen science section.)
- The simulations at PhET are fun to play with, and mostly educational.
- The what-if questions on the xkcd comic are well worth reading. (As is the comment on understanding quantum mechanics.)
- Perhaps less interesting, but still important, are the homepages for NST Mathematics and for the IB Physics resources.
- Wikipedia is becoming surprisingly good these days, especially for subjects about which lots of people know enough to fix errors. See, for example, the page on group velocity, with the helpful animation.
- I've made a list of links which I use myself; these will be of limited use to anyone else.
- There is an online textbook about Maxwell's equations which may be of help to IB Physics B students.
- Here are the Jesus College start-of-term IA Nat. Sci. maths test from Lent 2013, the (rather harder) tests from Lent 2014 and Lent 2015, and the (more reasonable) Lent 2016 test. There's also the Lent 2017 test, though this has a couple of mistakes: 1(d) is harder than I meant it to be, and 4(h) contains an unhelpful ambiguity. The Lent 2018 test is I think mistake-free, though it's still quite hard.
- There are some ramblings on partial differentiation that may be vaguely relevant to some of you. (But I'm not sure that I'd express myself quite like that if I were writing it again, especially as some of the nomencleture is at odds with that used in the course...)
- The advice on approaching difficult questions has been moved to the advice page.
Additional comments on the Lecture Notes
Derivation of Sound Wave Equation
In the IB Physics A Osciallations and Waves course, there is a derivation of the wave equation for sound waves in a gas. (The lecture handout is on the course web page, which also contains a good section of interesting links.) A number of groups have asked me about the approximation used in deriving equation 47. I think that the main confusion here comes from the use of p and \Psi_p, and I try to explain what's going on below.
In the run-up to equation 44, \Psi_p is introduced as the excess pressure due to the wave. Therefore the actual pressure in the gas, as a function of x and t, is p + \Psi_p, where p is the constant ambient pressure. This notation is used carefully and consistently to derive equations 44 and 45. In equation 46 and what follows, it's a bit sloppier. The thing that's being called p is really p+\Psi_p, the actual pressure. There is also an approximation going on: the small but finite changes in pressure and volume as the wave goes past, which are \Psi_p and \Delta V respectively, are being assumed to be in the same ratio as an infinitesimal dp and dV. This is a very good approximation, but it's important to be aware that an approximation is being made.
Thus what equation 46 is really trying to tell us is that \Psi_p = - \gamma (p + \Psi_p) times partial da/dx.
Once we come to take a derivative of equation 46, we need to use the product rule, because the thing written p is really the constant p plus the spatially-varying \Psi_p. Then the first p (after the \gamma) in the formula for the derivative ought to be p + \Psi_p, but we can legitimately ignore that \Psi_p, because it's then a small part of the equation that we've got. (The relies on \Psi_p's being a small correction to p, which is equivalent to the previous approximation.) In the next term, the partial dp/dx is really partial d(p + \Psi_p)/dx, which is partial d(\Psi_p)/dx. So we've now got a term with that derivative on both sides of the equation. In principle we should take them both to the LHS and divide by the 1 + \gamma (partial da/dx). This would make a vile equation. But here we can make a further approximation: if a is much smaller than \lambda, which it usually is, then partial da/dx is much smaller than 1.* As \gamma is of order 1, that means that the whole term can be neglected and the derivation proceeds as given.
* Since a has some sort of wave-like solution, partial da/dx is going to be something like ka, which is proportional to a/(\lambda). This seems to be what the lecturer is thinking, anyway. But really we have already decided that partial da/dx is small, because equation 46 tells use that it's a number of order 1 times \Psi_p/p, and the assumption that \Psi_p is small compared to p is one that we've been making since beginning to derive equation 46 itself.