= Translating-sphere acoustic thickness dipole
{title2=$\rho'_{\rm th}=\rho_0\mathcal V\,\mathbf n\cdot\ddot{\mathbf V}/(4\pi c_0^3r)$}
For a small rigid sphere translating with <velocity> $\mathbf V(t)$, the integrated thickness source $\rho_0\int\mathbf V\cdot\mathbf n_s\,dS$ vanishes. The first spatial moment is instead $\rho_0\int y_j\mathbf V\cdot\mathbf n_s\,dS=\rho_0\mathcal V V_j$, where $\mathcal V$ is the sphere volume. Expanding the <retarded time> across the compact source gives the displayed first nonzero thickness field. It is an <acoustic dipole>, at the same order as <acoustic loading noise> for a translating sphere. The <added mass of a sphere> supplies half this volume-mass coefficient through the loading term; their sum is $3\rho_0\mathcal V\mathbf n\cdot\ddot{\mathbf V}/(8\pi c_0^3r)$.
Back to article page
