 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 336

At leading order,
$$
u=R\cos\theta,
\qquad
u_t=-R\sin\theta,
$$
so
$$
f=-u(u_t)^2
=-R^3\cos\theta\sin^2\theta.
$$
The required period averages are
$$
\langle\cos\theta\sin^3\theta\rangle=0,
\qquad
\langle\cos^2\theta\sin^2\theta\rangle=\frac18.
$$
Hence
$$
R'=0,
\qquad
R\phi'=\frac{R^3}{8}.
$$
The initial data give $R(0)=1$ and $\phi(0)=0$, so
$$
\boxed{R(T)=1,
\qquad
\phi(T)=\frac T8}.
$$
Thus the perturbation produces a slow frequency shift but no leading amplitude decay:
$$
\boxed{u(t)\sim\cos\left(t+\frac{\epsilon t}{8}\right).}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a / i / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 2 / a / i / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)