A soap bubble is colourless stuff. The rainbow on it comes from arithmetic. Light striking the film reflects twice — once off the top surface, once off the bottom — and the two reflections travel back to your eye slightly out of step, because the second one crossed the film twice. Where they arrive in step, that colour is bright. Where they arrive half a wave apart, that colour vanishes. Tilt the bubble and the path changes, and so does the colour.

Two facts set the bookkeeping. First, the round trip is twice the thickness. Second, the reflection off the top face — going from air into something denser — flips by half a wavelength, while the reflection off the bottom face does not. That single flip is why the conditions look backwards from what you would guess. Bright in reflection: $2nt = \left(m + \tfrac{1}{2}\right)\lambda$. Dark in reflection: $2nt = m\lambda$.

Symbol by symbol: $t$ is the film thickness in metres (nanometres, in practice); $n$ is the index of the film, a naked number; $m$ is the order, a whole number counting how many wavelengths fit the round trip (0, 1, 2, … for the bright condition; 1, 2, 3, … for the dark one); and $\lambda$ is the wavelength in vacuum — the value you would measure out in the air. That last one is the chapter's most expensive trap.

Here is why. Inside the film the wave really is shorter, $\lambda_n = \lambda/n$ — you proved that two lessons ago. But the $n$ already sitting in $2nt$ is doing exactly that job: $2nt$ is the optical path length, the geometric round trip already converted. Divide $\lambda$ by $n$ as well and you have charged the index twice, and your film comes out wrong by a factor of $n$. Vacuum wavelength on the right, index on the left, once each.