PHY-217 | School of Physics and Astronomy

General
Aims and Objectives
Syllabus
Blog
Coursework
Marking Scheme

Welcome to the Vibrations and Waves (PHY-217) Home Page

Schedule

The schedule is for lectures to take place on Tuesday mornings 9–11 am (Fogg building, main lecture room), and for exercise classes to be held on Monday morning at 9–10 am and Thursday afternoon 1–2 pm (both in Francis Bancroft building, room 2.40).

Course books

There are two course books, both called Vibrations and Waves. If you want to buy just one book, I would recommend that by George King from Manchester University (Amazon link). The traditional book is by AP French from MIT (Amazon link). My feeling is that the book by King is closer to the way I want to teach this course than French (an example being the treatment of the damped harmonic oscillator, our second topic). On the other hand, being able to understand an alternative treatment is a good thing. There is not much difference in Amazon's price for a new copy of the paperback edition (£27.86 and £22.99 respectively).

Handout

The handout from the previous version of this course is available here. This is not a substitute for the books, particularly that by King. Lecture notes are available in the Coursework tab.

Coursework

The coursework tab includes all notes, exercise classes, and homeworks

Learning Outcomes

Aims

The aim of this course is to provide a basic understanding of the laws of vibrations and waves.

Learning Outcomes

By the end of the course the students should be able to: apply the laws of simple harmonic motion to various oscillating systems such as pendulum, the LC circuit, the Helmholtz resonator, etc; relate the driving force with aspects of resonance and to comprehend the driven simple harmonic motion; perform calculations of normal modes for coupled oscillators; deduce the 1D wave equation for a uniform continuous string; understand superposition of waves of same and different frequency; comprehend waves in gasses and solids.

Syllabus

Simple Harmonic Oscillator
Damped Harmonic Oscillator
Driven Harmonic Oscillator
Resonance
Superpositon of oscillations, beats
Two and three coupled oscillators and normal modes
Many coupled oscillators, the wave equation (of a string)
Travelling waves
Superposition of waves
Standing waves
Dispersion

Blog

Week 1

Week 1 is about laying down some foundations. We went quite slowly – hopefully not too slowly – because there were some concepts in here that absolutely need to become second nature very quickly. We followed the first 16 pages of the book by King.

The first is that the equation force = mass × acceleration will lead to a second order differential equation. Our differential equation right now is as easy as it gets. It will get a bit harder, but not so much. It will also get more complicated when we work with several variables rather than one, but we have book-keeping methods – matrices – to help us.

The second is that we obtain force as minus the spatial derivative of a potential energy function. Books often go straight to the force, because usually for simple cases it is intuitive. For example, if I pull on on a spring, the force to restore the spring to its equilibrium length will be in the direction opposite to that of the pull. And so most books run with this intuition. But later on when we deal with coupled systems and several equations, the scope for getting signs wrong multiplies. And if you get a sign wrong, that is the end of the calculation (not great in an exam). By all means use intuition, but what I am trying to do is prepare the ground for when it becomes safer not to do so.

The third is that we will always be working with a harmonic energy. For springs this works well, mostly because extending a spring doesn't actually generate too much strain on the metal in the coils. Almost always what we have is a Taylor expansion of the potential energy function in terms of the variable of interest, and typically we ignore everything higher than the quadratic (harmonic) term (note that there is no linear term in the expansion when the system is at equilibrium). The differential equations generated by the harmonic model are always solvable. However, this is not the case for models that include higher-order terms, so some approximation methods are needed (see this link for the solution of the simplest anharmonic equation).

The fourth is that most of our models will have oscillatory solutions, although in the next lecture we will actually meet cases where the pure exponential is what we need. So we noted some trivial points about cosine curves, including the fact tact we can have an arbitrary phase shift in our solutions. We will often work with real numbers (sines and cosines) but there are cases where things are much easier working with complex representation via Euler's formulae linking the complex exponential to the sine and cosine functions.

The fifth is that the equations have parameters that are either a solution or a boundary condition. In what we have done so far, the equations give explicit formulae for the angular frequencies, but the amplitude is set by us as a boundary equation. Note that the angular frequency for our spring systems is a function of the spring constant (the second differential of the energy) and the mass, but had no dependence on the amplitude. You may or may not have expected this, but it will always be the case (for the anharmonic oscillator described in this link, the frequency is modified from the harmonic result by the addition of a term that does depend explicitly on the amplitude).

The sixth is that we work in terms of the angular frequency, which is equal to 2π times the actual frequency. So beware never to forget the factor of 2π when comparing with data. Worse, often the natural parameter is the square of the angular frequency – this is what our equations gave us – and forgetting the 2π in this context will give you an error of a factor of 40, which is huge. You would be surprised at how often professional scientists make this mistake!

The seventh is to appreciate the numbers. Here we are talking about frequency. The basic unit is 1 Hz, but not so many things vibrate around 1 Hz. An example is that this scale or lower is the range of vibrations of the Earth. It is also a natural frequency scale for the human body, for example breathing, heart beating, swallowing, walking and swinging your arms. But we can't hear 1 Hz; instead out hearing range is from 20 Hz to 20 kHz. The note middle A is 440 Hz = 0.44 kHz. Radio waves (of the sort we use for communications), come in at around the MHz–GHz range, infrared radiation in the GHz–THz range, and atomic vibrations in the THz range. Light has an even higher frequency. It is good for these orders of magnitudes to become second nature.

The eighth is that a pure harmonic vibration will conserve energy, with periodic transfer of energy between kinetic and potential. For a harmonic oscillator, the average values of the potential and kinetic energies are equal. A moment's thought will tell you that this fact is intuitive, but we derived it in any case. It can be a very useful fact.

In the next lecture we will look at other simple systems, and then introduce damping, which allows energy to leak away from a vibration.

Coursework

Teaching materials for this course will be available here.

Homeworks

Homework sheet 1 download here. To be handed in by 5 pm on Thursday November 15th. This homework will count as double as part of your reading week task.

Homework solutions