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separates long hills, which radiate vertically propagating waves and carry energy upward, from short hills, whose response is evanescent and carries no mean vertical energy flux.

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / b / iv / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

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k > N / U

, then

m = i α

with

α > 0

chosen so that the disturbance decays as

z \to \infty

. The pressure amplitude is then ninety degrees out of phase with

w

, since

p / w

is purely imaginary. Hence

⟨ \overline{I}_{z} ⟩_{0} = \frac{1}{2} Re (p w^{*}) = 0.

(1)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / b / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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Horizontal momentum and incompressibility give

p = ρ_{0} \frac{ω}{k} u = - ρ_{0} \frac{ωm}{k ^{2}} w .

(1)

Therefore

⟨ \overline{I}_{z} ⟩_{0} = \frac{1}{2} Re (p w^{*}) = - \frac{ρ _{0} ωm}{2 k ^{2}} w_{0}^{2} .

(2)

For upward radiation take

ω = - U k

and

m = N^{2} / U^{2} - k^{2} > 0

, obtaining

⟨ \overline{I}_{z} ⟩_{0} = \frac{1}{2} ρ_{0} U^{2} k h_{0}^{2} N^{2} - U^{2} k^{2} > 0.

(3)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / b / ii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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The boundary condition gives complex amplitude

w_{0} = U k h_{0}

. Substitution of the internal gravity wave form

w = w_{0} e^{i (k x + m z - ω t)}

(1)

into the internal-wave equation gives

ω^{2} = \frac{N ^{2} k ^{2}}{k ^{2} + m ^{2}} .

(2)

Since

∣ ω ∣ = U k

m^{2} = \frac{N ^{2}}{U ^{2}} - k^{2} .

(3)

For

k < N / U

, choose the sign of

m

that gives upward energy propagation; with

ω = - U k

, this is

m > 0

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / b / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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In coordinates moving with the background fluid,

x^{'} = x - U t

. The stationary hill profile becomes

h = h_{0} sin (k x^{'} + k U t),

(1)

so in the convention

e^{i (k x^{'} - ω t)}

its frequency is

ω = - k U

; its physical angular frequency has magnitude

k U

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / b / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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For lattice spacing

2 a

, the first Brillouin zone is

- \frac{π}{2 a} \leq k \leq \frac{π}{2 a} .

(1)

At its boundaries

E_{\pm} = \pm t

, while at

k = 0

the energies are

E_{\pm} = \pm 5, t

. Thus the lower band runs from

- 5, t

- t

, the upper from

t

5, t

, and each has bandwidth

(5 - 1) t .

(2)

The requested plot consists of these two even branches, separated by a gap of size

2 t

at the zone boundaries.

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / b / ii / Solution by

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Using one even and one odd site per unit cell, Bloch theorem reduces the Hamiltonian to

H (k) = t (1 1 + e^{2 ika} 1 + e^{- 2 ika} - 1) .

(1)

Its eigenvalues are

E_{\pm} (k) = \pm t 1 + ∣1 + e^{2 ika} ∣^{2} = \pm t 1 + 4 cos^{2} (ka) .

(2)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / b / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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Translation preserves the alternating on-site term only when

(- 1)^{j + ℓ} = (- 1)^{j}

, so

[H_{t}, T_{ℓ}] = 0 ⟺ ℓ \in 2 Z .

(1)

The primitive lattice spacing is therefore

2 a

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / a / iii / Solution by

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The Floquet multipliers satisfy

λ^{2} - (tr F) λ + 1 = 0.

(1)

(tr F)^{2} < 4

, they are complex conjugates on the unit circle,

e^{\pm ika}

, and bounded Bloch waves exist:

E

lies in an allowed band. If

(tr F)^{2} > 4

, the multipliers are real reciprocal numbers, one growing and one decaying, so there is no extended bounded Bloch state and

E

lies in a gap.

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / a / ii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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For real

E

and real

V

, choose a real fundamental pair of solutions. The corresponding Floquet matrix has real entries, so

tr F (E)

is real. The trace is basis independent, so this holds for every fundamental pair.

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 34E / a / i / Solution by

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For two solutions of the Schrödinger equation, their Wronskian is constant because the equation has no first-derivative term. The determinant of the Floquet matrix is the ratio of the Wronskian at

x + a

to that at

x

, hence

det F (E) = 1.

(1)

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 33B / a / iii / Solution by

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For

- j \leq m < j

, the matrix element of

[O, J_{+}]

between

∣ j, m + 1 ⟩

and

∣ j, m ⟩

gives

C_{+} (j, m) (⟨ j, m + 1∣ O ∣ j, m + 1 ⟩ - ⟨ j, m ∣ O ∣ j, m ⟩) = 0.

(1)

The ladder coefficient is nonzero, so adjacent diagonal entries agree. Induction makes the diagonal entry independent of

m

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 Pinned article: Introduction to the OurBigBook Project

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Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
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. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
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This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
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Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
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Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
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Infinitely deep tables of contents:
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