Acoustic thickness noise (${\partial }_t({\rho }_0{v}_n{\delta }_s)$)

Thickness noise is the sound generated by displacement of fluid by a moving impermeable body, represented by the surface acoustic monopole term. For a rigid body, ${\int }_S{v}_{n}dS=d\mathcal{V}/dt=0$, so its leading acoustic compact-source approximation has no net-volume monopole. Its local source need not vanish: a moving finite-volume body can radiate through higher spatial moments. A thin-body or negligible-volume approximation is needed when all such radiation is to be omitted.

In the incompressible near field of a spherical bubble, $u_r=a^2\dot{a}/s^2$. Its volume-flux acoustic monopole is the displayed term at retarded time. The integrated surface pressure force vanishes by spherical symmetry, while the Lighthill stress tensor approximation ${\rho }_0{u}_i{u}_j$ has moment $S_{ij}=4\pi {\rho }_0{a}^3{\dot{a}}^2{\delta }_{ij}/3$. Its leading compact acoustic quadrupole contribution is ${\rho }_0(a^3{\dot{a}}^2{)}^{\prime \prime }/(3c_0^{4}r)$. For small fractional radius oscillation $\delta$ and compactness parameter $\alpha =\omega a_0/c_0$, its size relative to the leading monopole is $O(\delta {\alpha }^2)$.

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 $3{\rho }_0\mathcal{V}\mathbf{n}\cdot \ddot{\mathbf{V}}/(8\pi c_0^{3}r)$.