Conventions used everywhere

Wave physics has more arbitrary choices in it than most subjects: which way a phase rotates, where the $2\pi$ lives in a Fourier transform, which sign of the imaginary part of a refractive index means absorption. None of them changes the physics, and all of them change the formulas. Textbooks differ, and a student comparing two sources silently switching between conventions can lose a week to a stray minus sign.

So this course fixes its choices once, here, and never varies them. This page is the place to come back to when a formula elsewhere does not match one of ours.

The phase of a wave¶

Every complex representation of an oscillation or a wave requires one arbitrary decision: which sign of the exponent represents motion forward in time. Both choices appear in the literature:

	This course	The other common choice
Plane wave	$\psi(x,t) = \Real\!\left[A\,e^{\ii(kx - \omega t)}\right]$	$\psi = \Real\!\left[A\,e^{j(\omega t - kx)}\right]$
Time factor	$e^{-\ii\omega t}$	$e^{+j\omega t}$
Found in	physics: Hecht, Griffiths, Goodman, Georgi	engineering: circuits, signal processing
Phasors rotate	clockwise	counterclockwise
Converting	$\ii \leftrightarrow -j$	$j \leftrightarrow -\ii$

This course always uses the first column. The time factor is $e^{-\ii\omega t}$ , a wave with $k > 0$ travels toward $+x$ , and phasors rotate clockwise. It is the convention of all four of this course’s backbone texts, and it is the one under which the quantum-mechanical phase $e^{-\ii E t/\hbar}$ will later look familiar rather than backwards.

A worked example, so the sign is concrete

The oscillation $x(t) = A\cos(\omega t + \varphi)$ has the phasor $\hat{x} = A e^{-\ii\varphi}$ , because

\Real\!\left[\hat{x}\, e^{-\ii\omega t}\right] = \Real\!\left[A e^{-\ii(\omega t + \varphi)}\right] = A\cos(\omega t + \varphi).

(1)

One consequence you will meet in the oscillations modules: the steady-state response of a driven, damped oscillator is

\hat{x}(\omega) = \frac{F_0/m}{\wnat^2 - \omega^2 - \ii\gamma\omega},

(2)

whose argument at $\omega = \wnat$ is $+\pi/2$ — the displacement lags the drive by a quarter cycle at resonance. Under the other convention the same physics carries the opposite sign of the imaginary part. Check every result you derive against this one and you will never lose the sign.

The engineering convention appears symbolically on this page and nowhere else in the course. Everywhere else — including problem sets, quizzes and the laboratories — the first column is the only one in use. A project lint enforces this.

Fourier transforms¶

The transform pair carries two arbitrary choices: which direction gets the minus sign, and where the $2\pi$ goes. This course puts the minus sign in the forward transform and the $2\pi$ in the inverse:

F(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-\ii\omega t}\, \mathrm{d}t, \qquad f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega)\, e^{+\ii\omega t}\, \mathrm{d}\omega,

(3)

and identically in space:

A(k) = \int_{-\infty}^{\infty} \psi(x)\, e^{-\ii k x}\, \mathrm{d}x, \qquad \psi(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} A(k)\, e^{+\ii k x}\, \mathrm{d}k.

(4)

Three things make this the right set of choices for this course, rather than merely a set:

It is internally consistent with the phase convention. Assembling a wave packet from its spectrum gives $\psi(x,t) = \frac{1}{2\pi}\int A(k)\, e^{\ii(kx - \omega(k) t)}\, \mathrm{d}k$ — every component is a forward-travelling plane wave of the previous section, with no sign juggling.
It is what the code does. numpy.fft.fft puts the minus sign in the forward transform, so a formula on a page and the line of NumPy that checks it never disagree about signs.
The asymmetric $2\pi$ keeps $F(0) = \int f$ . The value of the transform at zero frequency is the plain area under the signal, with no stray prefactor.

Note that an inverse transform legitimately displays a $e^{+\ii\omega t}$ — the lint that polices the time factor knows about this page, and an inverse-transform display elsewhere carries an explicit exception marker.

The complex refractive index¶

In an absorbing medium the refractive index becomes complex. This course writes it

\tilde{n} = n + \ii\kappa, \qquad \kappa \ge 0 ,

(5)

and the sign is forced by the phase convention, not free: a wave travelling toward $+x$ through the medium is

e^{\ii \tilde{n} k_0 x} = e^{\ii n k_0 x}\, e^{-\kappa k_0 x},

(6)

which decays, as absorption must. Under the engineering time factor the same physics requires $\tilde{n} = n - j\kappa$ ; if you meet that spelling in another book, it is the same medium described in the other convention, not a disagreement about the physics.

