Support
[ \rho'(\mathbfx, t) \approx \fracx_i x_j4\pi c_0^4 r \frac\partial^2\partial t^2 \int T_ij(\mathbfy, t - r/c_0) d^3y ]
Here, (c_0) is the speed of sound in the ambient medium, and (T_ij) is :
[ \frac\partial \rho\partial t + \frac\partial\partial x_i(\rho u_i) = 0 ]
[ \frac\partial^2 \rho\partial t^2 - c_0^2 \nabla^2 \rho = \frac\partial^2 T_ij\partial x_i \partial x_j ]
[ \rho'(\mathbfx, t) \approx \fracx_i x_j4\pi c_0^4 r \frac\partial^2\partial t^2 \int T_ij(\mathbfy, t - r/c_0) d^3y ]
Here, (c_0) is the speed of sound in the ambient medium, and (T_ij) is :
[ \frac\partial \rho\partial t + \frac\partial\partial x_i(\rho u_i) = 0 ]
[ \frac\partial^2 \rho\partial t^2 - c_0^2 \nabla^2 \rho = \frac\partial^2 T_ij\partial x_i \partial x_j ]
