 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

2023

Paper 1

If $c_0\ll\omega a$, then $x\gg1$. The sphere is acoustically large, and
$$
\langle K\rangle\sim\langle W\rangle,
\qquad
|\langle I_r\rangle|
\sim c_0\langle K+W\rangle.
$$
Pressure and velocity are nearly in phase, so the near-surface field behaves locally like a progressive radiating wave. The power approaches
$$
2\pi a^2\rho_0\omega^2\epsilon^2c_0.
$$

If $c_0\gg\omega a$, then $x\ll1$. The sphere is acoustically compact, and
$$
\frac{\langle W\rangle}{\langle K\rangle}\sim x^2\ll1,
\qquad
\frac{c_0\langle K+W\rangle}{|\langle I_r\rangle|}
\sim\frac1{2x^2}\gg1.
$$
The surface field is mainly reactive kinetic energy: pressure and velocity are nearly in quadrature, energy is stored and returned during each cycle, and only a small mean <acoustic energy flux> is radiated. The mean power scales as
$$
\boxed{\langle\mathcal P\rangle
\sim\frac{2\pi\rho_0\omega^4\epsilon^2a^4}{c_0}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / iv / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / iv / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / iv / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)