Vol 6 - Landau+Lifschitz - Fluid Mechanics · Problem 70.2

Sound oscillations in a spherical cavity are governed by the wave equation, and the key is recognizing that central symmetry means the pressure depends only on radius $r$, not on angles. This reduces to a radial wave equation whose solutions must satisfy a boundary condition—the velocity normal to the wall vanishes at $r = a$. The allowed frequencies emerge as discrete resonances, determined by which radial standing-wave patterns fit inside the sphere while respecting this rigid boundary.

Sound oscillations in a spherical cavity are governed by the wave equation, and the key is recognizing that central symmetry means the pressure depends only on radius $r$, not on angles. This reduces to a radial wave equation whose solutions must satisfy a boundary condition—the velocity normal to the wall vanishes at $r = a$. The allowed frequencies emerge as discrete resonances, determined by which radial standing-wave patterns fit inside the sphere while respecting this rigid boundary.