The <adiabatic sound speed> and <Alfvén speed> are
$$
\boxed{v_s^2=\left(\frac{\partial p}{\partial\rho}\right)_s
=\frac{\gamma p}{\rho}},
\qquad
\boxed{\mathbf v_A=\frac{\mathbf B}{\sqrt{\mu_0\rho}},
\quad v_A=\frac{B}{\sqrt{\mu_0\rho}}}.
$$
The entropy is held fixed in the sound-speed derivative.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-314/3/a/solution