Real fields and intensity¶

The physical signal is always the real part. Complex exponentials are bookkeeping for amplitude and phase; nothing measurable is complex. Two consequences, fixed here once:

Nonlinear operations — energy, intensity, anything squared — act on the real signal, never on the complex representation. Take the real part first.
The time average of $\cos^2$ over a cycle is $\tfrac{1}{2}$ . The time-averaged intensity of a light wave of real amplitude $E_0$ in a medium of index $n$ is therefore
$I = \tfrac{1}{2}\, c\, \epsilon_0\, n\, E_0^2 ,$
(7)
and that factor of $\tfrac{1}{2}$ is stated here and never re-derived.

Units and constants¶

SI throughout. Metres, kilograms, seconds, and their combinations. Where a problem quotes electron-volts or ångströms, convert before computing.
Wavelengths are vacuum wavelengths. $\lambda_0$ quoted in nanometres; inside a medium the wavelength is $\lambda_0/n$ and the frequency is what stays fixed.
$c$ is exact. $c = 299\,792\,458\ \mathrm{m\,s^{-1}}$ by definition of the metre, and $\epsilon_0 = 1/(\mu_0 c^2)$ .
Angular frequency $\omega$ and frequency $\nu$ both appear. $\omega = 2\pi\nu$ always; formulas carry $\omega$ unless a measurement is being quoted in hertz.
The library works in plain SI floats. wavelab deliberately does not wrap quantities in a units package, so that a student can read omega0 = np.sqrt(stiffness / mass) and recognise it. Dimensional correctness is enforced in the test suite instead, where every formula is re-evaluated with pint quantities.

Epistemic labels: what kind of claim is this?¶

Wave physics mixes exact mathematics, empirical laws, modelling choices and numerical evidence, often within one paragraph. Blurring them is how students end up believing that a simulation “proved” an identity. Every boxed claim in this course therefore carries a label saying what kind of statement it is, and the boxes below are one live example of each.

Model specifications¶

Every simulation in the course — on a page or in a notebook — carries a seven-field block saying exactly what is being modelled. The fields never change and never reorder, so you can scan them. This is the one from the simple-harmonic-oscillator module:

Model specification

System: a point mass $m$ on a massless linear spring of stiffness $k$ , moving in one dimension; the observables are position, velocity and the energies.
Dynamics: Newton’s second law with the Hooke restoring force, $m\ddot{x} = -kx$ , integrated numerically by a velocity-Verlet scheme with a fixed time step.
Boundary: none — the mass moves on an infinite line and nothing is exchanged with anything.
Ensemble: a single deterministic trajectory per choice of $(x_0, v_0)$ ; where measurement noise is added it is Gaussian, independent, and seeded.
Ignored: spring mass, nonlinearity at large extension, internal damping, and the three-dimensional world the real spring hangs in.
Valid when: displacements stay in the linear regime of the real spring and the time step is well below the oscillation period.
Failure modes: amplitudes large enough to feel the spring’s nonlinearity, time steps near or above the stability limit, and any question about where the oscillation’s energy ultimately goes.

The two fields students skip are the two worth reading. Ignored tells you what the model is blind to, which is where its answers will be wrong. Failure modes names the regimes where it stops being a useful description at all — and an exam question that asks “when does this break down?” is asking you to reproduce that field.

The shape of a module¶

Every module page climbs in the same four steps, and you can enter and leave at the step that suits you.

Opening — a physical puzzle, and predictions you commit to before any equation. No prerequisites at all.
Core — the laboratory, the derivation, and the computational check. This is the exam-relevant spine, and it is the part to work through with a pen.
Consolidation — transfer to other settings, quizzes, exam-style problems, and reflection questions.
Advanced — deeper material, always marked with the box below and always last.

Simulation never proves¶

A simulation is a numerical experiment on a model. It can show you that a result is plausible, reveal where your intuition is wrong, and catch algebra errors. It cannot establish an identity, because it only ever examined finitely many cases of one particular model.

The division of labour this course keeps to:

Derivations establish. An identity is true because it follows from stated premises.
Simulations illustrate and check. They make a result visible and they catch mistakes.
Experiments decide. Whether a model describes the world is not a question mathematics or computation can answer.

When a page says “the fitted resonance frequency is $0.994\,\wnat$ ”, that is a numerical observation about a model. When it says “the amplitude of a driven oscillator peaks below its natural frequency whenever there is damping”, that is a theorem. The first is evidence for having implemented the second correctly. It is not why the second is true.